Franklin County Community School Corporation - Brookville, Indiana Curriculum Map Course Title: 6th Math Quarter: 1 Academic Year: 2011-2012 Essential Questions for this Quarter: 1. How do we use fractions, decimals, and percents in our daily lives? 2. How are fractions, decimals, and percents related? 3. How do I decide when to use fractions, decimals, or percents to make sense of quantities? 4. What in the world are negative numbers and how do we use them? 5. How can I use estimation to make sense of a situation?
Unit/Time Frame Standards Content Skills Assessment Resources
The Number System: 6.1.3a Whole Numbers Compare positive fractions, decimals Textbook Textbook: Glencoe Fractions, Decimals, 6.1.3b and mixed numbers using the Assignments McGraw-Hill Math and Percent’s 6.1.3c Mixed Numbers symbols <, >, or =. Connects Course 1 6.1.4a Worksheets (2012 ed.) Smart Start: 6.1.4b Fractions Plot points on a number line to A Plan for Problem 6.1.4c represent positive fractions, mixed Assignments connected.mcgraw- Solving 6.1.4d Decimals numbers and decimals. hill.com 6.1.4e Quizzes Add and Subtract 6.1.4f Percent’s Plot the approximate location of myskillstutor.com Decimals 6.1.5a positive fractions, decimals and mixed Tests 6.1.5b (6.7.7) Greatest Common Factor numbers on a number line. mathnook.com Greatest Common 6.1.7a Oral Responses Factor and Least 6.1.7b Least Common Multiple Convert fractions (proper and Teachers Helper Common Multiple 6.1.7c improper) and mixed numbers to 6.1.7d Estimation percents. Observations Saxon Math 7/6 Add and Subtract 6.1.7e Fractions 6.1.7f Unit Rate Convert fractions (proper and Skills Tutor Mailbox 6.1.7g improper) and mixed numbers to decimals. Measurement 6.1.7h Divisibility Acuity Hands On 6.2.3a Equations Convert decimals (including those Bar Graphs and Line 6.2.3b Discount greater than one) to percents. Graphs 6.2.3c Star Math edhelper.com 6.2.3d Tips Convert decimals (including those 6.2.3e Placement Test Brainpop greater than one) to fractions. 6.2.4a Interest Rate acuityathome.com 1 Franklin County Community School Corporation - Brookville, Indiana Curriculum Map Course Title: 6th Math Quarter: 1 Academic Year: 2011-2012 Essential Questions for this Quarter: 1. How do we use fractions, decimals, and percents in our daily lives? 2. How are fractions, decimals, and percents related? 3. How do I decide when to use fractions, decimals, or percents to make sense of quantities? 4. What in the world are negative numbers and how do we use them? 5. How can I use estimation to make sense of a situation?
Unit/Time Frame Standards Content Skills Assessment Resources
Chapter 1: Multiply and 6.2.5a ISTEP Divide Decimals 6.2.5b Principal Convert percents (including those Glencoe Study greater than 100%) to fractions. 6.2.8a Final Exam Guide and Practice 1.1 Multiply Decimals 6.2.8b Prime Workbook(previous Convert percents (including those 1.2 Divide Decimals 6.2.8c ed.) greater than 100%) to decimals. 1.3 Powers of 10 6.2.8d (6.2.9) Composite 6.2.9a (6.7.10) Name the equivalent decimal for a Scholastic Math Chapter 2: Multiply and 6.2.9b (6.7.10) Solution common fraction (halves, thirds, Divide Fractions 6.2.10a (6.7.9) fourths, fifths, tenths) using mental Book: Daily Math 6.2.10b (6.7.9) math. Practice, Grades 2-1 Multiply Fractions CC.6.RP.3c 6+ and Whole Numbers 6.5.10a Select an appropriate form of the 2-2 Multiply Fractions 6.5.10b number (fraction or decimal) to solve 2-3 Divide Fractions 6.5.10c problem situations. CC.6.RP.3b Chapter 4: Fractions, Define common factor and greatest Decimals and Percents common factor. Define common multiple and least 4-1 Fractions and common multiple. Decimals 4-2 Percents Determine the least common multiple 4-3 Compare and of two or more numbers. Order Fractions, Decimals, and Percents Find a common denominator for 2 Franklin County Community School Corporation - Brookville, Indiana Curriculum Map Course Title: 6th Math Quarter: 1 Academic Year: 2011-2012 Essential Questions for this Quarter: 1. How do we use fractions, decimals, and percents in our daily lives? 2. How are fractions, decimals, and percents related? 3. How do I decide when to use fractions, decimals, or percents to make sense of quantities? 4. What in the world are negative numbers and how do we use them? 5. How can I use estimation to make sense of a situation?
Unit/Time Frame Standards Content Skills Assessment Resources
4-4 Apply Percents adding and subtracting fractions by determining the least common Additional Lessons: multiple of two or more numbers.
Lesson 2: Solve Write or state the divisibility rules for Percent Problems 2, 3, 4, 5, 6 and 10 to assist in finding factors of numbers. Lesson 4: Factoring Determine the greatest common factor of two numbers.
Simplify fractions by determining the greatest common factor of two numbers. Determine the least common multiple of two or more numbers to solve problems
Multiply decimals up to thousandths by a whole number.
Multiply a decimal number (up to thousandths) by a decimal number (up to thousandths).
3 Franklin County Community School Corporation - Brookville, Indiana Curriculum Map Course Title: 6th Math Quarter: 1 Academic Year: 2011-2012 Essential Questions for this Quarter: 1. How do we use fractions, decimals, and percents in our daily lives? 2. How are fractions, decimals, and percents related? 3. How do I decide when to use fractions, decimals, or percents to make sense of quantities? 4. What in the world are negative numbers and how do we use them? 5. How can I use estimation to make sense of a situation?
Unit/Time Frame Standards Content Skills Assessment Resources
Divide whole numbers by decimals up to thousandths. Divide decimal numbers (up to thousandths) by a whole number. Divide a decimal number (up to thousandths) by a decimal number (up to thousandths). Explain orally or in writing how to multiply and divide fractions and perform the calculations. Explain why these procedures make sense. Solve problems involving addition, subtraction of positive fractions and explain why a particular operation was used for a given situation. Solve problems involving multiplication, and division of positive fractions and explain why a particular operation was used for a given situation. Calculate given percentages of a
4 Franklin County Community School Corporation - Brookville, Indiana Curriculum Map Course Title: 6th Math Quarter: 1 Academic Year: 2011-2012 Essential Questions for this Quarter: 1. How do we use fractions, decimals, and percents in our daily lives? 2. How are fractions, decimals, and percents related? 3. How do I decide when to use fractions, decimals, or percents to make sense of quantities? 4. What in the world are negative numbers and how do we use them? 5. How can I use estimation to make sense of a situation?
Unit/Time Frame Standards Content Skills Assessment Resources
quantity using both decimal and fractional representations of the percent. Solve problems involving discounts at sales given the original price and the discount rate. Solve problems involving simple interest earned given principal and interest rate. Solve problems involving tips given the cost of the service and the tip percentage. Estimate to decide whether an answer to a decimal computational problem (all four operations) is reasonable.
Estimate to decide whether the answer to a decimal problem is reasonable within the context of a given problem solving situation. (any of the four operations).
5 Franklin County Community School Corporation - Brookville, Indiana Curriculum Map Course Title: 6th Math Quarter: 1 Academic Year: 2011-2012 Essential Questions for this Quarter: 1. How do we use fractions, decimals, and percents in our daily lives? 2. How are fractions, decimals, and percents related? 3. How do I decide when to use fractions, decimals, or percents to make sense of quantities? 4. What in the world are negative numbers and how do we use them? 5. How can I use estimation to make sense of a situation?
Unit/Time Frame Standards Content Skills Assessment Resources
Compute mentally the solution to addition and subtraction problems using simple fractions (halves, thirds, fourths, fifths, sixths, eighths, tenths). Compute mentally the solution to addition and subtraction problems using simple decimals (including decimals added to and subtracted from whole numbers). Solve problems involving finding the whole, given a part and the percent. Note: This extends IAS 6.2.8. Add and subtract with money in decimal notation to solve problems. Multiply and divide with money in decimal notation. Interpret and/or round the results to the appropriate number of digits. Solve problems using multiplication and division with money in decimal notation.
6 Franklin County Community School Corporation - Brookville, Indiana Curriculum Map Course Title: 6th Math Quarter: 1 Academic Year: 2011-2012 Essential Questions for this Quarter: 1. How do we use fractions, decimals, and percents in our daily lives? 2. How are fractions, decimals, and percents related? 3. How do I decide when to use fractions, decimals, or percents to make sense of quantities? 4. What in the world are negative numbers and how do we use them? 5. How can I use estimation to make sense of a situation?
Unit/Time Frame Standards Content Skills Assessment Resources
Solve unit rate problems including those involving unit pricing and constant speed.
The Number System: 6.1.1a Integers Define negative numbers. Include the Textbook Integers 6.1.1b definition of integers and their Assignments 6.1.1c Absolute Value opposites. Chapter 7: Functions, 6.1.2a Worksheets Inequalities, and 6.1.2b Represent given situations using Integers 6.1.3d positive and negative numbers. Assignments 6.1.3e Solve problems involving the concept 7-3 Integers and the 6.1.3f Quizzes of integers (e.g., on a number line, in Coordinate Plane 6.1.3g counting, in temperature, in "owing"). Additional Lessons: 6.2.1a Tests Lesson 5 Compare and 6.2.1b Define and model the concept of Order Integers 6.2.1c absolute value for any integer, Oral Responses 6.2.1d positive or negative fraction, and 6.2.2a positive or negative decimal. 6.2.2b Observations 6.2.2c State and write the absolute value of 6.2.2d. any integer, positive or negative Skills Tutor fraction, and positive or negative decimal. Acuity
7 Franklin County Community School Corporation - Brookville, Indiana Curriculum Map Course Title: 6th Math Quarter: 1 Academic Year: 2011-2012 Essential Questions for this Quarter: 1. How do we use fractions, decimals, and percents in our daily lives? 2. How are fractions, decimals, and percents related? 3. How do I decide when to use fractions, decimals, or percents to make sense of quantities? 4. What in the world are negative numbers and how do we use them? 5. How can I use estimation to make sense of a situation?
Unit/Time Frame Standards Content Skills Assessment Resources
Compare integers and plot them on a Star Math number line. Compare integers, positive and Placement Test negative fractions, positive and negative decimals, and positive and ISTEP negative mixed numbers. Final Exam Plot integers, positive and negative fractions, positive and negative decimals, and positive and negative mixed numbers. Plot the approximate location of integers, positive and negative fractions, positive and negative decimals, and positive and negative mixed numbers on a number line. Using models, determine the results of using addition on integers by doing the following: adding two positives, adding two negatives, and adding a positive and a negative.
8 Franklin County Community School Corporation - Brookville, Indiana Curriculum Map Course Title: 6th Math Quarter: 1 Academic Year: 2011-2012 Essential Questions for this Quarter: 1. How do we use fractions, decimals, and percents in our daily lives? 2. How are fractions, decimals, and percents related? 3. How do I decide when to use fractions, decimals, or percents to make sense of quantities? 4. What in the world are negative numbers and how do we use them? 5. How can I use estimation to make sense of a situation?
Unit/Time Frame Standards Content Skills Assessment Resources
Analyze the results of addition on integers; state and justify the rules for adding integers. Using models, determine the results of using subtraction on integers by doing the following: subtracting two positives, subtracting two negatives, adding a positive and a negative. Analyze the results of subtraction on integers; state and justify the rules for subtracting integers. Using models, determine the results of using multiplication on integers by doing the following: multiplying two positives, multiplying two negatives, multiplying a positive and a negative.
Analyze the results of multiplication on integers; state and justify the rules for multiplying integers.
9 Franklin County Community School Corporation - Brookville, Indiana Curriculum Map Course Title: 6th Math Quarter: 1 Academic Year: 2011-2012 Essential Questions for this Quarter: 1. How do we use fractions, decimals, and percents in our daily lives? 2. How are fractions, decimals, and percents related? 3. How do I decide when to use fractions, decimals, or percents to make sense of quantities? 4. What in the world are negative numbers and how do we use them? 5. How can I use estimation to make sense of a situation?
Unit/Time Frame Standards Content Skills Assessment Resources
Using models, determine the results of using division on integers by doing the following: dividing two positives, dividing two negatives, and dividing a positive and a negative. Analyze the results of division on integers; state and justify the rules for dividing integers.
Standards for SMP1. Make sense of problems and Textbook Mathematical SMP2. persevere in solving them. Assignments Practice SMP3. SMP4. Reason abstractly and quantitatively. Worksheets SMP5. SMP6. Construct viable arguments and Assignments SMP7. critique the reasoning of others. SMP8. Quizzes Use appropriate tools strategically. 6.7.1a Attend to precision Tests 6.7.2a 6.7.3a Look for and make use of structure. Oral Responses 6.7.4a 6.7.5a Look for and express regularity in 6.7.6a repeated reasoning. Observations
10 Franklin County Community School Corporation - Brookville, Indiana Curriculum Map Course Title: 6th Math Quarter: 1 Academic Year: 2011-2012 Essential Questions for this Quarter: 1. How do we use fractions, decimals, and percents in our daily lives? 2. How are fractions, decimals, and percents related? 3. How do I decide when to use fractions, decimals, or percents to make sense of quantities? 4. What in the world are negative numbers and how do we use them? 5. How can I use estimation to make sense of a situation?
Unit/Time Frame Standards Content Skills Assessment Resources
6.7.7a Analyze problems by identifying 6.7.8a relationships, telling relevant from Skills Tutor 6.7.9a irrelevant information, identifying 6.7.10a missing information, sequencing and Acuity 6.7.11a prioritizing information, and observing patterns. Star Math Make and justify mathematical conjectures based on a general Placement Test description of a mathematical question or problem. ISTEP Decide when and how to divide a problem into simpler parts. Final Exam Apply strategies and results from simpler problems to solve more complex problems. Express solutions clearly and logically by using the appropriate mathematical terms and notation. Support solutions with evidence in both verbal and symbolic work.
Recognize the relative advantages of
11 Franklin County Community School Corporation - Brookville, Indiana Curriculum Map Course Title: 6th Math Quarter: 1 Academic Year: 2011-2012 Essential Questions for this Quarter: 1. How do we use fractions, decimals, and percents in our daily lives? 2. How are fractions, decimals, and percents related? 3. How do I decide when to use fractions, decimals, or percents to make sense of quantities? 4. What in the world are negative numbers and how do we use them? 5. How can I use estimation to make sense of a situation?
Unit/Time Frame Standards Content Skills Assessment Resources
exact and approximate solutions to problems and give answers to a specified degree of accuracy.
Select and apply appropriate methods for estimating results of rational- number computations. Use graphing to estimate solutions and check the estimates with analytic approaches. Make precise calculations and check the validity of the results in the context of the problem. Explain whether a solution is reasonable in the context of the original situation. Note the method of finding the solution and show a conceptual understanding of the method by solving similar problems.
12 Franklin County Community School Corporation - Brookville, Indiana Curriculum Map Course Title: 6th Math Quarter: 1 Academic Year: 2011-2012 Essential Questions for this Quarter: 1. How do we use fractions, decimals, and percents in our daily lives? 2. How are fractions, decimals, and percents related? 3. How do I decide when to use fractions, decimals, or percents to make sense of quantities? 4. What in the world are negative numbers and how do we use them? 5. How can I use estimation to make sense of a situation?
Unit/Time Frame Standards Content Skills Assessment Resources
13 Franklin County Community School Corporation - Brookville, Indiana COMMON CORE AND INDIANA ACADEMIC STANDARDS
INDIANA 2000 STANDARDS
Standard 1 Number Sense
Students compare and order positive and negative integers*, decimals, fractions, and mixed numbers. They find multiples* and factors*.
6.1.1 Understand and apply the basic concept of negative numbers (e.g., on a number line, in counting, in temperature, in “owing”). Example: The temperature this morning was -6º and now it is 3º. How much has the temperature risen? Explain your answer. 6.1.2 Interpret the absolute value of a number as the distance from zero on a number line and find the absolute value of real numbers. Example: Use a number line to explain the absolute values of -3 and of 7. 6.1.3 Compare and represent on a number line positive and negative integers, fractions, decimals (to hundredths), and mixed numbers. 5 Example: Find the positions on a number line of 3.56, -2.5, 1 6 , and -4. 6.1.4 Convert between any two representations of numbers (fractions, decimals, and percents) without the use of a calculator. 5 Example: Write 8 as a decimal and as a percent. 6.1.5 Recognize decimal equivalents for commonly used fractions without the use of a calculator. 1 1 2 Example: Know that 3 = 0.333… , 2 = 0.5, 5 = 0.4, etc. 6.1.6 Use models to represent ratios. Example: Divide 27 pencils to represent the ratio 4:5. 6.1.7 Find the least common multiple* and the greatest common factor* of whole numbers. Use them to solve problems with fractions (e.g., to find a common denominator to add two fractions or to find the reduced form for a fraction). 5 7 Example: Find the smallest number that both 12 and 18 divide into. How does this help you add the fractions 1 2 and 1 8 ? * positive and negative integers: …, -3, -2, -1, 0, 1, 2, 3, … * multiples: e.g., multiples of 7 are 7, 14, 21, 28, etc. * factors: e.g., factors of 12 are 1, 2, 3, 4, 6, 12 * least common multiple: e.g., the least common multiple of 4 and 6 is 12 * greatest common factor: e.g., the greatest common factor of 18 and 42 is 6
Standard 2 Computation
Students solve problems involving addition, subtraction, multiplication, and division of integers. They solve problems involving fractions, decimals, ratios, proportions, and percentages. 14
Franklin County Community School Corporation - Brookville, Indiana COMMON CORE AND INDIANA ACADEMIC STANDARDS
6.2.1 Add and subtract positive and negative integers. Example: 17 + -4 = ?, -8 – 5 = ?. 6.2.2 Multiply and divide positive and negative integers. Example: Continue the pattern: 3 2 = ?, 2 2 = ?, 1 2 = ?, 0 2 = ?, -1 2 = ?, -2 2 = ?, etc. 6.2.3 Multiply and divide decimals. Example: 3.265 0.96 = ?, 56.79 2.4 = ?. 6.2.4 Explain how to multiply and divide positive fractions and perform the calculations. 5 1 5 5 1 6 2 Example: Explain why 8 1 6 = 8 1 5 = 3 . 6.2.5 Solve problems involving addition, subtraction, multiplication, and division of positive fractions and explain why a particular operation was used for a given situation. 3 1 Example: You want to place a towel bar 9 4 inches long in the center of a door 27 2 inches wide. How far from each edge should you place the bar? Explain your method. 6.2.6 Interpret and use ratios to show the relative sizes of two quantities. Use the notations: a/b, a to b, a:b. Example: A car moving at a constant speed travels 130 miles in 2 hours. Write the ratio of distance to time and use it to find how far the car will travel in 5 hours. 6.2.7 Understand proportions and use them to solve problems. Example: Sam made 8 out of 24 free throws. Use a proportion to show how many free throws Sam would probably make out of 60 attempts. 6.2.8 Calculate given percentages of quantities and solve problems involving discounts at sales, interest earned, and tips. Example: In a sale, everything is reduced by 20%. Find the sale price of a shirt whose pre-sale price was $30. 6.2.9 Use estimation to decide whether answers are reasonable in decimal problems. Example: Your friend says that 56.79 2.4 = 2.36625. Without solving, explain why you think the answer is wrong. 6.2.10 Use mental arithmetic to add or subtract simple fractions and decimals. 1 1 Example: Subtract 6 from 2 without using pencil and paper.
Standard 3 Algebra and Functions
Students write verbal expressions and sentences as algebraic expressions and equations. They evaluate algebraic expressions, solve simple linear equations, and graph and interpret their results. They investigate geometric relationships and describe them algebraically.
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Franklin County Community School Corporation - Brookville, Indiana COMMON CORE AND INDIANA ACADEMIC STANDARDS
6.3.1 Write and solve one-step linear equations and inequalities in one variable and check the answers. Example: The area of a rectangle is 143 cm2 and the length is 11 cm. Write an equation to find the width of the rectangle and use it to solve the problem. Describe how you will check to be sure that your answer is correct. 6.3.2 Write and use formulas with up to three variables to solve problems. Example: You have P dollars in a bank that gives r% simple interest per year. Write a formula for the amount of interest you will receive in one year. Use the formula to find the amount of interest on $80 at 6% per year for one year. 6.3.3 Interpret and evaluate expressions that use grouping symbols such as parentheses. Example: Find the values of 10 – (7 – 3) and of 2(10 – 7)(3 + 1). 6.3.4 Use parentheses to indicate which operation to perform first when writing expressions containing more than two terms and different operations. Example: Write in symbols: add 19 and 34 and double the result. 6.3.5 Use variables in expressions describing geometric quantities. Example: Let l, w, and P be the length, width, and perimeter of a rectangle. Write a formula for the perimeter in terms of the length and width. 6.3.6 Apply the correct order of operations and the properties of real numbers (e.g., identity, inverse, commutative*, associative*, and distributive* properties) to evaluate numerical expressions. Justify each step in the process. Example: Simplify 3(4 – 1) + 2. Explain your method. 6.3.7 Identify and graph ordered pairs in the four quadrants of the coordinate plane. Example: Plot the points (3, -1), (-6, 2) and (9, -3). What do you notice? 6.3.8 Solve problems involving linear functions with integer* values. Write the equation and graph the resulting ordered pairs of integers on a grid. Example: A plant is 3 cm high the first time you measure it (on Day 0). Each day after that the plant grows by 2 cm. Write an equation connecting the height and the number of the day and draw its graph. 6.3.9 Investigate how a change in one variable relates to a change in a second variable. Example: In the last example, what do you notice about the shape of the graph? * commutative property: the order when adding or multiplying numbers makes no difference (e.g., 5 + 3 = 3 + 5), but note that this is not true for subtraction or division * associative property: the grouping when adding or multiplying numbers makes no difference (e.g., in 5 + 3 + 2, adding 5 and 3 and then adding 2 is the same as 5 added to 3 + 2), but note that this is not true for subtraction or division * distributive property: e.g., 3(5 + 2) = (3 5) r (3 2) * integers: …, -3, -2, -1, 0, 1, 2, 3, …
16 Franklin County Community School Corporation - Brookville, Indiana COMMON CORE AND INDIANA ACADEMIC STANDARDS Standard 4 Geometry
Students identify, describe, and classify the properties of plane and solid geometric shapes and the relationships between them.
6.4.1 Identify and draw vertical*, adjacent*, complementary*, and supplementary* angles and describe these angle relationships. Example: Draw two parallel lines with another line across them. Identify all pairs of supplementary angles. 6.4.2 Use the properties of complementary, supplementary, and vertical angles to solve problems involving an unknown angle. Justify solutions. Example: Find the size of the supplement to an angle that measures 122º. Explain how you obtain your answer. 6.4.3 Draw quadrilaterals* and triangles from given information about them. Example: Draw a quadrilateral with equal sides but no right angles. 6.4.4 Understand that the sum of the interior angles of any triangle is 180º and that the sum of the interior angles of any quadrilateral is 360º. Use this information to solve problems. Example: Find the size of the third angle of a triangle with angles of 73º and 49º. 6.4.5 Identify and draw two-dimensional shapes that are similar*. Example: Draw a rectangle similar to a given rectangle, but twice the size. 6.4.6 Draw the translation (slide) and reflection (flip) of shapes. Example: Draw a square and then slide it 3 inches horizontally across your page. Draw the new square in a different color. 6.4.7 Visualize and draw two-dimensional views of three-dimensional objects made from rectangular solids. Example: Draw a picture of an arrangement of rectangular blocks from the top, front, and right-hand side.
* vertical angles: angles 1 and 3 or 2 and 4 1 * adjacent angles: angles 1 and 2 or 2 and 3, etc. 4 2 * complementary angles: two angles whose sum is 90º 3 * supplementary angles: two angles whose sum is 180º (angles 1 and 2) * quadrilateral: a two-dimensional figure with four sides
* similar: the term to describe figures that have the same shape but may not have the same size
Standard 5 Measurement
Students deepen their understanding of the measurement of plane and solid shapes and use this understanding to solve problems. They calculate with temperature and money, and choose appropriate units of measure in other areas.
17 Franklin County Community School Corporation - Brookville, Indiana COMMON CORE AND INDIANA ACADEMIC STANDARDS
6.5.1 Select and apply appropriate standard units and tools to measure length, area, volume, weight, time, temperature, and the size of angles. Example: A triangular sheet of metal is about 1 foot across. Describe the units and tools you would use to measure its weight, its angles, and the lengths of its sides. 6.5.2 Understand and use larger units for measuring length by comparing miles to yards and kilometers to meters. Example: How many meters are in a kilometer? 6.5.3 Understand and use larger units for measuring area by comparing acres and square miles to square yards and square kilometers to square meters. Example: How many square meters are in a square kilometer? 6.5.4 Understand the concept of the constant π as the ratio of the circumference to the diameter of a circle. Develop and use the formulas for the circumference and area of a circle. Example: Measure the diameter and circumference of several circular objects. (Use string to find the circumference.) With a calculator, divide each circumference by its diameter. What do you notice about the results? 2 2 6.5.5 Know common estimates of π (3.14, 7 ) and use these values to estimate and calculate the circumference and the area of circles. Compare with actual measurements. Example: Find the area of a circle of radius 15 cm. 6.5.6 Understand the concept of significant figures and round answers to an appropriate number of significant figures. Example: You measure the diameter of a circle as 2.47 m and use the approximation 3.14 for π to calculate the circumference. Is it reasonable to give 7.7558 m as your answer? Why or why not? 6.5.7 Construct a cube and rectangular box from two-dimensional patterns and use these patterns to compute the surface area of these objects. Example: Find the total surface area of a shoe box with length 30 cm, width 15 cm, and height 10 cm. 6.5.8 Use strategies to find the surface area and volume of right prisms* and cylinders using appropriate units. Example: Find the volume of a cylindrical can 15 cm high and with a diameter of 8 cm. 6.5.9 Use a formula to convert temperatures between Celsius and Fahrenheit. Example: What is the Celsius equivalent of 100ºF? Explain your method. 6.5.10 Add, subtract, multiply, and divide with money in decimal notation. Example: Share $7.25 among five people. * right prism: a three-dimensional shape with two congruent ends that are polygons and all other faces are rectangles
18 Franklin County Community School Corporation - Brookville, Indiana COMMON CORE AND INDIANA ACADEMIC STANDARDS Standard 6 Data Analysis and Probability
Students compute and analyze statistical measures for data sets. They determine theoretical and experimental probabilities and use them to make predictions about events.
6.6.1 Organize and display single-variable data in appropriate graphs and stem-and-leaf plots*, and explain which types of graphs are appropriate for various data sets. Example: This stem-and-leaf diagram shows a set of test scores for your class: Stem Leaf 6 2 3 7 7 1 5 5 6 8 9 8 0 1 1 2 3 3 5 7 8 8 9 1 2 2 3 3 4 Find your score of 85 in this diagram. Are you closer to the top or the bottom of the class on this test? 6.6.2 Make frequency tables for numerical data, grouping the data in different ways to investigate how different groupings describe the data. Understand and find relative and cumulative frequency for a data set. Use histograms of the data and of the relative frequency distribution, and a broken line graph for cumulative frequency, to interpret the data. Example: A bag contains pens in three colors. Nine students each draw a pen from the bag without looking, then record the results in the frequency table shown. Complete the column showing relative frequency. Relative Color Frequency Frequency 2 Red 2 9 Green 4 Purple 3
6.6.3 Compare the mean*, median*, and mode* for a set of data and explain which measure is most appropriate in a given context. Example: Twenty students were given a science test and the mean, median and mode were as follows: mean = 8.5, median = 9, mode = 10. What does the difference between the mean and the mode suggest about the twenty quiz scores?
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Franklin County Community School Corporation - Brookville, Indiana COMMON CORE AND INDIANA ACADEMIC STANDARDS
6.6.4 Show all possible outcomes for compound events in an organized way and find the theoretical probability of each outcome. Example: A box contains four cards with the numbers 1 through 4 written on them. Show a list of all the possible outcomes if you draw two cards from the box without looking. What is the theoretical probability that you will draw the numbers one and two? Explain your answer. 6.6.5 Use data to estimate the probability of future events. Example: Teams A and B have played each other 3 times this season and Team A has won twice. When they play again, what is the probability of Team B winning? How accurate do you think this estimate is? 6.6.6 Understand and represent probabilities as ratios, measures of relative frequency, decimals between 0 and 1, and percentages between 0 and 100 and verify that the probabilities computed are reasonable. Example: The weather forecast says that the chance of rain today is 30%. Should you carry an umbrella? Explain your answer. * stem-and-leaf plot: the example under 6.6.1 shows 62, 63, 67, 71, 75, 75, 76, etc. * mean: the average obtained by adding the values and dividing by the number of values * median: the value that divides a set of data, written in order of size, into two equal parts * mode: the most common value in a given data set
Standard 7 Problem Solving
Students make decisions about how to approach problems and communicate their ideas.
6.7.1 Analyze problems by identifying relationships, telling relevant from irrelevant information, identifying missing information, sequencing and prioritizing information, and observing patterns. Example: Solve the problem: “Develop a method for finding all the prime numbers up to 100.” Notice that any numbers that 4, 6, 8, … divide into also divide exactly by 2, and so you do not need to test 4, 6, 8, … . 6.7.2 Make and justify mathematical conjectures based on a general description of a mathematical question or problem. Example: In the first example, decide that you need to test only the prime numbers as divisors, and explain it in the same way as for 4, 6, 8, … . 6.7.3 Decide when and how to break a problem into simpler parts. Example: In the first example, decide to find first those numbers not divisible by 2.
Students use strategies, skills, and concepts in finding and communicating solutions to problems.
6.7.4 Apply strategies and results from simpler problems to solve more complex problems. Example: In the first example, begin by finding all the prime numbers up to 10. 6.7.5 Express solutions clearly and logically by using the appropriate mathematical terms and notation. Support solutions with evidence in both verbal and symbolic work.
20 Franklin County Community School Corporation - Brookville, Indiana COMMON CORE AND INDIANA ACADEMIC STANDARDS
Example: In the first example, use a hundreds chart to cross off all multiples of 2 (except 2), then all multiples of 3 (except 3), then all multiples of 5 (except 5), etc. Explain why you are doing this. 6.7.6 Recognize the relative advantages of exact and approximate solutions to problems and give answers to a specified degree of accuracy. Example: Calculate the perimeter of a rectangular field that needs to be fenced. How accurate should you be: to the nearest kilometer, meter, centimeter, or millimeter? Explain your answer. 6.7.7 Select and apply appropriate methods for estimating results of rational-number computations. Example: Measure the length and height of the walls of a room to find the total area. Estimate an answer by imagining meter squares covering the walls. 6.7.8 Use graphing to estimate solutions and check the estimates with analytic approaches. Example: Use a graphing calculator to estimate the coordinates of the point where the straight line y = 8x – 3 crosses the x-axis. Confirm your answer by checking it in the equation. 6.7.9 Make precise calculations and check the validity of the results in the context of the problem. Example: In the first example, check whether some of the numbers not crossed out are in fact primes.
Students determine when a solution is complete and reasonable and move beyond a particular problem by generalizing to other situations.
6.7.10 Decide whether a solution is reasonable in the context of the original situation. Example: In the first example, decide whether your method was a good one — did it find all the prime numbers efficiently? 6.7.11 Note the method of finding the solution and show a conceptual understanding of the method by solving similar problems. Example: Use a hundreds chart to find all the numbers that are multiples of both 2 and 3. Ratios and Proportional Relationships RP INDIANA COMMON CORE STATE STANDARDS
Understand ratio concepts and use ratio reasoning to solve problems. 6.RP.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”
6.RP.2 Understand the concept of a unit rate a/b associated with a ratio a:b with b is not equal to 0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.” 1
21 Franklin County Community School Corporation - Brookville, Indiana COMMON CORE AND INDIANA ACADEMIC STANDARDS
6.RP.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. a. Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios. b. Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed? c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent. d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.1 Expectations for unit rates in this grade are limited to non-complex fractions. The Number System NS Apply and extend previous understandings of multiplication and division to divide fractions by fractions.
6.NS.1 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because ¾ of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi? Compute fluently with multi-digit numbers and find common factors and multiples.
Compute fluently with multi-digit numbers and find common factors and multiples.
6.NS.2 Fluently divide multi-digit numbers using the standard algorithm.
6.NS.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
6.NS.4 Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1-100 with a common factor as a multiple of a sum of two whole numbers with no common factor.
Apply and extend previous understandings of numbers to the system of rational numbers.
6.NS.5 Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.
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6.NS.6 Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. a. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., -(-3) = 3, and that 0 is its own opposite. b. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. c. Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.
6.NS.7 Understand ordering and absolute value of rational numbers. a. Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret –3 > –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right. b. Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write –3 oC > –7 oC to express the fact that –3 oC is warmer than –7 oC. c. Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. For example, for an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars. d. Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars.
6.NS.8 Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate. Expressions and Equations EE Apply and extend previous understandings of arithmetic to algebraic expressions.
6.EE.1 Write and evaluate numerical expressions involving whole-number exponents.
6.EE.2 Write, read, and evaluate expressions in which letters stand for numbers. a. Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation “Subtract y from 5” as 5 – y. b. Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms. c. Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s3 and A = 6 s2 to find the volume and surface area of a cube with sides of length s = 1/2.
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6.EE.3 Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.
6.EE.4 Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for. Reason about and solve one-variable equations and inequalities.
Reason about and solve one-variable equations and inequalities.
6.EE.5 Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
6.EE.6 Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
6.EE.7 Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.
6.EE.8 Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.
Represent and analyze quantitative relationships between dependent and independent variables.
6.EE.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time. Geometry G Solve real-world and mathematical problems involving area, surface area, and volume.
6.G.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
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6.G.2 Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.
6.G.3 Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.
6.G.4 Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems. Statistics and Probability SP Develop understanding of statistical variability.
6.SP.1 Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.
6.SP.2 Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.
6.SP.3 Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.
Summarize and describe distributions.
6.SP.4 Display numerical data in plots on a number line, including dot plots, histograms, and box plots.
6.SP.5 Summarize numerical data sets in relation to their context, such as by: a. Reporting the number of observations. b. Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. c. Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. d. Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered. Standards for Mathematical Practice
STANDARDS FOR MATHEMATICAL PRACTICE 25 Franklin County Community School Corporation - Brookville, Indiana COMMON CORE AND INDIANA ACADEMIC STANDARDS
The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education.
1. Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.
2. Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.
3. Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.
4. Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to
26 Franklin County Community School Corporation - Brookville, Indiana COMMON CORE AND INDIANA ACADEMIC STANDARDS plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose
5. Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.
6. Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.
7. Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.
8. Look for and express regularity in repeated reasoning. 27 Franklin County Community School Corporation - Brookville, Indiana COMMON CORE AND INDIANA ACADEMIC STANDARDS
Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.
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