Algebra 1 Unit 3: Systems of Equations s4

7th Grade Honors Unit 1 Foundations for Algebra

BY THE END OF THIS UNIT:

Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes. 7th Grade Honors Unit 1 Foundations for Algebra

Unit Plans Chapter and Section Suggested Homework Standard 7.EE.1;7.EE.2 1-1 Variables and Problems 1-5 Exs. 9-37 odd, 38-42, 44-58 Expressions

Standard 7.EE.4.a 1-2 Order of Operations Part 1 Problems 1-2 Exs. 9-33 and Evaluating Expressions Part 2 Problems 3-4 Exs. 6-24, 29-48 Standard 8.NS.1; 8.NS.2 1-3 Real Numbers and the Problems 1-5 Exs. 9-17 odd, 18-34 even, 35-47 odd, 48,49-59 Number Line odd, 60-64 even, 66-79 Standard 7.NS.1.d 1-4 Properties of Real Problems 1-4 Exs. 7-19 odd, 20-34 even, 35-45, 47-56 Numbers Standard 7.NS.1.a-c 1-5 Adding and Problems 1-4 Exs. 10-28 even, 29-39 odd, 41-53, 55-60, 61-76 Subtracting Real Numbers Standard 7.NS.2 1-6 Multiplying and Problems 1-4 Exs. 8-40 even, 41-50, 54-64, 66-74 Dividing Real Numbers Standard 7.EE.1; 7.EE.2 1-7 The Distributive Part 1 Problems 1-3 Exs. 3-37, 39 Property Part 2 Problems 4-5 Exs. 7-25 odd, 26-46 even, 47-57 Standard 7.NS.3; 8.NS.2 1-8 An Introduction to Problems 1-6 Exs. 8-32 even, 33-39 odd, 40-74 Equations Standard 7.RP.2 1-9 Patterns, Equations, Problems 1-3 Exs. 8-20, 23-33, 36-55 and Graphs

Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes. 7th Grade Honors Unit 1 Foundations for Algebra CORE CONTENT Cluster Title: Use properties of operations to generate equivalent expressions. Standard 7.EE.1: Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. Concepts and Skills to Master • Ability to understand linear expression terminology: sum, difference, term, product, factor, quotient, coefficient • Ability to factor by using division to express a linear expression by its factors; i.e., 2x – 6 = 2(x – 3) • Ability to expand by using multiplication to rewrite the factors in a linear expression as a product; i.e., 5(x+ 12) = 5x + 60 SUPPORTS FOR TEACHERS Critical Background Knowledge  Gathering data by conducting trials of an experiment or game: organizing data in tables and graphs I order to look for patterns and relationships  Identifying patterns in number and geometry  Analyzing maximum and minimum values in measurement  Organizing and displaying data in one and two dimensional graphs and tables  Developing operation algorithms for fractions, decimals and percents Academic Vocabulary Change, Dependent and independent variables, variable, coordinate pair, distance/time/rate of speed, relationship, rule Suggested Instructional Strategies Resources  Introduce by discussing variables. Model how  Textbook Correlation working with variables is the same as working ○ Algebra 1 Foundation Series – Ch1 Sections 1 & 7  with numbers. MARS Tasks o Short tasks Expressions and Equations  Discuss distributive property and how to factor o Answers to Short Tasks and expand with distributive property. o A12 Fencing MARs Assessment Lessons (MS)  Solving Real-Life Problems: Baseball Jerseys (Problems-Solving Section)  Steps to Solving Equations (Concept Development)  Solving Linear Equations in One Variable (Concept Development)

Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes. 7th Grade Honors Unit 1 Foundations for Algebra  Texas Instrument Lesson Scavenger Hunt

Sample Formative Assessment Tasks Skill-based task Problem Task  Expand the expression 2(x-3). Then demonstrate I purchase six pieces of candy and I buy three of each type for a your answer is correct by substituting a value in for total of $3.75 and one of the candies is priced at $.50 how much is the variable and showing the two expressions are the other candy per piece? equal in value.  Factor (3x-9). Then demonstrate your answer is Write an expression to demonstrate the problem. correct by substituting a value in for the variable and showing the two expressions are equal in value.

CORE CONTENT Standard 7.EE.2: Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.” Concepts and Skills to Master • Ability to utilize Properties of Operations in order to rewrite expressions in different forms •Ability to develop understanding of equivalent forms of numbers, their various uses and relationships, and how they apply to a problem

SUPPORTS FOR TEACHERS Critical Background Knowledge  Identifying patterns in number and geometry  Developing operation algorithms for fractions, decimals and percents Academic Vocabulary Percent of change, percent of error Suggested Instructional Strategies Resources  Introduce by comparing one object to another.  Textbook Correlation Then change which object is compared to the . Algebra 1 Foundation Series other. Chapter One Section 2  The concept of which is the whole or 100% is  MARS Tasks critical to understand o Short tasks Expressions and Equations  Texas Instrument Lesson o Scavenger Hunt Sample Formative Assessment Tasks Skill-based task Problem Task Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes. 7th Grade Honors Unit 1 Foundations for Algebra  54 is what percent of 50? If a pizza is purchased and then paid for with a sales tax of 5.5%  50 is what percent of 54? and then tip is added of 20% and the total paid, including the tip is $12.66 how much did the pizza originally cost? CORE CONTENT Cluster Title: Analyze proportional relationships and use them to solve real-world and mathematical problems. Standard 7.RP.2: Recognize and represent proportional relationships between quantities. Concepts and Skills to Master • Ability to recognize in a given proportional situation that the two “between ratios” and the two “within ratios” are the same • Ability to distinguish between additive and multiplicative situations • Ability to recognize that two equal ratios represent a proportion • Ability to recognize and represent the connection between equivalent ratios, values in a table, and graphed orderedpairs • Ability to express unit rates using a variety of representations, given a contextual situation SUPPORTS FOR TEACHERS Critical Background Knowledge  Understand what a variable is and the importance of defining the variable in an expression or equation.  Understand that a variable has some value and the same rules that apply to numbers also apply to variables  Understand the concept of an independent and dependent variable  Understand discrete versus continuous data.  Plot points on a graph  Construct graphs and tables Academic Vocabulary Proportional relationships; constant; unit rate; equivalent ratios Suggested Instructional Strategies Resources  Introduce the lesson by allowing students to  Textbook Correlation identify patterns through the use of tables. Algebra 1 Foundation Series – Ch 1 Section 9  MARS TASKS  Explore various ways to represent equations o Short Task for Ratios and Proportions (describe what made it easier for you to connect o Short Task Answers for Ratios and Proportions the concepts of tables and graphs to actual o E02 Counting Trees

Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes. 7th Grade Honors Unit 1 Foundations for Algebra equations.) o A12 Fencing o A21: Sale! o A23:T-Shirt Sale o A01: 25% Sale Sample Formative Assessment Tasks Skill-based task Problem Task The school needs to purchase TI-84 calculators. Each One cellular phone carrier charges $26.50 a month plus $0.15 a calculator is sells for $125.00. minute for local calls. Another carrier charges $14.50 a month and Part A. How many calculators can be purchased $0.25 a minute for local calls. with $5000? calculators Part B. Explain how you determined the number of Part A. For how many minutes is the cost of the plans the same? calculators can be purchased. Use an equation, graph and table in addition to minutes words, numbers, and/or symbols in your explanation. Part B. Use a table, graph and equation to justify your answer.

CORE CONTENT Cluster Title: Solve real-life mathematical problems using numerical and algebraic expressions and equations. Standard 7.EE.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. Standard 7.EE.4a. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers; solve equations of these forms fluently; compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. Concepts and Skills to Master  Ability to differentiate between algebraic solution and an arithmetic solution  Ability to solve double sided equations  Ability to create three representations to represent linear situations SUPPORTS FOR TEACHERS Critical Background Knowledge  Understanding constant rate of change and how to determine slope in real world situations.  Understanding the connection between different representations of linear situations.  Ability to solve double sided equations with and without distribution.  Ability to isolate the variable, when an unknown variable is specified in the formula.  Ability to construct patterns in a table and graph linear patterns.  Ability to use words to describe the change in dependent and independent variables. Academic Vocabulary Slope, rise, run, y-intercept, horizontal change, vertical change, x-intercept, intersecting lines, scale

Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes. 7th Grade Honors Unit 1 Foundations for Algebra Suggested Instructional Strategies Resources  Model multiple representations of the solutions. Prior to  Textbook Correlation: expecting students to respond to performance based tasks model . Algebra 1 Foundation Series problem solving using all three representations to determine Chapter One Section 2 solutions. Use the DVD Task as a preliminary task  MARS: Fencing http://www.insidemathematics.org/index.php/classroom-video- visits/public-lessons-comparing-linear-functions/274-comparing- linear-functions-faculty-debrief? Sample Formative Assessment Tasks Skill-based task Problem Task Your parents are willing to purchase movie passes and dinner for Environmental Connection: Did you know that every each friend that attends your birthday party. The movie tickets are drop of water that runs down the drain ends up flowing $7.50 for all ages and the dinner will cost $14.50 for each guest. If into the sewage system? Sending clean water down the both parents and the birthday person attends what is the drain fills up the sewer and allows dirty water and waste maximum number of guests that are allowed to attend. Write an overflow into the streams and oceans. Polluted water inequality or equation to represent the situation. can harm, and even kill, animals that drink it and live in the water. What can you do? www.futurefriendly.com Water Saving Tip - Turn off water when brushing your teeth. The average bathroom faucet runs at 2 gallons per minute. Turning it off when you brush your teeth is one of the easiest ways to save water. Use the number of members your household to write and equation and construct a linear representation of how much water could be saved in two weeks. Explain your linear representation and describe the variables used to support your explanation.

CORE CONTENT Cluster Title: Apply and extend previous understandings of operations with fractions to add, subtract, multiply and divide rational numbers. Standard 7.NS.1.a Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged. Standard 7.NS.1.b Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts. Standard 7.NS.1.c Understand subtraction of rational numbers a adding the additive inverse, p – q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. Standard 7.NS.1.d Apply properties of operations as strategies to add and subtract rational numbers. Concepts and Skills to Master  Ability to build on prior experience with positive and negative rational numbers  Ability to identify additive inverses using rational numbers.  Ability to apply knowledge of positive or negative values for fractions and decimals.  Ability to develop and apply the following properties: Commutative Property of Addition, Associative Property of Addition, Identify Property of Addition, and the Distributive Property. SUPPORTS FOR TEACHERS Critical Background Knowledge Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes. 7th Grade Honors Unit 1 Foundations for Algebra  Understand how to model addition and subtraction of integers using distance/direction on a number line and using chips.  Understand that the Commutative Property holds for addition of rational numbers.  Understand and use the relationship between addition and subtraction to simplify computation by changing subtraction problems to addition and vice versa.  Understand and use the relationship between addition and subtraction found in fact families.  Develop algorithms for adding and subtracting integers.  Recognize and solve problems involving addition and subtraction of integers.  Use the Distributive Property to solve problems.  Solve simple equations with missing facts by using fact families. Academic Vocabulary Additive inverse, rational numbers Suggested Instructional Strategies Resources  To introduce students to adding integers, discuss examples of  Textbook Correlation: saving and spending. o Algebra 1 Foundation Series  Start the lesson by showing students the addition of two positive Chapter One Sections 3,4,5,6 and 8 numbers. Then show that addition of two negative numbers  MARS Task: follow a similar pattern. Focus your presentation on integers with A11: Division different signs. Use the number line to model. E03: A Day Out  To introduce students to subtracting integers, ask them to explain E11: Taxi Cabs how they would subtract a greater number from a lesser number.  Texas Instruments Lessons:  To introduce subtraction of integers, relate negative integers to Adding Integers – A Modeling Approach borrowing money to buy something you do not have enough Adding Integers Exploration money for. Getting Negative Integer Subtraction – What’s the Difference? Integers Number Line Activity – Adding Integers  Other Resources: http://mathstar.lacoe.edu/lessonlinks/integers/integers_adding_main.h tml http://mathstar.lacoe.edu/lessonlinks/integers/integers_subtracting_m ain.html Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes. 7th Grade Honors Unit 1 Foundations for Algebra http://www.uen.org/Lessonplan/preview?LPid=23406

Sample Formative Assessment Tasks Skill-based task Problem Task A research team aboard an underwater research vessel Explain how you can tell without calculating the sum of a positive descends 1,500 feet beneath the surface of the water. They then rise 525 number and a negative number will be positive, negative, or zero. feet and descend again 350 feet. Write an addition expression to represent this situation. Then find the sum.

If the overnight temperature at the Arctic Circle was -14°F, but the temperature rose to 8°F during the day, what was the difference between these high and low temperatures? CORE CONTENT Cluster Title: Apply and extend previous understandings of operations with fractions to add, subtract, multiply and divide rational numbers.

Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes. 7th Grade Honors Unit 1 Foundations for Algebra Standard 7.NS.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. Standard 7.NS.2.a Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. Standard 7.NS.2.b Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then: -(p/q) = -p/q = p/-q Interpret quotients of rational numbers by describing real-world contexts. Standard 7.NS.2.c Apply properties of operations as strategies to multiply and divide rational numbers. Concepts and Skills to Master  Ability to identify and apply the following properties: Multiplicative Inverse, Commutative Property of Multiplication, Associative Property of Multiplication, Identity Property of Multiplication.  Ability to recognize that rules for multiplying signed numbers remain the same for all rational numbers.  Ability to explore and justify the result of division by 0.  Ability to apply and extend knowledge of addition and subtraction of integers (i.e., two color counters, arrows on a number line) to extend to multiplication and division.  Ability to use patterns and concrete models to devise a general rule for dividing integers.  Ability to identify and apply the following properties: Distributive Property, Associative Property, Commutative Properties, and Identity Properties. SUPPORTS FOR TEACHERS Critical Background Knowledge  Understand how to use a number line/motion model to develop the relationship between repeated addition and multiplication with integers.  Understand and use the relationship between multiplication and division found in fact families.  Develop and use algorithms for multiplying and dividing integers.  Examine number patters to confirm the algorithm for multiplication.  Recognize and solve problems involving multiplication and division of integers. Academic Vocabulary Additive Inverse, Algorithm, Commutative Property, Distributive Property, Integers, Negative Numbers, Order of Operation,

Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes. 7th Grade Honors Unit 1 Foundations for Algebra Positive Numbers, Rational Numbers Suggested Instructional Strategies Resources  Put the following problems on the board.  Textbook Correlation: 3 x -9 = -27 o Algebra 1 Foundation Series -2 x 5 = -10 Chapter One Sections 3,6 and 8 -6 x 0 = 0  MARS Task: -3 x -9 = 27 A11: Division -2 x -5 = 10 E03: A Day Out -20 ÷ 5 = -4 E11: Taxi Cabs -9 ÷ -3 = 3 20 ÷ -5 = -4  Texas Instruments Lessons: 9 ÷ 3 = 3 Dividing Fractions  Ask the students if they can spot any patterns. Dividing Notes Have students write their answers on the board. Division of Integers  Then, ask students if they can determine the rules Integers for multiplying and dividing integers? Multiplication of Integers

 Other Resources: On line game: http://www.math-play.com/multiplying-and- dividing-integers-game.html You Tube Video: http://www.youtube.com/watch? v=GkbXbptFnnw

Sample Formative Assessment Tasks Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes. 7th Grade Honors Unit 1 Foundations for Algebra Skill-based task Problem Task A sky diver’s elevation changes by -3600 feet in 4 minutes A farmer has 120 bushels of beans for sale at a farmer’s market. after the parachute opens. What is the average change He sells and average of 15 ¾ bushels each day. After 6 days, what in the sky diver’s elevation each minute? is the change in the total number of bushels the farmer has for sale at the farmer’s market? How does the number of bushels the The formula F = 9/5C + 32 changes a temperature reading farmer has change each day? Should the change be a positive or a from the Celsius scale C to the Fahrenheit scale F. What negative number? How do you know? What expression is the temperature measured in degrees Fahrenheit when represents the total number of bushels sold in 6 days? Evaluate the Celsius temperature is -25°C? your expression to determine the change in the total number of bushels the farmer has for sale at the farmer’s market. Is your answer reasonable? Explain why or why not.

Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes. 7th Grade Honors Unit 1 Foundations for Algebra CORE CONTENT Cluster Title: Apply and extend previous understandings of operations with fractions to add, subtract, multiply and divide rational numbers. Standard 7.NS.2.d Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. Concepts and Skills to Master  Ability to recognize that a terminating decimal or repeating decimal is a rational number.  Ability to recognize that when rational numbers in fractional form are converted to decimals, they either terminate or repeat. SUPPORTS FOR TEACHERS Critical Background Knowledge  Understand the connection between decimals and fractions.  Understands how to make connections between decimals and fractions represented in real world situations.  Understand terminating and repeating decimals.  Understand rational and irrational numbers.  Ability to convert fractions to decimals.  Ability to convert decimals to fractions.  Ability to apply algorithms for dividing integers. Academic Vocabulary Terminating decimals, repeating decimals, irrational numbers, rational numbers, positive numbers, negative numbers Suggested Instructional Strategies Resources  Before beginning this investigation, you may want to  Textbook Correlation: review: the difference between irrational and rational Algebra 1 Foundation Series – Ch1 – Sec: 3, 6, & 8 numbers; the difference between a terminating and a  MARS Task: repeating decimal; how to change a fraction to a E11: Taxi Cabs; A12: Fencing decimal and vice versa.  Texas Instruments Lessons: Repeating Decimals and Fractions

Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes. 7th Grade Honors Unit 1 Foundations for Algebra Fractions to Decimals Form

Sample Formative Assessment Tasks Skill-based task Problem Task Write as a decimal. You are tutoring a younger student. How would you explain Which is greater? -18/25 or -0.72727272… rational numbers, irrational numbers, and how are they different?

CORE CONTENT Cluster Title: Apply and extend previous understandings of operations with fractions to add, subtract, multiply and divide rational numbers. Standard 7.NS.3 Solve real-world and mathematical problems involving the four operations with rational numbers. Concepts and Skills to Master  Ability to describe and identify complex fractions.  Ability to apply knowledge of Order of Operations. SUPPORTS FOR TEACHERS Critical Background Knowledge  Understand the order of operations to order computation in problems.  Ability to use the Distributive Property to solve problems. Academic Vocabulary Distributive Property, Integers, Negative Numbers, Order of Operation, Positive Numbers, Rational Numbers Suggested Instructional Strategies Resources  Review with students the order of operations.  Textbook Correlation: Place the acronym on the board: PEMDAS. Have Algebra 1 Foundation Series – Ch 1 Section 8 students come up with their own words to  Texas Instruments Lessons: remember the acronym. Have students share with o Number Sense the class. o Oops! Order of Operations and the TI graphing calculator o Order Some Ops o What’s Your Address? Sample Formative Assessment Tasks Skill-based task Problem Task

Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes. 7th Grade Honors Unit 1 Foundations for Algebra What is the simplified form of (6² + 4) – 15? Consider the expression (1 + 5)² - (18 ÷ 3). Can you perform the operations in different orders and still get the correct answer? Explain your reasoning. CORE CONTENT Cluster Title: Know that there are numbers that are not rational, and approximate them by rational numbers. Standard 8NS.1: Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. Concepts and Skills to Master  Know that real numbers that are not rational are irrational.  Understand that finite decimal expansions of irrational numbers are approximations.  Show that rational numbers have decimal expansions that repeat eventually.  Convert a decimal expansion, which repeats eventually, into a rational number. SUPPORTS FOR TEACHERS Critical Background Knowledge  Understand the subsets of the real number system (natural numbers, whole numbers, integers, rational numbers).  Convert rational numbers to decimals using long division (terminating and repeating) (7.NS.2d). Academic Vocabulary Decimal expansion, repeating decimal, termination decimal, rational, irrational, square root, , п Suggested Instructional Strategies Resources Use the Pythagorean Theorem with Letn = 0.1  Textbook Correlation non-perfect squares tointroduceSo, 100n = 16. o Algebra 1 Foundation Series irrational numbers (8.G.7).Subtract - 10n = -1. Chapter One Section 3 Use the powers of ten technique:90n = 15 (explain to the right)÷90÷90  MARS Formative AssessmentLesson(MS) : n = Concept Development: Repeating Decimals  Texas Instrument 8.NS.1 Lessons: Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes. 7th Grade Honors Unit 1 Foundations for Algebra http://education.ti.com/calculators/downloads/US/Acti vities/Search/Standards

Sample Formative Assessment Tasks Skill-based task Problem Task Convert 0.35 to a fraction. Suppose you have a fraction with a denominator of 7. What is the longest string of non-repeating digits that will occur in the Group the following numbers based on what you know about decimal expansions of the number? (Hint: Use the long division algorithm to show that for a denominator of n, there are only n possible remainders, 0 to n-1.) the number system: 5.3, 1.,, 2, 𝜋, 4.010010001…

CORE CONTENT Cluster Title: Know that there are numbers that are not rational, and approximate them by rational numbers. Standard 8.NS.2: Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., п2). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations. Concepts and Skills to Master Compare and order irrational numbers. Place irrational numbers on a number line. Use approximations of irrational numbers to estimate the value of expressions. SUPPORTS FOR TEACHERS Critical Background Knowledge • Compare and place rational numbers on a number line. • Approximate irrational numbers as fractions or decimals. Academic Vocabulary rational, irrational, decimal expansion, square root, √, п, truncating, rounding Suggested Instructional Strategies Resources

 Construct the Wheel of Theodorus to create physical lengths  Textbook Correlation of the square roots of the counting numbers. Transfer those . Algebra 1 Foundation Series

Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes. 7th Grade Honors Unit 1 Foundations for Algebra lengths onto a number line. Chapter One Section 2

 Find increasingly accurate estimations for square roots of  Wheel of Theodorus Project: numbers by guess- and-check with a calculator.  Texas Instrument 8.NS.2 Lessons: http://education.ti.com/calculators/downloads/US/Acti vities/Search/Standards

Sample Formative Assessment Tasks Skill-based Task Problem Task Place the following numbers on a number line: 5.3, 1.7…,√10, Explain when each approximation of п (3.14, 3, 22/7) is useful 2, п/2 in calculating the circumference of a circle. Compare the answers you would get with each approximation. (Extension: Find between which two integers lies? 42 Research how different cultures have approximated pi.)

Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes.