GRADE 5 CURRICULAR PROGRESSION LESSONSalgebra.wceruw.org/documents/Grade 5 Lessons.pdf · GRADE 5...

Grade 5 Lessons Project LEAP (Blanton & Knuth, NSF DRK-12 #1207945)

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GRADE 5 CURRICULAR PROGRESSION LESSONS This document contains a draft of the Grade 5 lessons and associated item structures (identifying the key mathematical actions) for each lesson.


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ITEM STRUCTURES CORRESPONDING TO LESSONS: 1. Review relational understanding of equality (also includes preliminary implicit work

for both solving equations and development of Properties of Equality (reflexive)); Extends equations involving addition or multiplication to include those with subtraction.

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) as true or false

Identify solutions to missing value problems by reasoning from the structural relationship in the equation

2. Review Fundamental Properties involving addition: Additive Identity, Additive Inverse

(expressed as subtraction), Commutative Property of Addition, Associative Property of Addition

identify property in use represent the property in words and variables examine meaning of repeated variable or different variables in the same

equation identify generalization (property) in use (by doing computations or selecting

from cases where it is either used or not used) examine whether the property holds true for a given domain of numbers

(extend to fractions) 3. Review Fundamental Properties involving multiplication: Multiplicative Identity,

Commutative Property of Multiplication; Generalize Multiplicative Inverse identify property in use represent the property in words and variables examine meaning of repeated variable or different variables in the same

equation identify generalization (property) in use (by doing computations or selecting

from cases where it is either used or not used) examine whether the property holds true for a given domain of numbers

(extend to fractions) 4. Review: a × (b + c) = (a × b) + (a × c) (Distributive Property of Multiplication over

Division) identify properties in use (by doing computations or selecting from cases

where it is either used or not used) describe the property in words and variables develop a justification using an empirical argument, representation-based

argument, or argument based on algebraic use of number to support the conjecture’s truth

identify number domain on which conjecture is true


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examine meaning of different variables in same equation examine characteristic that generalization (property) is true for all values of

the variable in a given number domain 5. Generalizations about sums of four evens and/or odds (e.g., sum of three evens and

one odd) analyze information to develop a conjecture about the arithmetic relationship express the conjecture in words develop a justification by using an empirical argument, a representation-

based argument, and by reasoning with previously established generalizations about sums of evens and/or odds to support whether the conjecture is true or false; contrast the different types of arguments (?)

6. Generalization about whether multiplying two numbers always results in a larger

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

(counterexample, if it is false) o identify number domain on which conjecture is true

7. Review Generalizing Addition Property of Equations: If a = b, then a + c = b + c

(Addition Property of Equations – adding the same quantity to both sides of an equation preserves the relationship) and extend to Subtraction

Analyze computations to develop a conjecture about the arithmetic relationship

Express the conjecture in words Identify number domain on which conjecture is true Express the generalization (property) using variables Identify generalization (property) in use (by doing computations or selecting

from cases where it is either used or not used)

8. Generalize Subtraction Property of Equations: If a = b, then a - c = b - c (Subtraction Property of Equations – subtracting the same quantity from both sides of an equation preserves the relationship)

Analyze computation to develop a conjecture about the arithmetic relationship

Express the conjecture in words Identify number domain on which conjecture is true Express the generalization (property) using variables Identify generalization (property) in use (by doing computations or selecting

from cases where it is either used or not used) 9. Writing (linear) equations in one or two variables to model problem situations

Identify variable(s) to represent the unknown quantity or quantities


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Describe the algebraic expressions using variables Express the relationship of two equivalent expressions as an equation.

10. Transforming algebraic expressions into equivalent forms (factoring binomials)

Apply Fundamental Properties to transform simple algebraic expressions of the form ax + bx and ax - bx into the appropriate forms (a+b)x or (a-b)x.

Write an equation that shows the equivalence of the two expressions Explore why expressions are different than equations (i.e., expressions are

not to be ‘solved’) 11. Review solving two-step, repeated variable linear equations (e.g., 3x + 2x = 15)

Model problem situation to produce linear equation of the form ax + bx = c. Examine meaning of repeated variable in 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 unknown

12. Solving linear equations continued (e.g., 3(x+2)=18)

Model problem situation to produce linear equation of the form ax + b = c. 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

13. Writing (linear) inequalities to model problem situations (combined throughout lessons)

identify variables to represent unspecified quantities of different amounts examine meaning of different variables in same inequality, if appropriate describe the inequality relationship between the two quantities identify all possible ways to express the relationship

14. Review of Grades 3/4 key FT constructs

Linear function with two operations 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 to predict near data Identify a covariational relationship and describe in words Identify a function rule and describe in words and variables; use the rule to

determine far data points 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 scale and data representation Reversibility: given a value of the dependent variable and the function rule,

determine the value of the independent variable 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. Review of Grades 3/4 key FT constructs

Quadratic function (without linear and constant terms) and one-operation linear function (multiplicative for proportional reasoning)

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

object/quantity) and their role as varying quantity Identify recursive patterns and describe in words; use to predict near data Identify covariational relationships and describe in words Identify function rules and describe in words and variables for each function;

use the rule to determine far data points Develop a justification for why the function rules work 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

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

For linear 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)

Predict far data values by thinking intuitively about how the functions are changing (increasing/decreasing; how quickly?) from table and graph; check using the function rules

Describe growth patterns informally by looking at co-variation (how does dependent variable change given a unit change in independent variable) for each function; develop intuitive connections between growth pattern in function tables and shapes of graph (e.g., what does covariational relationship mean for shape of graph?)


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

TASK: Guinea Pig Problem Revisited 16. Exponential function (simple – form y=ax)

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; use function rule

to determine far data points 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

Construct a coordinate graph and attend to scale and data representation Predict far data values 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?)

TASK: Folding Paper 17. Simple exponential function, quadratic function (no linear term) and/or linear function (comparing two functions);

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

object/quantity) and their role as varying quantity Identify recursive patterns and describe in words Identify covariational relationships and describe in words Identify function rules and describe in words and variables for each function;

use the rule to determine far data points


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Develop a justification for why the function rule works by reasoning from the problem context or the function table for each function

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

Describe growth patterns informally by looking at co-variation (how does dependent variable change given a unit change in independent variable); use descriptions to predict or describe shape of each graph, including which graph might grower fastest, etc.

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

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

Compare growth patterns for the two functions (e.g., which grows faster and why?)

use graphs, function tables and function rules to describe differences in growth and why a particular function grows faster or slower

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

TASK: Rosie Gets Another Guinea Pig 18. Qualitative graph of piecewise linear function (2 or 3 pieces)

Examine nature of continuous variables and how data are represented Interpret behavior of function at various regions of the graph (e.g., what’s

happening at horizontal segments, what’s happening at segments that are not horizontal, what’s happening at corners)

Construct a narrative (story) to match the graph TASK: Sunflowers 19. Story narrative

Identify variables Construct axes and label with variables Construct a qualitative graph to match the story Describe how parts of graph are reflected in narrative and why shape of

graph accounts for narrative TASK: Angela’s Road Race


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Fifth-Grade Curricular Progression for each Big Idea Fifth Grade (EEEI): In 5th grade, students review their relational understanding of the equal sign by solving and interpreting tasks that use inverse operations (i.e., a task might include both multiplication and division or both addition and subtraction). They continue to model linear problem situations using algebraic expressions and equations and extend their work in transforming algebraic expressions to include applying the Distributive Property to expand or simplify algebraic expressions. They extend their understanding of identifying the Fundamental Properties in computational work to identifying properties used when transforming algebraic expressions. They strengthen their understanding of solving simple equations, and extend this to more complex equations. As in previous grades, the emphasis on solving equations is on analyzing the structure of an equation as a way to determine a solution rather than on applying formal procedures. As a precursor to the study of formal procedures for solving equations in middle grades, students explore and generalize Properties of Equations for addition and subtraction (i.e., if a = b, then a + c = b + c; if a = b, then a – c = b – c). Finally, students strengthen their understanding of inequalities by comparing algebraic expressions 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; linear equations with repeated variables (e.g., ax + bx = c); linear equations in two variables Inequalities: linear forms comparable to those used in equations and expressions Core Actions: Equality solve and interpret equations (number sentences) written in various formats (e.g.,

other than a + b = c) identify solutions to missing value problems by interpreting the equal sign

relationally and reasoning from the structural relationship in the equation Expressions apply Fundamental Properties to transform simple algebraic expressions of the

form ax + bx and ax – bx and (a+b)x or (a-b)x into the appropriate forms. explore why expressions are different than equations (i.e., expressions are not to be

‘solved’) identify variable(s) to represent the unknown quantity or quantities in a problem

situation represent the quantity as an algebraic expression using variables interpret an algebraic expression in the context of the problem


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explore why expressions are different than equations (i.e., expressions are not to be ‘solved’)

Equations model problem situations using (linear) equations in one or two variables (e.g., ax + b = c) or repeated variables (e.g. ax + bx = c) identify variable(s) to represent an unknown quantity or quantities represent the relationship of two equivalent expressions as an equation solve two-step, three-step and repeated variable linear equations by examining the structure of the equation (e.g., 3x + 2x = 15) examine the meaning of a repeated variable in the same equation check the solution of an equation by substituting the value of the variable in original equation or determine if the solution is reasonable given the context of the problem informally examine the role of variable as fixed or varying unknown (Properties of Equations) analyze computations to develop a generalization about the Addition Property of

Equations (If a = b, then a + c = b + c or “adding the same quantity to both sides of an equation preserves the relationship”) and the Subtraction Property of Equations (If a = b, then a - c = b – c, or “subtracting the same quantity from both sides of an equation preserves the relationship”)

express generalizations about properties of equations in words identify number domain on which the generalizations are true express the generalizations (properties) using variables identify generalizations (properties) in use in computations Inequalities identify variables to represent two unspecified (unmeasured) quantities of different

amounts examine meaning of different variables in the same inequality represent the inequality relationship between two quantities or algebraic

expressions identify all possible ways to express the relationship


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Fifth Grade (FT): In 5th grade, students continue to strengthen their understanding of different types of relationships, representations of these relationships, and justifications of generalized relationships. They extend their work with linear and quadratic functions to include exponential relationships. They deepen and extend their knowledge of functions by interpreting and predicting the behavior of a linear, quadratic or exponential function; examining growth differences in different types of functions (linear, quadratic and exponential) by looking at their graphs, function tables, function rules, and connections among these representations; interpreting graphs of two functions of different types (e.g., one linear and one quadratic) in order to solve mathematical situations; and interpreting the behavior of functions represented through qualitative graphs. Ideas are sequenced so that students initially interpret and predict function behavior and growth in the context of more familiar linear and quadratic functions. These ideas are then explored using simple exponential functions of the form y = ax. Comparisons and interpretations of function behavior are made across all three function types, continuing previous work by comparing growth patterns evidenced by graphs and tables, and extending this by predicting shapes of graphs based on growth patterns in function tables. As part of this, students are asked to think about problem situations in which functions might be increasing or decreasing, and to determine this behavior from graphs or tables. Finally, qualitative graphs are introduced to strengthen students’ ability to coordinate two variables in interpreting function behavior and to introduce the transition from discrete to continuous variables. Types of Functions: y=ax, y=ax+b, y=x2, y=x2+b, y=ax (for positive integers a, b) Core Actions: 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 to predict near data; identify a covariational relationship and describe in words; identify a function rule and describe in words and variables; use the function rule to determine far data points; 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;

construct a coordinate graph and attend to scale and data representation;


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reversibility: use either the function table or function rule to determine the value of the independent variable given the value of the dependent variable;

for linear functions, use multiplicative relationship to reason proportionally about data (e.g., If 2 pieces of candy cost 10 cent, how much would 4 pieces cost?);

predict a far data value by thinking intuitively about how the linear 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 in linear, quadratic, and exponential functions (How does the dependent variable change given a unit change in independent variable?); use descriptions to predict or describe shape of each graph, including which graph might grow fastest, etc.

check predictions of shapes of graphs for linear, quadratic and exponential functions by comparing to actual graphs;

develop qualitative connections between growth pattern in function table and shape of graph for linear, quadratic, and exponential functions (e.g., what does covariational relationship mean for shape of graph?);

predict far data values by thinking informally about how linear, quadratic, and exponential functions are changing (increasing/decreasing; how quickly?) from table and graph; check using the function rules;

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

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

examine the nature of continuous variables and how data are represented in qualitative graphs of piecewise functions;

interpret the behavior of a piecewise function represented by a qualitative graph at various regions of the graph (e.g., what’s happening on horizontal segments, what’s happening on segments that are not horizontal, what’s happening at corners);

construct a narrative (story) to match a qualitative piecewise graph given a story (narrative) of linear and/or nonlinear motion, construct a qualitative

graph to match the motion depicted in the story; identify variables and describe how parts of the graph are reflected in the narrative and why the shape of the graph accounts for the narrative.


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Fifth-Grade Progression (GA): Students review previously established Fundamental Properties, including their symbolic forms. Fundamental Properties are re-examined in terms of their validity on number domains broader than those previously used. In particular, students examine whether Fundamental Properties are valid for fractions. They also continue to use Fundamental Properties to simplify computations and identify these properties in use. They continue to explore other arithmetic generalizations, extending their work with generalizations about classes of numbers (e.g., evens and odds) to include generalizations about outcomes of calculations (e.g., Does multiplying two numbers always produce a larger result?). Finally they continue to develop their understanding of different types of arguments to support generalizations and compare and contrast the strength of more general arguments with empirical arguments. Core Actions: analyze information to develop a conjecture about the arithmetic relationship express the conjecture in words and, if appropriate, variables 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

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

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

examine meaning of repeated variables in same equation examine meaning of different variables in same equation examine any constraints on values of variable (i.e., a cannot be 0 to avoid division

by 0) express generalizations (equations) in different equivalent forms (e.g., a = b + c is

equivalent to a – c = b) examine characteristic that generalization (property) is true for all values of the

variable in a given number domain identify generalization/property in use when doing computational work


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Fifth Grade (VAR): In 5th grade, students continue to refine their understanding of the concept of variable. In particular, they continue to use variables to represent arithmetic generalizations; algebraic expressions, equations, and inequalities; and functional relationships. They continue to explore contexts in which a variable may act as a fixed but unknown quantity, a generalized number, or a varying quantity. They reinforce their understanding of variable as the measure or amount of an object rather than the object itself and of the meaning of repeated variables or different variables in an algebraic expression, equation, inequality, or function rule. They also continue to interpret the meaning of a variable within a problem context. Finally, they extend their understanding of variable by an introduction to representations of continuous variables in coordinate graphs. GA use variables to represent arithmetic generalizations examine the meaning of a repeated variable in the same equation (e.g., a – a = 0) examine the meaning of different variables in the same equation (e.g., a + b = b + a) EEEI use variables to represent a fixed but unknown quantity or 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) understand the meaning of repeated variables or different variables in an

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

FT use a variable 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 examine graphical representations of continuous variables


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Grade 5 (PR): Students in this grade level will solve missing-value proportion problems by iterating and/or partitioning with ratios increasing in complexity. They will focus on sense-making strategies as opposed to rote application of the cross-multiplication algorithm. Students will investigate multiplicative relationships between two quantities by extending this same relationship to other pairs of quantities. The use of the ratio table as problem-solving tool will continue to be used at this level to help students organize and think through their iterating/partitioning work as well as their work with equivalent fractions. Generating multiple equivalent ratios will help students develop the idea of rate as an infinite class of equivalent ratios. Students will also consider the idea of ratio as a measure and how changing one quantity in a ratio impacts the ratio as a whole. Lastly, they will begin applying division computational rules as they relate to proportional reasoning, preparing them to think about the concept of unit rate.

1. Solve missing-value proportion problems by iterating and/or partitioning with ratios Generate a class of equivalent ratios including those involving decimals or

fractions beyond “easy” ratios (e.g., those found by doubling) develop idea of an infinite equivalence class (rate) TASKS: 15 missing-value proportion tasks in Kaput & West (1994, pp. 267-269)

ranked by difficulty for 6th graders “To make coffee, David needs exactly 8 cups of water to make 14 small cups

of coffee. How many small cups of coffee can he make with 12 cups of water?” (Clark et al., 2003)

“On a certain map, the scale indicates that 5 cm represents the actual distance of 9 miles. Suppose the distance between two cities on this map measures 2 cm. Explain how you would find the actual distance between these two cities.” (Weinberg, 2002)

“The label on a box of cheese crackers tells consumers that 6 crackers contain 70 calories. How many calories are in 20 crackers” (Ellis, in press, p. 44)—ask students to solve using unit ratio approach

“Suppose that 18 wheat crackers have 167 calories. How many calories are in 11 crackers?” (Ellis, in press, p. 45)—ask students to solve using unit ratio approach with messier numbers

See Misailidou & Williams (2003) appendices for validated missing-value proportion items used in grades 5-8

2. Ratio as measure o Identify the attribute a particular ratio is measuring o Identify how changing one quantity in a ratio impacts the ratio as a whole

o TASKS:


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How could we measure the steepness of a wheelchair ramp? Then: What happens when individual quantities (length, height) change? How does each change impact steepness? (Ellis, in press, p. 25)

3. Division and Proportional Reasoning • students work with division and proportional reasoning problems in preparation

of rate TASKS:

• Best Buy on Cat Food (Fosnot, c.t., Jacob, B. 2007 Best Buys, Ratios and Rates)


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Grade 5 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 other operations (x, ÷, -).

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. • 20 = 8 + 12 • 8 + 12 = 20 + 0 • 10 = 30 ÷ 3 • 8 + 12 = 9 + 12 • 6 - 5 = 7 - 6 • 6 x 5 = 30 x 0 • 4× 5 = 2× 10 • 20 ÷10 = 40 ÷20 • 20 × 10 = 40 × 20 • 20 -10 = 40 – 20 • 20 +10 = 40 +20 • 3 × 8 = 4 × 6 × 1 • 63 = 63 • 30 + 10 = 10 + 30 • 30 – 10 = 10 – 30 • 30× 10 = 10 × 30 • 30÷10 = 10 ÷ 30

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? Explain your thinking


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• 8 × 12 = ___ × 12 • 16 - 5 = ___ - 6 • 5 + 8 = ___ + 9 • 123 + 3 = ___ + 2 • 8× 4 = ___ × 2 • 20 ÷10 = 40 ÷___ • 79 + 15 = ___ + 14 • 16 + ___ = 15 + 4 • ___ × 2 = 12 × 6 • 24 = ___ • 0 + ___ = 257 • 1 × ___ = 257 • 20 x 10 = 40 x 20 • 20 ÷10 = __ ÷20


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Lesson 2 (GA): Review and Extend Fundamental Properties involving Addition

Additive Identity, Additive Inverse (expressed as subtraction), Commutative Property of Addition, Associative 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 x 0 0=37-37 237+104=104+237

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

___ + 237 = 237 + 395 0 x 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 equation a different way? (e.g., a + 0 = a, a = a + 0, 0 + a = a, etc) E. For what numbers is your conjecture true? Does your conjecture hold for fractions? How about decimal numbers? Use numbers, pictures (e.g. a number line), or words to explain your thinking.

0 a


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Additive Inverse A. What can you say about what happens when you subtract a number from itself? 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 equation a different? E. For what numbers is your conjecture true? Is it true for all numbers? Does your conjecture hold for fractions? How about decimal 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 in an equation using variables. Why did you use different variables? E. Can you express your equation a different way using the same variables? F. For what numbers is your conjecture true? Is it true for all numbers? Does your conjecture hold for fractions? How about decimal numbers? Use numbers, pictures, or words to explain your thinking. Associative Property of Addition: Suppose you have 3 numbers (a, b, and c) to add together. Discuss what the following means: (a + b) + c or a + (b + c) Will you get the same result regardless of which two numbers you add first? How do you know? Group Work: 1. Application: Samantha has 24 cupcakes for her birthday party. Her mother buys some extra cupcakes. At the party, Samantha’s friends eat the 24 cupcakes, but not the extras. Write an equation that represents this situation. What can you say about the number of cupcakes left over?


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Discuss how this problem uses the Additive Identity property. 2. Application: Callie and her two friends, Carter and Nadine, are collecting box tops for school. Callie has 26, Carter has 34, and Nadine has 20. How many do they have all together? How did you find your answer? Write an equation that represents this situation. Can you get your answer a different way? Discuss how this problem uses propoerties such as the Associative Property of Addition. 3. Compute 37 + 43 – 13 without using an algorithm. Discuss how students’ strategies used the Fundamental Properties.


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Lesson 3 (GA): Review and Extend Fundamental Properties involving Multiplication

Brief review of Multiplicative Identity, Zero Property of Multiplication, Commutative Property of Multiplication;

Develop Multiplicative Inverse (expressed as division) Jumpstarts: 1. Complete the following: 13 x ___ = ___ x 13

___ = 239 x 1 2

3x ___ = 2

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2. 0 = 0 x ____. What numbers will make this equation true? 3. 2 ÷ ___ = 1 What number makes this equation true? Draw a picture that

explains your thinking. 4. Compute the following without using an algorithm: 90 + 10 x 0 + 17

Discuss how decomposing quantities and the fundamental properties can be used to make computation more efficient. Commutative Property of Multiplication Marta says that she can multiply two numbers in any order. What does she mean? Write an equation that shows Marta’s claim for any two numbers. Do you agree with her? For what numbers do you think her claim this true? Do you think her claim is true for fractions? How about for decimal numbers? Explain your thinking using words or pictures. G. Compute the following without using an algorithm.

10 x 3 x 200 (Discuss the use of Commutative Property of Multiplication)


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Multiplicative Identity

1 x ____ = z. For what values of z is this true? (Talk about the different number domains appropriate to 5th grade where

this is true) Draw a picture that shows that 1 times any number is that number.

Multiplicative Inverse (expressed as division) 1. Will has 4 video games. He shares his games equally with his friends so that all 4 friends (including Will) have the same number of games. After Will shares, how many games does each person have? How did you get your answer? Draw a picture to explain your result. Write an equation to show how you got your answer. 2. There are 23 students in Ms. Stephens’ class. Suzanne brought 23 cupcakes to class. She shared her cupcakes so that everyone had the same number of cupcakes. How many cupcakes did each person get? Draw a picture to explain your result. Write an equation to show how you got your answer. 3. Carter has 10 dog treats for her dog, Flossie. She divides the treats so that Flossie gets the same amount each day. If all of Flossie’s treats are gone at the end of the 10th day, how many treats did she get each day? Write an equation to show how you got your answer. 4. What do you notice about each of the equations you wrote above? How would you describe this in words?


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5. Suppose you have some cookies that you distribute so that each person in your group has one cookie. What can you say about the number of cookies and the number of people in your group? Write an equation that shows this relationship.

Does this relationship hold for any number of cookies or people you have in your group? That is, is the relationship true for any number of cookies and people when

each person always ends up with one cookie?


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Lesson 4 (GA): Review of Distributive Property and Mathematical Arguments

Jumpstarts – TBD 1. Describe the following property in your own words: a × (b + c) = (a × b) + (a × c), for numbers a, b and c 2. If brownies cost c each, what does 4×c represent? Review of Distributive Property Mrs. Gardiner asked her students to come up with an argument that convince her that this property was true. Two students, Seth and J’ana, came up with the following arguments: Seth’s Argument: I tried this with some different numbers. First, I tried 3, 4 and 6. Since 3 x (4 + 6) = 3 x 10 = 30, and (3 x 4) + (3 x 6) = 12 + 18 = 30, then it’s true. Then I tried it with some really big numbers – 20, 50 and 100. Since 20 x (50 + 100) = 20 x 150 = 3000 and since (20 x 50) + (20 x 100) = 1000 + 2000 = 3000, then it’s true. So, I think it’s always going to be true because it worked for these cases. J’Ana’s argument I drew the following picture. The letters a, b and c represent any lengths for my rectangles.

b c

a


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I know that the area of any rectangle is the length times the width. If I look at the big rectangle in my picture, then it’s length is a and its width is b + c. So the area has to be a x (b + c). But I could also look at the areas of the two smaller rectangles. Their areas are each a x b and a x c. I can think of the area of the big rectangle as the sum of the areas of these two smaller rectangles. So the area of the big rectangle is (a x b) + (a x c). So, I have two ways to represent the area of my big rectangle: a x (b + c) and (a x b) + (a x c). Since these have to be the same areas, then they must be equal. So a x (b + c) = (a x b) + (a x c). Which do you think is a better argument? Seth’s or J’Ana’s? Why?


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Lesson 5: Products of Evens and Odds Multiplying Evens and Odds: A. Do you get an even or an odd number when you multiply any two even numbers? State a conjecture that describes your thinking.Why do you think your answer is correct? Draw a picture to justify your thinking. B. What happens if you multiply three even numbers? State a conjecture that describes your thinking.Why do you think your answer is correct? Draw a picture to justify your thinking. C. What happens if you multiply four even numbers? State a conjecture that describes your thinking. Why do you think your answer is correct? Draw a picture to justify your thinking. D. What if you had a bunch of even numbers, but didn’t know how many. If you multiplied these numbers, would you get an even number or an odd number for your result? Why? State a conjecture that describes your thinking. E. Mitch said that if you multiply an even number by an odd number you will also get an even number. Do you agree? Why?


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Lesson 6: Exploring Arithmetic Generalizations

Jumpstarts: Caroline told her teacher that a + b – b = a. Do you agree with her? Why? Graphing Function Table involving Even/Odd numbers Relational Understanding of Equal Sign Exploring Generalizations A. Marta said that any time you multiply two numbers, the result is always bigger than either of the numbers? Do you agree with her? Why? State a conjecture that describes your thinking. B. Develop an argument that you might use to convince someone that your conjecture is true. C.If students agree with Marta, introduce counterexamples that would show the conjecture is false: What if I multiply 2 and 1

2?

D. Explore number domains for which students’ conjectures hold true or fail to hold true: For what sets of numbers are the following conjectures true:

The product of two numbers is always larger than either factor The product of two numbers is always smaller than either factor The product of two factors is always the same as one of the factors


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Lesson 6: Properties of Equations (Reviewing Addition)

Jumpstarts: Graphing (qualitative) Inequality Relational Understanding of Equal Sign

Reviewing Addition

A. Mick says that because 379 = 369 + 10 is true, he knows that 379 + 236 = 369 + 10 + 236 is also true. Do you agree with Mick? Why or why not?

B. Do you think this will hold for any number you add to both sides of the equation 379 = 369 + 10? That is, will the result always be a true equation? How do you know? Explore this with your partner.

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

D. Develop an argument that shows your conjecture is true. Use words or pictures to explain your thinking.

E. For what numbers is your conjecture true?

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. How would you complete the following equations for numbers a, b and c?

a + ___ + c = ___ + b + ____

If a = b, then a + c = b + ___

H. If a = 457, then a + 10 = ____. How do you know? If 7 + x = ___ + x, what number goes in ____? How do you know?


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Lesson 7: Properties of Equations - Exploring Subtraction Jumpstarts: Properties of equations - Addition Consider 84 + 20 + 4 = x. Find x given that 84 + 20 = 104. How did you find x? What number for d will make the equation 347 = 342 + d true? Extending to Subtraction A. Elizabeth has a set of cubes on a balance scale. She took a photo, looking down, of the cubes on the scale:

Do you think the scale is balanced? Why? Elizabeth wants to remove some of the cubes. What can you say about how she can remove cubes so that the scale remains balanced? Suppose Elizabeth removes 2 cubes from each side of the scale. Write an equation that illustrates or models what Elizabeth does.


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B. Consider the equation above: 379= 369 + 10. If we subtract 236 from 379, is the result a true equation? That is, is 379 – 236 = 369 + 10 a true equation? Why? C. What would would you need to do to make it a true equation? Why? D. State a conjecture that describes your thinking. E. For what numbers is your conjecture true? F. Suppose that a is equal to b. How would represent this as an equation? Trini’s teacher asked her to use this (equation) to write the conjecture “Any time I subtract the same number from both sides of an equation, the result is a true equation” in an equation using variables. Trini wrote: If a = b, then a – c = b. Do you agree with what Trini wrote? Why? G. If a = 457, then a - 10 = ____. How do you know?

If 7 - x = ___ - x, what number goes in ____? How do you know?


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Lesson 8: Modeling Problem Situations with Linear Expressions (Review)

Jumpstarts: TBD Graphing Inequality Properties of Equations Modeling Problem Situations with Linear Expressions A. Jay, Kevin, and Marcia are washing cars to earn money for their school vacation. We don’t know how many cars Jay has washed, but we do know that Kevin has washed twice as many cars as Jay. Maria has washed one more car than Jay Represent the number of cars that Jay, Kevin and Maria have each washed. What does your variable (or variables) represent? B. Suppose Kevin said he counted the number of cars he washed and the total is 25 cars. Do you agree? Why? What can you say about the total number of cars Kevin has washed? How do you know? C. What can you say about the total number of cars Maria has washed? How do you know? D. How would you represent the number of cars Kevin and Maria washed all together? E. If Kevin and Maria (correctly) counted the total number of cars they washed, what can you say about the kind of number this would be? Why? F. How could you use what you have learned above to build an argument that the total number of washes that Kevin and Maria will always be an odd number?


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Lesson 9: Modeling Problem Situations with Linear Equations (Lesson 8 Continued)

Jumpstarts: Mrs. Gardiner asked her students to construct an argument that the sum of two even numbers is always even. She wrote two of her students’ arguments on the board: Janice’s argument: I added a bunch of different even numbers together, like this: 4 + 8 = 12, 12 + 10 = 22, 0 + 4 = 4 and 16 + 10 = 26. Every time I did this, I got an even number. So I think that every time you add two even numbers, the answer will be an even number. Mark’s argument: I drew a picture to show this. Let’s suppose I have two rods whose lengths are even numbers. Then I can cut each rod into two equal pieces in the following way:

If I join the rods

I still have two sections of length a and two sections of length b. I can rearrange the cut pieces:

a a

b

a a b b

b

a b a b


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Now, I have two sections of length a + b. So the total length is even. Which do you think is the better argument? Why? Modeling Problem Situations with Linear Equations. Recall how we represented the number of cars Jay, Kevin, and Maria each washed. A. Write an inequality that shows the relationship between the number of cars Jay washed and the number of cars Kevin washed. Can you represent your inequality in a different way? B. On the following open number line depicting the number of cars, x, that Jay washed, indicate the number of cars Kevin and Maria each washed in relation to this. C. Given how we represented the number of cars Maria washed, and if we know that Maria washed 27 cars, write an equation that represents this situation. D. Use your equation to find the number of cars Jay washed.

x


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Lesson 10: Combining Simple Algebraic Expressions

Jumpstarts: Mrs. Gardiner wrote another argument on the board that the sum of two even numbers is even. Kiera’s argument: I know that I can write any even number as 2 times some other number. For example, I can write 24 as 2 x 12. If I represent any number with, say, the letter b, then I can write any even number as 2 x b. I can represent a different even number as, say, 2 x c. So if I add my two even numbers together, I get 2 x b + 2 x c. But by the Distributive Property, I know that 2 x b + 2 x c = 2 x (b + c). But 2 x (b + c) is just 2 times some number, so it has to be even. So the sum of two even numbers is even. Which of the three arguments do you think is better? Why? Combining Simple Expressions: A. Suppose you have two rectangles. Each has width 3 ft. One has length 5 ft and one has length 7 ft. Draw a picture to represent this situation. Find the total areas of the rectangles. B. If you join the rectangles end-to-end at their width, find the area of the new larger rectangle. C. Write an equation that shows the relationship between the total areas of the separate rectangles and the combined larger rectangle. D. Explain how your equation illustrates the Distributive Property. E. Suppose we don’t know the width of the rectangles (assume their lengths stay the same). How could you represent the width of each rectangle? Draw a picture to represent your two rectangles with the new width. How would you represent the total area of the new rectangles?


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F. If you join the rectangles end-to-end at their width, how would you represent the area of the new larger rectangle? G. Write an equation that shows the relationship between the total areas of the separate triangles and the combined larger rectangle. H. Explain how your equation illustrates the Distributive Property. I. Using your observations in the above tasks, suppose you had two rectangles, one with area 4 x c and one with area 12 x c. What would the combined area of the rectangles be? J. Write an equation that shows the relationship between areas of the two smaller rectangles and the larger rectangle. Draw a picture that represents this equation.


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Lesson 11: Modeling and Solving Problem Situations that Involve Linear Equations wth Repeated Variables

Jumpstarts: 1. Find the following: 3×a + 5×a 3a + 5a 27b + 27b 10×2 – 3×2 Graphing (qualitative) Inequality Relational Understanding of Equal Sign Lesson: Port City Coffee Shop sells ice cream sandwiches. Their top selling flavor is key lime, their medium selling flavor is strawberry, and their lowest selling flavor is chocolate. One summer, they sold twice as many strawberry sandwiches as chocolate sandwiches, and four times as many key lime sandwiches as chocolate sandwiches. A. Represent the relative amounts of ice cream sandwiches sold on the open number line below. B. Write an inequality to show the relative relationships between the number of ice cream sandwiches sold. Can you represent your inequality in a different way? C. In order to complete his inventory, the owner is interested in the total number of key lime and strawberry sandwiches sold. How would you represent this amount?


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D. If the owner calculated that a total of 240 key lime and strawberry ice cream sandwiches had been sold, write an equation that depicts this relationship. What does the variable in your equation represent? Why did you use the same variable? E. Use your equation to find the number of chocolate sandwiches sold. Does your solution seem reasonable


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Lesson 12: Modeling and Solving Problem Situations that Involve Linear Equations (Continued)

Jumpstarts: 1. If 12 = 3×a, what is a? How do you know? 2. If Port City Coffee Shop sold twice as many strawberry ice cream sandwiches as chocolate sandwiches, construct a graph that might represent the number of sandwiches sold, for each type, over the summer. Consider the following two rectangles, joined on their length. The blue rectangle has width h ft and length 10 ft. The yellow rectangle has length 10 ft and width 2 ft.

A. Write an expression that represents the area of the blue and yellow retangles. B. Write an inequality that represents the relationship between the two areas. Write your inequality in a different way. C. Write an expression that represents the combined area of the rectangles D. If their total area is 100 ft, what is the area of each rectangle?

10

h

2


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Lesson 14: Review of Function Concepts

Jumpstarts – TBD 1. Review Cutting a One-loop String String Problem Revisited: Cutting a Two-Loop String A. Fold a piece of string to make two loops. While it is folded, make one cut (see

figure). How many pieces of string do you have? Fold another piece of string to make two loops. Make two cuts and find the number of pieces of string that results. 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? (Explore the notion that the quantities vary and what values they might be.) 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 describe your relationship in words? Describe your relationship using variables. E. Why did you use different letters to represent the two different quantities?

cut line


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F. Why do you think your relationship is true? How can you justify your relationship from the problem context? G. If you folded a piece of string and cut it 65 times, how many pieces of string would you get? 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 61 pieces of cut string. How many cuts did your friend

make in order to get 15 pieces of string? How did you get your answer?

J. Using your table, describe how the number of pieces of string changes in relation to the number of cuts?

K. How would you describe the shape of the graph for this type of relationship in the

table?


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Lesson 15 Jumpstarts - TBD Guinea Pig Diet Revisited A. Rosie has two skinny guinea pigs, Oreo and Peanut, that she feeds two different

diets. She wants to see which diet is the best for gaining weight, so she weighs them each day for a week and records their weights in hectograms. One morning, 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? C. What relationships do you see in the data? Use these to predict Oreo’s and Peanut’s weight on the eighth day of the diet. D. For each guinea pig, find a relationship between his weight and the day he was weighed. How would you describe your relationship in words? Describe your relationship using variables. E. Why did you use different letters to represent the different quantities? F. Why do you think your relationship is true?

1 3 4 5 7

1 9 25 36

Peanut’s Diet

1 2 3 5 6

2 4 8 10 14

Oreo’s Diet


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G. 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? H. Using your tables, how would you describe how the weight of each guinea pig changes in relation to the number of days the weight is measured? How would you describe the shape of each graph in relation to the data in each table? I. Using your table, which guinea pig do you think grows faster? Why? Looking at your graphs for the two guinea pigs’ weights, how can you determine which guinea pig grows faster? J. If the two guinea pigs continue to grow at this rate, what would you estimate their weight to be at the end of the second week? Use your tables or graphs to justify your estimate. Use your function rules to check your estimate. How close was your prediction? K. Which do you think is the better diet? Why? Use your tables and graphs to justify your answer.


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Lesson 16 – Exponential Functions Jumpstarts - TBD Paper Folding (adapted from Blanton, 2008)

A. Take a piece of paper and fold it in half to create two regions (see figure). Fold the paper in half again. How many regions does this create? Continue this, folding the paper in half a third time. How many regions does this third fold create? 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 regions that would result from 5 folds. D. Find a relationship between the number of folds and the number of regions that result. How would you describe your relationship in words? Describe your relationship using variables. Use this relationship to predict the number of regions you would get after 10 folds. E. Why do you think your relationship is true? F. Construct a graph that shows the number of regions for the corresponding number of folds.

1 fold yields 2 regions


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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? G. Using the table, how would you describe how the number of regions is changing in comparison to the number of folds? Using your graph, how would you describe how the number of regions is changing in comparison to the number of folds? Use your graph to estimate the number of regions for 12 folds. Check your estimate using the relationship you identified. H. How would you describe the shape of the graph given the type of relationship depicted in the table?


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Lesson 17

Jumpstarts - TBD Rosie Gets Another Guinea Pig A. Rosie gets another guinea pig. She names him Rascal. She decides to feed him a

new and improved diet she purchased at the pet store. This diet is different than the food she gave Oreo and Peanut, so she wants to compare Rascal’s weight gain to the other guinea pigs. To make her task simpler, she decides to compare weights only for Rascal and Oreo.

She can’t find the information on Oreo’s weight, but she does remember that on the first day of her experiment, Oreo weighed 2 hectograms. She also remembers that he had a steady weight gain of 2 hectograms every day. B. Using this information, can you reconstruct a table showing Oreo’s weight for each

day of Rosie’s one-week experiment? What do the variables in your table represent?

Rosie notices that Rascal weighs the same as Oreo on the first day of her experiment. However, as she continues to weigh Rascal, she notices that each day Rascal’s weight increases by a factor of 2. For example, if Rascal weighs 4 hectograms on one day, then on the next day his weight is 4x2, or 8, hectograms. Use this to construct a table to depict Rascal’s weight for Rosie’s one-week experiment. What do the variables in your table represent? C. For each guinea pig, what relationships do you see in the data? Use this to predict both Oreo’s and Rascal’s weight for the eighth day of the experiment. D. For each guinea pig, find a relationship between the guinea pig’s weight and the day on which the weight was taken. How would you describe your relationship in words? Describe your relationships using variables. Why did you use different letters to represent the different quantities? E. Why do you think your relationship is true?


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F. Use these relationships to predict each guinea pig’s weight at the end of 2 weeks. G. Using the table, how would you describe how each guinea pig’s weight changes over the length of the experiment? Based on this, what would you predict about the shape of the graph of each guinea pig’s weight gain during the experiment? Which guinea pig do you think is growing the fastest? Why? H. For each guinea pig, draw a graph that shows the guinea pig’s weight for the

corresponding day. Draw all 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? Compare your predictions about the shape of the graphs to the graphs you drew. G. Looking at your graph, how would you describe the shape of the graph that results from data in each table? I. Which guinea pig do you think grows faster? Use your tables and graphs to justify your answer. How can you use your function rules to support your answer? J. If the two guinea pigs continue to grow at this rate, what would their weight be at the end of the second week? K. Which do you think is the better diet? Why? Use your tables, graphs, or function rules to justify your answer.


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Lesson 18 Jumpstarts: TBD 5-5: Sunflowers (adapted from Leinhardt et al, 1990) A. Ana is growing a sunflower in a new pot and decides to chart its growth. After observing the height of the graph each day for the length of her experiment, she draws the following graph to reflect what she found. B. Describe the growth of her sunflower for each of the regions labeled A, B and C. C. How would you describe how the sunflower grew? Write a story that might

characterize its growth. D. Why do you think Ana drew a line (as opposed to discrete points) to represent her

observations?

time

height

A

B C


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Lesson 19

Jumpstarts: TBD

5-6: Angela’s Road Race Angela ran a road race for Hunger Awareness. Her friend, Tim, followed her to track her progress. At the end of the race, he sent a text to Angela’s daughter to describe the race: “Hi – your Mom did great in the race today! She started out very fast and ran at a steady pace for about half of the race. She slowed down a little, and about three-fourths of the way through the race, stopped to rest. She finished the last fourth of the race running at a steady pace that was faster than her pace at the beginning of the race, so the rest must have been good for her!” A. Construct a graph that corresponds to how Angela ran the race. Do not label the

graph with any units. Label the axes with variables representing the appropriate quantities.

B. Describe how the shape of the graph corresponds to the different parts of the

narrative.

GRADE 5 CURRICULAR PROGRESSION LESSONSalgebra.wceruw.org/documents/Grade 5 Lessons.pdf · GRADE 5...

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