1 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 fieldtested through a one-year classroom teaching experiment at grade 4. Grade 4 Lessons Project LEAP (Blanton & Knuth, NSF DRK-12 #1207945)
2 ITEM STRUCTURES CORRESPONDING TO LESSONS 1 : 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 representationbased 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 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) 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.
3 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 representationbased 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.
4 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 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) 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
5 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 function rule. That is, when function variables are substituted with corresponding values from the table, the result must be a true equation 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 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 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
6 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 = ax 2 - without linear and constant terms; y = ax 2 + 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 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 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 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
7 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 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; 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?)
8 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 representationbased 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 4 th 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
9 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 o o review different meanings of =, including as expressing a relationship between quantities interpret equations (number sentences) written in various formats (e.g., other than a + b = c) 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
10 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 4 th 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 = x 2 and extending this more generally to those of the form y = x 2 + 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 = x 2, y =x 2 + a (for positive integers a, b)
11 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?);
12 Fourth Grade (VAR): In 4 th grade, students refine their understanding of the concept of variable introduced in 3 rd 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
13 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.
14 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?
15 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
16 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. 0 a Additive Inverse A. What can you say about what happens when you subtract a number from itself? Describe your conjecture in words.
17 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.
18 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.)
19 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 4 x = 1 4 1 x = b. For what values of b is this true? (Talk about the different number domains appropriate to 4 th 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 4 th 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
20 (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)
21 Lesson 4 (GA): Products of Evens and Odds Jumpstarts: 1. What numbers will make the following equations true? 39 441 = 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 4 x 12 3 = 3, what is 3 4 x 12 3 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 3 4 Number of pairs created Number of cubes left over
22 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?
23 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:
24 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.
25 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)
26 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?
27 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 4 x 12 3 = 3, what is 3 4 x 12 3 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
28 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.
29 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 ).
30 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)
31 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.
32 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?
33 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?
34 (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?
35 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?
36 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)