Mental Math: Estimate Sums and Differences

RETEACH Workbook

Grade 4 P

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NNL_SEGr4.inddL_SEGr4.indd template1template1 66/7/07/7/07 33:04:36:04:36 PPMM Unit 1: UNDERSTAND WHOLE NUMBERS Unit 2: MULTIPLICATION AND DIVISION AND OPERATIONS FACTS

Chapter 1: Understand Place Value Chapter 4: Multiplication and Division Facts 1.1 Place Value Through 4.1 Algebra: Relate Hundred Thousands ...... RW1 Operations ...... RW19 1.2 Model Millions ...... RW2 4.2 Algebra: Relate Multiplication 1.3 Place Value Through and Division ...... RW20 Millions ...... RW3 4.3 Multiply and Divide 1.4 Compare Whole Numbers. ....RW4 Facts Through 5 ...... RW21 1.5 Order Whole Numbers...... RW5 4.4 Multiply and Divide 1.6 Problem Solving Workshop Facts Through 10 ...... RW22 Strategy: Use Logical 4.5 Multiplication Table Reasoning ...... RW6 Through 12 ...... RW23 4.6 Patterns on the Chapter 2: Add and Subtract Whole Multiplication Table ...... RW24 Numbers 4.7 Problem Solving Workshop 2.1 Algebra: Relate Addition Skill: Choose the and Subtraction ...... RW7 Operation ...... RW25 2.2 Round Whole Numbers 4.8 Algebra: Find Missing Through Millions ...... RW8 Factors ...... RW26 2.3 Mental Math: Estimate Sums and Differences ...... RW9 Chapter 5: Algebra: Use Multiplication and 2.4 Mental Math Division Facts Strategies ...... RW10 5.1 Multiplication 2.5 Problem Solving Workshop Properties ...... RW27 Skill: Estimate or 5.2 Multiplication and Division Exact Answer ...... RW11 Expressions ...... RW28 2.6 Add and Subtract 3-Digit and 5.3 Order of Operations ...... RW29 4-Digit Numbers...... RW12 5.4 Multiplication and Division 2.7 Choose a Method...... RW13 Equations ...... RW30 5.5 Problem Solving Workshop Chapter 3: Algebra: Use Addition and Strategy: Predict and Test ... RW31 Subtraction 5.6 Explore Inequalities ...... RW32 3.1 Addition Properties ...... RW14 5.7 Patterns: Find a Rule ...... RW33 3.2 Write and Evaluate Expressions ...... RW15 3.3 Addition and Subtraction Equations ...... RW16 3.4 Problem Solving Workshop Strategy: Work Backward ... RW17 3.5 Patterns: Find a Rule ...... RW18

GG4-NL4-NL TOC_TE.inddTOC_TE.indd 1 66/7/07/7/07 33:44:02:44:02 PPMM Unit 3: TIME, TEMPERATURE, AND DATA Unit 4: MULTIPLY BY 1-DIGIT AND 2-DIGIT NUMBERS

Chapter 6: Time and Temperature 6.1 Telling Time ...... RW34 Chapter 9: Multiply by 1-Digit Numbers 6.2 Elapsed Time ...... RW35 9.1 Mental Math: Multiplication 6.3 Elapsed Time on Patterns ...... RW57 a Calendar ...... RW36 9.2 Mental Math: Estimate 6.4 Algebra: Change Units Products ...... RW58 of Time ...... RW37 9.3 Model 2-Digit by 1-Digit 6.5 Problem Solving Workshop Multiplication ...... RW59 Skill: Sequence 9.4 Record 2-Digit by 1-Digit Informaiton ...... RW38 Multiplication ...... RW60 6.6 Temperature ...... RW39 9.5 Multiply 3-Digit and 4-Digit 6.7 Explore Negative Numbers .. RW40 Numbers and Money ...... RW61 9.6 Multiply with Zeros ...... RW62 Chapter 7: Collect and Organize Data 9.7 Problem Solving Workshop 7.1 Collect and Organize Data .. RW41 Skill: Evaluate 7.2 Venn Diagrams ...... RW42 Reasonableness ...... RW63 7.3 Find Mean, Median, and Mode ...... RW43 Chapter 10: Multiply by 2-Digit Numbers 7.4 Line Plots ...... RW44 10.1 Mental Math: Multiplication 7.5 Choose a Reasonable Patterns ...... RW64 Scale and Interval ...... RW45 10.2 Multiply by Tens ...... RW65 7.6 Problem Solving Workshop 10.3 Mental Math: Estimate Skill: Make Products ...... RW66 Generalizations ...... RW46 10.4 Problem Solving Workshop Strategy: Solve a Chapter 8: Interpret and Graph Data Simpler Problem ...... RW67 8.1 Bar Graphs ...... RW47 10.5 Model 2-Digit by 2-Digit 8.2 Make Bar and Double-Bar Multiplication ...... RW68 Graphs ...... RW48 10.6 Record 2-Digit by 2-Digit 8.3 Circle Graphs ...... RW49 Multiplication ...... RW69 8.4 Algebra: Use a Coordinate 10.7 Multiply 2-Digit and 3-Digit Grid ...... RW50 Numbers and Money ...... RW70 8.5 Line Graphs ...... RW51 10.8 Choose a Method ...... RW71 8.6 Make Line Graphs ...... RW52 10.9 Problem Solving Workshop 8.7 Problem Solving Workshop Skill: Multistep Problems .... RW72 Strategy: Make a Graph ..... RW53 8.8 Choose an Appropriate Graph ...... RW54 8.9 Algebra: Graph Relationships ...... RW55 8.10 Problem Solving Workshop Skill: Identify Relationships ... RW56

GG4-NL4-NL TOC_TE.inddTOC_TE.indd 2 66/7/07/7/07 33:44:07:44:07 PPMM Unit 5: DIVIDE BY 1-DIGIT AND Unit 6: FRACTIONS AND DECIMALS 2-DIGIT DIVISORS

Chapter 15: Understand Fractions and Mixed Chapter 11: Understand Division Numbers 11.1 Divide with Remainders ..... RW73 15.1 Read and Write Fractions ... RW94 11.2 Model 2-Digit by 1-Digit 15.2 Model Equivalent Division ...... RW74 Fractions ...... RW95 11.3 Record 2-Digit by 1-Digit 15.3 Compare and Order Division ...... RW75 Fractions ...... RW96 11.4 Problem Solving Workshop 15.4 Read and Write Mixed Strategy: Draw a Diagram ... RW76 Numbers ...... RW97 11.5 Mental Math: 15.5 Compare and Order Division Patterns ...... RW77 Mixed Numbers ...... RW98 11.6 Mental Math: 15.6 Problem Solving Workshop Estimate Quotients ...... RW78 Skill: Sequence 11.7 Place the First Digit ...... RW79 Information ...... RW99

Chapter 12: Practice Division Chapter 16: Add and Subtract Fractions and 12.1 Problem Solving Workshop Mixed Numbers Skill: Interpret the 16.1 Model Addition ...... RW100 Remainder ...... RW80 16.2 Model Subtraction ...... RW101 12.2 Divide 3-Digit Numbers 16.3 Record Addition and Money ...... RW81 and Subtraction ...... RW102 12.3 Zeros in Division ...... RW82 16.4 Problem Solving Workshop 12.4 Choose a Method ...... RW83 Strategy: Write an Equation ...... RW103 Chapter 13: Divide by 2-Digit Divisors 16.5 Add and Subtract 13.1 Estimate Quotients ...... RW84 Mixed Numbers ...... RW104 13.2 Model Division by 2-Digit 16.6 Model Addition and Divisors ...... RW85 Subtraction of Unlike 13.3 Record Division ...... RW86 Fractions ...... RW105 13.4 Adjusting Quotients ...... RW87 13.5 Problem Solving Workshop Chapter 17: Understand Decimals and Place Skill: Too Much/Too Little Value Information ...... RW88 17.1 Relate Fractions and Decimals ...... RW106 Chapter 14: Number Theory and Patterns 17.2 Decimals to Thousandths ... RW107 14.1 Factors and Multiples ...... RW89 17.3 Equivalent Decimals ...... RW108 14.2 Divisibility Rules ...... RW90 17.4 Relate Mixed Numbers 14.3 Prime and Composite and Decimals ...... RW109 Numbers ...... RW91 17.5 Compare and 14.4 Number Patterns ...... RW92 Order Decimals ...... RW110 14.5 Problem Solving Workshop 17.6 Problem Solving Workshop Strategy: Find a Pattern ..... RW93 Skill: Draw Conclusions .....RW111

GG4-NL4-NL TOC_TE.inddTOC_TE.indd 3 66/7/07/7/07 33:44:12:44:12 PPMM Chapter 18: Add and Subtract Decimals and Unit 8: MEASUREMENT AND PROBABILITY Money 18.1 Round Decimals ...... RW112 18.2 Estimate Decimal Sums Chapter 22: Customary and Metric Measurement and Differences ...... RW113 22.1 Measure Fractional Parts ... RW139 18.3 Model Addition ...... RW114 22.2 Algebra: Change 18.4 Model Subtraction ...... RW115 Customary Linear Units .....RW140 18.5 Record Addition 22.3 Weight ...... RW141 and Subtraction ...... RW116 22.4 Customary Capacity ...... RW142 18.6 Make Change ...... RW117 22.5 Problem Solving 18.7 Problem Solving Workshop Workshop Strategy: Strategy: Compare Compare Strategies ...... RW143 Strategies ...... RW118 22.6 Metric Length ...... RW144 22.7 Algebra: Change Metric Unit 7: GEOMETRY Linear Units ...... RW145 22.8 Mass ...... RW146 22.9 Metric Capacity ...... RW147 Chapter 19: Lines, Rays, Angles, and Plane 22.10 Problem Solving Workshop Figures Strategy: Make a Table .....RW148 19.1 Points, Lines, and Rays ...... RW119 19.2 Measure and Chapter 23: Perimeter, Area, and Volume Classify Angles ...... RW120 23.1 Estimate and Measure 19.3 Line Relationships ...... RW121 Perimeter ...... RW149 19.4 Polygons ...... RW122 23.2 Algebra: Find Perimeter ....RW150 19.5 Classify Triangles ...... RW123 23.3 Area of Plane Figures ...... RW151 19.6 Classify Quadrilaterals ...... RW124 23.4 Algebra: Find Area ...... RW152 19.7 Circles ...... RW125 23.5 Problem Solving Workshop 19.8 Problem Solving Skill: Use a Formula ...... RW153 Workshop Strategy: 23.6 Relate Perimeter Compare Strategies ...... RW126 and Area ...... RW154 23.7 Estimate and Find Volume Chapter 20: Motion Geometry of Prisms ...... RW155 20.1 Congruent and Similar Figures ...... RW127 Chapter 24: Probability 20.2 Turns and Symmetry ...... RW128 24.1 List All Possible 20.3 Transformations ...... RW129 Outcomes ...... RW156 20.4 Problem Solving Workshop 24.2 Problem Solving Workshop Strategy: Act It Out ...... RW130 Strategy: Make an 20.5 Tessellations ...... RW131 Organized List ...... RW157 20.6 Geometric Patterns ...... RW132 24.3 Predict Outcomes of Experiments ...... RW158 Chapter 21: Solid Figures 24.4 Probability as a Fraction ....RW159 21.1 Faces, Edges, and Vertices .. RW133 24.5 Experimental Probability ... RW160 21.2 Draw Figures ...... RW134 24.6 Combinations and 21.3 Patterns for Solid Figures .. RW135 Arrangements ...... RW161 21.4 Different Views of Solid Figures ...... RW136 21.5 Problem Solving Workshop Strategy: Make a Model ....RW137 21.6 Combine and Divide Figures ...... RW138

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1 Name Name Place Value Through Hundred Thousands Model Millions You can use a place-value chart to read and write large numbers. One million is the counting number that comes after 999,999. Each group of three digits is called a period. When you write a One million is written as 1,000,000. number, separate each period with a comma. Use base-ten blocks. How many thousands are in 1,000,000? THOUSANDS PERIOD ONES PERIOD Place Hundreds Tens Ones Hundreds Tens Ones Value 100,000 10,000 1,000 100 10 1 One 5 1,000 Think 100 10 1 100 10 1 thousands thousands thousand ones ones one Write four hundred seven thousand, fifty-one Ten 5 10,000 in two other forms. To write a number in standard form, write the number using digits. 407,051 One hundred 5 100,000

To write a number in word form, write the number in each period and follow it with the name of the period. Do not name the ones period. RW1-RW2 One thousand 5 1,000,000 Remember to use commas to separate each period of numbers.

407,051 is four hundred seven thousand, fifty-one. So, there are 1,000 thousands in 1,000,000.

To write a number in expanded form, write the sum of the value of each digit. Solve. 0 Do not include digits for which the value is . 1. How many tens are in 1,000? 2. How many thousands are in 10,000?

407,051 5 400,000 1 7,000 1 50 1 1

Write each number in two other forms. 3. How many hundreds are in 1,000,000? 4. How many tens are in 10,000? 1. 52,321

5. How many ten thousands are in 6. How many hundreds are in 1,000? 1,000,000 2. 965,143 ?

Tell whether the number is large enough to be in the millions or more. Write yes or no. 3. 90,000 1 6,000 1 200 1 80 1 1 7. the number of people in your school 8. the number of hairs on your body

4. one hunded twenty thousand, fifty-six 9. the distance in feet around the world 10. the number of pebbles on a hill

5. six hundred thousand, ninety-four 11. the number of days in a year 12. the number of channels on TV 77/19/07 5:14:40 PM / 1 9 /

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2 Name Name Place Value Through Millions Compare Whole Numbers Look at the digit 3 in the place value chart below. It is in the hundred millions place. Use less than ( < ), greater than ( > ), or equal to ( = ) to compare numbers. In word form, the value of this digit is three hundred million. Compare. Write < , >, or = . $631,328 $640,009 In standard form, the value of the digit 3 is 300,000,000. or

PERIOD • Write one number under the other.

MILLIONS THOUSANDS ONES Line up like places. Hundreds Tens Ones Hundreds Tens Ones Hundreds Tens Ones $631,328 $640,009 3 6 5, 0 0 9, 0 5 8

• Compare the digits, beginning with the greatest place value position. Read the number shown in the place value chart. $631,328 In word form, this number is written as three hundred sixty-five million, nine thousand, fifty-eight. $640,009 You can also write the number in expanded form: • Circle the first digits that are different. RW3-RW4 300,000,000 + 60,000,000 + 5,000,000 + 9,000 + 50 + 8 $631,328 Write each number in two other forms. $64 0,009 1. 600,000,000 1 1,000,000 1 300,000 1 2,000 1 300 1 70 1 8

• Since 4 ten thousands > 3 ten thousands, then $640,009 > $631,328.

Compare. Write < , >, or = for each .

1. 4,092,332 993,793 2. $2,092,092 $2,102,002 2. three million, twenty-one thousand, four hundred

3. 337,493 377,943 4. $5,000,000 $500,000

3. 428,391,032 5. 7,192,322 1,797,322 6. 4,002,384 4,020,000

7. 8,344,475 8,344,475 8. $5,492,000 $492,000

Use the number 46,268,921. 4. Write the value of the digit 6. 5. Write the digit in the ten millions 9. 4,928,388 4,923,488 10. 7,084,122 7,187,084 place.

11. 1,203,437 1,203,445 12. $528,807,414 $5,699,001 77/19/07 5:15:03 PM / 1 9 /

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3 Name Name Order Whole Numbers Problem Solving Workshop Strategy: When you put whole numbers in order, you find which numbers have Use Logical Reasoning the lowest and highest values. What if a basketball team has a score that is a 2-digit number. You can compare each number’s digits to order them from least to greatest. The sum of the digits is 8. The difference between the digits is 2. Write the numbers in order from least to greatest. The tens digit is less than the ones digit. What is the score? $55,997; $57,000; $56,038 Read to Understand

• Align the numbers under each other. 1. What are you asked to find? $55,997 $57,000 $56,038 Plan 2. How can using logical reasoning help you solve the problem? • Compare the digits beginning with the greatest place value position. $55,997 • Start with the digit in the ten thousands place. $57,000 • The digit 5 is the same in all 3 numbers • The next digit is in the thousands place. Solve

RW5-RW6 $56,038 3. What is the score? Describe how you used logical reasoning. • Circle the first group of digits that are different.

$55,997 • Compare the digits 5, 7, and 6. $57,000 • Seven is greater than 5 and 6, so 57,000 is the greatest number. $56,038 • Five is less than 6, so 55,997 is less than 56,038.

Since 5 < 6, and 6 < 7, the numbers in order from least to greatest are: 4. Write your answer in a complete sentence. $55,997; $56,038; $57,000.

Write the numbers in order from least to greatest. Check

1. 787,925; 1,056,000; 789,100 5. Look at the problem. Does the answer make sense for the problem? Explain.

2. 94,299; 82,332; 100,554 Solve by using logical reasoning. 6. Mr. Lee’s class sold 140 muffins and 7. Shari’s soccer team won every game of 3. $1,354,299; $1,942,332; $1,300,554 235 cookies at the bake sale. The the season. Her team played every number of cupcakes sold was between Saturday and Sunday for three weeks the number of muffins and cookies each month. Suppose they played this sold. What is the greatest number of schedule for three months. How many 7,234,000; 6,311,094; 2,102,444 4. cupcakes the class could have sold? games did Shari’s soccer team win? 77/19/07 5:15:25 PM / 1 9 /

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4 Name Name Algebra: Relate Addition and Subtraction Round Whole Numbers Through Millions You can solve addition and subtraction sentences by using When you round a number, you replace it with a number that is easier to work with 7 8 15 8 7 15 related facts. For example, + = and + = are but not as exact. You can round numbers to different place values. related addition facts; 15 – 8 = 7 and 15 – 7 = 8 are related subtraction facts. Round 3,476,321 to the place of the Round 8,851,342 to the nearest underlined digit. thousand. Write a related fact. Use it to complete the number sentence. • Identify the underlined digit. • Identify the digit in the thousands place. 2 9 5 6 j The underlined digit is a 4 in the j The 1 to the left of the comma is Use an inverse operation. hundred thousands place. in the thousands place. • • • An inverse operation is an opposite operation. Addition is the inverse, or Look at the number to the right of the Look at the number to the right of the the opposite, operation of subtraction. Subtraction is the opposite of addition. underlined digit. digit in the thousands place.

0 4 j 0 4 1 2 9 5 6 j If that number is - , the If that number is - , the stays underlined digit stays the same. the same. Use inverse operations to write the problem as an addition sentence. j If that number is 5-9, the j If that number is 5-9, the 1 increases 9 1 6 5 15 underlined digit increases by 1. RW7-RW8 by 1. Use a fact family to find the missing number. j The number to the right of the j The number to the right of the thousands place is a 3, so the 9 1 6 5 15 6 1 9 5 15 underlined digit is a 7, one in the thousands place will so the underlined digit, 4, will stay the same. 15 2 6 5 9 15 2 9 5 6 increase by one; 4 1 5 + = . • Change all the digits to the right of the So, 15 2 9 5 6. • Change all the digits to the right of the thousands place to zeros. hundred thousands place to zeros. Write a related fact. Use it to complete the number sentence. So, 3,476,321 rounded to the nearest So, 8,851,342 rounded to the nearest hundred thousand is 3,500,000. thousand is 8,851,000. 1. 6 1 5 13 2. 9 2 5 733. 1 5 11 4. 16 2 5 7

Round each number to the place value of the underlined digit. 1. 3,452 2. 180 3. $72,471 4. 5,723,000 5. 1 8 5 16 6. 10 2 5 4 7. 1 12 5 15 8. 2 4 5 13

Round each number to the nearest ten, thousand, hundred thousand, and million.

9. 12 1 5 2010. 15 2 5 8911. 2 5 3512. 1 5 21 5. 2,472,912 6. 1,333,456 66/18/07 5:00:56 PM / 1 8

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5 Name Name Mental Math: Estimate Sums and Differences Mental Math Strategies When you estimate, you use numbers that are not as exact as the There are strategies that can help you add or subtract mentally. The Break Apart original numbers but are easier to use. To find a difference that is Strategy, the Friendly Number Strategy, and the Swapping Strategy are three close to the exact number, sum of you can estimate. of the strategies you can use. Add or Subtract mentally. Tell the strategy you used. Use rounding to estimate. Break Apart Strategy Friendly Number Strategy Swapping Strategy Estimate the sum. 867 2 425 867 2 425 145 1 213 Round each number to the nearest thousands place. • Break apart the • Friendly numbers end • Swap digits with the Add the estimated numbers. place values. with a zero. same place value. 9,000 • Subtract each • Add to make one of your • Add to make one of 9,362 place value numbers end with zero. your numbers end with __1 5,781 __1 6,000 separately. zero. 15,000 867 1 3 5 870 800 2 400 5 400 145 1 5 5 150 870 is a friendly number. So, the estimated sum is 15,000. 60 2 20 5 40

RW9-RW10 • Adjust the other 7 2 5 5 2 • Adjust the other number by number by subtracting Use front-end estimation to estimate. adding the same amount. • Add the differences. the same amount. 425 1 3 5 428 213 2 5 5 208 Subtract the front digits. 400 1 40 1 2 5 442 • Subtract the new numbers. • Add the new numbers. Change all the other digits to zeros. 870 2 428 5 442 150 1 208 5 358 70,000 73,206 So, 867 2 425 5 442. __2 21,358 __2 20,000 145 1 213 5 358. 887 2 425 5 442. So, 50,000 So, You can also use this 50,000 method for addition So, the estimated difference is . You can only use this method for You can only use this and subtraction. subtraction. method for addition. Use rounding to estimate. 7,000,115 54,823 640 7,980 Add or subtract mentally. Tell the strategy you used. 78,402 2. 3. 4. 5. 1. __1 91,113 ___1 4,000,212 __2 36,911 __1 480 __2 1,341 714 + 224 322 138 83 61 150 350 293 168 1. 2. + 3. – 4. + 5. –

Use front-end estimation to estimate. 1,642 6,789

32,116 17,450 22,001 6. 7. 8. 9. 10. 6. 428 – 364 7. 69 + 81 8. 654 – 270 9. 36 – 22 10. 187 + 250 __2 834 __1 71,930 __1 81,942 __2 4,663 __2 926

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6 Name Name Problem Solving Workshop Skill: Add and Subtract 3-Digit and 4-Digit Numbers Estimate or Exact Answer You can use place value to add and subtract large numbers. An estimate will help you know if your answer is reasonable. To be an airline pilot, you must fly a total of at least 1,500 hours. Estimate. Then find the sum. Dan flew 827 hours last year and 582 hours this year. How many 1,000 1 3,000 5 4,000 726 more hours must Dan fly to be an airline pilot? Round the numbers to estimate: . __1 2,643 So, 726 1 2,643 is about 4,000.

1. What are you asked to find? Add the ones. Add the tens. Add the hundreds Add the thousands. Regroup 13 hundreds.

1 1 726 726 726 726 2. Should you use an estimate or an exact answer? __1 2,643 __1 2,643 __1 2,643 __1 2,643 9 69 369 3,369 3. Solve the problem in the space below. What operations did you use?

Since 3,369 is close to the estimate of 4,000, the answer is reasonable. W1RW12 RW11 Solve: Operations Used: Estimate. Then find the difference. 6,000 2 4,000 5 2,000 6,000 Round the numbers to estimate: . __2 4,275 So, 6,000 2 4,275 is about 2,000.

Regroup 6 thousands Regroup 10 hundreds Regroup 10 tens as Subtract. as 5 thousands, as 9 hundreds, 9 tens, 10 ones. 10 hundreds. 10 tens. 9 99 99 4. How many more hours must Dan fly to be an airline pilot? 5 10 510 10 5 1010 10 5 10 10 10 Write your answer in a complete sentence. 6, 0 0 0 6, 0 0 0 6,000 6,000 2224, 2 7 5 4, 2 7 5 4,275 24,275 1, 7 2 5 5. How can you tell if your answer is correct? Since 1,725 is close to the estimate of 2,000, the answer is reasonable.

Estimate. Then find the sum or difference. 2,314 613 415 5,796 1,427 1. 2. 3. 4. 5. Explain whether to estimate or find an exact answer. 1__ 342 1__ 3,986 __1 123 __1 427 1__ 4,967 Then solve the problem. 6. Jane took a math test. She answered 7. Lance spent 180 days in Mr. Lee’s 134 items correctly in the first section class and 176 days in Mrs. Mac’s 113 and items correctly in the second class. About how many days did 7,000 1,376 4,931 5,000 6. 7. 2,527 8. 9. 10. section. How many items did Jane Lance spend in Mr. Lee’s and __2 251 2__ 1,761 __2 149 2__ 3,187 2__ 4,291 answer correctly in all? Mrs. Mac’s classes in all? 77/19/07 5:15:48 PM / 1 9 /

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7 Name Name Choose a Method Addition Properties Choose the method that works best for solving an addition Commutative Property or a subtraction problem. Think about the question being asked, The Commutative Property states that you can add numbers in any order and then use one of the following three methods. and the sum will be the same. For example, 2 1 3 5 5 and 3 1 2 5 5. Paper and Pencil The order in which you add numbers does not change the sum. Paper and pencil can be used with greater numbers Find the missing number. and when the answer has to be exact. 47 1 j 5 56 1 47 56 1 47 uses the same numbers 47 1 j Calculator as , but in a different order.

A calculator can be used with greater numbers that need So, j 5 56. 47 1 56 5 56 1 47 to be regrouped and when the answer has to be exact. Associative Property Mental Math The Associative Property states that the way addends are grouped does not change Mental math can be used when using rounded or smaller numbers. the sum. For example, (4 1 2) 1 5 and (5 1 4) 1 2 both equal 11. The way you group numbers does not change the sum. Find the difference. 945,322 2 461,070 RW13-RW14 Find the missing number. Since numbers are Work from right to left. Continue subtracting. j 1 (31 1 18) 5 (24 1 31) 1 18 (24 1 31) 1 18 greater and you need an Regroup when needed. uses the same numbers as j 1 (31 1 18), except exact answer, use pencil 2112 2148 2 945,322 grouped together differently. 945,322 and paper. j 5 24 So, . 24 1 (31 1 18) 5 (24 1 31) 1 18 2461,070 2461,070 945,322 __ __ 2__461,070 252 484,252 Identity Property

The Identity Property states that when you add zero to any number, So, 945,322 2 461,070 5 484,252. the sum is that number. For example, 45 1 0 5 45.

Find the sum or difference. Write the method used. Find the missing number. 62 1 j 5 62

1,420,300 620,000 3,721,682 Adding a zero to 62 does not change 62. 62 1 0 5 62 378,452 1. 2. 3. 4. So, j 5 0. __ 2 193,511 ___ 1 1,678,500 __ 2 419,500 ___ 1 6,161,248

Find the missing number. Name the property you used. 1. 39 1 j 5 39 2. (7 1 j ) 1 8 5 7 1 (2 1 8)

6,280,000 701,000 7,573,150 621,899 5. 6. 7. 8. ___ 1 1,200,000 __ 1 213,655 ___ 1 1,300,000 __ 2 107,000 3. j 1 40 5 40 1 22 4. 3 1 (14 1 6) 5 (3 1 j ) 1 6

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8 Name Name Write and Evaluate Expressions Addition and Subtraction Equations An expression is part of a number sentence that has numbers An equation is a number sentence stating that two amounts and signs such as “1,” “2,” and “3,” but it does not have an “5” are equal. sign. You can find the value of an expression by adding, Write an equation. Use 1, 2 or 5 for each . subtracting, multiplying, and/or dividing. When there are Emil has 18 stamps. He uses some parentheses in an expression, you need to do the operation in stamps, s, and now has 12 stamps left. parentheses first. 18 12 Find the value of the expression. s 18 2 12 13 2 n if n 5 6 • Since the variable, s, represents s the number of stamps Emil used, 6 13 2 6 • Replace n with in the equation. you will subtract. Solve the expression. 13 2 6 5 7 • • Since 12 is the number of stamps 18 2 s 5 12 So, the value of the expression is 7. Emil has left, you will use the equal sign. Write an expression with a variable. Tell what the variable represents. 18 2 5 12 RW15-RW16 So, the equation is s . Alisha found some shells on the beach. She gave 4 of them to Lance. Solve the equation. • Choose a variable to represent the unknown Let s represent the shells 13 2 5 4 number in the problem. Alisha found. n 13 2 5 4 • Alisha gave away 4 shells, so 4 were • Use mental math to find which number n represents. n 13 4 subtracted from her total. Think: “ minus what number equals ?” 9 13 2 9 5 4 • Write the expression. s 2 4 • Replace n with in the equation. 4 5 4 So, the expression is s 2 4, and the variable s • The answer is correct. represents the shells Alisha found. So, n 5 9. Write an equation for each. Use 1, 2 or 5 for each . Find the value of each expression. 1. Tim has 16 socks. After he loses 2. There are 30 bowls on the shelf. 12 1 if 5 8 5 1 (7 1 7) 14 2 (5 1 3) 1. j j 2. 3. some of his socks, s, he has 10 socks There are some white bowls, b, and left. 12 green bowls. 16 5 10 30 12 4. j 1 6 if j 5 4 5. (11 1 2) 2 4 6. 15 1 (3 1 7) s b

Write an expression with a variable. Tell what the variable represents. Solve the equation. 8 1 j 5 20 2 9 5 8 4 1 5 12 7. Karl has some books. He gives 8. Mary has 5 pens. She finds 3. 4. n 5. k 3 to Joe. some more.

6. y 1 5 5 13 7. 6 1 j 5 12 8. 11 2 j 5 8 77/19/07 5:16:09 PM / 1 9 /

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9 Name Name Problem Solving Workshop Strategy: Patterns: Find a Rule Work Backward The input and output numbers in a table are related by a rule. Find the relationship between the two numbers. Many volunteer teams must patrol and clean the lion preserve. Find a rule. Write your rule as an equation. Twelve teams leave the preserve on patrol. Seven teams arrive Use the equation to extend your pattern. to clean. There are 23 teams at the preserve now. How many volunteer teams were there originally? Input a 12 25 31 43 59 Read to Understand Output b 20 33 39 jj 1. What are you asked to find? • Find a relationship between each input number and output number. 12 1 8 5 20 25 1 8 5 33 31 1 8 5 39 Plan • The rule is to add 8 to each input number to get the output number. Write the rule. 2. How can you use the work background strategy to solve this problem? a 1 8 5 b Use the rule to complete the table.

RW17-RW18 • Solve 43 1 8 5 51 59 1 8 5 67 3. Solve the problem. Show your work below. Input a 12 25 31 43 59 Output b 20 33 39 51 67 • The rule is add 8 to a, or a 1 8 5 b. So, use 51 and 67 to extend your pattern.

4. How many volunteer teams were there originally? Find a rule. Use your rule to find the next two ordered pairs. Check 1. Input c 62 58 47 jj 2. Input e 31 46 59 73 90 5. Is there another strategy you could use to solve the problem? Output d 57 53 42 31 24 Output f 42 57 70 jj

Work backwards to solve. 4:30 7. There were some cars in a parking 6. Pete got home at P.M. It took him 3. Input g 18 24 57 jj 4. Input j 14 17 22 34 56 20 minutes to walk from school to the lot in the morning. Later, 15 more cars Output h 9 15486284 Output k 4712jj library. He was at the library for 1 hour. arrived then 9 cars left. There are now It took Pete 5 minutes to walk from the 60 cars in the parking lot. How many library at home. At what time did Pete cars were originally in the parking lot? leave school? 77/19/07 5:16:33 PM / 1 9 /

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1 0 Name Name Algebra: Relate Operations Algebra: Relate Multiplication and Division When you add or multiply, you join equal-size groups. Multiplication and division are opposite operations, or inverse operations. A fact family When you subtract or divide, you separate into equal-size is a set of multiplication and division sentences that use the same numbers. groups or find out how many are in each group. Write the fact family for the set of numbers. Write the related multiplication sentence. 2, 3, 6 Draw a picture that shows the sentence. 2 3 3 5 6 6 4 3 5 2 3 1 3 1 3 1 3 1 3 5 15 Divide 6 triangles into 3 equal groups.

Start with 0 on the number line. Skip count by 3’s five times. Stop at 15. 2 rows of 3 triangles each. There are 2 triangles in each group. So, 2 3 3 5 6 . So, 6 4 3 5 2. ' ( ) * + , - . / 0 (' (( () (* (+ (, 3 3 2 5 6 6 4 2 5 3 5 groups of 3 is the same as 5 3 3. 6 2 So, the related multiplication sentence is 5 3 3 5 15. Divide triangles into equal groups. RW19-RW20 Write the related division sentence. Draw a picture that shows the sentence. There are 3 triangles in each group. 3 2 9 2 3 2 3 2 3 5 0 rows of triangles each. So, 3 3 2 5 6. So, 6 4 2 5 3. Start with 9 on the number line. Subtract 3 three times. Stop at 0. So, the fact family for 2, 3, 6 is 2 3 3 5 6, 3 3 2 5 6, 6 4 3 5 2, 6 4 2 5 3.

Write the fact family for the set of numbers. ' ( ) * + , - . / 0 1. 2, 9, 18 2. 3, 5, 15 A group of 9 divided into equal groups of 3 is the same as 9 4 3. So, the related division sentence is 9 4 3 5 3.

Write the related multiplication or division sentence. Draw a picture that shows the sentence. 3. 1, 8, 8 4. 2, 8, 16 1. 2 1 2 1 2 1 2 1 2 5 10 2. 15 2 3 2 3 2 3 2 3 2 3 5 0

5. 5, 8, 40 6. 3, 8, 24

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1 1 Name Name Multiply and Divide Facts Through 5 Multiply and Divide Facts You can use a multiplication table to help you find products and quotients. Through 10 Find the product. Find the product or quotient. Show the strategy you used. ' ( ) * + , 50 4 10 3 3 3 ' '''''' • Solve using inverse operations, so use multiplication facts you know. • The first factor is 3. Think: 10 3 5 5 50 Find the row labeled 3. ( '()*+, ) ')+-/(' • So, 50 4 10 5 5. • The second factor is 3. Find the column labeled 3. * ' * - 0 () (, + '+/()(-)' 3 3 9 • Trace with your fingers across row 3 and down column 3. , ' , (' (, )' ), • Draw a rectangular array that is 3 units wide and 9 units long. • Where row 3 and column 3 meet is the - ' - () (/ )+ *' product, 9. . ' . (+ )( )/ *, 3 3 3 5 9 / ' / (- )+ *) +' RW21-RW22 • So, . • Count the squares in the array. 0 ' 0 (/ ). *- +, • There are a total of 27 squares. Find the quotient. 3 3 9 5 27 ' ( ) * + , • So, . 15 4 5 ' '''''' • The divisor is 5. Find the row labeled 5. ( '()*+, Find the product or quotient. Show the strategy you used. • The dividend is 15. Follow across row 5 ) ')+-/(' 5 3 4 5 12 3 2 5 until you find the number 15. 1. 2. * ' * - 0 () (, • From 15, follow your finger up to the + '+/()(-)' 30 4 5 5 81 4 9 5 top of the column. 3. 4. , ' , (' (, )' ), • The number you find is the quotient, 3. - ' - () (/ )+ *' 5. 7 3 8 5 6. 7 3 7 5 • So, 15 4 5 5 3. . ' . (+ )( )/ *, / ' / (- )+ *) +' 7. 24 4 6 5 8. 6 3 9 5 0 ' 0 (/ ). *- +,

9. 5 3 9 5 10. 9 3 7 5 Find the product or quotient.

1. 8 4 2 2. 5 3 3 3. 4 3 2 4. 25 4 5 5. 24 4 3 11. 6 3 3 5 12. 28 4 4 5

13. 44 4 4 5 14. 4 3 0 5 6. 4 3 6 7. 20 4 5 8. 3 3 6 9. 24 4 4 10. 7 3 4

15. 32 4 8 5 16. 64 4 8 5 77/19/07 5:16:52 PM / 1 9 /

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1 2 Name Name Multiplication Table Through 12 Patterns on the Multiplication Table You can use a multiplication table to multiply and divide numbers through 12. When a whole number is multiplied by another whole number, the product is called a multiple. For example, 5 3 3 5 15, so 15 is a multiple of 5 and 3. Use the multiplication table to column find the product of 7 3 11. When a number is multiplied by itself, the product is called a square number. ' ( ) * + , - 0/. (' (( () For example, 4 3 4 5 16, so 16 is a square number. 7 • The first factor is . ' ' ' ' ' ' ' ' ' ' ' ' ' ' column 7 ( ' ( ) * + , - . / 0 (' (( () Find the row labeled . ' ( ) * + , - 0/. (' (( () ) ' ) + - / (' () (+ (- (/ )' )) )+ 11 Which multiples have only even numbers? ' '''''''''' ' ' ' • The second factor is . * ' * - 0 () (, (/ )( )+ ). *' ** *- ( '()*+,-./0 (' (( () Find the column labeled 11. + ' + / () (- )' )+ )/ *) *- +' ++ +/ Look at the rows in the multiplication table. ) ')+-/('()(+(-(/ )' )) )+ * ' * - 0 () (, (/ )( )+ ). *' ** *- ' , (' (, )' ), *' *, +' +, ,' ,, -' • Trace your finger across row 7 , Look for a row where all the multiples are even + '+/() (- )' )+ )/ *) *- +' ++ +/ row - ' - () (/ )+ *' *- +) +/ ,+ -' -- .) , ' , (' (, )' ), *' *, +' +, ,' ,, -' 11 - ' - () (/ *')+*- +) +/ ,+ -' -- .) and down column . . ' . (+ )( )/ *, +) +0 ,- -* .' .. /+ numbers. row . ' . (+ )( )/ *, +) +0 ,- -* .' .. /+ • Where row 7 and column 11 / ' / (- )+ *) +' +/ ,- -+ .) /' // 0- All the multiples for the numbers 2, 4, 6, 8, 10, / ' / (- )+ *) +' +/ ,- -+ .) /' // 0- 0 ' 0 (/ ). *- +, ,+ -* .) /( 0' 00 ('/ 0 ' 0 (/ ). *- +, ,+ -* .) /( 0' 00 ('/ meet is the product, 77. and 12 are even. (' '(')'*'+','-'.'/'0' ('' ((' ()' (' ' (' )' *' +' ,' -' .' /' 0'('' ((' ()' (( ' (( )) ** ++ ,, --.. // 00 ((' ()( (*) • So, 7 3 11 5 77. (( '( ( )) ** ++ ,, --.. // 00 ((' ()( (*) () ' () )+ *- +/ -' .) /+ 0-('/ ()' (*) (++ () RW23-RW24 ' () )+ *- +/ -' .) /+ 0-('/()'(*) (++ column Use the rule to find the missing numbers in the table at the right. Divide the input by 12. Use the multiplication table to find the ' ( ) * + , - 0/. (' (( () ' '''''''''' ' ' ' 11 3 11. • Find the row for the divisor, 12 . Input Output square number for ( '()*+,-./0 (' (( () ) ')+-/('()(+(-(/ )' )) )+ 12 36 Find the column and row for the number 11. * ' * - 0 () (, (/ )( )+ ). *' ** *- • Trace your finger across row + '+/()(-)')+)/*)*- +' ++ +/ to find the dividend, 36. 72 6 Use your finger to move across row 11 and , ' , (' (, )' ), *' *, +' +, ,' ,, -' - ' - () (/ )+ *' *- +) +/ ,+ -' -- .) row 36 3 108 9 down column 11. . ' . (+ )( )/ *, +) +0 ,- -* .' .. /+ • Since is in the column for , / ' / (- )+ *) +' +/ ,- -+ .) /' // 0- 3 is the output. 144 12 Your fingers meet at 121. 0 ' 0 (/ ). *- +, ,+ -* .) /( 0' 00 ('/ (' '(')'*'+','-'.'/'0' ('' ((' ()' • So, 36 4 12 5 3. 11 3 11 = 121, so 121 is the square number. (( '( ( )) ** ++ ,, --.. // 00 ((' ()( (*) () ' () )+ *- +/ -' .) /+ 0-('/ ()' (*) (++

Find the product or quotient. Show the strategy you used. Find the square number.

1. 2 3 12 2. 12 4 3 3. 4 3 11 4. 66 4 6 1. 5 3 5 2. 8 3 8 3. 2 3 2 4. 4 3 4

ALGEBRA Use the rule. 5. 7 3 7 6. 10 3 10 7. 9 3 9 8. 12 3 12 5. Multiply the input by 12. 6. Divide the input by 11.

Input Output Input Output 4 33 Use the multiplication table.

60 77 9. Do any numbers have only 10. What pattern do you see in the multiples odd-numbered multiples? of 5? 7 8

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1 3 Name Name Problem Solving Workshop Skill: Algebra: Find Missing Factors Choose the Operation You can draw models and use fact families to find missing factors. The hiking trip is on Friday. On Thursday, the high temperature Find the missing factor. was 678F. Russell hopes that it will be 8º warmer on Friday. 10 3 g 5 70 What temperature is Russell hoping for? OOOOOOOOOO Use a model to find the value of g. OOOOOOOOOO 1. What are you asked to find? Use the product, 70, and the factor, 10, to draw 70 counters in rows of 10. OOOOOOOOOO OOOOOOOOOO Count the number of rows to find the missing factor. OOOOOOOOOO 2. Will you separate a group into smaller groups to solve? Why or why not? There are 7 rows of 10, so the missing factor, g, is 7. OOOOOOOOOO So, 10 3 7 5 70, or g 5 7. OOOOOOOOOO Find the missing factor. 3. Which operation will you use to answer the question? 3 11 5 132 RW25-RW26 Q Use a related division sentence to find the missing factor, Q. Think of the fact family and write the 4. What was the high temperature on Thursday? How many degrees will you related division sentence. 132 4 11 5 Q add to Thursday’s temperature to get Friday’s temperature? Solve for Q. 132 4 11 5 12, or Q5 12 The missing factor is 12. 5. What temperature is Russell hoping for on Friday? Write a number sentence So, 12 3 11 5 132, or Q5 12. to solve. Find the missing factor. 1. 7 3 n 5 28 2. 5 3 p 5 30 3. m 3 5 5 45 4. 6 3 n 5 48 6. How can you check your answer?

Tell which operation you would use to solve the problem. 5. 8 3 g 5 72 6. w 3 11 5 55 7. 4 3 j 5 32 8. 9 3 v 5 27 Then solve the problem. 7. Maria has 24 apples. She wants to 8. Gina has 5 boxes. She places give an equal number of apples to 6 books in each box. How many each of her 4 friends. How many books does Gina have in all? apples should Maria give each 9. 4 3 g 5 12 10. 9 3 v 5 81 11. 3 3 n 5 36 12. 10 3 z 5 80 friend? 66/18/07 5:02:49 PM / 1 8

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1 4 Name Name Multiplication Properties Multiplication and Division Expressions Use properties and mental math to find the product. Some multiplication and division expressions have variables that Associative Property Distributive Property represent numbers. A variable is a letter or symbol that stands for any number. Examples of multiplication and division expressions The Associative Property states that the The Distributive Property states that you are 9 3 k and h 4 3. way factors are grouped does not change can break apart one of the factors and the product. multiply its parts by the other factor. Write an expression that matches the words. A handful of keys, k, divided equally and put on 4 key chains. 8 3 3 3 3 7 3 11 Write a variable to represent the unknown Let k represent the keys. 8 3 3) 3 3 11 • ( uses the same numbers as • Break apart so you can easily use number in the problem. 8 3 (3 3 3), except they are grouped mental math. The keys are being separated into four groups, 4 4 together differently. 11 5 10 1 1 k so the operation is division. • Choose which set of factors you can • Multiply both parts by 7, and then add So, the expression is k 4 4. easily use mental math to solve. the products. 7 3 10 5 70 8 3 (3 3 3) or (8 3 3) 3 3 Find the value of the expression. 7 3 1 5 7 8 3 9 5 72 24 3 3 5 72 s 4 3 if s 5 27 RW27-RW28 70 1 7 5 77 So, 8 3 3 3 3 5 72. The variable s represents 27. s 4 3 , 7 3 11 5 77. So Replace s with 27. 27 4 3 Commutative Property Solve using division. 27 4 3 5 9 The Commutative Property states you can multiply numbers in any order and the product So, the value of s 4 3 is 9, if s 5 27. will be the same. The order in which you multiply numbers does not change the product. For example, 10 3 3 and 3 3 10 both have a product of 30. Write an expression that matches the words. Zero Property Identity Property 1. 7 times more books than before 2. the price of some pens, at $1 each The Zero Property states that the product The Identify Property states that the of any number and zero is zero. For product of any number and 1 is that example, 12 3 0 5 0, 56 3 0 5 0, and number. For example, 9 3 1 5 9, 864 3 0 5 0 56 3 1 5 56, 864 3 1 5 864 . and . 3. 40 plates divided equally and put on 4. some shoes divided equally and put into 8 boxes Use the properties and mental math to find the product. some tables 1. 6 3 2 3 4 2. 8 3 12 3. 6 3 0 3 4

Find the value of the expression. 4. 2 3 13 3 1 5. 5 3 14 6. 5 3 5 3 3 5. b 3 5 if b 5 4 6. 60 4 k if k 5 10 7. 8 3 w if w 5 6

7. 4 3 6 3 3 8. 2 3 0 3 3 9. 8 3 1 3 5 8. m 4 4 if m 5 36 9. s 3 9 if s 5 5 10. h 4 7 if h 5 49

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1 5 Name Name Order of Operations Multiplication and Division Equations To solve problems that have more than one type of operation, you Write an equation. Choose a variable for the unknown. need to know the order of operations. This is the set of rules Tell what the variable represents. that tells which operation to do first. First, solve any operations in parentheses. The total number of rings divided equally Next, multiply and divide from left to right. Then, add and subtract from left to right. among 4 friends is 2 rings for each friend. Follow the order of operations to find the value of each expression. • Decide on a variable to represent The variable, r, will represent the 3 3 (6 2 2) 1 4 4 2 the unknown amount. total number of rings. Perform the calculations 3 3 (6 2 2) 1 4 4 2 • Choose an operation to solve The rings are being separated into inside the parentheses first. the problem. four equal groups, so the operation 3 3 4 1 4 4 2 is division. Think: 6 2 2 5 4 The rings are divided into equal r 4 4 5 2 Multiply and divide from left to right. • 3 3 4 1 4 4 2 groups of 2 rings. Think: 3 3 4 5 12 and 12 1 2 So, the equation is r 4 4 5 2, and the variable, r, represents the total number of rings. 4 4 2 5 2.

RW29-RW30 Solve the equation. Add and subtract from left to right. 12 1 2 a 3 7 5 63 Think: 12 1 2 5 14 • Use mental math to find which What number times 7 equals 63? 14 number is represented by a. So, 3 3 (6 2 2) 1 4 4 2 5 14 • You know that 9 3 7 5 63. a 5 9 • Replace a with 9. 9 3 7 5 63 Follow the order of operations to find the value of the expression. 63 5 63 1. (12 1 8) 4 (4 2 2) 2. (8 3 3) 2 6 3. 4 3 (3 3 2) So, the value of a is 9.

Write an equation for each. Choose a variable for the unknown. Tell what the variable represents. 4. 14 2 (5 1 2) 5. (2 3 4) 1 (5 3 1) 6. 5 1 10 4 2 1. 5 bowls with a number of grapes in 2. $40 divided equally among a each is 30 grapes. number of friends is $10 each.

7. 7 3 6 1 6 8. (8 3 4) 4 (3 1 1) 9. 2 3 (12 4 4) 2 3 Solve the equation. 3. b 3 7 5 35 4. 60 4 k 5 10 5. 9 3 j 5 45

10. 9 1 (6 3 2) 4 2 11. 10 4 2 3 5 12. (8 3 5) 2 8 4 4

6. 36 4 m 5 6 7. s 3 3 5 12 8. h 4 6 5 7 66/18/07 5:03:13 PM / 1 8

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1 6 Name Name Problem Solving Workshop Strategy: Explore Inequalities Predict and Test Remember: $ means “greater than or equal to,” and # means “less than or equal to.” Marc likes to solve word scrambles and mazes. Yesterday he solved 10 word scrambles and mazes in all. He solved 2 more word scrambles Which of the numbers 2, 3, 5, and 6 make the inequality true? than mazes. How many word scrambles did Marc solve yesterday? m 2 2 $ 3 a , 7 Read to Understand • Substitute each number for m. Try each of the numbers to see 1. What are you asked to find? • Substitute 2 for m. 2 2 2 $ 3. which ones make the inequality true. 0 $ 3, so 2 does not work. Try 2. 2 , 7. Since 2 , 7, 2 Plan • Substitute 3 for m. 3 2 2 $ 3. works. Try 3. 3 , 7. 2. How can you use the predict and test strategy to help you solve the problem? 1 $ 3, so 3 does not work. Since 3 , 7, 3 works. • Substitute 5 for m. 5 2 2 $ 3. Try 5. 5 , 7.

RW31-RW32 3 $ 3, so 5 works. 5 , 7 5 Solve Since , works. Substitute 6 for m. 6 2 2 $ 3. 3. Use the table to help you solve the problem. • Try 6. 6 , 7. 4 $ 3, so 6 works. Since 6 , 7, 6 works. Predict Test Does it Check? So, 5 and 6 make the inequality true. So, 2, 3, 5, and 6 make the inequality true. Difference Total 6 scrambles 4 mazes Which of the numbers 2, 5, 8, and 9 make the inequality true? 5 scrambles 1. b # 7 2. n 3 2 $ 14 3. a , 12 5 mazes

Which of the numbers 4, 6, 8, and 10 make the inequality true? Number of word scrambles Marc solved: 4. 5 $ k 5. h 3 2 $ 13 6. 10 # 4 1 w Check

4. Does the answer make sense for the problem? Explain.

Predict and test to solve. Which of the numbers 1, 3, 7, and 9 make the inequality true? 5. The product of two numbers is 42. 6. Sue has three times as much money 7. 7 # y 8. r 3 3 # 25 9. 3 # h Their sum is 13. What are the numbers? as Dan. Together they have $40. How much money does each person have? 66/18/07 5:03:28 PM / 1 8

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1 7 Name Name Patterns: Find a Rule Telling Time When you are asked to to find a rule, look for a pattern in the On a digital clock, the colon (:) separates the hours from the numbers. See how different pairs of numbers are related. minutes. The number to the left of the colon tells the hour. The number to the right of the colon tells the minutes. Find a rule. Write your rule as an equation. Use your rule to find the missing numbers. Write the time as shown on a digital clock.

Input, b 90 70 60 50 30 20 10 47 minutes after twelve 20 minutes before four Output, c 976 • Determine the hour. • Determine the hour. Look for a pattern in the numbers to find a rule. • Since it is 47 minutes after twelve, the It is 20 minutes before four, so it is not Study how the input numbers relate to the output numbers. • • hour is still 12:00. four yet. The hour is 3:00. The first pair is 90 and 9. 90 4 10 5 9. • Determine the minutes. Determine the minute. The second pair is 70 and 7. 70 4 10 5 7. • • The rule is “divide by 10.” • 47 minutes have gone by, so the • There are 60 minutes in an hour. In 20 minute is 0:47. minutes, it will be 4:00. • Write your rule as an equation. RW33-RW34 You are dividing b by 10 to get c, so the equation is b 4 10 5 c. • So, the time is 12:47. • Subtract: 60 minutes 2 20 minutes 5 40 minutes, or 40 minutes have Find the missing numbers in the table using the equation. • passed. The minute is 0:40. 50 4 10 5 5; 30 4 10 5 3; 20 4 10 5 2; 10 4 10 5 1 So, the time is 3:40. So, the rule is “divide by 10,” the equation for the rule is b 4 10 5 c, and the missing • numbers in the table are 5, 3, 2, and 1. Write the time as shown on a digital clock. Find a rule. Write your rule as an equation. Use your rule to find the missing numbers. 1. 5 minutes after seven 2. 50 minutes after three 3. 22 minutes before five 1. 2. Input, r 23568 Input, a 367910

Output, s 18 27 45 Output, b 12 24 28

4. 46 minutes after nine 5. 5 minutes after two 6. 20 minutes after eight

3. Input, r 6 9 18 21 33 4. Input, p 24 689

Output, s 236 Output, m 12 24 36 7. 11 minutes after twelve 8. 30 minutes before 9. 25 minutes after four three

5. 6. 5 10 48 Input, e 50 30 25 10 Input, w 80 70 40 1 0 . minutes before 11. minutes before 12. minutes after seven eleven one Output, f 10 6 5 1 Output, z 40 35 20 10 5

77/19/07 5:17:12 PM / 1 9 /

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1 8 Name Name Elapsed Time Elapsed Time on a Calendar

Elapsed time tells how much time has passed from the beginning to the Li went to the dentist on January 15. If today January end of an activity. The end time is the time when an activity ends. is February 29, how many weeks and days is Sun Mon Tue Wed Thu Fri Sat it since she saw the dentist? Find the elapsed time. Use a clock to find the end time. 12345 15 15 12:00 3:07 From January to February is one month. start: A.M. start: A.M. 6789101112 end: 3:07 P.M. elapsed time: 8 hr 55 min Then count the days from Friday, February 15 to Friday, February 29. 13 14 15 16 17 18 19 • Count the hours. Be sure to check the • Add the elapsed time to the start time. A.M. and P.M. That is 14 days. 20 21 22 23 24 25 26 8 hr 55 min • Begin at 12:00 A.M. (start time) and 1 3 hr 7 min So, it has been 6 weeks and 3 days since 27 28 29 30 31 12:00 12 ___ count to P.M. That is hours. 11 hr 62 min Li saw the dentist. 12:00 P.M. to 3:00 P.M. is 3 hours. 12 1 3 5 15 hours. • Regroup minutes to hours. • Count the minutes. • There are 60 minutes in 1 hour. Frank finished building a model ship on RW35-RW36 62 5 1 2 3:00 P.M. to 3:07 P.M. is 7 minutes. min hr min February • February 26. If he worked on it for 8 weeks, 11 1 1 1 2 5 12 2 Sun Mon Tue Wed Thu Fri Sat • Add the minutes to the hours. • hr hr min hr min. when did he start the ship? 12:02 12 • 15 hours 1 7 minutes 5 15 hours 7 • So, the end time is P.M. Count the weeks backwards from Tuesday, minutes. February 26 to Tuesday, January 1. 3456789 15 7 • So, the elapsed time is hours Eight weeks backwards from February 26 is 10 11 12 13 14 15 16 minutes, or 15 hr 7 min. January 1. 17 18 19 20 21 22 23 So, January 1 is when Frank started the ship. Find the elapsed time. 24 25 26 27 28 29 1. start: 1:05 P.M. 2. start: 9:14 P.M. 3. start: 2:10 A.M. 4. start: 6:45 P.M. end: 2:05 P.M. end: 11:28 P.M. end: 2:35 A.M. end: 8:20 P.M. Use the calendars above. 1. Terry goes on vacation from 2. Marty’s class studies Australia for 6 21 3 5. start: 7:00 P.M. 6. start: 6:50 A.M. 7. start: 10:30 P.M. 8. start: 1:10 A.M. January to January . For weeks. The class begins the unit 14 end: 12:10 A.M. end: 4:50 P.M. end: 3:20 A.M. end: 7:35 P.M. how many days is Terry on vacation? of study on January . When does the unit of study end?

Find the end time. 3. There is snow on the ground in Marsha’s 4. Simone takes a 5-week class at the 9. start: 3:07 A.M. 1 0 . start: 6:29 P.M. 11. start: 9:47 A.M. 12. start: 5:30 A.M. yard from January 13th to February 9th. local college. Her class ends on February elapsed time: elapsed time: elapsed time: elapsed time: For how many days was there snow on 23. When does Simone’s 7 hr 5 min 1 hr 12 min 7 hr 30 min 48 min the ground? class begin? 66/18/07 5:04:03 PM / 1 8

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1 9 Name Name Algebra: Change Units of Time Problem Solving Workshop Skill: When you change time from longer units (such as years) to shorter Sequence Information units (such as days), you multiply. When you change time from shorter 10:15 units (such as minutes) to longer units (such as hours), you divide. Jo’s class arrived at the science museum at A.M. to study the invention of the airplane. The students have Complete. Tell whether you multiply or divide. Units of Time 3 hours 30 minutes in all at the museum. The class wanted 40 minutes for lunch and 25 minutes in the gift shop after 1 minute = 60 seconds 3 years 5 j days lunch. What was the latest time they could stop for lunch? • Years are longer than days, so multiply. 1 hour = 60 minutes 1. What are you asked to find? 1 year 5 365 days. Multiply the number of years by 365. • 1 day = 24 hours • 3 3 365 5 1,095. • So, you multiply to show 3 years 5 1,095 months. 1 week = 7 days 2. 1 year = 12 months At what time will the students leave the museum? Explain. 120 hours 5 j days 1 year = 365 days RW37-RW38 • Hours are shorter than days, so divide. 1 day 5 24 hours. Divide the number of hours by 24. • 1 year = about 52 weeks • 120 4 24 5 5 3. What is the latest time the students can stop for lunch? Write your answer in a 120 5 5 • So, you divide to show hours days. 1 leap year = 366 days complete sentence.

4. How can you check your answer to this problem? Complete. Tell whether you multiply or divide. 1. 5 weeks 5 j days 2. 24 months 5 j years

3. 3 hours 5 j minutes 4. 168 hours 5 j days Solve the problem. 5. Jim’s father was born in 1920. Jim’s 6. Tina mows the lawn and cleans out mother was born in 1924. Jim’s mother the garage today. She starts at 9:00 A.M. was 30 years old when Jim was born. and it takes a total of 3 hours. First, she 5. 3 years 5 j weeks 6. 5 hours 5 j minutes Jim’s father was 32 years old when mows the lawn. Then, she takes a Jim’s sister Pat was born. Which child is 15 minute break. Next, she cleans the older: Jim or Pat? garage, which takes her 1 hour 7. 2 years 5 j days 8. 1,460 days 5 j years 30 minutes. At what time does Tina take her break?

9. 9 hours 5 j minutes 10. 1,440 minutes 5 j hours 66/18/07 5:05:20 PM / 1 8

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220 0 Name Name Temperature Explore Negative Numbers Some thermometers show temperature in degrees Fahrenheit ( 8F ). Negative numbers are numbers less than zero, such as –3 or –16. Others show temperature in degrees Celsius ( 8C). Some Negative numbers have a small minus (–) sign next to them. thermometers use both Fahrenheit and Celsius. Positive numbers are numbers greater than zero, such as 5 and 12. Zero is neither a positive nor a negative number. Use the thermometer to find the temperature 30 shown by each letter. Name the numbers represented by points D and E on the number line. C 25 ED C 20

• The thermometer uses degrees Celsius. 15 20 10 0 10 20 • The letter C is at the line for 258 Celsius. 10 • First, find the letter on the number line. 5 Then, count from 0 to see which number is at the point for each letter. So, the temperature at the letter C is 258C. 0 Point D is located at 12. There is a plus sign before the number. When there is a D • 5 plus sign, the number is positive. Point D is at 112, or positive 12.

• The thermometer uses degrees Celsius. – 10 • Point E is located at 16. The minus sign (–) in front of the number shows that the –

RW39-RW40 0 • The letter D is below the number , so it is 15 number is negative. Point E is at 16, or negative 16. a negative temperature. 20 D So, point D is at 112 and point E is at –16. – • • D is at the line for 208 Celsius. 25 –208 30 So, the temperature at the letter D is C. Name the number represented by each letter on the number line below. DCKEM AH QFJP G RNB L °C

–30 –20 –10 0 +10 +20 +30 Use the thermometer below to find the temperature shown by each letter.

80 1. A 2. B 3. C 4. D 1. C 2. D 3. E G 75 70 65 60 K 55 50 5. E 6. F 7. G 8. H 45 40 35 4. F 5. G 6. H 30 25 20 15 9. J 10. K 11. L 12. M 10 5 J 0 E D 5 10 H 7. I 8. J 9. K 15 13. N 14. P 15. Q 16. R 20 F 25 30

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2 1 Name Name Collect and Organize Data Venn Diagrams You can use a frequency table to organize data. You can use a Venn diagram to sort and describe data. Remember the section where the circles overlap shows data the circles have in common. Use the Field Day Participants frequency table. Field Day What label could you use for Section B? If 2 sixth graders decide not to participate and Participants Frequency A B C 5 • Section A contains multiples of 2. fifth graders sign up, how many total students Grade (Number of Students) will participate from the fifth and sixth grades? • Section C contains multiples of 3. K 45 • Adjust the number of sixth graders. • So, the label for Section B should be MULTIPLES MULTIPLE MULTIPLES 2 Two decide not to participate, so subtract . 1 42 multiples of 2 and 3. OF 2 OF OF 50 2 2 5 48. 2 54 6 12 Why are the numbers and sorted in 2 4 6 3 9 • Adjust the number of fifth graders. Section B of the diagram? 8 12 15 Five sign up, so add five. 3 58 • The number 6 is a multiple of both 2 and 3. 14 10 55 1 5 5 60. 4 41 • The number 12 is a multiple of both 2 and 3. • Add the number of fifth and sixth graders. 5 55 So, the numbers 6 and 12 are in Section B RW41-RW42 48 1 60 5 108 6 50 because both are multiples of 2 and 3. So, 108 fifth and sixth graders will participate.

For 1–2, use the Venn diagram at the right. For 1–2, use the Field Day Participants frequency table above. 1. What label should you use for A B C section B? 1. How many total students will participate 2. How many more students will from the second grade and third grade? participate from sixth grade than from Kindergarten? NUMBERS NUMBERS 2. What are two numbers that LESS THAN GREATER THAN For 3–5, use the Favorite Drawing Tool frequency table below. could go in Section B? 60 50 Tell whether each statement is true or false. Explain.

3. More students chose markers than Favorite Drawing Tool 3 4 color pencils. Drawing Tool Frequency For – , use the Venn diagram at the right. Color pencil 3 3. What are two numbers that A B C could go in Section B? Crayon 8 Marker 7 EVEN MULTIPLES 15 NUMBERS OF 5 4. More students chose color pencils 5. A total of 20 students responded 4. Can the number go in than crayons. to the survey about favorite drawing Section B? Explain. tools. 66/18/07 5:05:48 PM / 1 8

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2 2 Name Name Find Mean, Median, and Mode Line Plots The mean is the average number in a set of data. The median is the You can use a line plot to graph data. The range is the difference middle number in a set of data. The mode is the number that occurs between the lowest and highest values in the data. A clump is the most often in a set of data. where data is concentrated in one place. A hole is where there is no data. Find the median. Use the Tourist Time Survey data to answer the questions below. Rainfall • List the numbers in order from What is the range of the data? Month Apr May Jun Jul Aug Sep least to greatest. Tourist Time Survey • On the line plot, find the greatest 3, 4, 5, 7, 8, 9 Question: How many number of hours spent visiting. Inches 8 75943 hours did you spend • Find the number in the middle of • 5 is the greatest number. the list. visiting Kennedy • Find the least number of hours Space Center Visitor 5 7 Find the mean. • Both and are the middle on the line plot. Complex? numbers. So, add the two middle 0 Add the numbers in the chart. • is the least number. 2, 3, 4, 1, • numbers and divide by 2. Responses: • Find the difference between 2, 5, 1, 3, 2, 1, 2, 1, 8 1 7 1 5 1 9 1 4 1 3 5 36 (' )*+, • • 5 1 7 5 12; 12 4 2 5 6. the greatest and the least numbers. 3, 3, 2, 1, 0, 3, 4, 5, 1 RW43-RW44 ?flijJg\ekM`j`k`e^ Count how many numbers are in the set. • So, the median is 6. • 5 2 0 5 5. 6 • There are numbers. Find the mode. So, the range is 5. Divide the sum by 6. • • All of the numbers appear once. How many hours did most tourists spend? How can you tell? 36 4 6 5 6 • Look at the line plot to see which hour has the most responses. • So, the mean is 6. So, there is no mode. • The line plot has the most Xs at 1. Find the mean, median, and mode. So, most people visited for 1 hour because the line plot has the most Xs at 1. For 1–4, use the Sports Survey Data at the right. 1. Games Played 2. Books Read Sports Data Survey 1. Make a tally table to show the data. Month Jan Feb Mar Apr Month Jan Feb Mar Apr Question: How many hours do you spend playing sports each week? Games 7 5 7 9 Books 10 5 4 5 Responses: 1, 4, 4, 5, 2, 6, 6, 4, 1, mean median mode mean median mode 0, 5, 4, 2, 4, 1

3. What is the range of the data?

3. Science Club 4. Thunderstorms Age 7 8 9 10 11 Month May Feb Mar Apr 2. Make a line plot to show the data. 4. For how many hours do most Frequency 5 1 4 6 4 Games 8 3 6 9 people play sports? mean median mode mean median mode How can you tell? 77/19/07 5:17:30 PM / 1 9 /

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2 3 Name Name

Choose a Reasonable Scale and Interval Problem Solving Workshop Skill: -AXIMUM4ARGET (EARTRATE The interval of a graph is the difference between one number Make Generalizations and the next on the scale of a graph. Graphs use different intervals based on the data being graphed. Use the table at the right. Make a generalization. Miyu is 25 years old. She does aerobics for a half 5 10 25 100 Choose , , , or as the most reasonable hour and then measures her heart rate. If it is (EARTBEATSPER-INUTE interval for each set of data. 180 beats per minute, should Miyu slow her 200, 350, 100, 250, 500 exercising, or can she continue at the same !GEINYEARS The numbers are all in the hundreds. An interval of 5 or 10 would be too small to fit all pace? Explain your answer. the data on the graph. An interval of 100 would show the data clearly on the graph. 1. What are you asked to find? So, an interval of 100 would be most reasonable. 25, 79, 50, 45, 90 2. Look at the information in the graph. Make a generalization. The numbers are all less than 100, so an interval of 100 would be too big. An interval of 5 would be too small to fit all the data. An interval of 10 would show the data clearly

RW45-RW46 on the graph. 10 So, an interval of would be most reasonable. 3. Find a relationship between Miyu’s heart rate and the information in the chart. What conclusion can you draw about Miyu’s heart rate? Choose 5, 10, or 100 as the most reasonable interval for each set of data. Explain your choice.

1. 180, 195, 183, 210, 200 2. 8, 13, 12, 6, 9 4. Should Miyu slow her exercising or can she continue at the same pace? Explain.

3. 19, 13, 11, 7, 5 4. 35, 65, 72, 21, 83

5. How can you check your answer? 5. 90, 130, 200, 150, 310 6. 560, 620, 720, 890, 360

6 7 7. 8, 1, 3, 7, 4, 6 8. 93, 66, 25, 75, 71 For – , use the table above. Make a generalization. Then solve the problem. 6. What if the graph showed a maximum 7. Sid is 45 years old. While target heart rate of 145? For what age exercising, his heart beats 190 times would that rate be? per minute. Should Sid slow his 52, 60, 34, 88, 69 757, 283, 891, 362, 549 9. 10. exercising? 66/18/07 5:06:18 PM / 1 8

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2 4 Name Name Bar Graphs Make Bar and Double-Bar Graphs You can use bar graphs to compare data. In vertical bar graphs, You can use double-bar graphs to compare Favorite Time of Day the bars run from the bottom of the graph to the top of the graph. similar kinds of data. Double-bar graphs have Time Boys Girls In horizontal bar graphs, the bars run from left to right. two bars for each category. Morning 6 8 Afternoon 20 17 Use the Camp Choices bar graph. Camp Choices Use the data in the table at the right to make two bar graphs. Then make a double-bar graph. Evening 33 31 Which camp was chosen by the fewest students? 25 20 The shortest bar is the one for drama. So, drama camp 15 10 • Make a bar graph for the boys’ data. Favorite Time of Day (Boys) was chosen by the fewest number of students. 5 35 0 • Decide on a scale for the bar graph. 30

Number of Students 25 How many students chose space camp? 20 15 • Title and labels will be the same as in the table. 10 Space 20 Drama Sports 5 The bar for space camp goes up to the line for . Outdoor 0 Adventure Camp • Use the numbers shown in the table to draw a So, 20 students chose space camp. Morning Afternoon Evening Number of Students bar for each time of day. Time of Day

1 2 For – , use the Favorite Fruit bar graph. Favorite Time of Day (Girls) RW47-RW48 35 1. Which fruit do most • Use the same steps to make a bar 30 25 Favorite Fruit 20 students prefer? graph for the girls’ favorite time of day. 15 10 5 • Use the same scale and interval. 0 25 Morning Afternoon Evening Number of Students 20 Time of Day 15 10 2. How many students prefer 5 • Make a double-bar graph for both sets if data. Favorite Time of Day oranges? 0 Girls

Number of Students Key: • Use the numbers shown in the table. Draw a Boys Apple Orange Grapes Peach 35 30 bar for each time of day for the girls and the boys. 25 Fruit 20 15 • Use one color for the boys and another 10 5 for the girls. 0 Number of Students For 3–4, use the Favorite Sport bar graph. Morning Afternoon Evening • Insert a key to show what each color represents. Time of Day 3. Which two sports received the same Favorite Sport number of votes? Use the data in the table at the right to make Victories Team April May June two bar graphs. Then make a double-bar graph. Jays 20 Use the space provided below. 839 15 Robins 7128 10 5 4. How many votes did 0 hockey receive in all? Number of Students Football Hockey Soccer Sport 66/18/07 5:06:29 PM / 1 8

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2 5 Name Name Circle Graphs Algebra: Use a Coordinate Grid You can use circle graphs to show data in different categories. Each color in the graph stands for a category. Each category is You can plot points on a coordinate grid using ordered pairs. An divided into sections to show the data. ordered pair has two numbers in parentheses, like this, (3,6). Use the Favorite Breakfast graph. The first number tells how far to move horizontally across the grid. Which type of breakfast received the least number The second numbers tells how far to move vertically up the grid. of votes? Write the point for each ordered pair on the • The least number of votes is represented by the coordinate grid. smallest section. Favorite Breakfast 2 4 10 ( , ) 9 • In the graph, the smallest section is labeled eggs. G • Start at 0. The first number in the ordered pair is 2, 8 • So, the type of breakfast with the least number of Wafﬂ es Eggs 7 H so move across two units to the right. D votes is eggs. 6

-axis 5

4 y F Which type of breakfast received the greatest • The second number is , so move up 4 A number of votes? four units. You end at point A. 3 C • The greatest number of votes is represented by the 2 4 2 • So, the point for ( , ) is A. 1 largest section. Cereal E (3,8) 023145789106 RW49-RW50 • In the graph, the largest section is labeled cereal. x-axis • So, the type of breakfast with the greatest number of • Start at 0. Move 3 units to the right and 8 units up. votes is cereal. • So, the point for (3,8) is G. For 1–2, use the Favorite Subject graph. Use the coordinate grid below. Write the point for each ordered pair. 1. Which subject received the least Favorite Subject (1,5) (0,4) number of votes? 1. 2. 10 Science 6 Math 2 9 P D N I Reading 3 8 7 J H 2. Which subject received almost half 6 G of the votes? C E L

-axis 5

y F 3. (7,2) 4. (9,9) 4 Writing 8 3 M 2 A For 3–4, use the Favorite Animal graph. 1 BK R 3. How many people said that dogs Favorite Animal 023145789106 were their favorite animal? x-axis Monkey 8 Horse 15 5. (2,0) 6. (9,1) 7. (6,5) 8. (0,3)

4. How many votes did horses receive?

Dog 11 Cat 4 66/18/07 5:06:50 PM / 1 8

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2 6 NameName Name Line Graphs Make Line Graphs Line graphs show how data changes over time. Trends refer to Use the data in the table to make a line graph. parts of the line graph where data stays the same, increases, or Plant Growth decreases over a period of time. 20 What was the height of Height of Water in a Glass Outdoors Plant Growth 15 the water in the glass on End of Week 1234 12 10 Thursday? 10 Height (in 6 9 13 15 5 8 inches) 6 • Find the line labeled for Height (in Inches) 0 4 Thursday on the bottom of the 2 1432 0

graph. Follow the line up to the Height (in Centimeters) End of Week Mon Tue Wed Thu Fri point. See which horizontal line Day • Write a title for the graph: Plant Growth. the point touches. Between which two days did the water level stay • Choose a scale and interval. Since the smallest number in the table is 6 and the • Follow the point accross to the the same? greatest number is 15, you can use a scale of 5. Write the label and scale numbers left. See where the line • When a line between two points does not go up or along the left side of the graph. touches the scale. RW51-RW52 down, it means that the data stays the same. The • Write the labels for the weeks along the bottom of the graph. • The line touches the scale at line between the points for Tuesday (Tue) and 6 • Plot the points. Then draw line segments to connect the points from left to right. cm, so the height of the Wednesday (Wed) stays level. So, the water level water in the glass on stayed the same between Tuesday and For 1–2, use the data to make line graphs in the boxes below. Thursday was 6 cm. Wednesday. 1. 2. Haley’s Afternoon Bikeride Bianca’s Writing Progress For 1–2, use the Plant Growth graph. For 3–4, use the Games Won graph. Time (in 30 60 90 120 Total Pages 13911 minutes) Time (in Plant Growth Games Won Distance (in 15 30 45 60 9 162127 minutes) miles) 14 10 12 8 10 8 6 6 4 4 2 2 Number of Games 0 0 Height (in centimeters)

Oct Nov Dec Jan Feb Mar Apr Oct Nov Dec Jan Feb Mar Apr Day Day

1. How tall was the plant in December? 3. During which 3 months did the team win the same number of games?

2. During which two months did the 4. How many games did the team win in plant’s height increase the most? April? 66/18/07 5:07:26 PM / 1 8

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2 7 Name Name Problem Solving Workshop Strategy: Choose an Appropriate Graph You can use a variety of graphs to present data. Make a Graph Places Manuel’s Class The graph you use depends on the type of data you are showing. Manuel’s class voted for their favorite places to Would Like to Visit Place Girls Boys visit. Make a double-bar graph to show the data. Lake 2 1 bar graph or double-bar graphs circle graphs Then find which place had the greatest difference Ocean 4 2 Use to show and compare data about Use to compare parts of a group to a in votes between girls and boys. National Park 2 7 different categories or groups. Example: whole group. Example: the favorite Read to Understand Amusement Park 6 8 cars sold in June, July, and August. subjects of a group of students. 1. What are you asked to find? line graphs line plots pictographs Use to show how data Use to show the Use to show and compare change over time. frequency of the data along data about different Plan Example: growth of a tree a number line. Example: categories or groups. over one year. number of students who Example: number of shirts, 2. How can making a graph help you solve this problem? have 1, 2, or 3 siblings. shoes, and pants in a closet.

RW53-RW54 Solve

3. Complete the graph below. Which place had the greatest difference between boys Choose and explain the best type of graph or plot for the data. and girls. 5 Places We’d Like to Visit the number of inches of rain over days 10 • This data focuses on how the amount of rain changed 8 Girls 6 or stayed the same over the 5 days. 4 Boys 2 0 • A line graph would be the best type of graph for this data Number of Students Lake Ocean National Amusement because a line graph shows how data changes over time. Park Park Place Choose and explain the best type of graph or plot for the data.

1. the types of shoes that were most 2. how a family spent $100 Check often purchased in April

4. How can you check to see that your answer is correct?

3. how much a vine grew each year 4. how many home runs a team hit at each of 5 games Use the Winter Snowfall table to make a graph.

5. Winter Snowfall Month Inches of Snow 5. the number of students playing 6. the temperature in a small town over Dec 25 different sports at recess 3 months Jan 30 Feb 15 66/18/07 5:07:40 PM / 1 8

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2 8 Name Name Algebra: Graph Relationships Problem Solving Workshop Skill: A function is an equation that tells you the value of a number, y, Identify Relationships in terms of another number, x. For example: y 5 x 1 9. A Robin is fencing in a square garden that is 2 yards on function table connects an input value, x, to an output value, y. 18 each side. José is fencing in a square garden that is 16 Find a rule. Write your rule as an equation. Use the equation to 4 yards on each side. How much more fencing does 14 12 find the missing numbers. José need than Robin? Use the graph to the right. 10 axis 8 Yards, x 1 2 3 4 5 6 j y-

Perimeter 6 Feet, y 3 j 9 12 15j 21 4 1. What are you asked to find? 2 • Look for a pattern in the table to find a rule. 023145 x-axis Try to figure out how the input numbers relate to the output numbers. Length of Side • The value of y is always 3 times more than the value of x. So, the rule is to multiply x by 3. 2. How is the length of each side related to the perimeter? • Write the rule as an equation: y 5 3x.

RW55-RW56 • Use the equation to find the missing numbers in the table. 2 3 3 5 6 6 3 3 5 18 7 3 3 5 21 3. How much fencing does José need? How much fencing does Robin need? So, the rule is y 5 3x, and the missing numbers in the table are 6, 18, and 7.

Find a rule. Write your rule as an equation. Use the equation to extend the pattern. 4. How much more fencing does José need than Robin? 1. Pennies, x 10 20 j 40 j 60 j Dimes, y 1 j 3 4 5 6 7

5. How can you check your answer?

2. Input, x 100 j 80 70 60 j 40 Output, y 50 45 j 35 j 25 20 Use the Pipes table to solve.

6. How many pounds would an 18-foot 7. Dan needs 23 feet of pipe to fix his length of pipe weigh? drain. He buys two 10-foot sections 3 Pipes and one -foot section. What will be 3. Input, x 1 j 3 4 j 67 Length, ft 61012 the total weight of the pipes? Output, y 7 14212835jj Weight, lb 18 30 36 66/18/07 5:07:52 PM / 1 8

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2 9 Name Name Mental Math: Multiplication Patterns Mental Math: Estimate Products You can use basic facts and mental math to multiply whole You can use rounding and compatible numbers to estimate products. 10 100 1,000 numbers by multiples of , , . Round the greater factor. Then use mental math to estimate the product. Use mental math to complete the pattern. 6 3 325

• 9 3 6 is a basic math fact. 9 3 6 5 54 • Round 325 to the nearest hundred. 325 rounds to 300 • Use the basic fact to find 9 3 60. • Use mental math. 6 3 3 5 18 60 9 3 60 5 540 Since has one zero, add one zero to the product. 6 3 30 5 180 9 3 600 • Use the basic fact to find . 6 3 300 5 1,800 Since 600 has two zeros, add two zeros to the product. 9 3 600 5 5,400

Use mental math to complete the pattern. So, 6 3 325 is about 1,800. • 4 3 5 is a basic math fact. 4 3 5 5 20 Estimate the product. Write the method. Use the basic fact to find 4 3 50. • 7 3 $7.59 50 4 3 50 5 200 RW57-RW58 Since has one zero, add one zero to the product. • Use a compatible number that is easy to compute mentally. 7 3 $7.59 • Use the basic fact to find 4 3 500. Since 500 has two zeros, add two zeros to the product. 4 3 500 5 2,000 7 3 $8

Use mental math to complete the pattern. • Use mental math. 7 3 8 5 56 1. 7 3 5 5 35 2. 3 3 4 5 12 3. 9 3 8 5 72 7 3 50 5 3 3 40 5 9 3 80 5 7 3 80 5 560 7 3 500 5 3 3 400 5 9 3 800 5 7 3 800 5 5,600 • Add a decimal and dollar sign. 7 3 $8.00 5 $56.00 4. 8 3 6 5 48 5. 2 3 7 5 14 6. 4 3 4 5 16 So, 7 3 $7.59 is about $56.00. 8 3 5 480 2 3 5 140 4 3 5160 8 3 600 5 2 3 700 5 4 3 400 5 Round the greater factor. Then use mental math to estimate the product.

1. 6 3 316 2. 5 3 $2.89 3. 4 3 $3.20

7. 6 3 3 5 8. 7 3 7 5 9. 4 3 6 5 6 3 30 5 7 3 70 5 4 3 60 5 6 3 5 1,800 7 3 5 4,900 4 3 5 2,400 Estimate the product. Write the method. 6 3 3,000 5 7 3 7,000 5 4 3 6,000 5 4. 3 3 508 5. 7 3 22 6. 8 3 3,061 66/18/07 5:08:05 PM / 1 8

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3 0 Name NameName Model 2-Digit by 1-Digit Record 2-Digit by 1-Digit Multiplication Multiplication You can use arrays to multiply 2-digit numbers by 1-digit numbers. Use partial products and place value with regrouping to multiply. Use grid paper to model the product. 14 Use partial products. Record your answer. Multiply. 81 3 5. 9 3 14 81 80 80 3 5 5 400 9 Estimate the product. rounds to ; . or 1 9 14 1 1 5 81 • Draw a by array. Multiply the one, or , by . 3_ 5 3_ 5

5 8 80 5 81 or 80 410 • Make two smaller arrays. Use products you know. Multiply the tens, or , by . 3_ 5 _3 5

400 9

RW59-RW60 Add the partial products. 5 1 400 5 405 So, 81 3 5 5 405. Since 405 is close to the estimate of 400, it is reasonable. 9 3 10 9 3 4 Use place value and regrouping. 9 3 10 5 90 • Find the product of the two smaller arrays. Multiply. 57 3 4. 9 3 4 5 36 Estimate the product. 57 rounds to 60; 60 3 4 5 240. 2 7 4 57 4 3 7 5 28 • Find the sum of the products. 90 1 36 5 126 Multiply the ones by . ones ones 3 4 So, 9 3 14 5 126. Regroup the 2 tens. _ 28 2 Use grid paper to model the product. 5 5 7 4 3 5 5 20 Multiply the tens by tens tens Record your answer. Add the regrouped tens. _ 3 4 20 tens 1 2 tens 5 22 tens 1. 3 3 13 2. 6 3 16 228

So, 57 3 4 5 228. Since 228 is close to the estimate of 240, it is reasonable.

Estimate. Then record the product. 1. 63 2. 83 3. 36 4. 67 3_ 3 3_ 5 3_ 8 3_ 6 3. 5 3 17 4. 4 3 14

5. 59 6. 72 7. 18 8. 42 3_ 3 3_ 4 3_ 4 3_ 7

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3 1 Name Name Multiply 3-Digit and 4-Digit Numbers and Money Multiply with Zeros When you multiply 3-digit and 4-digit numbers, you may need to regroup. Estimate. Then find the product. When you multiply money, ignore the dollar sign and the decimal, $10.20 and add them to your final product. 3 7 Estimate. Then find the product. Estimate the product. $10.20 rounds to $10; $10 3 7 5 $70 $13.24 Multply the 0 one, or 0, by 7. 1020 0 3 7 3 7 Place the 0 in the ones place. 3 7 0 0 Estimate the product. $13.94 rounds to $10; $10 3 7 5 $70 4 7 2 1 Multiply the ones by . $13.24 Multiply the 2 tens, or 20, by 7. 1020 2 2 4 3 7 5 28 Regroup the tens. 3 7 Regroup 1 hundred. 3 7 3 7 8 40 14

2 7 21 1 Multiply the tens by . 0 0 7 RW61-RW62 $13.24 Multiply the hundreds, or , by . 1020 0 Add the regrouped tens. 2 3 7 5 14 3 7 Add the regrouped hundred to the 3 7 3 7 Regroup 1 hundred. 14 1 2 5 16 68 product: 0 1 1 5 1. 140 0

3 7 2 21 Multiply the hundreds by . $13.24 Add the regrouped hundreds. 3 3 7 5 21 1 3 7 1 1,000 7 1020 1 Regroup 2 thousands. 21 1 1 5 22 Multiply the thousand, or , by . 268 3 7 3 7 7140 7 Multiply the 1 thousand by 7. 212 $13.24 Add the regrouped thousands. 1 3 7 5 7 $71.40 3 7 Add the decimal and the dollar sign to the final product. 9268 7 1 2 5 9 So, $10.20 3 7 5 $71.40. Since $71.40 is close to the estimate of $70, it is reasonable.

Add a dollar sign and a decimal point $92.68 to your answer. Estimate. Then find the product. 4,046 7,040 $13.09 5,008 So, $13.24 3 7 5 $92.68. 1. 2. 3. 4. 3 2 3 3 3 7 3 2 Since $92.68 is close to the estimate of $70, it is reasonable.

Estimate. Then find the product. 1. $3,184 2. 828 3. $26.37 4. 6,916 3 2 3 2 3 2 3 7 77/19/07 5:17:50 PM / 1 9 /

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3 2 Name Name Problem Solving Workshop Skill: Mental Math: Multiplication Patterns Evaluate Reasonableness You can use basic math facts and patterns to help you multiply whole numbers by multiples of 10, 100, and 1,000. The Franklins have a breakfast budget. Each week, they spend $5 for cereal, $7 for bacon, $4 for bread, $2 for eggs, Use patterns and mental math to find the product. and $3 for juice. How much will they spend on breakfast in 12 3 6,000 4 weeks? • 12 3 6 is a basic math fact. 12 3 6 5 72 1. What are you asked to find? • Use the basic math fact to find 12 3 6,000. • Since 6,000 has three zeros, add 12 3 6,000 5 72,000 three zeros to the product. So, 12 3 6,000 5 72,000. 2. Which operations will you use to solve this problem? 11 3 1,000

• 11 3 1 is a basic math fact. 11 3 1 5 11

RW63-RW64 3. How much will they spend on breakfast in 4 weeks? Estimate. Then solve. • Use the basic math fact to find 11 3 1,000. • Since 1,000 has three zeros, add 11 3 1,000 5 11,000 three zeros to the product.

4. Does your answer seem reasonable? Why or why not? So, 11 3 1,000 5 11,000.

Use patterns and mental math to find the product.

1. 22 3 10 2. 35 3 100 3. 67 3 100 4. 48 3 1,000

Solve the problem. 5. Each month Mario spends $5 on sports 6. Amy’s school is made up of 524 cards, $12 on snacks, and $9 on school students. The principal gave 5. 40 3 100 6. 31 3 1,000 7. 49 3 10 8. 50 3 10 supplies. If Mario spends the same each student 3 pencils. Amy says amount each month, how much does 1,048 pencils were handed out. Is Mario spend all together in 3 months? Amy’s answer reasonable? Why or why not? 9. 80 3 20 10. 10 3 300 11. 42 3 1,000 12. 90 3 900

13. 19 3 10 14. 93 3 100 15. 60 3 100 16. 17 3 1,000 77/19/07 5:18:09 PM / 1 9 /

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3 3 Name Name Multiply by Tens Mental Math: Estimate Products The multiples of 10 are 10, 20, 30, 40, and so on. You can use compatible numbers and rounding to estimate products. When one of the factors in a multiplication problem is a Use rounding and mental math to estimate the product. 10 multiple of , you can find the product by solving a $23 3 62 simpler problem and writing a zero. • Round both numbers to the nearest ten. $23 rounds to $20. 62 rounds to 60. Choose a method. Then find the product. 78 3 60 • Rewrite the problem using the rounded numbers. $20 3 60 78 3 6 78 3 6 5 468 • Solve the simpler problem . • Use mental math. $2 3 6 5 $12 • There is one zero in 60. $20 3 6 5 $120 Write one zero in the product. 4,680 $20 3 60 5 $1,200 So, $23 3 62 is about $1,200. • Since 78 3 6 5 468, then 78 3 60 5 4,680 78 3 60 5 4,680. Use compatible numbers and mental math to estimate products.

RW65-RW66 So, 41 3 178 41 3 178

Choose a method. Then find the product. • Use compatible numbers that are easy to compute mentally. 40 3 200 90 3 18 46 3 50 50 3 32 1. 2. 3. 4 3 2 5 8 • Use mental math. 4 3 20 5 80 4 3 200 5 800 40 3 200 5 8,000 So, 41 3 178 is about 8,000.

4. 22 3 40 5. 60 3 28 6. 36 3 30 Estimate the product. Choose the method. 1. 78 3 21 2. $ 46 3 59 3. 81 3 33 4. 67 3 102

5. $42 3 88 6. 51 3 36 7. 73 3 73 8. $44 3 99

7. 12 3 20 8. 47 3 40 9. 66 3 60

9. 92 3 19 10. 26 3 37 11. 193 3 18 12. 58 3 59

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3 4 Name Name Problem Solving Workshop Strategy: Model 2-Digit by 2-Digit Multiplication Solve a Simpler Problem You can use arrays to multiply 2-digit numbers by 2-digit numbers. When Maggie goes on birding watching trips, she sees at 18 Use the model and partial products to solve. least 45 birds. She has taken 12 bird watching trips a 19 3 18 year over the past 16 years. How many birds has Maggie seen? The array is 19 units wide and 18 units long. 19

Read to Understand

1. What are you asked to find?

10 8 Plan Divide the array into 4 smaller arrays. 2. How can you use the solve a simpler problem strategy to solve this problem? Use products you know. 10 RW67-RW68

Solve

3. Solve a simpler problem. Show your work below. 12 3 16 5 192 Find the products of smaller arrays. 10 3 10 5 100 10 3 8 5 80 192 3 45 5 192 3 ( 1 ) 10 3 9 5 90 5 (192 3 ) 1 (192 3 ) 1 (192 3 ) 9 3 8 5 72 5 1 1 Find the sum of the products. 100 1 80 1 90 1 72 5 342 5 So, 19 3 18 5 342. Check Use the model and partial products to solve. 4. How did this strategy help you solve the problem? 1. 21 3 25 2. 16 3 14 3. 24 3 15

Solve a simpler problem. 5. Newcastle’s population is 6. Cindy shoots 50 free throws a day, 385,296 people. Dunn’s population 7 days a week. How many free is 194,825 people. How many more throws does she shoot in 5 weeks? people live in Newcastle than Dunn?

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3 5 Name Name Record 2-Digit by 2-Digit Multiplication Multiply 2-Digit and 3-Digit Numbers and Money Estimate. Then find the product. When you multiply money, it is the same as multiplying whole numbers. $ 89 Just add the decimal ( . ) and dollar sign ( ) to your answer. 3_ 47 Estimate. Then find the product.

$3.99 Estimate the product. 89 rounds to 90; 47 rounds to 50 3 30 90 3 50 5 4,500 Estimate the product. $3.99 rounds to $4; 30 stays at 30. 6 Multiply the 9 ones by the 7 ones. 89 $4 3 30 5 $120 6 Regroup the tens. 3 47 7 3 9 5 63 399 0 3 ones ones ones Multiply by ones. $3.99 3 30 399 3 0 ones 5 000 000 8 7 6 Multiply the tens by the ones. 89 7 ones 3 8 tens 5 56 tens 399 3 30 22 Add the regrouped tens. 3 47 Multiply by tens, or . $3.99 623 56 tens 1 6 tens 5 62 tens Add the regouped tens. 3 30

RW69-RW70 000 399 3 30 5 11,970 9 4 40 3 Multiply the ones by the tens, or . 89 11970 Regroup the 3 tens. 3 47 623 9 ones 3 40 5 360 Add the partial products. $3.99 60 3 30 000 1 11970 3 Multiply the 8 tens by the 4 tens, or 40. 89 11970 Add the regrouped tens. 3 47 8 tens 3 4 tens 5 32 tens Add the decimal. There will always be 2 623 $119.70 3560 32 tens 1 3 tens 5 35 tens digits to the right of the decimal in money. Add a dollar sign to your answer.

Add the partial products. 89 3 47 So, $3.99 3 30 5 $119.70. 623 Since $119.70 is close to the estimate of $120, it is reasonable. 1 3560 4183 So, 89 3 47 5 4,183. Since 4,183 is close to the estimate of 4,500, it is reasonable. Estimate. Then find the product. $4.19 386 126 241 1. 2. 3. 4. Estimate. Then choose either method to find the product. 3_ 21 3_ 33 3__ 42 3_ 37 76 24 14 64 1. 2. 3. 4. 3_ 31 3_ 35 3_ 28 3_ 56

$2.45 406 $6.20 187 5. 6. 7. 8. 3__ 16 3_ 24 3__ 44 3_ 29

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3 6 Name Name Choose a Method Problem Solving Workshop Skill: The numbers you multiply help you decide whether to use Multistep Problems mental math, paper and pencil, or a calculator. 1962 800 The Space Needle, built in for the Seattle World’s Fair, is 325 605 feet tall. The Petronas Towers, an office building in 3_ 30 3_ 24 Malaysia, has 88 stories. Each story is about 14 feet high. These are easy numbers with which to These numbers would be harder to About how much taller is the Petronas Towers than the Space Needle? work. Use mental math to solve. multiply using mental math. Use paper and pencil to solve. 1. What are you asked to find? Estimate: 800 3 30 5 24,000 300 3 20 5 6,000 Think: 8 3 3 5 24 Estimate: 325 80 3 3 5 240 Think: 3_ 24 2. Can you compare the heights of the two structures using just one step? 800 3 3 5 2,400 1300 Why or why not? 800 3 30 5 24,000 6500 7800 So, 800 3 30 5 24,000. RW71-RW72 So, 325 3 24 5 7,800. Since 24,000 is close to the estimate of Since 7,800 24,000, it is reasonable. is close to the estimate of 6,000, it is reasonable. 3. Which operation, or operations, do you need to use to solve the problem?

Estimate. Then find the product. Write the method you used. 4. About how much taller is the Petronas Tower than the Space Needle?

49 50 $24 60 1. 2. 3. 4. 3_ 62 3_ 20 3_ 64 3_ 40 5. How can you check your answer?

Solve the problem. 6. Gina owns a music store. On Saturday, 7. Dan and Peter both enjoy hiking her store sold 56 CDs for $15.00 each. through the woods. Dan hiked On Sunday her store sold 32 CDs for 6 miles a day for 5 days. Peter 45 93 361 799 5. 6. 7. 8. $8.50 each. How much money did hiked 9 miles a day for 4 days. 3_ 25 3_ 79 3_ 83 3_ 45 Gina’s store make in all on Saturday How many more miles did Peter

and Sunday? hike than Dan? 66/18/07 5:09:45 PM / 1 8

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3 7 Name Name Divide with Remainders Model 2-Digit by 1-Digit Division When a number cannot be divided evenly, there is an amount left You can use base–ten blocks to model division of a 2-digit over. This leftover amount is called the remainder. number, such as 26, by a 1-digit number, such as 3. Use counters to find the quotient and remainder. Use base–ten blocks to find the quotient and remainder. 24 4 5 7 qw 84 • You are dividing the dividend, 24, so use 24 counters. • Show 84 as 8 tens and 4 ones. Then draw 7 circles, since you are dividing 84 by 7. • Since you are dividing 24 by 5, draw 5 circles. Then divide the 24 counters into 5 equal-sized groups. • Place an equal number of tens in each circle. If there are any tens left over, regroup them as ones. Now place an equal number of ones in each group.

• There are 4 counters in each circle, so the quotient is 4. There are 4 counters left over, so the remainder is 4.

RW73-RW74 So, 24 4 5 5 4 r4. • Count the number of tens and ones in each circle to find the quotient. 1 2 Use counters to find the quotient and remainder. There is ten and ones in each circle. 42 4 5 10 1 2 5 12, so the quotient is 12.

• You are dividing the dividend, 42, so use 42 counters. • There are no leftover blocks, so there is no remainder. 84 4 7 5 12 • Since you are dividing 42 by 5, draw 5 circles. • So, . Divide 42 counters into 5 equal-sized groups. Use base–ten blocks to find the quotient and remainder.

1. 71 4 8 2. 35 4 8 3. 19 4 6

• There are 8 counters in each circle, so the quotient is 8. There are 2 counters left over, so the remainder is 2. So, 42 4 5 5 8 r2. Use counters to find the quotient and remainder.

1. 19 4 6 2. 14 4 3 3. 29 4 9

4. 6q w 49 5. 5q w 28 6. 7q w 35

4. 31 4 7 5. 65 4 8 6. 44 4 5 66/18/07 5:09:56 PM / 1 8

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3 8 Name Name Record 2-Digit by 1-Digit Division Problem Solving Workshop Strategy: You can use long division to divide 2-digit numbers by 1-digit Draw a Diagram numbers. Sometimes the numbers will divide evenly. Other times there will be a remainder. Ari runs a training school for pet actors. Last year he trained 3 times as many dogs as cats. If the total number of dogs Choose a method. Then divide and record. and cats he trained last year is 84, how many cats did he 5 qw 85 1 train? Use Long Division. 5 qw 85 Read to Understand Begin in the tens place, 2 5 1. What are you asked to find? divide 8 by 5. 3 8 4 5 5 1 Write 1 in the tens place. Plan 1 3 5 5 5 Multiply. . 2. How can drawing a diagram help you solve this problem? Subtract. 8 2 5 5 3. Compare. 3 , 5. 17 RW75-RW76 5 5 qw 85 Solve Then bring down the ones and 2 5 divide 35 by 5. 35 3. Complete and label the diagram below to solve the problem. 2 35 Total pets 35 4 5 5 7 0 dogs is 84 pets. Write 7 in the ones place. 21 Multiply. 7 3 5 5 35. 4. How many cats did Ari train? Subtract. 35 2 35 5 0. Compare. 0 , 5. So, 85 4 5 5 17. Check 5. Does the answer make sense for the problem? Explain. Choose a method. Then divide and record.

1. 7 82 2. 6 74 3. 3 57

Draw a diagram to solve.

6. A crate has 5 times as many apples as 7. Julio has 6 times as many green oranges. If there are 120 pieces of fruit marbles as red marbles. If he has in the crate, and there are only apples 210 marbles altogether, how many 4. 7 95 5. 9 qw 89 6. 6 qw 36 and oranges in the crate, then how marbles are green? many oranges are there? 66/18/07 5:10:09 PM / 1 8

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3 9 Name Name Mental Math: Division Patterns Mental Math: Estimate Quotients Use mental math to complete the pattern. You can use compatible numbers and rounding to estimate quotients.

64 4 8 5 Estimate the quotient. 640 4 8 5 147 4 3 6,400 4 8 5 Use compatible numbers to estimate. 15 4 3 • Solve for the basic math fact. 64 4 8 5 8 • Look at the numbers in the problem. 150 4 3 • In 640 4 8, there is one zero added to Think of some basic math facts you know 640 4 8 5 80 150 is close to 147 the dividend. So there will be one zero added to that are similar to the numbers in the problem. the quotient. 15 4 3 5 5 6,400 4 8 5 800 • Use mental math to divide. • In 6,400 4 8, there are 2 zeros added to 150 4 3 5 50 the dividend, so there will be 2 zeros So, a good estimate for 147 4 3 is 50. added to the quotient. Estimate the quotient. So, 64 4 8 5 8; 640 4 8 5 80; and 6,400 4 8 5 800. 4,522 4 9

RW77-RW78 Use mental math to complete the pattern. Use rounding to estimate. 90 4 9 5 • Round the dividend to the nearest thousand. 4,522 rounds to 5,000 900 4 9 5 Round the divisor to the nearest ten. 9 rounds to 10 9,000 4 9 5 • Rewrite the problem using the rounded numbers. 5,000 4 10 90 4 9 5 10 • Solve for the basic math fact. • Use mental math to divide. 5 4 1 5 5 900 4 9 • In , there is one zero added to 5,000 4 10 5 500 the dividend. So there will be one zero added to 900 4 9 5 100 4,522 4 9 500 the quotient. So, a good estimate for is . 9,000 4 9 2 • In , there are zeros added to 9,000 4 9 5 1,000 Estimate the quotient. the dividend, so there will be 2 zeros 1. 409 4 4 2. 212 4 5 3. 7,839 4 2 4. 311 4 6 added to the quotient. So, 90 4 9 5 10; 900 4 9 5 100; and 9,000 4 9 5 1,000.

Use mental math to complete the pattern. 2,103 4 4 731 4 7 239 4 4 403 4 8 1. 25 4 5 5 2. 63 4 9 5 3. 56 4 7 5 5. 6. 7. 8. 250 4 5 5 630 4 9 5 560 4 7 5

2,500 4 5 5 6,300 4 9 5 5,600 4 7 5

9. 119 4 3 10. 5,788 4 6 11. 787 4 4 12. 273 4 4 4. 18 4 3 5 5. 36 4 4 5 6. 30 4 3 5

180 4 3 5 360 4 4 5 300 4 3 5

1,800 4 3 5 3,600 4 4 5 3,000 4 3 5 66/18/07 5:10:22 PM / 1 8

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4 0 Name Name Place the First Digit Problem Solving Workshop Skill: When you use long division, it is important to place the first digit of the Interpret the Remainder quotient in the correct place. You can use estimation and place value to help you find where to place the first digit. Solve. Write a, b, or c, to explain how to interpret the remainder. Tell where to place the first digit. Then divide. a. Increase the quotient by 1. 6 qw 126 b. Quotient stays the same. Drop the remainder. • Use compatible numbers to estimate. 12 4 6 5 2 126 4 6 is about the same as 120 4 6. 120 4 6 5 20 c. Use the remainder as the answer. 9 • 120 4 6 5 20, so the first digit is in the tens place. Guides lead groups of people on biking tours in the park. 96 21 There are people who decided to go on the tours. • Now place the first digit in the How many people will be on a tour that is not full? quotient in the tens place and divide. 6q w 126 2 12 1. What are you asked to find? 06 2 6 So, the first digit is the tens place, and 126 4 6 5 21. 0 RW79-RW80 Tell where to place the first digit. Then divide. 2. Divide 96 by 9. How many groups of 9 people will the guides lead on biking tours? 889 4 8 • See if the number in the hundreds place of 889 is larger than 8. • If the number in the hundreds place is smaller than the divisor, 111 r1 then you begin at the next place to the right: the tens place. 8q w 889 3. What is the remainder? How would you interpret the remainder? 2 8 • If the number in the hundreds place is larger than or equal to the 08 divisor, then you can place the first digit in the hundreds place. 2 8 8 8 09 • Here, is equal to , so the first digit will be in the hundreds place. 2 8 Now divide. 1 4. How many people are on a tour that is not full? • So, the first digit is in the hundreds place, and 889 4 8 5 111 r1.

Tell where to place the first digit. Then divide.

Solve. Write a, b, or c or to explain how to interpret the remainder. 1. 8 qw 286 2. 5q w 743 3. 536 4 4 4. 647 4 9 5. Each van holds 9 people. There are 6. Each car in an amusement park 82 people traveling. How many ride can hold 6 people. There are vans will be completely full? 34 people waiting in line. How many people will be on a car that is not full? 66/18/07 5:10:35 PM / 1 8

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4 1 Name Name Divide 3-Digit Numbers and Money Zeros in Division You can use multiplication to check division problems. Divide and check. Divide and check. 9 927 924 4 8 or 8 924 Estimate to see where to place the first digit. 927 is about 900 • Since there is 1 hundred, place the 9 is about 9 • Estimate to see where to place the first digit. 924 is about 900 first digit in the hundreds place. 900 4 9 5 100 Since there is 1 hundred, place the first digit in the 8 is about 9 hundreds place. 900 4 9 5 100 • Divide. Place the 1 in the hundreds place. 1 1 1 1 • Multiply. Place the 9 below the 9 hundreds. 9 927 9 9 • Divide. Place the 1 in the hundreds place. 8 924 or 8 9 29 29 28 • Subtract. Bring down the 2 tens. or • Multiply. Place the 8 below the 9 hundreds. 28 02 0 12 1 • Subtract. Bring down the 2 tens. Repeat the steps. 10 0 • Divide. Place the 0 in the tens place. 9 927 9 2 29 Repeat the steps. • Multiply. Place the 0 below the 2 tens. or 20 11 1 7 02 2 1 8 924 8 12 • Subtract. Bring down the ones. 20 RW81-RW82 • Divide. Place the in the tens place. or 8 2 28 28 27 • Multiply. Place the below the tens. 12 4 • Subtract. Bring down the 4 ones. Repeat the steps. 28 103 3 44 • Divide. Place the 3 in the ones place. 9 927 9 27 Repeat the steps. • Multiply. Place the 27 below the 27 ones. 2 9 or 227 115 5 02 • Subtract. There is no remainder. 0 • Divide. Place the 5 in the ones place. 8 924 or 8 44 20 927 4 9 5 103 40 44 28 240 So, . 27 • Multiply. Place the below the ones. 12 4 • Subtract. The remainder is 4. 227 Ϫ8 0 So, 924 4 8 5 115 r4. 44 Check. Ϫ40 103 3 9 5 927 4 • Multiply the quotient by the divisor. Check. • Add the product and the remainder. 927 1 0 5 927 • Since the sum is the dividend, your answer is correct. • Multiply the quotient by the divisor. 115 3 8 5 920 • Add the product and the remainder. 920 1 4 5 924 • Since the sum is the dividend, your answer is correct. Divide and check.

5 524 2. 7 910 3. 4 815 Divide and check. 1.

$582 7 397 3 858 1. 6 2. 3. 66/18/07 5:20:52 PM / 1 8

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4 2 Name Name Choose a Method Estimate Quotients You can use paper and pencil, a calculator, or mental math to divide. You can use compatible numbers or basic math facts to estimate quotients. Divide. Write the method used. Write the numbers you would use to estimate each quotient. Then estimate. 2,198 4 3 3 2,198 or Estimate. Estimate. 732 r2 23 188 or 188 4 23 3,600 4 37 • Use paper and pencil. 3 2198 221 • Find compatible numbers. • Find compatiable numbers. 188 4 23 09 3,600 4 37 So, 2,198 4 3 5 732 r2. 29 08 180 4 20 26 3,600 4 36 2 • Use mental math. • Use a basic fact and a pattern of 18 4 2 5 9 $26.08 10 8 or $26.08 4 8 multiples of . 180 4 20 5 9 • Use a calculator to solve. 36 4 36 5 1 • So, a reasonable estimate for

RW83-RW84 188 4 23 9. 360 4 36 5 10 26.08 4 8 5 3.26 is 3,600 4 36 5 100

So, $26.08 4 8 5 $3.26. • So, a reasonable estimate for 3,600 4 37 is 100.

7 4,270 4,270 Ϭ 7 or Write the numbers you would use to estimate the quotient. • Use mental math for numbers that are easy to work with. Then estimate. 7 4,270 4 7 Here, the dividend contains multiples of , the divisor. 5,893 703 4 68 38 2,121 1. 32 2. 3. So, use mental math to solve. 4,200 4 7 5 600 70 4 7 5 10 • Add the quotients. 600 1 10 5 610 610 So, 4,270 4 7 5 610, or 4 4,270 . 4,146 4 59 4,825 655 4 80 4. 5. 69 6. Divide. Write the method used.

8 3,729 5 3,075 8 4,888 4 1,472 1. 2. 3. 4.

2,994 8,245 4 92 51 5,024 7. 73 8. 9. 66/18/07 5:21:02 PM / 1 8

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4 3 Name Name Model Division by 2-Digit Divisors Record Division You can model division by using base-ten blocks. You can use long division or partial quotients to divide. Use base-ten blocks to divide. Divide and record. 99 4 31 35 841 First, model the dividend, 99, as 9 tens and 9 ones. • Estimate to place the first digit. 800 4 40 5 20 and 840 4 40 5 12 In both estimates, the first digit is in the tens place, so you will place the first digit in the tens place. • Regroup 8 hundreds 4 tens as 84 tens. The divisor is 31, so make as many groups of 31 as you can. Then divide the 84 tens by 35. 2 • Divide. 84 4 35 35 841 270 • Multiply. 35 3 2 14 • Subtract. 84 2 70 • Compare. 14 , 35 RW85-RW86 • Bring down the ones. Divide 141 by 35. 3 31 3 24 There are groups of . So, is the quotient. • Divide. 141 4 35 35 841 • Multiply. 35 3 4 270 6 6 141 2 140 141 There are ones left over, so is the remainder. • Subtract. Ϫ140 1 , 35 So, 99 4 31 5 3 r6 • Compare. 1 1 1 Use base-ten blocks to divide. There is left over so the remainder is . So, 841 4 35 5 24 r1 1. 65 4 21 2. 96 4 47 Divide and record.

1. 36 775 2. 27 361

3. 117 4 19 4. 175 4 58

3. 689 4 16 4. 978 4 61 66/18/07 5:21:23 PM / 1 8

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4 4 Name Name Adjusting Quotients Problem Solving Workshop Skill: When an estimate is too high or too low, you must adjust the quotient. Too Much/Too Little Information Adjust the estimated digit in the quotient, if needed. A radio station plays 49 minutes of music each hour. The rest Then divide. of the hour, commercials are played. In 2 hours, the station 32 3 played songs. How many minutes of commercials does the 3 45 210 station play in hours? • Try the estimate, 3. 1. What are you asked to find? 3 3 45 5 135 75 . 45 3 Since the remainder is 45 210 2. What information is given in this problem? larger than the divisor, 2135 the estimate is too low. 75

• Try 4. 3. What information do you need to solve this problem? RW87-RW88 4 3 45 5 180 4 45 210 • So, 210 4 45 5 4 r30. 2180 4. Is there any extra information given in this problem? If yes, why is the information 30 not needed?

Adjust the estimated digit in the quotient, if needed. Then divide. 531 5. How many minutes of commercials does the station play in 3 hours? 1. 27 138 2. 39 171 3. 16 134

Decide if the problem has too much or too little information. Then, solve if possible. If there is too little information, identify the missing information. 6. Peter earns $9 per hour. He worked on 7. There are 25 students in the class, 666 Tuesday and Thursday. How 12 boys and 13 girls. If each student 4. 41 222 5. 33 256 6. 48 306 much money did Peter earn? has 6 pencils, how many pencils are in the classroom? 66/18/07 5:21:37 PM / 1 8

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4 5 Name Name Factors and Multiples Divisibility Rules A factor is a number multiplied by another number to find a You can use special rules to find out if a number is divisible by product. For example, 3 and 5 are factors of 15. The number 15 another number. is a multiple of 5 and 3, because 3 3 5 5 15.

Use arrays to find all the factors of 6. List the first twelve multiples of 8. Number Divisibility Rule Example

Make an array with 6 squares. Multiply 8 by the numbers 1 through 12. 2 The last digit must be even. 456. The last digit is an even number (0, 2, 4, 6, 8). 3 List each product. 5 The last digit must be 0 or 5. 45. The last digit is a 5. 10 The last digit must be 0. 90. The last digit is a 0. 2 1 3 8 5 8 7 3 8 5 56 2 3 8 5 16 8 3 8 5 64 25 The last digits must be 00, 25, 50, or 75. 3,250. The last two digits are 50. This array contains 2 rows of 3 squares 3 3 8 5 24 9 3 8 5 72 Tell whether 480 is divisible by 2, 5, 10, or 25. each. So, 2 and 3 are both factors of 6. 4 3 8 5 32 10 3 8 5 80 480 is divisible by 2 because the last digit is even. 5 3 8 5 40 11 3 8 5 88 • Make a different array with 6 squares. • 480 is divisible by both 5 and 10 because the last digit is a 0. 6 6 3 8 5 48 12 3 8 5 96 RW89-RW90 480 is not divisible by 25 because the last digits are not 00, 25, 50, or 75. 1 • So, the first twelve multiples of 8 are: So, 480 is divisible by 2, 5, and 10. An array with 1 row of 6 contains 6 8, 16, 24, 32, 40, 48, squares. So, 1 and 6 are both factors of 6. 56, 64, 72, 80, 88, 96. Tell whether the number is divisible by 2, 5, 10, or 25. 6 1 2 3 6 So, the factors of are , , , and . 1. 675 2. 100 3. 320 4. 126

Use arrays to find all the factors of each product. 1. 10 2. 16 3. 21

5. 250 6. 254 7. 30 8. 835

9. 400 10. 975 11. 104 12. 56 List the first twelve multiples of each number. 4. 3 5. 5

13. 470 14. 250 15. 248 16. 575 6. 7 7. 10 66/18/07 5:21:52 PM / 1 8

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4 6 Name Name Prime and Composite Numbers Number Patterns A prime number has only two factors: 1 and itself. The number To find number patterns, find a rule. If a number pattern 3 is prime because it has only 1 and 3 as its factors. A composite increases, try addition or multiplication. If a number number has more than two factors. The number 15 is pattern decreases, try subtraction or division. composite because its factors are 1, 3, 5, and 15. Find a rule. Then find the next two numbers in your pattern. Make arrays to find the factors. Write prime or composite for each number. 18, 28, 38, 48, j , j 11 Look at the first two numbers: 18 and 28. What rule changes 18 to 28? Make all the arrays you can with 1 3 11 11 square tiles. 11 3 1 Since 28 is larger than 18, try addition or multiplication. 1 row of 11 tiles 11 rows Try “add 10” because 18 1 10 5 28. of 1 tile each List the factors. Test “add 10”: 28 1 10 5 38. 38 1 10 5 48. 11 1 11 The factors of are and . The rule works. 11 2 1 11 Since only has factors, and , it is prime. Find the next two numbers. RW91-RW92 Think: 48 1 10 5 58; 58 1 10 5 68. 21 So, the rule is “add 10,” and the next two numbers in the pattern are 58 and 68. Make all the arrays you can with 21 square 7 3 3 3 3 7 21 3 1 Find a rule. Then find the next two numbers in your pattern. 7 3 tiles. rows of rows of 21 rows 1. 25, 20, 15, 10, j , j 2. 13, 19, 25, 31, j , j 3 7 tiles tiles of 1 tile each 1 3 21 3. 30, 33, 32, 35, 34, 37, j , j 4. 10, 21, 32, 43, j , j List the factors. 1 row of The factors of 21 are 1, 3, 7, and 21. 21 tiles Since 21 has more than 2 factors, 1, 3, 7, and 21, it is composite. 5. 2, 4, 8, 16, j , j 6. 3, 5, 10, 12, 24, 26, j ,j Make arrays to find the factors. Write prime or composite for each number.

1. 16 2. 19

7. 100, 80, 60, 40, j , j 8. 7, 12, 10, 15, 13, 18, j , j

9. 800, 400, 200, 100, j , j 10. 3, 2, 12, 11, j , j 66/18/07 5:22:03 PM / 1 8

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4 7 Name Name Problem Solving Workshop Strategy: Read and Write Fractions Find a Pattern A fraction is a number that names part of a whole or part of a set. You use fractions every day. You can eat a The table below shows the number of windows on each floor of a fraction of a banana. You can read a fraction of a book. skyscraper. How many windows might be on the tenth floor? Write a fraction for the shaded part. Floor 1 2 3 4 5 6 7 8 9 Find the denominator, or bottom number. Find the numerator, or top number. Number of Windows 4 8 6 108 12101412 The denominator is the total number of The numerator is the number of Read to Understand equal parts in the figure. The rectangle is shaded parts. There are seven parts divided into ten equal parts, so 10 is the shaded, so 7 is the numerator. 1. What are you asked to find? denominator.

numerator __7 , o 10 or seven tenths f the rectangle is shaded. Plan denominator

2. How would the find a pattern strategy help you solve the problem? Write a fraction for the unshaded part. RW93-RW94 Find the denominator, or bottom number. Find the numerator, or top number. The denominator is 10. The numerator is the number of unshaded parts. There are three parts unshaded, so 3 is the Solve numerator. 3. What is the pattern for the table? How many windows might be on the tenth floor? numerator __3 , 10 or three tenths of the rectangle is unshaded. denominator Write a fraction for the shaded part. Write a fraction for the unshaded part. Check 1. 2. 3. 4. Does the answer make sense for the problem? Explain.

shaded shaded shaded Find a pattern to solve. 5. Peter started an exercise program. On 6. Jan rode her bike for 10 minutes the unshaded unshaded unshaded the first day he did 5 pushups. On the 12 first day, minutes the second day, 4. 5. 6. second day he did 10 pushups. On the and 14 minutes the third day. If the third day he did 15 pushups. If the pattern continues, how many pattern continues how many pushups minutes will Jan ride her bike on will Peter do on the sixth day? the fifth day? shaded shaded shaded

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4 8 Name Name Model Equivalent Fractions Compare and Order Fractions Equivalent fractions are two or more fractions that name You can compare two fractions to find out which is larger, the same amount. There are several ways to model even when the two fractions have different denominators. equivalent fractions. You can also compare a group of fractions and put them in order by size. Write two equivalent fractions. 3 1 __ __ , . 5 __4 __1 ___9 _1 _1 Compare 4 and 2 . Write , or . Order the fractions , , and from • Line up one 4 bar with the bar for 1 to show 4 . 6 3 12 _1 _1 least to greatest. • Line up 8 bars to show the same amount as 4 . _2 1_ • 8 shows the same amount as 4 . __1 _1 • Line up 12 bars to show the same amount as 4 . __3 _1 . • 12 shows the same amount as 4 _2 __3 _1 _1 • So, 8 and 12 are equivalent to 4. Line up three 4 bars with the bar for 1 _3 Write two equivalent fractions. to show 4. Start with the fraction bar for 1. 1 1 0 1 2 3 4 5 6 _ _ _4 _1 __9 RW95-RW96 , , • The number lines start at 0 6 6 6 6 6 6 6 Line up one 2 bar to show 2. Line up fraction bars for 6 3 and 12. 1 and end at . Arrange the fraction bars in order from Arrange the fraction bars in order 0 1 2 3 longest to shortest. • When the number lines are lined up, the 3 3 3 3 from shortest to longest. _2 _1 __4 fraction lines up with the fractions and . _3 _1 6 3 12 4 is a longer row than 2. So, the order from least to greatest _1 __4 _2 • So, and are equivalent to . 0 1 2 3 4 5 6 7 8 9 10 11 12 _3 _1 _1 _4 __9 3 12 6 12 12 12 12 12 12 12 12 12 12 12 12 12 So, 4 > 2. is 3 , 6 , 12 .

Write two equivalent fractions for each model. Compare. Write <, >, or = for each . 5 1 2 3 6 4 1. 2. 1. __ __ 2. __ __ 3. __ __ 8 3 4 6 7 5

Order the fractions from least to greatest.

__3 1 9 ___5 1 3 __3 __4 ___4 4. ,, __ ___ 5. ,,__ __ 6. , , 3. 4. 4 4 10 12 2 8 5 9 12 ' ( ) * + , ' ( ) * + , - . / 0 (' , , , , , , (' (' (' (' (' (' (' (' (' (' ('

' ( ) * + , - . / 0 (' (' (' (' (' (' (' (' (' (' (' (' ' ( ) * + , , , , , , , 77/19/07 5:18:30 PM / 1 9 /

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4 9 Name Name Read and Write Mixed Numbers Compare and Order Mixed Numbers A mixed number is a number made up of a whole number You can use models to compare and order mixed numbers. and a fraction. 6 _1 6 _1 , . 5 Compare 6 and 2. Use , , or . Write a mixed number for the model at the right. • Draw a number line divided into sixths. _1 The circle on the left represents 1 whole figure shaded. Locate 6 . 6 - ( ) * + , . _5 - - - - - The circle on the right represents 6 shaded. • Draw a number line divided into halves. 6 _1 1 _5 Locate 2. So, 6 figures are shaded. ( - ) . 5 _5 6 _1 6 _1 Rename 6 as a fraction. • Since 2 is to the right of 6 on the number line, _1 _1 5 _5 then 6 < 6 . Model 6. 6 2 1 _7 1 _1 1 __12 Order the mixed numbers 9, 2, 18 from greatest to least. • Draw a number line divided 1 _7 1 2 3 4 5 6 7 8 2 _6 into ninths. Label 9. 1 9 9 9 9 9 9 9 9 Rename each 1 whole as 6 . Draw a number line divided RW97-RW98 • 1 _1 1 1 2 into halves. Label 2. 2 • Draw a number line divided 1 __12 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 2 The total number of sixths is the numerator of the fraction. into eighteenths. Label 18. 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 The numerator is 35. 1 _7 • Since 9 is farthest to the right, it is the greatest number. 5 _5 __35 . So, 6 renamed as a fraction is 6 1 _1 • Since 2 is farthest to the left, it is the least number. 1 _7 , 1 __12 , 1 _1 Write a mixed number for each picture in standard form and word form. • So, the order from greatest to least is 9 18 2.

1. 2. 3. Compare the mixed numbers. Use < , >, or = for each .

1. 2. 1 1 2 3 4 5 6 7 2 1 2 3 4 5 8 8 8 8 8 8 8 3 6 6 6 6 6 4

1 2 3 4 1 2 3 4 5 6 7 1 5 5 5 5 2 3 8 8 8 8 8 8 8 4

__3 __2 3 4 Rename the fraction as a mixed number and each mixed number as a fraction. 1 1 3 __ 3__ 8 5 6 8 You may wish to draw a picture. Order the mixed numbers from greatest to least. 5 3 _ 3 _ 7 4 1 1 10 3 3 3 2 1 2 3 4. 3 5. 4 2 __ , 2__ , 3__ 4 __ , 4___ , 4__ 2__ , 3 __ , 1__ 5 __ , 6__ , 6 __ 3. 9 5 3 4. 5 11 4 5. 5 7 8 6. 5 5 8 77/19/07 5:18:49 PM / 1 9 /

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5 0 Name Name Problem Solving Workshop Skill: Model Addition Number lines can help you add fractions. Fractions that have the same Sequence Information denominator, or bottom number, are called like fractions. When 1 _1 Gloria gives her youungest cat 2 cups of food. She gives fractions have the same denominator, you can add the numerators, _7 the oldest cat 8 cup of food and her second-oldest cat or top numbers. The denominator always stays the same. 1 _3 4 cups of food. Draw a number line to represent the amounts Find the sum. of food. Which cats gets the most food? __1 1 1__ 1. What are you asked to find? 2 2

2. What are the three amounts of food that you are comparing? 1 2 0 2 2 3. Label the three number lines below for each amount of food. • The denominator tells the number of equal parts the number line is divided into. • The denominator is 2, so the number line is divided into 2 equal parts. _1 _2 0 1 2 • Label the number line with 0, 2 , and 2 . RW99-RW100 _1 _1 _1 _2 • Shade the part from 0 to 2 . To add 2 , move from 2 to 1, or 2 . 012 • Since there are 2 equal parts in all, that means there are _2 2 out of 2 equal parts, or 2 . _1 _1 _2 • So, 1 5 , or 1. 0 1 2 2 2 2

Find the sum. 4. Order the food amounts from greatest to least. 1. 2. 3. ( ) * + , 0 1 2 3 4 5 6 7 8 9 1 ( ) * + , - 5. Which cat gets the most food? ' - - - - - ( 10 10 10 10 10 10 10 10 10 ' ...... ( 5 2 6. How can you check your answer? __5 1 __1 ___ 1 ___ __3 1 __2 6 6 10 10 7 7

7. Mike has three movies and wants to 8. Joe, Lara, and Mike share a bucket of 2 _2 watch the longest one. The first movie popcorn. Joe eats 3 cups, Lara eats 2 _1 2 _3 2 _1 is 3 hours long, the second movie is 6 cups, and Mike eats 5 cups. Who 1 _7 Model the sum. Record your answer. 8 hours long, and the third movie eats the most popcorn? Who eats the _2 is 2 hours long. Order the length of 5 least? Order the amounts of popcorn 4. 5. 6. the movies from longest to shortest. from least to greatest. 3 4 2 __1 1 __1 __ 1 __1 __ 1 __ 3 3 7 7 7 7 77/19/07 5:19:42 PM / 1 9 /

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5 1 Name Name Model Subtraction Record Addition and Subtraction Number lines can be used to subtract fractions. When you add or subtract fraction with like denominators, Fractions with common denominators are called like add or substract only the numerators. fractions. When you have like fractions, you only need to Find and record the sum. subtract the numerators. The denominator stays the same. 4 __ 1 __3 Find the difference. 6 6 __1 2 1__ 2 2

15 0 1 2 2 2 • The denominator is 2 so draw a number line and divide it into 2 equal parts. _1 • Begin at 2 . 1 RW101-RW102 _ 1 • Subtract 2 by counting back part on the number line. 4 parts shaded 3 parts shaded 7 parts shaded ______15______0 _0 6 equal parts 6 equal parts 6 equal parts • You stopped on so, 2 of the number line remains. _1 _1 _0 That means 2 2 2 5 2 , or 0. 4 3 Add the numerators. 4_____ 1 3 7__ __1 __ 1 __ , or 1 Find the difference. 6 6 Write the denominator. 6 6 6 4 3 7 1 So, __ 1 __ 5 __ , or 1__ . 1. 2. 3. 6 6 6 6 ( ) * + 0 1 2 3 4 5 6 7 8 9 1 ( ) * + , ' , , , , ( 10 10 10 10 10 10 10 10 10 ' - - - - - ( Find and record the sum or difference. 4 1 __ 2 __ ___ 6 2 ___2 2__ 2 1__ 3 4 3 1 1 7 8 3 5 5 1. __ 1 __ 2. _ _ 2 __ 3. _ _ 1 __ 4. ___ 2 ___ 10 10 6 6 5 5 4 4 9 9 11 11

Model the difference. Record your answer. 10 5 2 1 5 4 6 4 5. ___ 2 ___ 6. __ 1 __ __ 1 __ ___ 2 ___ 4. 5. 6. 12 12 3 3 7. 6 6 8. 10 10

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5 2 Name Name Problem Solving Workshop Strategy: Add and Subtract Mixed Numbers Write an Equation You can add and subtract mixed numbers with like denominators. Add or subtract the fractions first. Then add or subtract the whole numbers. Jen takes a dance class of hip-hop and jazz. The class lasts You can check your answer by making a model of the equation. __9 __4 hour. Hip-hop lasts hour. Which part lasts longer? 1 3 10 10 3__ 1 1__ Model and record the sum. 6 6 Read to Understand Start with the fractions. Add the 1 3 4 1. What are you asked to find? Write the question as a fill-in-the-blank sentence. __ + __ = __ numerators first. Keep the denominators 6 6 6 the same. 15 Add the whole numbers. 3 + 1 = 4 If needed, change any improper 1 3 4 Plan 3 __ 1 1__ 54__ fractions in the sum to a mixed number. 6 6 6 2. Choose a variable. What does the variable represent? 4 2 1 3 4 2 Write the answer in simplest form. 4 __ 5 4__ 3 __ 1 1__ 5 4__ , or 4__ 6 3 6 6 6 3 RW103-RW104 Model and record the difference. 4 2 1__ 2 __ 5 5 Solve Start with the fractions. Subtract the 4 __ 2 __ 2 5 2__ 3. Write an equation to find the length of the jazz class. Which part of the 5 5 5 numerators. Keep the denominators the class lasts longer? same. Subtract the whole numbers. 1 2 0 5 1 4 2 2 2 5 Check If needed, change any improper 1__ 2 __ 5 1 __ 5 5 5 4. How can you check your work? fractions in the difference to a mixed number.

2__ 4 2 2 Write the answer in simplest form. 1 1__ 2 __ 5 1__ 5 5 5 5 Model and record the sum or difference. _5 __6 5. Of George’s books, 8 are about animals. 6. Maria spends 14 of her day in 1 __2 4 __1 6 __5 3 __7 _2 __2 1. 5 2. 3 3. 6 4. 8 There are 8 books in his animal school. She studies math for 14 of collection about dogs. What fraction of her time at school. What fraction of 1 1 1__ 1 4 2__ 2 2__ 1 2 2 6__ 5 3 6 8 the books are about other animals? Maria’s day is spent studying other Write an equation to solve. subjects in school? Write an equation to solve.

2 ___7 3 __4 2 ___8 4 __8 5. 10 6. 7 7. 11 8. 9 1 3 ___5 2 2 __4 1 2 ___8 1 3 __5 10 7 11 9 77/19/07 5:13:54 PM / 1 9 /

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5 3 Name Name Model Addition and Subtraction Relate Fractions and Decimals of Unlike Fractions A decimal is a number with one or more digits to the right of the decimal point. Write the fraction and decimal shown by the model. Unlike fractions are fractions that have different denominators, 0 5 10 or bottom numbers. You can use a model to add or subtract 10 10 10 unlike fractions. 0 0.5 1 Model to find the sum. The number line is divided into 10 equal parts.

2 1 First, label the parts below the number line using tenths __ 1 __ 3 6 Use pattern blocks to model each fraction. starting at 0 and stopping at 1.

_2 Then, label the parts above the number line using fractions Model . __0 __10 3 starting at 10 and ending at 10 . _1 Use two 3 pattern blocks. 0 1 2 3 4 5 6 7 8 9 10 10 10 10 10 10 10 10 10 10 10 10

_2 RW105-RW106 Use pattern blocks to find a fraction equivalent to 3 . 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

2 4 Locate the point on the number line. __ 5 __ __6 0.6 3 6 The fraction 10 and the decimal names the point. __ 6 So, the fraction and the decimal shown by the model is and 0.6. _1 10 Model . 6 Write the fraction as a decimal. _1 Use one 6 pattern block. ____5 Read the fraction aloud as five hundredths. 100 _2 _4 You found that 3 is equivalent to 6. Write a zero in the ones place, and 0.jj _4 _1 Since 6 and 6 have the same denominator, you can add the fractions. then write the decimal point. Write a zero in the tenths place. 0.0 j 4 1 5 Add. __ 1 __ 5 __ Write a 5 in the hundredths place. 0.05j 6 6 6 ___5 So, 100 written as a decimal is 0.05. , _2 1 _1 5 _5 . So 3 6 6 Write the fraction and decimal shown by each model. 1. 0 5 10 2. 0 5 10 Model to find the sum or difference. 10 10 10 10 10 10 3 __1 1 __1 5 __2 1 __ 5 ___7 2 __ 1 5 010.5 010.5 1. 3 4 2. 4 8 3. 10 5

2 ___3 7 2 1 __2 Write each fraction as a decimal. You may draw a picture. 4. __ 1 5 5. __ 2 __ 5 6. __ 1 5 5 10 9 3 6 3 ____3 __ 4 ____20 ___9 3. 100 4. 8 5. 100 6. 10 77/19/07 5:20:05 PM / 1 9 /

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5 4 Name Name Decimals to Thousandths Equivalent Decimals Write 0.303 as a fraction. Write 0.007 as a fraction. You can use models to show equivalent decimals. Read the decimal aloud. Read the decimal aloud. Use a hundredths model. Are the two decimals equivalent? 5 ﬁ Think: The decimal is “three hundred three Think: The decimal is “seven Write or . 0.71 0.17 thousandths.” thousandths.” Model and . Write the numbers to the right of the Write the number to the right of the Shade in the appropriate number of squares in each hundredths model. decimal as the numerator. decimal as the numerator. Think: The number to the right of the Think: The number to the right of the decimal is 303. decimal is 7. The denominator is the place value of the The denominator is the value of the digit 0.71 71 0.17 17 digit farthest right. which is farthest right. is hundredths. is hundredths. Shade in 71 squares. Shade in 17 squares. Think: The digit farthest right is in the Think: The digit farthest right is in the thousandths place so the denominator thousandths place so the denominator is See if the models have the same number of squares shaded. RW107-RW108 0.71 0.17 is 1,000. 1,000. has more squares shaded than . 0.303 ____303 . The decimal models show that 0.71 is not equivalentt to 0.17. So, written as a fraction is 1,000 ____7 . So, 0.007 written as a fraction is 1,000 So, 0.71 ﬁ 0.17.

Write each decimal as a fraction. Write an equivalent decimal for 0.20. You may use a decimal model. 1. 0.725 2. 0.238 3. 0.093 4. 0.512 Shade 20 squares of a hundredths model to show 0.20. There are two columns shaded.

Shade two columns of a tenths model. 5. 0.784 6. 0.056 7. 0.207 8. 0.003 The decimal models show that 0.2 is equivalent to 0.20. So, 0.2 5 0.20.

Use a tenths model and a hundredths model. Are the two 9. 0.692 10. 0.555 11. 0.413 12. 0.708 decimals equivalent? Write 5 or ﬁ.

1. 0.3 and 0.30 2. 0.72 and 0.27 3. 0.60 and 0.6 4. 0.05 and 0.50

13. 0.312 14. 0.009 15. 0.475 16. 0.902 Write an equivalent decimal for each. You may use decimal models. 5. 0.9 6. 0.2 7. 0.50 8. 0.7 66/18/07 5:23:42 PM / 1 8

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5 5 Name Name Relate Mixed Numbers and Decimals Compare and Order Decimals Write an equivalent mixed number and a decimal for the model. Use the number line to order the decimals from least to greatest. The number line below is divided into 10 equal parts. $2.90, $2.09, $2.50, $2.55 110 5 110 Locate the dollar amounts on the number line below. 10 10 10

1.0 1.5 2.0 $2.09 $2.50 $2.55 $2.90 First, label the parts below the number line using tenths 1.0 2.0 starting at and stopping at . 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90 3.00 Then, label the parts above the number line using fractions __0 __10 starting at 1 and stopping at 1 . 10 10 The decimals on a number line go in order from left to right. 10 111111111 2 3 4 5 6 7 8 119 10 10 10 10 10 10 10 10 10 10 10 10 The decimal that is farthest to the left is the least. The decimal that is farthest to the right is the greatest. 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.09 is the farthest left, so it is the least. Locate the point on the number line. RW109-RW110 2.90 is the farthest right, so it is the greatest. 1.8 1 __8 The decimal is and the mixed number is 10. So, the order of the decimals from least to greatest is $2.09, $2.50, $2.55, $2.90. 1 __8 1.8 So, the mixed number and the decimal shown by the model is 10 and . Compare. Write <, >, or 5 for the . Write the decimal for the mixed number. Then write the word form. 2 _3 $1.64 $1.46 4 _3 100 _3 5 _____3 3 25 5 ___75 Write one decimal above the other. 1.64 • Write the fraction 4 using a denominator of . 4 4 3 25 100 Be sure to line up the decimal points. 1.46 ___75 0.75 • Write 100 as a decimal. Both decimals start with one. Continue comparing. • Write the whole number. 2.75 Compare the next digit to the right, the tenths place. 1.64 2 _3 2.75 So, 4 written as a decimal is , or two and seventy-five hundredths. Since 6 > 4, that means 1.64 > 1.46. 1.46 Write an equivalent mixed number and a decimal for each model. So, $1.64 > $1.46. 1. 3 0 3 5 310 2. 10 10 10 Use the number line above to order the decimals from least to greatest. 3.0 3.5 4.0 1. $2.20, $2.21, $2.02, $2.10 2. $2.33, $2.43, $2.34, $2.40

Write an equivalent mixed number and a decimal for each. Then write the word form. You may use a model. Compare. Write <, >, or 5 for each . 5.1 2 ___7 3. 4. 100 3. 0.43 0.34 4. 2.36 2.09 5. $0.57 $1.57

6. $1.80 $1.83 7. 2.27 3.25 8. 1.03 1.30 77/19/07 5:20:35 PM / 1 9 /

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5 6 Name Name Problem Solving Workshop Skill: Round Decimals Draw Conclusions When you round a decimal to the nearest tenth, the rounded number will stop at the tenths place. For example, when 4.73 is 2.4 Megan lives miles from Bob’s Bookstore. Josh lives rounded to the nearest tenth, it becomes 4.7. 2.15 miles away, and Sam lives 2.09 miles away. Who lives closest to Bob’s Bookstore? Round 5.55 to the nearest tenth. Round $48.92 to the nearest dollar.

1. What are you asked to find? • Identify the tenths place. • When you round to the nearest dollar, you round to the ones place. Identify Think: The 5 to the right of the decimal the ones place. point is in the tenths place. Think: The 8 2. How can you use information in the problem to draw a conclusion? • Look at the digit to the right of the is in the ones place. tenths place. • If the number to the right of the ones If that number is less than 5, the digit in place is less than 5, the digit in the the tenths place stays the same. If it is ones place stays the same. If it is 5 or 5 or more, the digit is increased by 1. more, the digit is increased by 1. RW111-RW112 3. Use the information in the problem to order the distances from nearest to farthest Think: The digit to the right of the tenths Think: 9 is greater than 5, so round the distance from Bob’s Bookstore. place is 5, so round the number in the number in the ones place up. Round tenths place up. Round 5.55 to 5.6. $48.92 to $49. So, 5.55 rounded to the nearest tenth So, $48.92 rounded to the nearest dollar 4. Who lives closest to Bob’s Bookstore? Write your answer in a complete sentence. is 5.6. is $49.

Round each number to the nearest tenth and each money 5. How can you check your answer? amount to the nearest dollar.

1. 1.92 2. $21.10 3. 56.16 4. $24.60

Solve the problem. 5. 6.07 6. 28.34 7. 6.37 8. 55.92 6. Dane spent $12.50 on school supplies. 7. Sarah leaped 2.1 meters. Kelly leaped Peter spent $12.05. Marshall spent 2.23 meters. Linda leaped $12.25. Which student spent the least 2.19 meters. Who leaped the farthest on school supplies: Dane, Peter, or distance: Sarah, Kelly, or Linda? $81.99 8.35 48.48 8.77 Marshall? 9. 10. 11. 12.

13. 39.94 14. $6.03 15. 18.26 16. 77.98 66/18/07 5:24:06 PM / 1 8

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5 7 Name Name Estimate Decimal Sums and Differences Model Addition You can use rounding to estimate decimal sums and differences. You can use decimal models to help you add decimals.

Estimate the difference. Use a model to find the sum.

$91.24 1.6

__2 $39.87 $91.24 $91 _1 1.0

__2 $39.87 __2 $40 Round each number to the nearest dollar. Step 1: Shade squares to represent 1.6. : 6 $91 Think Shade all the squares in one model and squares in another Estimate by subtracting the rounded numbers. __2 $40 hundredths model. So, an estimate for $91.24 2 $39.87 is $51. $51 Estimate the sum.

4.0 4.0 4

8.9 8.9 9 RW113-RW114 1_ 1.4 1_ 1.4 _ 1 1 Step 2: Shade in squares to represent 1.0. Round each number to the nearest whole number. 4 Think: Add one whole model to the model of 1.6. Estimate by adding the rounded numbers. 9 _1 1 So, an estimate for 4.0 1 8.9 1 1.4 is 14. 14 Estimate the sum or difference.

3.9 $52.35 5.02 1. 2. 3. _1 1.8 __2 40.22 4.96 Step 3: Count all the squares. 1 1.78 __ Think: There are 2 whole models shaded and 6 squares of a hundredths model, or 2.6.

$31.99 81.25 23.92 1.6 1 1.0 5 2.6 4. 5. 6. So, . __1 12.12 __2 29.50 __2 11.03 Use models to find the sum. 1. 2.1 1 0.59 2. 1.4 1 0.22

3.95 58.75 $1.87 7. 8. 9. 7.11 __2 45.11 __1 1.09 1__ 2.18

6.67 8.67 $72.99 3. 1.27 4. 0.81 10. 11. 12. __1 2.25 5.89 __2 32.87 __1 1.15 __1 0.43

1__ 6.92

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5 8 Name Name Model Subtraction Record Addition and Subtraction You can use decimal models to help you subtract decimals. Equivalent decimals can help add or subtract numbers that do not have the same number of digits after the decimal point. Use a model to find the difference. Estimate. Then record the sum. 1.1

_2 0.4 $23.44 __1 $19.85

Step 1: Shade 11 columns on decimal models to show 1.1. Estimate the sum by rounding to the nearest dollar. $23.44 $23 1 Shade all the columns on one model and column on another model. __1 $19.85 __1 $20 Add the estimates. $43 11 Line up the decimal points and add to find the sum. $23.44 __1 $19.85

$43.29 Since $43.29 is close to the estimate of $43, it is reasonable.

RW115-RW116 4 0.4 Step 2: Draw Xs on of the shaded columns to show . So, $23.44 1 $19.85 5 $43.29. Estimate. Then record the difference.

67.1 2__ 9.98

Estimate the difference by rounding to the 67.1 67 nearest whole number. __2 9.98 _2 10 Step 3: Count the shaded columns that do not have Xs on them. There are 7 columns that do not have Xs on them. 57 So, 1.1 2 0.4 5 0.7. Line up the decimal points. Place zeros to the 67.10 right of the decimal point so each number has __2 9.98 Use models to find the difference. the same number of digits after the decimal point. 5161010 1. 1.4 2 0.61 2. 1.6 2 1.08 Subtract as you would with whole numbers. 67.10 Place the decimal point in the difference. __2 9.98 Since 57.12 is close to the estimate of 57, it is reasonable. 57.12 So, 67.1 2 9.98 5 57.12. Estimate. Then record the difference.

0.84 1.39 3. 4. 1. 7.5 2. $4.55 3. 24.3 4. 71.7 __2 0.17 __2 1.14 _2 2.3 1 1.97 2__ 18.18 __1 2.34 __

5. 8.2 6. $45.36 7. 61.7 8. 52.99 _1 3.5 __2 19.55 __1 12.84 __1 65.20 77/19/07 5:21:01 PM / 1 9 /

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5 9 Name Name Make Change Problem Solving Workshop Strategy: Make change. List the bills and coins. Compare Strategies Count On Subtract The vending machine has 10 drinks left. Some are juice and some are Cost: $6.57 Cost: $3.34 water. There are more cans of juice left than bottles of water. What Paid with: Paid with: possible combinations of water and juice could be left in the machine? Step 1: Count on from the cost using coins Step 1: Subtract the cost from the Read to Understand and bills. amount paid. 1. What are you asked to find? $6.57 Think: The item cost . Count on: $5.00 Think: __2 $3.34 $1.66 Plan $6.58 $6.59 $6.60 $6.70 $6.75 $7.00 2. What strategy could you use to solve this problem? Step 2: Count on to make the change, $1.66. Start with the bills. RW117-RW118 Think: $1.00, $1.25, $1.50, $1.60, Solve $8.00 $9.00 $10.00 $1.65, $1.66. 3. Show how you use your strategy to solve the problem. What possible combinations Step 2: Count the coins and bills. of water and juice could be left in the machine? So, the change is 1 $1 bill, Think: 3 pennies, 1 dime, 1 nickel, 1 quarter, 2 quarters, 1 dime, 1 nickel, and 3 $1 bills 5 $3.43. 1 penny. So, the change is $3.43.

Make change. List the bills and coins.

1. Cost: $5.95 2. Cost: $2.28 Paid with: Paid with: Check 4. How can you check to see if your answer is reasonable?

Choose a strategy to solve.

5. Jen has a red shirt, a blue shirt, and a 6. The mail comes at 1:00 P.M. on 3. Cost: $13.39 4. Cost: $4.63 green shirt. She has black pants and Monday, 2:00 P.M. on Tuesday, Paid with: Paid with: blue pants. How many different 1:00 P.M. on Wednesday, and shirt-pant combinations can Jen make? 2:00 P.M. on Thursday. What time will Tell which strategy you used. the mail come on Friday? Tell which strategy you used. 77/19/07 5:21:28 PM / 1 9 /

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6 0 Name Name Points, Lines, and Rays Measure and Classify Angles Term Definition Example Name a geometric term An angle is formed when two rays or two line segments share the same endpoint, or that represents each vertex. An acute angle measures greater than 0º and less than 90º. An object. names an exact location in obtuse angle measures greater than 90º and less than 180º. A straight angle point A space flagpole forms a straight line and measures 180º. A right angle measures 90º and forms a square corner. • A flagpole does not line part of a line; has two continue without end in Use a protractor to measure the angle. Then write acute, right, straight, or obtuse. segment endpoints DE both directions. The tops and bottom of the pole are like endpoints. straight path of points that L F So, a flagpole can be line continues without end in both • H BC M directions represented by a line segment. G part of a line that has one laser beam N RW119-RW120 ray endpoint and continues FG without end in one direction • A laser beam comes out • When you measure an angle using a of a box or pointer and protractor, remember to place the center shines without end in flat surface that continues L point of the protractor on the vertex of plane K M one direciton. without end in all directions the angle. • /LMN measures 90º. An angle that • So, a laser beam can be • /FGH measures 60º. measures 90º is a right angle. represented by a ray. So, /FGH is an acute angle. So, /LMN is a right angle. Name a geometric term that represents the object. Use a protractor to measure the angle. Then write acute, right, straight, or obtuse. 1. highway 2. tip of a marker 3. screwdriver 1. A 2. L 3. B C J

M K 4. football field 5. grain of salt 6. two way arrow N L

7. meadow 8. piece of rope 9. flashlight beam SRQ D 4. 5. X 6. Z E F 10. extension cord 11. farm field 12. railroad track Y 77/19/07 5:21:55 PM / 1 9 /

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6 1 Name Name Line Relationships Polygons Pairs of lines may be intersecting, parallel, or perpendicular. Polygons are closed plane figures that are formed by 3 or more straight Name any line relationships you see sides that are line segments. A regular polygon’s sides Type of Line Deﬁ nition Example in the figure below. Write intersecting, are all the same length, and its angles are all the same measure. intersecting lines that cross parallel, or perpendicular. Name the polygon. Tell if it appears each other at one Sides and to be regular or not regular. point and form A obtuse Polygon Example B Angles four angles acute D 3 sides C triangle 3 angles 3 vertices parallel lines that never 4 sides intersect and are D The lines dividing the black and white quadrilateral 4 angles The polygon has 3 sides, so it is a triangle. always the same = • 4 vertices • A stripes are parallel lines because the distance apart 5 sides > • The sides of the triangle are not all the same lines never intersect. pentagon 5 angles length, so the triangle is not regular. 5 vertices • The lines that form the top of the kite RW121-RW122 6 sides perpendicular lines that inter- are intersecting lines because they hexagon 6 angles sect and form M cross at one point and form an angle. 6 vertices four right angles 8 sides K L right • The lines where the kite poles octagon 8 angles intersect the stripes are perpendicular 8 vertices • The polygon has 8 sides, so it is an octagon. N lines because they intersect and form 10 sides The angles of the octagon are all the same right angles. decagon 10 angles • 10 vertices measure, and the sides are all the same So, the figure contains parallel, intersecting, and perpendicular lines. length, so the octagon is regular. Name any line relationships you see in each figure. Name the polygon. Tell if it appears to be regular or not regular. Write intersecting, parallel, or perpendicular. 1. 2. 3. 1. 2.

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6 2 Name Name Classify Triangles Classify Quadrilaterals Classify each triangle. Write A quadrilateral is a polygon with 4 sides and 4 angles. Triangle Description Example isosceles, scalene, or equilateral. Classify each figure in as many of the Then write right, acute, or obtuse. Quadrilateral Description following ways as possible. Write 2 cm 2 cm equilateral 3 equal sides quadrilateral, parallelogram, rhombus, - 2 pairs of parallel sides 9 ft parallelogram rectangle, square, or trapezoid. 2 cm 6 ft - opposite sides are equal

isosceles two equal sides 3 cm 3 cm 7 ft - 2 pairs of parallel sides square - 4 equal sides 2 cm • Each side is a different length, so • The figure has 4 sides, so it is a - 4 right angles 4 cm the triangle is scalene. quadrilateral. It has 2 pairs of parallel sides, scalene no equal sides 2 cm so it is a parallelogram. All 4 sides are 3 cm • There is one obtuse angle, so - 2 pairs of parallel sides equal, so it is a rhombus. the triangle is obtuse. rectangle - opposite sides are equal Draw an example of a quadrilateral - 4 right angles RW123-RW124 right 1 right angle that has no parallel sides. 9 yd 9 yd - 2 pairs of parallel sides rhombus - 4 equal sides acute 3 acute angles 9 yd - exactly 1 pair of parallel • All three sides are equal, so the trapezoid sides This figure has 4 sides and none of the triangle is equilateral. • obtuse 1 obtuse angle sides are parallel. • There are 3 acute angles, so the triangle is acute. Classify each figure in as many of the following ways as possible. Write quadrilateral, parallelogram, rhombus, rectangle, square, or trapezoid. Classify each triangle. Write isosceles, scalene, or equilateral. Then write right, acute, or obtuse. 1. 2. 3. 1. 2. 3. 12 cm 12 cm 7 ft 8 in. 8 in. 5 ft

8 in. 4 cm 6 ft Draw an example of each quadrilateral. 4. It has 2 pairs of parallel 5. It has 2 equal sides. 6. It has only 1 pair of sides. parallel sides. 6 6 8 m 4. 12 5. yd yd 6. 3 m m

9 9 m 8 m yd 10 m 77/19/07 5:22:32 PM / 1 9 /

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6 3 Name Name Circles Problem Solving Workshop Strategy: A circle is a closed figure made up of points that are the same distance from the center. Circles have many parts. Compare Strategies Construct circle J with a Part of George wants to design a swimming pool that has at least two Description Example 2-centimeter radius. Label each ACB Circle sets of parallel sides and at least two sets of equal sides. of the following. Which of the swimming pool diagrams at the right would look B a line segment a. chord: EF like George’s design? ED whose two A b. diameter: BC Read to Understand chord endpoints are on P the circle 1. What are you asked to find? chord: AB J C Plan a chord that B passes through 2. What strategy would you use to solve this problem? diameter CD E the center of the P

RW125-RW126 circle F diameter: CD • Radius JC is 2 cm. a line Solve segment with • Circle J has chord EF on it. Chord 3. Complete the Venn diagram and table below to solve the problem. one endpoint at EF is a line segment whose two radius K the center and P endpoints are on the circle. Use Logical Reasoning Make a Table one endpoint on Figure ABCDE the circle radius: PK • Diameter BC passes through the A center of the circle. 2 sets 2 sets 2 sets of parallel sides yes no parallel equal sides sides Construct circle P with a 3-centimeter Construct circle L with a 1-inch radius. 2 sets of equal sides yes no radius. Label each of the following. Label each of the following. 1. chord: MK 4. chord: XY 4. Which of the swimming pool diagrams would look like George’s design? 2. radius: PB 5. radius: LA Check 3. diameter: AD 6. diameter: CN 5. Which strategy was more helpful: use logical reasoning or make a table? Why?

Solve. 6. Pete’s sandbox has at least 1 right 7. A piece of fabric has at least 1 set angle. Which figure from above looks of parallel sides and at least 2 like the sandbox? obtuse angles. Which figures from above match this description? 66/18/07 5:25:22 PM / 1 8

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6 4 Name Name Congruent and Similar Figures Turns and Symmetry Congruent figures have the exact same size and shape. If you draw a line to divide a figure in half, and the two halves are Similar figures have the same shape but different sizes. identical to one another, then the figure has line symmetry. If a Tell whether the two figures are congruent and similar, similar, or neither. figure can be rotated around a center point and still look the same in at least two positions, it has rotational symmetry. Tell whether the figure appears to have line symmetry, rotational symmetry, both, or neither.

• The figures both have the exact same • The figures are the same shape, so they shape. This means they are similar. are similar. AB AB • The figures both have the exact same • Both of the figures are the same size. • In figure A, the penny does not look size. Since they are also the exact same Because they also have the same shape, • In figure A, the bug is exactly the same exactly the same on both sides of the on each side. So, the bug has line line. So, the penny does not have line RW127-RW128 shape, the figures are congruent. they are congruent. symmetry. symmetry. • So, the figures are both congruent and • So, the figures are both congruent and similar. similar. • In figure B, the bug has been rotated. It • In figure B, the penny has been rotated. It looks different from figure A. So, the bug now looks different from figure A. So, the does not have rotational symmetry. penny does not have rotational Tell whether the two figures are congruent and similar, similar, or neither. symmetry. • So, the bug has line symmetry. 1. 2. 3. • So, the answer is neither. Tell whether the figure appears to have line symmetry, rotational symmetry, both, or neither. 1. 2. 3.

4. 5. 6.

7. 8. 9.

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6 5 Name Name Transformations Problem Solving Workshop Strategy: Act it Out A transformation is the movement of a figure. There are three Buddy used pattern blocks to make the two rabbits kinds of transformations. A translation moves a figure by sliding shown at the right. Are the rabbits congruent? it in a straight line. A rotation moves a figure by turning it around a Explain how you know. single point. A reflection moves a figure by flipping it over a Read to Understand line. Tell how the figure was moved. Draw figures to show a translation, a rotation, 1. What are you asked to find out in this problem? Write translation, rotation, or and a reflection of the figure. reflection.

reflection Plan 2. How would you use the act it out strategy to solve the problem?

• If the figure was a reflection, or flipped over, it would look like two

RW129-RW130 arrows pointing in opposite translation directions. Solve

• If the figure was a translation, the 3. Solve the problem. Describe the strategy you used. arrows would be in different places but both pointing in the rotation same direction. • The figure has been turned, or rotated. So, the figure is a rotation. translation rotation reflection Tell how each figure was moved. Write , , or . 4. Are the rabbits congruent? 1. 2. 3.

Check

5. What other strategy could you use to solve the problem? Draw figures to show a translation, a rotation, and a reflection of the figure.

4. Act it out to solve. 6. Tim made the sun below using pattern 7. Mara made the leaf below using blocks. Does Tim’s sun have pattern blocks. Does Mara’s leaf line symmetry? have line symmetry?

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6 6 Name Name Tesselations Geometric Patterns A tessellation is a repeating pattern of closed figures that covers Geometric patterns are patterns based on color, size, shape, position, a surface. There are no open spaces or gaps in a tessellation, and number of figures. A geometric pattern repeats over and over again. and none of the figures in the tessellation ever overlap. If a figure Find a possible pattern. Then draw the next two figures in your pattern. can be used to make a tessellation, the figure is said to “tessellate.”

Tell whether the figure will tessellate. Write yes or no.

• The pattern uses large and small circles. • The pattern uses large triangles and small The large and small circles alternate: 1 triangles. large, 1 1 1 small, large, small, and so on. • Each large triangle is followed by 3 small • The pointed part of the figure fits • When you use the figure above to try to • The dot on each pair of circles alternates triangles. Then the pattern repeats again: perfectly into the indented part. Because make a tessellation, there are gaps and the between being on the top of each pair of one large triangle, three small triangles, of this, there are no gaps, and the figure surface is not covered. circles and being on the right side of each and so on. RW131-RW132 does not overlap. pair of circles. • So, the next two figures would be: • So, the next two figures would be:

• So yes, the figure will tessellate. • So no, the figure will not tessellate. Find a possible pattern. Then draw the next two figures in your pattern

1. 2. Tell whether the figure will tessellate. Write yes or no. 1. 2. 3.

3. 4.

4. 5. 6.

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6 7 Name Name

Faces, Edges, and Vertices face Draw Figures • The face is a polygon that is a flat surface. You can use square dot paper to draw three-dimensional figures. • The edge is where two faces meet. edge What solid figure does the number cube 1 3 at the right look like? • A vertex is where or more edges meet. vertex 3 2 • The base is the flat surface on which a solid figure rests. base The number cube has 6 faces that are square. It has 12 edges and A cube 8 vertices. A solid figure with 6 square faces, 12 edges and 8 vertices has 12 edges. is a cube or a rectangular prism. Use dot paper to draw the number cube. A rectangular Remember: all sides of a cube are the same length. prism has 12 edges. Draw a square Draw slanted line segments Connect the end points from the vertices. and draw hidden lines. A triangular pyramid

RW133-RW134 has 6 edges.

A square pyramid has 8 edges.

A rectangular For 1–2, use dot paper to draw each figure. pyramid has 8 edges. 1. What solid figure does Name a solid figure that has 8 edges. the soup can look like? A square pyramid and a rectangular pyramid both have 8 edges.

Name a solid figure that is described. 1. 5 faces 2. 4 faces 3. all square faces

2. What solid figure does the shoebox look like?

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6 8 Name Name Patterns for Solid Figures Different Views of Solid Figures A net is a pattern that can be folded to make a three-dimensional figure. You can identify solid figures by the way they look from different views. For example, Draw a net that can be cut to make a model of each solid figure. you can look at the object from the top, from the front, and from the sides. Name the solid figure that has the following views. A square pyramid has 5 faces. top view front view side view

The base of a square pyramid is a square. Draw the base.

The other faces of a square pyramid are triangles. • The top view shows that the figure has a rectangular base. Draw the other 4 faces of the solid figure. The crossed lines show that the sides come together to form a point. • The front and side views show that the faces are triangles. • A solid figure with a rectangular base and triangular faces is a rectangular pyramid. W3 RW136 RW135 So, the figure is a rectangular pyramid. A cube has 6 faces. Name the solid figure that has the following views.

The base of a cube is a square. 1. top view front view side view Draw the base.

Draw the other 5 faces.

2. top view front view side view

Draw a net that can be cut to make a model of each solid figure. 1. 2. 3.

3. top view front view side view 77/19/07 5:23:15 PM / 1 9 /

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6 9 Name Name Problem Solving Workshop Strategy: Combine and Divide Figures Make a Model You can make new figures by dividing a figure. You can also make a new figure by combining two figures together. Alicia used the fewest possible cubes to make a building whose views are shown below. How many cubes did Alicia use? Draw a picture to show the new figure. Draw a picture to show the new figure. Divide a rectangle to make two triangles. Combine two rectangles to make a square. top view front view side view Step 1: Draw a rectangle. Step 1: Step 2: Draw a line that divides the Draw a rectangle with the width rectangle in half. twice as long as the length. Step 2: Think: Draw the line from one vertex to Draw a second rectangle that another so that the rectangle is touches the first one. It should Read to Understand divided into two triangles, rather be exactly the same size as the first one. 1. What are you asked to find? than two squares. Think: Draw the rectangles side by side, W3 RW138 RW137 so that their longer sides are Plan touching. This forms a square. 2. How would you use the make a model strategy to help solve the problem?

Solve 3. Use centimeter cubes to build the top view. Then add more cubes to create the Draw a picture to show the new figure. front view and the side view. How many cubes did Alicia use? 1. Combine two squares to make a 2. Divide a rectangle to make a triangle rectangle. and a trapezoid. Check

4. How can you check your answer to the problem?

Make a model to solve. 3. Combine two trapezoids to make a 4. Divide a square to make 4 triangles. parallelogram. 5. James uses centimeter cubes to make 6. Pam uses blocks to make a square. a rectangular prism. The prism is Each side of the square is 4 blocks 7 cubes wide, 2 cubes high and 2 cubes long and one block high. Then Pam long. How many cubes does James pulls out 4 blocks in the center of the use? square. How many blocks are left? 66/18/07 5:26:27 PM / 1 8

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7 0 Name Name Measure Fractional Parts Algebra: Change Customary Linear Units The units that are usually used to measure length are inch (in.), You can use multiplication and division to change from one unit foot (ft), yard (yd), and mile (mi). of length to another. You multiply in order to change from a _1 larger unit to a smaller unit, such as feet to inches. You divide in Estimate to the nearest inch. Then measure to the nearest 2 inch. order to change from a smaller unit to a larger unit, such as feet to yards.

Complete. Tell whether you multiply or divide. Customary Units of Length 48 ft 5 j yd

• Think about how the two units are related. 1 foot (ft) 5 12 inches (in.) • 3 feet 5 1 yard. 1 23 • Think: should I multiply or divide? inches 1 yard (yd) 5 3 feet (ft), RW139-RW140 • Feet are smaller than yards, so divide. or 36 inches (in.) • Estimate the length of the fower bud to the nearest inch. 48 4 3 5 16 1 5 5,280 • The length of the flower bud is between 2 and 3 inches. 48 feet feet in 1 yard total yards mile (mi) feet (ft), or 1,760 yards (yd) • To the nearest inch, the flower bud is 3 inches long. So, you divide to show 48 ft 5 16 yd. • Measure the flower bud to the nearest half inch. _1 Complete. Tell whether you multiply or divide. • The length of the flower bud is between 2 2 inches and 3 inches. 2 _1 1. 120 ft 5 j yd 2. 3 ft 5 j in. 3. 27 yd 5 j ft • The length of the flower bud is closer to 2 inches. _1 So, the length of the flower bud to the nearest half inch is 2 2 inches. _1 Estimate to the nearest inch. Then measure to the nearest 2 inch. 1. 2. 4. 48 in. 5 j ft 5. 8 ft 5 j in. 6. 60 ft 5 j yd

7. 13 ft 5 j in. 8. 3,520 yd 5 j mi 9. 2 mi 5 j ft

3. 4.

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7 1 Name Name Weight Customary Capacity The customary units for measuring weight are ounces (oz), Capacity tells how much liquid a container can hold. The customary pounds (lb), and tons (T). There are 16 ounces in one pound. units for measuring capacity are teaspoon (tsp), tablespoon (tbsp), There are 2,000 pounds in one ton. You can use multiplication cup (c), pint (pt), quart (qt), gallon (g), and fluid ounce (fl oz). and division to change from one unit to another. 1 cup = 8 fluid ounces 1 pint = 2 cups 1 tablespoon = 3 teaspoons Complete. Tell whether you multiply or divide. 1 gallon = 4 quarts 1 quart = 2 pints 7 lb 5 j oz Customary Units of Weight Complete the table. Change the units. • Tell how the two units are related. • Determine how many pints are in 1 gallon. • 1 pound 5 16 ounces. Gallons, g 5 j 8 • 1 gallon 5 8 pints • Think: should I multiply or divide? 1 pound (lb) 5 16 ounces (oz) Pints, pt j 56 j • Multiply or divide by 8. • Pounds are larger than ounces, so multiply. 5 g 3 8 5 40 pt 7 3 16 5 112 Gallons, g 578 56 pt 4 8 5 7 g RW141-RW142 1 5 2,000 7 lb oz in 1 lb total oz ton (T) pounds (lbs) 8 3 8 pt 5 64 pt Pints, pt 40 56 64 So, you multiply to show 7 lb 5 112 oz. So, 5 g 5 40 pt, 7 g 5 56 pt, and 8 g 5 64 pt. Complete the table. Change the units. Complete. Tell whether you multiply or divide. • Determine how many cups are in 1 quart. 1. 2 T 5 j lb 2. 64 lb 5 j oz 3. 8,000 lb 5 j T Cups, c 28 40 j • 4 cups 5 1 quart Quarts, qt jj16 • Multiply or divide by 4. 28 c 4 4 5 7 qt 4. 32 oz 5 j lb 5. 3 lb 5 j oz 6. 4 T 5 j lb Cups, c 28 40 64 40 c 4 4 5 10 qt 16 qt 3 4 5 64 c Quarts, qt 71016 So, 28 c 5 7 qt, 40 c 5 10 qt, and 64 c 5 16 qt.

7. 10,000 lb 5 j T 8. 64 oz 5 j lb 9. 2,000 lb 5 j T Complete each table. Change the units. 1. 2. Teaspoons, t 615 Gallons, g 68 Tablespoons, s 7 Quarts, qt 8 10. 144 oz 5 j lb 11. 7 T 5 j lb 12. 11 lb 5 j oz

3. 4. Cups, c 12 18 Cups, c 64 96 13. 160 oz 5 j lb 14. 16,000 lb 5 j T 15. 13 lb 5 j oz Pints, pt 7 Gallons, g 1 66/18/07 5:26:51 PM / 1 8

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7 2 Name Name Problem Solving Workshop Strategy: Metric Length Compare Strategies Some metric units of length are millimeter (mm), centimeter (cm), decimeter (dm), meter (m), and kilometer (km). A coin is An angler is a person who fishes for fun. Mia is an angler. about 1 millimeter thick. Your index finger is about 1 centimeter 6 Mia caught a fish that weighed pounds. How many wide. A CD is a little more than 1 decimeter wide. A baseball bat ounces did the fish weigh? is about 1 meter long. A kilometer is about 4 city blocks long. Read to Understand Choose the most reasonable unit of measure. Write mm, cm, dm, m, or km. 1. What are you asked to find? home school

Plan 2. Which strategy would you use to solve this problem: make a table or draw a diagram? Why? • The map shows a long distance between • The picture shows a short distance

RW143-RW144 school and home, so km would be the across a coin, so mm or cm would be most reasonable unit of measure. the most reasonable units of measure. Solve Choose the most reasonable unit of measure. Write mm, cm, dm, m, or km. 3. How will you solve the problem using the strategy you chose? Show your work. 1. 2.

4. How many ounces did Mia’s fish weigh?

3. 4. Check 5. How can you check to see if your answer is correct?

5. 6. Choose a strategy to solve. 6. Mary’s jump rope is 7 feet long. How 7. A pole is 5 feet long. A baton is many inches long is the rope? 36 inches long. How much longer is the pole than the baton? 77/19/07 5:23:44 PM / 1 9 /

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7 3 Name Name Algebra: Change Metric Linear Units Mass To change metric linear units, multiply or divide by 10, 100, or Mass is how much matter an object contains. Two metric units 1,000. You multiply in order to change larger units to smaller of mass are the gram (g) and kilogram (kg). A dollar bill has a units, such as meters to centimeters. You divide in order to mass of about 1 gram. A textbook has a mass of about 1 change smaller units to larger units, such as meters to kilometers. kilogram. One kilogram is equal to 1,000 grams. You divide in order to change grams to kilograms. You multiply in order to Complete. change kilograms to grams. Tell whether you multiply or divide. Metric Units of Length Complete. Tell whether you multiply or divide. 566 mm 5 j dm 3 kg 5 j g j kg 5 10,000 g • Think about how the two units are related. 1 centimeter (cm) 5 10 millimeters (mm) • Think about how the two units are • Think about how the two units are related. • 1 decimeter 5 100 millimeters. related. • 1 kilogram 5 1,000 grams. • Think: should I multiply or divide? • 1 kilogram 5 1,000 grams. 1 5 10 • Think: should I multiply or divide? • Millimeters are smaller than decimeters, decimeter (dm) centimeters (cm) • Think: should I multiply or divide? • Grams are smaller than kilograms, RW145-RW146 so divide. Kilograms are larger than grams, so divide. 566 4 100 5 5.66 • so multiply. 1 5 1,000 10,000 4 1,000 5 10 millimeters mm in 1 dm total meter (m) millimeters (mm) 3 3 1,000 5 3,000 decimeters grams g in 1 kg total kilograms g in 1 kg total kilograms So, you divide to show 566 mm 5 5.66 dm. 1 kilometer (km) 5 1,000 meters (m) grams So, you multiply to show 3 kg 5 3,000 g. So, you divide to show 10,000 g 5 10 kg. Complete. Tell whether you multiply or divide. Complete. Tell whether you multiply or divide. 1. 40 cm 5 j mm 2. 500 mm 5 j dm 3. 6 km 5 j m 1. 5,000 g 5 j kg 2. 7 kg 5 j g 3. j g 5 2 kg

4. 5,000 cm 5 j m 5. 4 m 5 j dm 6. 200 mm 5 j cm 4. 8 kg 5 j g 5. 9,000 g 5 j kg 6. 3,000 g 5 j kg

600 5 j 8,000 5 j 30 5 j 7. cm dm 8. m km 9. dm m 7. 11,000 g5 j kg 8. 22,000 g 5 j kg 9. 13 kg 5 j g

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7 4 Name Name Metric Capacity Problem Solving Workshop Strategy: Make a Table Capacity is the amount of liquid a container can hold. The metric Juan is making keychains to sell at a craft fair. He plans to units for capacity include liter (L) and milliliter (mL). A dropper sell them for $4 each. How much money will Juan make if he holds about 1 milliliter. An average-sized water bottle holds about sells 14 keychains? 1 1 5 1,000 liter. liter milliliters. You can use multiplication and Read to Understand division to change metric units of capacity. 1. Write the question as a fill-in-the-blank sentence. Complete. Tell whether you multiply or divide.

12,000 mL 5 j L 3 L 5 j mL Plan • Think about how the two units • Think about how the two units 2. How can making a table help you solve the problem? are related. are related. • 1 liter 5 1,000 milliliters. • 1,000 milliliters 5 1 liter • Think: should I multiply or divide? • Think: should I multiply or divide? Solve RW147-RW148 • Milliliters are smaller than liters, so • Liters are larger than milliliters, so multiply. 3. Complete the table to solve the problem. divide. 3 3 1,000 5 3,000 12468101214 12,000 4 1,000 5 12 Keyrings Sold liter mL in 1 L total milliliters mL in 1 L total milliliters Amount Earned $4 $8 $16 liters 3 5 3,000 So, you multiply to show L mL. 4. How much money will Juan make if he sells 14 keychains? Write your answer in a 12,000 5 12 So, you divide to show mL L. complete sentence.

Complete. Tell whether you multiply or divide. 1. 5 L 5 j mL 2. 2,000 mL 5 j L 3. 7 L 5 j mL Check

5. How can you check to see if your answer is correct?

4. 4,000 mL 5 j L 5. 8 L 5 j mL 6. 9 L 5 j mL

Make a table to solve. 6. Dave practices shooting free throws 7. Kelly stacks CDs on her 2 out of every 5 days. How many shelves. Each shelf holds 15 CDs. 12 5 j 15,000 5 j 10,000 5 j 7. L mL 8. mL L 9. mL L days will Dave have practiced after How many CDs will Kelly have 40 school days? stacked if she filled 5 shelves?

10. 25,000 mL 5 j L 11. 6 L 5 j mL 12. 11,000 mL 5 j L 66/18/07 5:27:23 PM / 1 8

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7 5 Name Name Estimate and Measure Perimeter Algebra: Find Perimeter Perimeter is the distance around a figure. You can use string You can find the perimeter of a figure by adding the lengths of and a ruler to find the perimeter of an object. the sides, or by using a formula. Use string to estimate and measure the perimeter of your desk. Find the perimeter. 110 m • Take a long piece of string and wrap it all the way around your desk. )'Zd Make sure that the string is tight against the sides of your desk. If the string is too loose, your estimate and measurement will be less accurate. 42 m 42 m (-Zd • Mark or cut the string to show the perimeter of the desk. (+Zd 110 m • Lay the string on the ground so that it is straight. 5 (2 3 ) 1 (2 3 ) Now estimate: about how many inches long is the string? • Use the formula P l w . This means Perimeter 5 (2 3 length) 1 (2 3 width). • Measure the string with a ruler. Record the perimeter in inches. (,Zd Substitute the measurements for the length and 96 • • A typical school desk has a perimeter of about inches. width into the formula.

RW149-RW150 Find the perimeter of the figure at the right. • Add the lengths of all of the sides. P 5 (2 3 l) 1 (2 3 w) 20 • The shape is a rectangle. • The figure has sides that are cm, 16 15 14 P 5 (2 3 110) 1 (2 3 42) Count the number of units on each side of cm, cm, and cm long. P 5 220 1 84 the rectangle. • 20 1 16 1 15 1 14 5 65. 5 304 • 4 units 1 3 units 1 4 units 1 3 units 5 14 units So, the perimeter of the figure is P 65 304 • So, the figure has a perimeter of 14 units. cm. So, the perimeter of the figure is m.

Use string to estimate and measure the perimeter of each object. Find the perimeter. 1. a textbook 2. a folder 3. a box of crayons 1. 2. ,Zd 3. +Zd /d (,]k Find the perimeter of each figure. .Zd 4. 5. 6. ()d (,]k

4. -d -d 5. 6. *d

7. 8. 9. ,d` (+d -d ((d (*d

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7 6 Name Name Area of Plane Figures Algebra: Find Area Area refers to how many square units are needed to cover a You can use multiplication and formulas in order to find the area surface. of a figure. You must know the figure’s length and width. Estimate the area of each figure. Each unit is 1 sq cm. Find the area. Find the area. 9 mi 6 m

9 mi 16 m

• Identify the figure’s length. The length is • Use the formula A 5 l 3 w. This means the distance across the figure. The length to multiply the figure’s length by its • Count the squares. • Count the squares. of the figure is 9 miles. width. 24 • There are 16 full squares. • There are full squares. Identify the figure’s width. The width is • The figure is 16 m long and 4 • There are 4 almost-full squares. • • There are almost-full squares. the distance from the top of the figure to 6 m wide.

RW151-RW152 8 • There are 6 half-full or almost half-full • There are half-full or almost half-full 8 4 2 5 4 the bottom of the figure. The width of the 5 3 squares. 6 4 2 5 3 squares. about full squares A l w about full squares figure is 9 miles. • Add all of the squares you counted. A 5 16 3 6 • Add all of the squares you counted. • Multiply the length by the width. 24 1 4 1 3 5 31 16 1 4 1 4 5 24 A 5 96 9 3 9 5 81 So, the area of the figure is about So, the area of the figure is about 96 24 24 So, the area is 81 square miles. So, the area is sq m. 31 square centimeters, or 31 sq cm. square centimeters, or sq cm.

Find the area. Estimate the area of each figure. Each unit is 1 sq cm. 1. 2. 3. 3 mi 1. 2. 3. 8 cm 5 yd 11 mi 8 cm 2 yd

4. 5. 6. 4. 5. 6. 4 in. 7 ft 13 km 12 in.

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7 7 NameName Name Problem Solving Workshop Skill: Relate Perimeter and Area Use a Formula The perimeter and area of a figure are related. Sometimes when 10 in. you change the perimeter, the area also changes. Similarly, you A diagram of Ryan’s sand castle is shown at the may change the perimeter of a figure when you change the area. right. He decides to line the perimeter of his castle 18 in. Find the perimeter and area of the figures at the right. with shells. If each shell is two inches long, how 28 in. Then draw another figure that has the same perimeter but a different area. many shells does Ryan need? 9 1 1 1 9 1 1 5 20 1. What are you asked to find? 10 in. • The figure’s perimeter is 20 mi. 9 in. 1 in. 24 in. • 9 3 1 5 9 2. What is the missing length of Ryan’s castle? 10 in. The figure’s area is 9 square in. 24 in. 2 10 in. 5 20 • The figure you draw must have a perimeter of , 10 mm 9 3. How can you use the formula below to find the and an area that is not equal to . ? perimeter of the castle? Find the perimeter of each figure. Then draw another RW153-RW154 P 5 a 1 b 1 c 1 d 1 e 1 f figure that has the same area but a different perimeter. 12 mm 5 1 1 1 1 1 P 10 1 12 1 10 1 12 5 44 24 in. • P 5 inches The figures perimeter is 44 mm. • 12 3 10 5 120 The figure’s area is 120 mm. 120 4. How can you use the perimeter of the sand castle to find the number of shells • So, the figure you draw must have an area of , 44 Ryan needs? Solve the problem. and a perimeter not equal to . Find the perimeter and area of each figure. Then draw another figure that has the same perimeter but a different area.

Use the formula to solve. 1. 2. 3. 5. The diagram below shows a 6. The diagram below shows a 6 m 8 ft 4 cm swimming pool. The owner needs to field divided into four sections. place 1-inch tiles around the pool’s What is the area of Section C? 8 m perimeter. How many tiles will the 8 ft 6 cm owner need? 10 m 20 in.

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7 8 Name Name Estimate and Find Volume of Prisms List All Possible Outcomes Volume is the amount of space a solid figure occupies. Volume is Use the table. measured in cubic units, such as cubic inches, cubic List all the possible outcomes for the experiment. centimeters, and cubic feet. You can find the volume of a solid Sally’s Experiment figure by counting or multiplying. Roll a Number Cube and Toss a Coin Count or Multiply Number Coins 123456 Heads Tails 3 • Make an organized list. • Write the possible outcomes for the coin: heads, tails. • Write the possible outcomes for the number cube: 1, 2, 3, 4, 5, 6. 4 • Pair the possible outcomes of the coin with the possible outcomes of the number 6 cube. Heads, 1; Heads, 2; Heads, 3; Heads, 4; Heads, 5; Heads, 6; RW155-RW156 • Count all of the cubes that are used to • The prism is 6 units long, 4 units wide, Tails, 1; Tails, 2; Tails, 3; Tails, 4; Tails, 5; Tails, 6. 3 make the prism. and units high. So, the possible outcomes for tossing the coin and rolling the number cube are • There are total of 8 cubes. • Multiply length 3 width 3 height. Heads, 1; Heads, 2; Heads, 3; Heads, 4; Heads, 5; Heads, 6; Tails, 1; Tails, 2; Tails, 3; Tails, 4 Tails, 5; Tails, 6. 6 3 4 3 3 5 72

So, the prism has a volume of So, the figure has a volume of For 1, use the pictures below. List all the possible outcomes of the experiment. 8 cubic units. 72 cubic units. 1. tossing the cube labeled 1 to 6 and Count or multiply to find the volume. spinning the pointer on the spinner 1. 2. 3. 6 3 2

For 2 – 3, use the table below.

2. How many possible outcomes are there? Todd’s Experiment 4. 5. 6. Spin the Pointer and Toss a Coin Number Coins 12 34 5 3. How many times did the outcome Heads Tails, 5 occur? Tails 66/18/07 5:29:51 PM / 1 8

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7 9 Name Name Problem Solving Workshop Strategy: Predict Outcomes of Experiments Make an Organized List You can predict the likelihood of some events. Some events are certain, which means they will always happen. Some events are Yuri is playing a game using a coin and a spinner. impossible, which means they will never happen. An event is likely He tosses the coin and spins the pointer. List all if it has a greater than even chance of happening. An unlikely event the possible outcomes. has a less than even chance of happening. Two events can be Read to Understand equally likely, which means that they have the same chance of 1. What are you asked to find? happening.

Tell whether each event is likely, unlikely, certain, or impossible. Plan tossing a number greater than 1 on a cube labeled 1 to 6 2. How can making an organized list help you solve this problem? The number cube has the numbers 1, 2, 3, 4, 5, and 6. Five of these numbers are greater than 1, so it is likely that you will toss a 2, 3, 4, 5, or 6.

Solve So, the event is likely. RW157-RW158 3. Make an organized list. Show your work below. spinning a multiple of 4 on a spinner with 4 equal parts labeled 4, 8, 12, and 16 heads, 1 heads, 2 , , • Each of the numbers on the spinner is a multiple of 4. , 1 , , 3 tails, 4 Therefore, the pointer will always land on a multiple of 4.

• Events that will always happen are certain. 4. How many possible outcomes are there? So, the event is certain. Check likely unlikely certain impossible 5. Does the answer make sense for the problem? Explain. Tell whether the event is , , , or . 1. pulling a red marble from a bag that 2. tossing a number greater than 2 on contains 6 green marbles, 4 white a number cube labeled 1 to 6 marbles, and 2 yellow marbles Make an organized list to solve. 6. Tom spins the pointer below and tosses 7. Mary spins the pointer below and a number cube labeled 1 to 6. List all also tosses a coin. How many 3. spinning an odd number on a spinner 4. pulling a white marble from a bag that the possible outcomes. possible outcomes are there? with four equal parts labeled contains 12 green marbles, 2 white 3, 5, 7, and 9 marbles, and 14 yellow marbles YZ [ 5. tossing a number less than 6 on a 6. spinning a multiple of 6 on a number cube labeled 1 to 6 spinner with five equal parts labeled 21, 24, 31, 35, and 47 66/18/07 5:30:31 PM / 1 8

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8 0 Name Name Probability as a Fraction Experimental Probability You can use numbers to express the mathematical probability of an The experimental probability of an event can be found by event happening. If an event will never happen, its mathematical conducting repeated trials. Compare the number of times the event probability is 0. If the event is certain to happen, its mathematical actually occurs to the total number of times you repeat the activity. probability is 1. All other mathematical probabilities are written as Use the spinner at the right and the table below. fractions. The numerator tells the number of favorable outcomes, The spinner has equal sections. and the denominator tell the number of possible outcomes. Use the spinner. The spinner has equal sections. Write the Maryellen’s Results probability of not spinning gray as a fraction. Outcomes White Gray Black Spotted • There are 2 white sections, 3 black sections, and 1 gray Tally section. So there are 6 possible outcomes. • A favorable outcome would be to land on a section that is not What is the experimental probability of the pointer not stopping on gray? gray. There are 5 favorable outcomes: landing on the 2 white • To find the experimental probability, look at the results in Maryellen’s tally table. sections and on the 3 black section’s. • experimental probability: RW159-RW160 number of times the pointer does not stop on gray __32 _4 • Replace the information in the fraction below. ______5 , or 2 3 total number of times Maryellen spins the pointer 40 5 favorable outcomes ( white, black) __5 ______Probability of not gray 5 5 possible outcomes (1 gray, 2 white, 3 black) 6 __32 _4 , . _5. So, the experimental probability of the pointer not stopping on gray is 40 or 5 So, the probability of not spinning gray is 6 1 3 4 6 For – , use the spinner below. The For – , use the spinner below. The What is the mathematical probability of the pointer not stopping on gray? spinner has equal sections. Write the spinner has equal sections. Write the probability as a fraction. probability as a fraction. • To find the mathematical probability, look at the possible outcomes on the spinner. • mathematical probability:

favorable outcomes (black, white, spotted) _3 ______5 total possible outcomes (black, white, grey) 4 _3 1. spinning white 4. not spinning white So, the mathematical probability of the pointer not stopping on gray is 4 . Use the number cube and the table below.

1. What is the experimental probability Results 4 2. not spinning gray 5. spinning gray or white of tossing ? What is the mathematical Outcomes 1 23456 probability? Tally

2. What is the experimental probability of tossing 3. spinning gray or dotted 6. not spinning white or black an even number? What is the mathematical 6 probability? 3 2 66/18/07 5:31:18 PM / 1 8

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8 1 Name Combinations and Arrangements You can use tree diagrams to find combinations and arrangements. A combination is a choice in which the order does not matter. For example, in a blue shirt/black pants combination, the order in which you list the two items does not matter. In an arrangement, though, the order of the item does matter. Make a tree diagram to list and find the number of possible combinations.

Sundae Choices Flavor Topping Combinations Flavor: vanilla, strawberry, chocolate caramel vanilla, caramel Topping: caramel, cherry vanilla cherry vanilla, cherry • List the flavor choices in the left column. caramel strawberry, caramel • strawberry List the topping choices in the center cherry strawberry, cherry column. caramel chocolate, caramel • Draw lines to connect each flavor to chocolate cherry chocolate, cherry each topping.

RW161 • Write each combination in the third column. • Count the total number of combinations • There are a total of 6 different combinations.

Make a tree diagram to list and find the number of possible combinations. 1. Salad choices 2. Decorating choices Lettuce: iceberg, romaine Rug Color: brown, gray, tan Dressing: ranch, garlic, French Wall Color: white, tan, yellow, green 66/18/07 5:31:54 PM / 1 8

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