grade 4 curricular progression lessonsalgebra.wceruw.org/documents/grade 4 lessons.pdf · grade 4...

Grade 4 Lessons Project LEAP (Blanton & Knuth, NSF DRK-12 #1207945)

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GRADE 4 CURRICULAR PROGRESSION LESSONS This document contains a draft of the Grade 4 lessons and associated item structures (identifying the key mathematical actions) for each lesson. The lessons were field-tested through a one-year classroom teaching experiment at grade 4.


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ITEM STRUCTURES CORRESPONDING TO LESSONS1: 1. Relational understanding of equality (also includes preliminary implicit work for both

solving equations and development of Properties of Equality (reflexive)); Extends third grade work with the operation of addition in equations to include multiplication.

Understand the meaning of ‘=’ as expressing a relationship between two equivalent quantities

Interpret equations written in various different formats (e.g., other than a + b = c)

Solve missing value problems by reasoning from the structural relationship in the equation.

2. Review Additive Identity, Additive Inverse (expressed as subtraction), Commutative

Property of Addition, Multiplicative Identity, Zero Property of Multiplication analyze information to develop a conjecture about the arithmetic relationship express the conjecture in words develop a justification or argument to support the conjecture’s truth; identify values for which the conjecture is true express the generalization (property) using variables examine meaning of repeated variable or different variables in the same

equation examine characteristic that generalization (property) is true for all values of

the variable in a given number domain identify generalization (property) in use (by doing computations or selecting

from cases where it is either used or not used) 3. a + (b + c) = (a + b) + c; (Associative Property of Addition)

analyze information to develop a conjecture about the arithmetic relationship express the conjecture in words and variables develop a justification using an empirical argument and a representation-

based argument identify number domain on which conjecture is true examine meaning of different variables in same equation examine characteristic that generalization (property) is true for all values of


from cases where it is either used or not used) 4. a × b = b × a (Commutative Property of Multiplication)

identify generalization (property) in use (by doing computations or selecting from cases where it is either used or not used)

1 Lessons may span more than one week.


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analyze information to develop a conjecture about the arithmetic relationship express the conjecture in words and variables develop a justification using an empirical argument and a representation-

based argument identify number domain on which conjecture is true examine meaning of different variables in same equation examine characteristic that generalization (property) true for all values of the

variable in a given number domain identify generalization (property) in use (by doing computations or selecting

from cases where it is either used or not used) 5. Generalizations about products of evens and/or odds (two terms)

analyze information to develop a conjecture about the arithmetic relationship express the conjecture in words develop a justification to support whether the conjecture is true or false

justification (using an empirical argument, representation-based argument, and verbal generalized argument; constrast the three types of arguments)

develop concept of counterexample 6. Writing (linear) algebraic expressions and inequalities to model problem situations

Identify variable(s) to represent an unknown, varying quantity or quantities Describe a quantity as an algebraic expression using variable notation Interpret the algebraic expression in terms of the problem context Understand whether an arbitrary amount must be even or odd (or if not

enough information is given to determine parity) Represent an inequality relationship between two quantities Identify different equivalent) ways to write the expression or inequality

relationship (application of properties – e.g., Commutative Prop; development of mathematical convention – e.g., 3x vs x3; x < 2x or 2x > x)

7. Modeling problem situations using (linear) equations and inequalities in one variable;

Solving problem situations involving one-step linear equations Identify variable(s) to represent the unknown quantity or quantities Describe the algebraic expressions using variable notation Model the problem situation with a linear equation or inequality Analyze the structure of the problem to determine value of variable. Check the solution or determine if the solution is reasonable given the

context of the problem Informally examine role of variable as unknown, fixed quantity

8. Solving linear equations (two step, repeated variable)

Identify variable(s) to represent the unknown quantity or quantities Describe the algebraic expressions using variable notation Model problem situation with a linear equation.


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examine meaning of repeated variables in the same equation Analyze the structure of the problem to determine value of variable. Check the solution or determine if the solution is reasonable given the

context of the problem Informally examine role of variable as fixed, unknown quantity

9. Generalizations about Properties of Equations: Adding any number to both sides of

a true equation results in a true (and equivalent) equation. analyze information to develop a conjecture about the arithmetic relationship express the conjecture in words develop an argument to support the conjecture’s truth identify number domain on which conjecture is true

10. a × (b + c) = (a × b) + (a × c) (Distributive Property of Multiplication over Addition)

analyze information to develop a conjecture about the arithmetic relationship express the conjecture in words and variables develop a justification using an empirical argument or representation-based

argument to support the conjecture’s truth; contrast the two types of arguments

identify the number domain on which conjecture is true examine meaning of different variables in the same equation examine characteristic that generalization (property) is true for all values of


from cases where it is either used or not used) 11. (Review of Grade 3 key FT constructs) Linear function with one operation

(multiplicative; y = mx) (Lessons 11 and 12) Generate data and organize in a function table Identify variables (including as number of object/magnitude of quantity, not

object/quantity) and their role as varying, unknown quantity Identify a recursive pattern and describe in words; use pattern to predict

near data Identify a covariational relationship and describe in words Identify a function rule and describe in words and variables Use function rule to predict far function values Examine the meaning of different variables in a function Develop a justification for why the function rule works by reasoning from the

problem context or the function table; Recognize that corresponding values in a function table must satisfy the

function rule. That is, when function variables are substituted with corresponding values from the table, the result must be a true equation


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Construct a coordinate graph and examine issues of scale and data representation

Use multiplicative relationship to reason proportionally about data (e.g., if 2 pieces of candy cost 10 cent, how much would 4 pieces cost)

Reversibility: given a value of the dependent variable and the function rule for a one operation function, determine the value of the independent variable

12. Comparing two linear functions with one or two operations (additive and/or

multiplicative) (Lesson 13) Generate data and organize in a function table Identify variables (including as number of object/magnitude of quantity, not

object/quantity) and their role as varying quantity Identify a recursive pattern and describe in words; use pattern to predict

near data Identify a covariational relationship and describe in words Identify a function rule and describe in words and variables Examine the meaning of different variables in a function Develop a justification for why the function rule works by reasoning from the

problem context or the function table Recognize that corresponding values in a function table must satisfy the


For an appropriate function, use multiplicative relationship to reason proportionally about data (e.g., if 2 pieces of candy cost 10 cent, how much would 4 pieces cost)

Reversibility: given a value of the dependent variable and the function rule (for a two-operation function), determine the value of the independent variable

Construct coordinate graphs and attend to scale and data representation Interpret graphs of two linear functions to solve a problem situation

13. Linear function with two operations (additive and multiplicative) (Lesson 14)

Generate data and organize in a function table Identify variables (including as number of object/magnitude of quantity, not


near data Identify a covariational relationship and describe in words Identify a function rule and describe in words and variables Use function rule to predict far function values Examine the meaning of different variables in a function Develop a justification for why the function rule works by reasoning from the

problem context or the function table


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Recognize that corresponding values in a function table must satisfy the function rule. That is, when function variables are substituted with corresponding values from the table, the result must be a true equation

Construct a coordinate graph and attend to issues of scale and data representation

Reversibility: given a value of the dependent variable and the function rule for a two-operation function, determine the value of the independent variable

Describe growth patterns informally by looking at co-variation (how does dependent variable change given a unit change in independent variable; how is this reflected in the graph)

Develop intuitive connections between growth pattern in function table and shape of graph (e.g., what does covariational relationship mean for shape of graph?)

14. Quadratic functions (y = ax2 - without linear and constant terms; y = ax2 + c -

without linear term and with constant term); Lessons 15-16 Generate data and organize in a function table Identify variables (including as number of object/magnitude of quantity, not


near data Identify a covariational relationship and describe in words Identify a function rule and describe in words and variables Examine the meaning of different variables in a function Develop a justification for why the function rule works by reasoning from the



Construct a coordinate graph and attend to scale and data representation Predict a far data value by thinking intuitively about how the function is

changing (increasing/decreasing; how quickly?) from table and graph; check using the function rule

Describe growth patterns informally by looking at co-variation (how does dependent variable change given a unit change in independent variable)

Develop intuitive connections between growth pattern in function tables and shapes of graph (e.g., what does covariational relationship mean for shape of graph?)

15. Quadratic function (without linear and constant terms); linear function with one

operation; Lesson 17 Generate data and organize in function tables Identify variables (including as number of object/magnitude of quantity, not

object/quantity) and their role as varying quantity


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Identify recursive patterns and describe in words Identify covariational relationships and describe in words Identify function rules and describe in words and variables Examine the meaning of different variables in a function Develop a justification for why the function rules work by reasoning from the



Construct coordinate graphs for each function on the same set of axes; attend to scale and data representation

Describe growth patterns for linear and quadratic functions by looking at covariational relationship

Compare growth patterns for the two functions (e.g., which grows faster and why?); use graphs and function tables to explain differences in growth and why a particular function grows faster

Interpret function behavior depicted in tables or graphs to solve a problem situation (e.g., which is better diet for caterpillar and why?)


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Fourth-Grade Progression (GA): Students review previously established Fundamental Properties, including their symbolic forms. They continue generalizing new Fundamental Properties, focusing primarily on properties involving multiplication. All Fundamental Properties are examined within a grade-appropriate number domain. Students also continue to use Fundamental Properties to simplify computations and identify these properties in use. They continue their work with other arithmetic generalizations involving classes of numbers (e.g. evens and odds). In particular, they extend their understanding of generalizations about sums of even numbers and odd numbers to include generalizations about products. Additionally, they examine generalizations that incorporate previously established generalizations (e.g., ‘a + b – b = a’ incorporates the Fundamental Properties, b – b = 0 and a + 0 = a, addressed in Grade 3). They also continue to develop their understanding of types of arguments used to justify generalizations, extending these forms from empirical arguments and representation-based arguments to arguments based on reasoning with previously established generalizations. Finally, in order to begin establishing the limitations of empirical arguments used to justify arithmetic generalizations, they begin to compare and contrast the strengths of empirical arguments with more general arguments. Core Actions: o analyze information to develop a conjecture about the arithmetic relationship o express the conjecture in words and, if appropriate, variables o develop a justification using, as appropriate, an empirical argument, algebraic use

of number argument, representation-based argument, or argument based on reasoning with previously established

o examine the limitations of an empirical argument o identify number domain on which conjecture is true, including extending number

domains for which generalizations were previously established to examine whether generalization still holds

o examine meaning of repeated variables in same equation o examine meaning of different variables in same equation o examine constraints on values of variable (i.e., a cannot be 0 to avoid division by 0) o examine characteristic that generalization (property) is true for all values of the

variable in a given number domain o identify generalization/property in use when doing computational work Fourth Grade (EEEI): In 4th grade, students continue to develop a relational understanding of the equal sign by solving equations and interpreting ‘=’ in tasks that involve addition, subtraction, multiplication, or division. They continue to model problem situations using algebraic


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expressions and equations. They continue to develop an informal understanding of Properties of Equality, focusing on the symmetric and transitive properties. They also extend work on solving equations to include solving equations with a repeated variable. The emphasis in solving equations is on analyzing the structure of the equation rather than on applying routine procedures for solving equations. Finally, students continue to develop their understanding of inequalities by comparing algebraic quantities and using inequalities to express their relative magnitudes. Types of Equations: single variable one-step or two-step linear equations of the form x + a = b, ax = b, or ax + b = c, two-step linear equations with repeated variables, linear equations in two variables Expressions and Inequalities are in linear forms comparable to equations. Core Actions: Equality o review different meanings of ‘=’, including as expressing a relationship between

quantities o interpret equations (number sentences) written in various formats (e.g., other than a

+ b = c) o solve missing value problems by interpreting the equal sign relationally and

reasoning from the structural relationship in the equation (Develop informal understanding of Symmetric Property of Equality) o develop an equation that expresses the relationship between two unspecified quantities of equal amounts o identify all possible ways to express the relationship o describe generalization informally (that if a = b, then b = a) Expressions o identify variable(s) to represent the unknown quantity or quantities in a problem situation o represent the quantity as an algebraic expression using variables o interpret an algebraic expression in the context of the problem o explore why expressions are different than equations (i.e., expressions are not to be ‘solved’)

Equations o model problem situations to produce a linear equations* of the form ax = b or linear equations with a variable repeated (e.g., ax + x = b). o model problem situations to produce linear equations in two variables


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o identify variable(s) to represent the unknown quantity or quantities in a problem situation o represent the relationship of two equivalent expressions as an equation o for linear equations with repeated variables, examine the meaning of a repeated variable in the same equation o analyze the structure of the equation to determine the value of variable. o check the solution of an equation by substituting the value of the variable in the original equation or determine if the solution is reasonable given the context of the problem o informally examine role of variable as a fixed or varying unknown

Inequalities o identify variables to represent two unspecified (unmeasured) quantities of different amounts o examine meaning of different variables in same inequality o represent the inequality relationship between two quantities or algebraic expressions o identify all possible ways to express an inequality relationship

Fourth Grade (FT): In 4th grade, students strengthen their understanding of different types of relationships of linear functions of the form y = mx + b, representations of relationships in words, variables, tables, and graphs, and justifications of generalized relationships. They extend their work with functions to include quadratic relationships, focusing initially on relationships of the form y = x2 and extending this more generally to those of the form y = x2 + b (for positive integers b). Students continue to informally connect functions and equations by examining functions for which the dependent variable is a specific value. Through this, they continue to develop an intuition about reversibility, as a precursor to inverse operations, by actions of ‘doing and un-doing’ on function rules. They deepen their knowledge of functions by learning to interpret and predict the qualitative behavior of a single function (linear or quadratic) by inspecting its graph and function table; examining qualitative growth differences in functions by looking at their graphs and function table; and interpreting graphs of two linear functions in order to solve mathematical situations (e.g., Best Deal). Concepts are sequenced so that students first informally explore these ideas using more familiar linear functions. The work of predicting and interpreting function behavior, examining growth patterns in tables and graphs, and interpreting the graphs of two functions to solve mathematical situations, is then extended to include simple quadratic relationships. Types of Functions: y=ax, y=ax + b, y = x2, y =x2+ a (for positive integers a, b)


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Core Actions: a. generate data and organize in a function table; b. identify variables (including as number of object/magnitude of quantity, not

object/quantity) and their role as varying quantity; c. identify a recursive pattern and describe in words; use pattern to predict near data; d. identify a covariational relationship and describe in words; e. identify a function rule and describe in words and variables; f. use function rule to predict far function values; g. examine the meaning of different variables in a function; h. develop a justification for why the function rule works by reasoning from the

problem context or the function table; i. recognize that corresponding values in a function table must satisfy the function

rule. That is, when function variables are substituted with corresponding values from the table, the result must be a true equation;

j. construct a coordinate graph and examine issues of scale and data representation; k. use multiplicative relationship to reason proportionally about data; l. reversibility: use either the function table or function rule to determine the value of

the independent variable given the value of the dependent variable; m. predict a far data value by thinking intuitively about how the function is changing

(increasing/decreasing; how quickly?) from table and graph; check using the function rule;

n. interpret graphs of two linear functions to solve a problem situation; o. describe growth patterns informally for linear and quadratic functions by looking at

co-variation (how does the dependent variable change given a unit change in independent variable);

p. compare growth patterns for linear and quadratic functions (e.g., which grows faster and why?); use graphs and function tables to explain differences in growth and why a particular function grows faster;

q. identify qualitative connections between the growth pattern in the function table and the shape of the graph for linear and quadratic functions (e.g., what does the co-variational relationship observed in the function table mean for the shape of the graph?);

r. interpret function behavior for linear and quadratic functions depicted in tables or graphs to solve a problem situation (e.g., which is the better diet and why?);


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Fourth Grade (VAR): In 4th grade, students refine their understanding of the concept of variable introduced in 3rd grade as they re-examine these concepts in more advanced situations. In particular, they continue to use variables to express arithmetic generalizations; simple algebraic expressions, equations, and inequalities; and simple functional relationships. They continue to examine situations in which a variable may act as a fixed but unknown quantity, a generalized number, or a varying quantity. They continue to explore the meaning of repeated variables or different variables in an algebraic expression, equation, inequality, or function rule. They also continue to explore the meaning of variable as the measure or amount of an object rather than the object itself and to interpret the meaning of a variable within a problem context.

GA • use variables to represent arithmetic generalizations • examine the meaning of a repeated variable in an equation (e.g., a – a = 0) • examine the meaning of different variables in an equation (e.g., a + b = b + a) EEEI • identify variables to represent either a fixed, unknown or varying, unknown quantity • understand that a variable represents the measure or amount of an object rather

than the object itself • interpret the meaning of a variable within the problem context (e.g., understand that

‘x’ represents the number of pieces of string) • understand the meaning of a repeated variable or different variables in an

expression, equation, or inequality • use variables when writing algebraic expressions, equations, and inequalities

FT • use variables to represent a varying quantity • understand that a variable represents the measure or amount of an object rather than the object itself • interpret the meaning of a variable within a problem context (e.g., understand that ‘x’ represents the number of pieces of string) • describe a function rule using variables; • examine the meaning of different variables in a function rule Proportional Reasoning: Students in this grade will begin to investigate and understand that a proportion is a relationship of equality between two ratios. In addition, students will notice that while


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individual quantities may change their ratios will still remain constant. They will build units (i.e. calculate the unit price for a product, compare unit prices, or identify information needed for comparison shopping) through their maintenance of proportional relationships. Continuing with the build-up strategy, students will transition into being able to unitize or use equivalence to separate off equal amounts by means of cognitive chunking or regrouping of a given quantity into manageable or conveniently sized parcels (i.e. 24 cola cans can now be represented as 2-12 packs of cola cans). Finally, students will be encouraged to use multiplicative thinking rather than repeated addition. At this grade level students are also exposed to a ratio table as a graphic organizing tool to evaluate given information within a problem context.


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GRADE 4 LESSONS

Lesson 1 (EEEI): Relational Understanding of the Equal Sign (Review and Extend)

Objective:

• Continue to develop a relational understanding of the equal sign by interpreting equations written in various formats (other than, e.g., a+b=c) as true or false and by solving missing value problems. Extend equations using only addition to those using multiplcation as well.

Jump Start: How would you describe what the symbol ‘=’ means? Developing a Relational Understanding of the Equal Sign (Review)

A. Which of the following equations are true? Explain. • 8 + 12 = 20 • 8 + 12 = 20 + 0 • 30 = 10 3 • 8 + 12 = 0 + 20 • 6 + 5 = 11 + 4 • 6 + 5 = 7+ 4 • 4 5 = 2 10 • 20 = 8 + 12 • 3 8 = 4 6 1 • 20 = 20 • 8 + 12 = 20 + 2 • 8 + 12 = 8 + 12 • 19 27 = 19 27 • 8 + 12 = 12 + 8 • 6 7 = 7 6

B. Write three true or false number sentences (Focus on students’ examples that use operations other than just addition)

C. What numbers will make the following equations true?

×

× ×

× × ×

× ×

× ×


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• 8 12 = ___ 12 • 5 + 8 = ___ + 9 • 123 + 3 = ___ + 2 • 8 4 = ___ 2 • 79 + 15 = ___ + 14 • 16 + ___ = 15 + 4 • ___ 2 = 12 6 • 24 = ___ • 0 + ___ = 257 • 1 ___ = 257

× ×

× ×

× ×

×


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Lesson 2 (GA): Developing Fundamental Properties (Review) (Additive Identity, Additive Inverse, Commutative Property of Addition)

Jumpstarts (Review of Additive Identity, Additive Inverse, and Commutative Property of Addition, Relational Understanding of Equal Sign): 1. Are these equations true or false? Explain.

8=8+0 0=37-37 23+17=17+23

1. What numbers or values make the following number sentences true?

___ + 237 = 237 + 395 0 + 15 = ___ 384 – ____ = 384

Additive Identity: A. What happens when you add zero to a number? Describe your conjecture in words. C. Represent your conjecture using a variable. Why did you use the same variable? What does it mean to use the same variable in an equation? D. Can you express your conjecture a different way using the same variable and number? (e.g., a + 0 = a, a = a + 0, 0 + a = a, etc) E. For what numbers is your conjecture true? Is it true for all numbers? Use numbers, pictures (e.g. a number line), or words to explain your thinking.

Additive Inverse A. What can you say about what happens when you subtract a number from itself? Describe your conjecture in words.

0 a


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C. Represent your conjecture using a variable. Why did you use the same variable? What does it mean to use the same variable in an equation? D. Can you express your conjecture a different way using the same variable and number? E. For what numbers is your conjecture true? Is it true for all numbers? Use numbers, pictures, or words to explain your thinking. Commutative Property of Addition: A. What can you say about the order in which you add two numbers? Describe your conjecture in words. D. Represent your conjecture using variables. Why did you use different variables? What does it mean to use different variables in an equation? E. Can you express your conjecture a different way using the same variables? F. For what numbers is your conjecture true? Is it true for all numbers? Use numbers, pictures, or words to explain your thinking. Group Work: 1. Application: Jenna has 83 pencils. Her mother gives her some more pencils. The next day, she gives her friend Mark the pencils her mother gave her. How many pencils does Jenna have now? Write an equation that represents this situation. Discuss how this problem uses the Additive Identity property. 2. Application: Callie’s mother has some juice boxes in her pantry. Callie’s friends come over to play and her mother gives everyone a juice box. She doesn’t have any left. Write an equation that represents this situation. Discuss how this problem uses the Additive Inverse Property. 3. Compute 10 + 47 – 5 without using an algorithm. Marianne said she solved a) in the following way: I wrote 10 as 5 + 5, so 10 + 47 – 5 = 5 + 5 + 47 – 5 = 5 + 47 + 5 – 5. Since 5 – 5 is just zero, I know that 10 + 47 – 5 = 5 + 47. But 5 is 2 + 3, so I know that 5 + 47 = 2 + 3 + 47. Since 3 + 47 is 50, then 5 + 47 is 2 + 50, or 52. My answer is 52.


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Discuss Marianne’s strategy and how she used the Additive Identity, Additive Inverse and Commutative Property of Addition. (This might be a student’s strategy from class – focusing on this would be ideal. The goal is to look at the steps and think about why a step is justifiable in terms of Fundamental Properties. E.g., 5 – 5 = 0 because any number subtracted from itself is zero, or Additive Inverse.)


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Lesson 3 (GA): Developing Fundamental Properties Brief review of Multiplicative Identity, Zero Property of Multiplication;

Introduction of Commutative Property of Multiplication, Associative Property of Addition

Jumpstarts: 1. (Multiplicative Identity) Complete the following: 13 x ___ = 13

___ = 239 x 1 1

4x ___ = 1

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1 x ____ = b. For what values of b is this true? (Talk about the different number

domains appropriate to 4th grade where this is true) Draw a picture that shows that 1 times any number is that number.

1. (Zero Property of Multiplication)

0 = 0 x ____. What numbers will make this equation true? (If students give a specific value, move the discussion towards the concept that “any number” works here. Talk about the different number domains appropriate to 4th grade where this is true)

3. Compute the following without using an algorithm: 95 + 39 – 39 + 12

Discuss how decomposing quantities and the fundamental properties can be used to make computation more efficient. Associative Property of Addition Compute:

• 43 + 21 + 9 • 45 + 21 + 5 • 70 + 25 + 5 • 10 + 5 -15-5 + 16 • 10 + 5 + 35

Discuss the strategies students used and identify ways that fundamental properties are used to make computation more efficient (rather than using an algorithm) Have students make up one on their own


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(Notice where students do 43 + 21, then add 9, or 21 + 9 then add 43 – it’s more efficient to do the latter. If they understand the associative property, they will know that they can add any two first, they don’t have to go left to right). Discuss that (43 + 21) + 9 is the same as 43 + (21 + 9) and it’s easier to do it the second way.) Commutative Property of Multiplication A. Find 3 x 5. Draw an array to show your answer is true. B. Find 5 x 3. Use your array to show your answer is true. C. What do you notice about 3 x 5 and 5 x 3? How can you use your array to show this? D. Do you think this works for other numbers? What can you say about the order in which you multiply any two numbers? Describe your conjecture in words. E. Represent your conjecture using variables. Why did you use different variables? What does it mean to use different variables in an equation? F. For what numbers is your conjecture true? Is it true for all numbers? Use an array to explain your thinking. G. Compute the following without using an algorithm.

5 x 17 x 20 (Discuss the use of Commutative Property of Multiplication)


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Lesson 4 (GA): Products of Evens and Odds Jumpstarts: 1. What numbers will make the following equations true? 39441

× ____ = 0 (Zero Property of Multiplication)

1/2 = 1/2 + 2/3 − ___ (Additive Inverse) m = 1 × ___ (Multiplicative Identity) 345 + ___ = ____ × 345 (Additive and Multiplicative identities) 2. Calvin’s Number: Calvin is thinking of a number. If he multiplies the number by 1, adds 0, then subtracts the result, what does he get? Write an equation that describes what he did. How do you know your equation is correct? What kind of number could Calvin have been thinking of? 3. Challenge: If 3

4x 123

= 3, what is 34

x 123

x 2? How do you know?

Multiplying Evens and Odds: A. “How Many Pairs?” Use cubes to answer the following questions: How many pairs of cubes are in the number 6? How many cubes are left over after you’ve made your pairs? Use your cubes to complete the following table for the given numbers.

Number Number

of pairs created

Number of cubes left over

3

4


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5

6

7

What do you notice? What kinds of numbers have no cubes left over after all pairs are made? What kinds of numbers have a cube left over? Write a sentence to describe each of your observations. B. Jackson is multiplying an even number and an odd number. Do you think his result will be even or odd? C. State a conjecture in words about what happens when you multiply an even number and an odd number. C. Do you think your conjecture is always true? Why? Use cubes or draw a picture (such as an array) to explain your thinking. D. (Developing notion of a counter example) Mitch said if you multiply any two even numbers numbers together you will always get an odd number. Do you agree? Why?


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Lesson 5 (GA): Products of Evens and Odds Which is a Better Argument?

Jumpstarts: Sam did the following problems. 2 + 1 = 3 4 + 1 = 5 6 + 1 = 7 He made a conjecture that when he adds 1 to any whole number, his answer will always be odd. a) Do you agree with Sam? Explain why? Mrs. Gardiner asked her students to come up with an argument that the sum of two even numbers is always even. She shared 3 different arguments that students made with the class: Marta wrote: 4 + 6 = 10 and 10 is even. 12 + 8 = 20 and 20 is even. 14 + 4 = 18 and 18 is even. So every time you add two even numbers together, you get an even number. Jackson wrote: If I have two sets of cubes, with an even number of cubes in each set, then if I put all the cubes together every cube will have a partner and there will be no cubes left over. It doesn’t matter how many cubes I have in each set as long as it is an even number of cubes. I won’t have any cubes left over. For example, if I have a set of 4 cubes and a set of 6 cubes, I can pair them like this:


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Carter wrote: I know from the definition of an even number that any even number can be written as 2 times a number. For example, 12 can be written as 2 x 6. So if I add two even numbers together, then it’s like adding 2 times a number plus 2 times another number. For example, 200 is 2 x 100 and 100 is 2 x 50. So adding 200 + 100 is like adding 2 x 100 plus 2 x 50. But this is the same as 2 times those two numbers added together: 2 x 100 plus 2 x 50 is the same as 2 times 150. But 2 times 150 has to be even because it’s 2 times a number. 1. Discuss each of the arguments in your group. Which do you think is a better argument? Why? Record what you group thinks makes a good argument. 1. Use Jackson’s or Carter’s method from (1) to construct an argument for the

following conjecture: The sum of an odd number and an even number is odd.


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Lesson 6 (EEEI): Writing (linear) algebraic expressions to model problem situations

Jumpstarts: 1. Compute the following without using an algorithm: 56 + 34 + 23 – 23 125 x 2 + 340 – 40 Discuss the properties students used to compute the above efficiently Kara’s teacher asked her to compute 56 + 34 + 23 – 23. Kara wrote the following: 56 + 34 + 23 - 23 = 56 + 34 + 0 = 56 + 34 = 50 + 6 + 30 + 4 = 50 + 30 + 6 + 4 = 80 + 10 = 90 Explain why Kara’s thinking is correct. 2. Draw an array to show that 5 x (10 + 10) = (5 x 10) + (5 x 10) Recycling Bottles (Task 4-2)

A. Brady and Evelynne are collecting bottles to recycle. Brady has some and is keeping them in his room. How can you represent the number of bottles that Brady has? Evelynne has twice as many bottles as Brady. How would you represent the number of bottles that Evelynne has? Express your answers in both words and variables.

B. Who has more bottles, Brady or Evelynne? How do you know? Write an inequality to represent the relationship between the number of bottles Brady and Evelynne each have. (Have them write this in multiple ways – e.g., as x < 2x and x > 2x)

C. Josh is also collecting bottles, and he has one more bottle than Evelynne. How

would you represent the number of bottles that Josh has? Express your answer in both words and variables. Use an inequality to express the relationships between the number of bottles Brady, Josh, and Evelynne have.

(Make sure they understand that the variable represents the quantity, not the object)


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D. Evelynne counted her bottles and found that she has 11. Do you think she

counted correctly? (Develop the connection that Evelynne must have an even number of bottles, so she could not have 11.) Josh counted his bottles and found that he has 14. Do you think he counted correctly? (Develop the connection that Josh must have an odd number of bottles, so he could not have 14.) E. What can you say about the number of bottles that each person has?

(Might get a variety of answers. Develop notion that they can have any number of bottles – i.e., the variable is a varying quantity, but there is a relationship between the number of bottles each has.)

F. Suppose Josh (correctly!) counts his bottles and finds that he has 15. Write an equation that describes what you know about the number of bottles Josh has.

G. What does the variable in your equation represent? What value of the variable in

your equation will make the equation true?


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Lesson 7 (EEEI): Review Solving Problem Situations Using Linear Equations and Inequalities (Recycling Bottles)

Jumpstarts: 1. Is a + b – b = a? How do you know? 2. If 3

4x 123

= 3, what is 34

x 123

x 2? How do you know?

3. Use arrays to show that 2 × 4 is the same as (2×1) + (2×3) 4-5: Recycling Bottles Part 3 Recall:

• Brady has some bottles, but we don’t know how many. • Evelynne has twice as many bottles as Brady • Josh has one more bottle than Evelynne.

How did we represent the number of bottles each person has?

A. Evelynne counts her bottles and finds that she has 20. Write an equation that describes the relationship between this amount and the expression describing the number of bottles she has.

B. Find the value of the variable in your equation.

C. How did you get your solution? How do you know your solution is correct?

D. Given that Evelynne has 20 bottles, what can you say about the value that the

variable could be? Can it be more than one value? How is that different than previously, when we didn’t know that she had 20 bottles, only twice the number that Brady had?

(Develop understanding that variable acts as unknown – there is only one value it could be once we know she has 20 bottles; previously, there wasn’t enough information to be able to say that the variable had a single value only, so it could have been any range of values. Thus, the context of the problem is important!!)

4-8: Which is Larger? Recycling Bottles Revisited


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Recall the expressions representing the number of bottles Brady, Evelynne and Josh have. Answers to the following problem depend on the fact that the value of the variable is a positive number. While you might not get into a discussion of negative numbers, it is important for students to realize from the problem context that the value of the variable must be greater than or equal to zero (can’t have a negative number of bottles).

A. In the bottles problem, what did the variable A (or whatever variable was used) represent?

B. What can you say about the values this variable might be? (Students need to understand that the problem context dictates that the value –because it represents the number of bottles - must be 0, 1, 2, 3, 4, ….)

Recall the representations for the number of bottles Brady, Evelynne and Josh each have. The following questions refer to this. C. Which is larger? A or 2xA How do you know? Draw a picture or use arrays to support your reason. D. Which is larger? A or 2xA + 1? How do you know? Draw a picture or use arrays to support your reason. E. Which is larger? 2xA or 2xA + 1 How do you know? Draw a picture or use arrays to support your reason. F. Write an inequality that represents the relationship between the number of bottles Brady and Evelynne have. Can you represent this relationship in a different way? G. Write an inequality that represents the relationship between the number of bottles Brady and Josh have. Can you represent this relationship in a different way? H. Suppose we that Charlotte has more bottles than Brady but fewer bottles than Evelynne. If Charlotte has K bottles, write an inequality that shows this relationship.


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Lesson 8 (EEEI): Solving linear equations (two step, repeated variable) Jumpstarts: 1. If A < B and B < C, write an inequality that shows the relationship between A and C. Do you think this relationship is always true? Why? 2. Since 3 + 4 = 7 is true, is 3 + 4 – 2 = 7 – 2 also true? How do you know? (Do students need to compute each side, or do they see that you’re just subtracting the same amount from each side, so the equation is still true?) 3. Use arrays to show that 3 × 9 is the same as (3×2) + (3×7) (Preliminary to Distributive Property) 4-6 Recycling Bottles Part 4 Recall:

• Brady has some bottles, but we don’t know how many. • Evelynne has twice as many bottles as Brady • Josh has one more bottle than Evelynne.

How did we represent the number of bottles each person has?

A. Brady and Evelynne combine their bottles. How would you represent the total amount of bottles they have?

B. They count their bottles and find they have a total of 30 bottles. Write an

equation that describes the relationship between this amount and the expression describing the number of bottles they have.

C. The variable appears more than once in this equation. What can you say about the value of the variable each time it appears?

(Develop notion that value must be the same)

D. Find the value of the variable. Show how you got your answer. (We are not concerned that they combine like terms. If they do, that’s fine. They might test a set of values, or think of it as “what number can I add to twice itself to get 30”).


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E. How would you convince your friend that your solution is correct? (Develop notion that solution needs to satisfy the equation.) F. What does the value of this variable represent in terms of Brady and Evelynne’s

bottles ? (Should understand that the value of the variable is the number of bottles Brady has) How can you use this to find the number of bottles that Evelynne has? G. What can you say about the value that the variable could be in your equation?

Can there be more than one value (solution) for the variable? (Develop understanding that variable acts as unknown and there is only one value it could be)


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Lesson 9 – Properties of Equations Jumpstarts: 1. Compute 563x0 + 341 + 273 – 273 without using an algorithm. Explain your thinking. 2. Are these equations true or false? Explain.

b = b + 0 0= y – y r + t = t + r

3. Use arrays to show that 4 × 8 is the same as (4×3) + (4×5) (Preliminary to Distributive Property) 4. If A<B, is B<A? Explain your thinking. What might A and B represent? Exploring Properties of Equations

A. Ian says that because 37 + 10 = 47 is true, he knows that 37 + 10 + 24 = 47 + 24 is also true. Do you agree with Ian? Why or why not?

B. Do you think this will hold for other numbers you add to both sides of the equation 37 + 10 = 47? Will the result always be a true equation? How do you know? Explore this with your partner.

C. Do you think this will hold for any number you add to both 37 + 10 and 47? That is, will the result always be a true equation? How do you know?

D. Do you think this holds for any equation (not just 37 + 10 = 47)? Explain your thinking.

E. Develop a conjecture in words that describes what happens when you add a number to both sides of an equation. (e.g., If you add the same number to both sides of an equation, the result is still a true equation.)

F. What do you think happens if you add different numbers to each side of a true equation? Is the result still a true equation? Explain your thinking.

G. (Challenge) Do you think your generalization (from D) holds for other operations (e.g., subtraction, multiplication)? Explain your thinking.


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Lesson 10 – Distributive Property of Multiplication over Division Jumpstarts: 1. Mrs. Gardiner’s Number: Mrs. Gardiner is thinking of a number. If she doubles the number and then subtracts the result, what does she get? Write an equation that describes what she did. 2. Consider 6 x A = 12 x B Find values for A and B to make 3 true equations. What is the relationship between the values you chose for A and B? Distributive Property A. Review our work on the following: 1. Use arrays to show that 2 ×(1+3) is the same as (2×1) + (2×3) 2. Use arrays to show that 3 ×(2+7) is the same as (3×2) + (3×7) 3. Use arrays to show that 4 ×(3+5) is the same as (4×3) + (4×5) B. Do you notice any patterns here? (This will be really hard to say in natural language!) Do you think this will happen for any numbers? C. Using what you’ve noticed above, how would you complete the following: 2 x (b + c) = _______________? (this might be a little easier – limiting to two variables for now….) D. Why does this rule use two different variables (b and c)? E. How would you use your rule to multiply 2 x 107 without an algorithm?

F. If we change 2 to a different number, do you think your rule will still work? Do you think it will work if we replace 2 with any number? How could you represent this? G. Can you write an equation to represent your new rule? H. Can you use arrays to show your rule is true for any numbers?


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Lesson 11 - Review of Functional Thinking (grade 3 constructs) Jumpstarts: 1. Find 3 x 120. (Notice whether they use Distributive Property and highlight this) 2. Mia’s teacher asked her to find 3 x (10 + 7). She said this was the same as 3 x 10 + 7, or 37. Do you agree? Use arrays to explain your thinking. 2. Janice needs to multiply 8 × 49. This is what she writes on her paper:

8 × 49 = 8 × (40 + 9) = (8 × 40) + (8 × 9)

= 320 + 72 = 392

Do you think she is correct? Why? Which do you think is a better way to find 8 x 49: Janice’s way or 49 (standard algorithm)? Why? x 8 4-1: Raymond’s Reward (adapted from Brizuela and Earnest, 2008) A. Raymond has some money. As a reward, his grandmother offers to triple the

amount of money he has. If he originally had $3, how much money would he have after his grandmother tripled this amount? What if he had $4? $5? $6? What can you say about the amount of money Raymond has before he receives any from his grandmother? (develop notion that we can’t say it is a specific amount – it’s unknown and the amount could be anything, could vary) What about the amount he has after?

B. Organize your information in a table. What do the variables represent?


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(Do they see variables as representing object or quantity) C. What relationships do you see in your table? Use this to find the amount of money Raymond would have after his grandmother gave him his reward if he had $10 initially. D. Find a rule that represents the relationship between the amount of money Raymond

had before and after he received his grandmother’s reward. How would you describe this relationship in words? (e.g., The amount of money he has after he gets his grandmother’s reward is 3 times what he started with.)

Use your variables to represent this relationship. E. Why did you use different letters to represent the two different quantities?


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Lesson 12 – Review of Functional Thinking continued – focus on graphing (Grade 3 constructs)

Jumpstarts: 1. Recall that A represented the number of bottles Brady has and 2 x A represented the number of bottles Evelynne has. What is the relationship between A and 2 x A? Write an inequality to represent this. Can you express your inequality in a different way? 2. Recall equations from last week:

(1) 2 x A + 1 = 15 (2) 15 = 2 x A + 1 (3) 15 = A x 2 + 1

Are these equations the same? How do you know? (in class discussion, identify the properties used: (1) and (2) are the same because if a = b, then b = a; (2) and (3) are the same because 2 x A = A x 2 by the comm. prop. of mult.) 3. What is the value of n in the following equation? How do you know?

3 x n + 2 = 14

4. Write an equation that represents the relationship depicted in the following arrays:

is the same as

LESSON Recall: Recall the rule for Raymond’s amount, y = 3z, where z represents his initial amount and y represents the amount he has after his grandmother triples what he started with. F. Why do you think this relationship is correct?


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G. If Raymond started with $90, how much would he have after his grandmother paid him? H. Construct a graph that shows the amount of money Raymond had before and after

he received his grandmother’s reward. How did you represent your data? How did you label the axes? Could more points be represented on your graph? How far could you extend your graph? (Notice how they handle issues of scale and unit size – are spacings equal?) I. If Raymond started with $7 and ended up with $21, how much money would he end

up with if he started with $9? How did you get your solution? H. Suppose Raymond counted his money after his grandmother gave him his reward.

He found that he had $60. How much money did he start with? How did you get your answer? How could you use your table to get your answer? How could you use your rule? How could you use your graph?

(Notice whether and how they can use different representations – table, graph, rule – to answer reversibility questions)


37

Lesson 13 - Comparing two linear functions with one or two operations Jumpstarts: TBD 4-3: Raymond’s Reward: Best Deal (Adapted from Brizuela and Earnest, 2008) Recall: Raymond’s Reward A. Raymond has some money. As a reward, his grandmother now offers him two deals: Deal 1: She will triple the amount of money he has. Deal 2: She will double the amount of money he has and add 2 dollars. Question: Raymond wants to choose the best deal. What should he do to figure this out? (See if students come up with a plan for figuring this out – hopefully they will generate data, organize, find a rule and/or graph, and use this….! If not, the following sets this up.) B. For each deal, find the amount of money Raymond would have after he received his grandmother’s reward if he originally had $3. What if he had $4? $5? 6$? Organize your information in a table. What do your variables represent?

C. For each deal, what relationships do you see in the data? Use these relationships to predict how much money Raymond would have for each deal after he received his grandmother’s reward if he had $10 to start with. Which deal is better if Raymond has $10 to start with? Why? D. For each deal, construct a graph to represent the amount of money Raymond would have before and after he received his grandmother’s reward. Construct your graphs on the same coordinate axes. How did you represent your data? How did you label the axes? Could more points be represented on your graph? How far could you extend your graph?


38

E. For each deal, find a relationship between the amount of money Raymond had before and after he received his grandmother’s reward. How would you describe these relationships in words?

Use variables to represent your relationships. F. Why did you use different letters to represent the two different quantities? G. Why do you think your relationship is true? H. Suppose Raymond has $5 and is considering Deal 1. If he starts with $5 and ends with $15, how much would he end with if he started with $6? Did you use your table, graph, or rule to find the solution? Explain your thinking using one of these representations. I. Suppose that Raymond selected Deal 2 and had a total of $22 after his grandmother paid him. How much money did he start with? How do you know? How can you use your table to figure this out? How can you use your rule? J. Using your graphs, determine which deal is better. Explain your thinking.


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Lesson 14 - Linear Function with Two Operations (Note this was the final task given in Grade 3; Lesson 15 reviews and extends that lesson.) Jumpstarts:

1. Raymond’s grandmother offers him a new deal where she will multiply the amount of money Raymond has by a secret number. If Raymond starts with $9, he ends with $54 with his grandmother's deal. How much money would Raymond end up with if he started with $12? What is the secret number? Explain how you got your solution.

2. Maddie thinks that since 20 + 10 = 30 is true, then 20 + 10 – 5 = 30 is also true. Do you agree with Maddie? Why?

3. Finish the following equation. Write your answer as a product of 12:

12x(3 + 16) = 12x3 + ______

How do you know your answer is correct? String Problem: Cutting a One-Loop String (adapted from a MTMS article on functions) A. Fold a piece of string to make one loop. While it is folded, make 1 cut (see figure).

How many pieces of string do you have? Fold another piece of string to make one loop. Make 2 cuts and find the number of pieces of string. Repeat this for 3, 4, and 5 cuts. What can you say about the number of cuts? What can you say about the number of pieces of string?

cut line


40

B. Organize your information in a table. What do the variables represent? C. What relationships do you see in the data? Use this to predict the number of pieces of string you would have after 8 cuts. D. Find a relationship between the number of cuts and the number of pieces of string. How would you represent your relationship in words? Use variables to represent your relationship. E. Why did you use different letters to represent the two different quantities? F. Why do you think your relationship is true? Can you explain your rule in terms of the problem context? (notice whether and if students can use the problem context and table of values to justify the rule) G. If you folded a piece of string and cut it 80 times, how many pieces of string would you get? How do you know? H. Construct a graph that shows the number of pieces of string for each number of cuts. How did you represent your data? How did you label the axes? Could more points be represented on your graph? How far could you extend your graph? I. Suppose your friend had 15 pieces of cut string. How many cuts did your friend

make in order to get 15 pieces of string? How do you know? How can you use your table to find your answer? How can you use your rule? How can you use your graph?

J. If your friend said she counted 16 pieces of cut string after she made her cuts, what

would you say?

K. How would you describe how this function grows? Use your table to justify your thinking.

(Notice whether they say things like the functon will grow at a steady rate, or than one more cut will always yield two more pieces of string)


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L. What do you notice about how the values in the function table change and the resulting shape of the graph?

(Notice whether students see the relationship between the changes in the function table and that the resulting graph is linear. E.g., do they see that one more cut always yields two more pieces of string, and that this type of growth is represented by a line)


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Lesson 15 –Quadratic functions (without linear and constant terms) Jumpstarts:

1. Chris thinks that 2 x (4 + 5) = 2x4 + 5. Do you agree? Why? Draw an array to justify your thinking.

2. Mrs. Gardiner wants to purchase brownies for her students. If she has $8, she can get 4 boxes of brownies. How many boxes of brownies can she get if she has $12? Explain how you got your solution.

3. Carter thinks that if a = b, then a - 2 = b - 2. Do you agree? Why?

Squaring a Region

A. Find the number of small squares in each of the square regions shown below.

B. Organize your information in a table. What do your variables represent? C. What relationships do you see in the data? Use these relationships to predict the number of squares that would be in Figure 4.

Fig. 1 Fig. 2

Fig. 3


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D. Find a relationship between the figure number and the number of squares in each region. How would you describe your relationship in words? Use your variables to represent this relationship. E. Why did you use different letters to represent the two different quantities? F. Why do you think your relationship is true? G. Construct a graph that compares the figure number and number of squares in each

region. How did you represent your data? How did you label the axes? Could more points be represented on your graph? How far could you extend your graph? H. From the graph, how would you describe how the number of small squares in each

square region is changing? From the table, how would you describe how the number of small squares changes?

(Notice whether students say things like the functon will grow at a steady rate or see that it does not in this situation. Introduce ideas about growth rate that is not constant, but increasing.) Use this to predict the number of squares in Figure 10. Check your prediction using your rule. I. What do you notice about how the values in the table change and the shape of the

graph?

(Notice whether students see that the graph is not that of a line and that the table does not reflect a linear growth rate.)


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Lesson 16 - Quadratic function with constant term 4-6: Growing Snake (Angela’s problem, in Blanton 2008)

A. Angela has a snake and she wants to look at how he is growing. She draws a

picture of the snake each day. If she measures the snake’s length by the number of triangles (including his head), how long is the snake on Day 1? Day 2? Day 3?

B. Organize your information in a table. What do the variables represent? C. What relationships do you see in the data? Use this to predict the length of the snake on Day 5. D. Find a relationship between length of the snake and the day he was measured. How would you describe your relationship in words? Use variables to represent your relationship. E. Why did you use different letters to represent the two different quantities?

• Day 1

• Day 2

Day 3

•


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F. Why do you think your relationship is true? G. Construct a graph that shows the length of the snake for the corresponding day. How did you represent your data? How did you label the axes? Could more points be represented on your graph? How far could you extend your graph? H. Using your table, how would you describe how the length of the snake changes in relation to the days his length is measured? How would you describe the growth of the snake over time? I. How would you describe the shape of the graph for this type of relationship in the table? (Do students see the connection between a nonlinear relationship in the table (growth rate is not constant) and a nonlinear graph?)


46

Lesson 17 – Comparing Linear and Quadratic Functions: (quadratic function of the form y = ax2; linear function of the form y=mx)

Jumpstarts: 1. Which of the following equations are always true (for all values of the variable(s))? Explain your thinking.

(a) a + b = b + c (b) 25 = a + 10 (c) a – a = 0 x b

2. If it costs $10 to ride a roller coaster 5 times, how much does it cost to ride the roller coaster 12 times? How do you know? 4-7: Guinea Pig Diet A. Rosie has two newborm guinea pigs, Oreo and Peanut, that she feeds two different

milk-based diets. She wants to see which diet is the best for gaining weight, so she weighs Oreo and Peanut each day for a week and records their weights in hectograms. At the end of the week, she drew the following table to organize the information she found, but had to go to school before she could finish it. Can you help her finish the table?

B. What variables did you use in your table headings? What do the variables represent?

1 3 4 5 7

1 4 9 36

Peanut’s Diet

1 2 3 5 6

2 4 8 10 14

Oreo’s Diet


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C. What relationships do you see in the data for each diet? Use this to predict Oreo’s and Peanut’s weights on the tenth day. D. For each guinea pig, find a relationship between his weight and the number of days. How would you describe your relationship in words? Use variables to represent your relationship. E. Why did you use different letters to represent the different quantities? F. Why do you think your relationship is true? H. For each diet, draw a graph that shows the guinea pig’s weight for the

corresponding day. Draw both graphs on the same axes. How did you represent your data? How did you label the axes? Could more points be represented on your graph? How far could you extend your graph? I. Using your tables, how would you describe how the weight of each guinea pig changes in relation to the day measured? That is, how would you describe each guinea pigs growth over time? Which guinea pig grows faster? How can you tell from your table? How can you tell from your graph? J. If the guinea pigs need to gain weight as quickly as possible, which do you think is the better diet? The diet Oreo is eating, or the diet Peanut is eating? Why? Use your tables and graphs to justify your answer. I. If guinea pigs needed to grow at a steady rate as newborns, which diet would you recommend? Why?

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