Number of (Nested) Shopping Carts Three nested shopping carts are shown.

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Christine Flowers
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1 Randy must fit shopping carts into an area that has a length of 82 feet and a width of one shopping cart. He made some measurements necessary for his computations. The table shows the length of a set of shopping carts as they are nested together. Number of (Nested) Shopping Carts Length in Inches Three nested shopping carts are shown. Randy has decided to use an algebraic expression to determine the length of nested shopping carts Determine a function rule for the length, in inches, of a set of nested carts in terms of the number of nested shopping carts. How do the numbers in the function rule relate to the physical shopping carts? 3. What is the length of 50 nested shopping carts? 4. How many carts fit into an area with a length of 82 feet and a width of one shopping cart? Use algebraic methods and verify your solution using a table or a graph. Linear Functions, Equations, and Inequalities 133

2 Materials: Notes One graphing calculator per student Algebra TEKS Focus: (A.5) Linear functions. The student understands that linear functions can be represented in different ways and translates among their various representations. The student is expected to: (C) use, translate, and make connections among algebraic, tabular, graphical, or verbal descriptions of linear functions. (A.7) Linear functions. The student formulates equations and inequalities based on linear functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is expected to: (A) analyze situations involving linear functions and formulate linear equations or inequalities to solve problems; Scaffolding Questions What do you know about the shopping carts? How long is one cart? What is the total length of two nested carts? If one cart is 37.5 inches long, how could you find the additional length for each nested cart? Will the model for these data be linear? Why or why not? How can you compute the rate of change for the situation? Complete this new table with the missing values. Number of (Nested) Shopping Carts Process n Sample Solutions Length in Inches 1. Determine a function rule for the length, in inches, of a set of nested carts in terms of the number of nested shopping carts. The length for one cart is 37.5 inches. The rate of change is inches for 7 carts or inches for 1 cart. 134 Linear Functions, Equations, and Inequalities

3 The total length is 37.5 plus for every additional shopping cart. L = (n 1) or L = n where n is the number of carts and L is the length of the set of carts. The function may also be represented by a table or a graph. 2. How do the numbers in the function rule relate to the physical carts? The inches represents the nested length of the carts; that is, it is the length of the cart that slides into the cart in front of it each time. The inches is the amount that hangs out for each new cart. Additional Algebra TEKS: (A.4) Foundations for functions. The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations and inequalities in problem situations. The student is expected to: (A) find specific function values, simplify polynomial expressions, transform and solve equations, and factor as necessary in problem situations; Number of Shopping Carts Process Length in Inches (1) (2) (3) 60 Texas Assessment of Knowledge and Skills: Objective 3: The student will demonstrate an understanding of linear functions. Objective 4: The student will formulate and use linear equations and inequalities (4) (5) 82.5 n n Linear Functions, Equations, and Inequalities 135

4 Graphing calculator table: Enter Y1 = x Graph: 3. What is the length of 50 nested shopping carts? When there are 50 carts, substitute 50 for n in the formula: L = (50) = inches This value can be determined using the calculator table. 136 Linear Functions, Equations, and Inequalities

5 This value can also be determined using the calculator graph. 4. How many carts fit into an area with a length of 82 feet and a width of one shopping cart? Use algebraic methods and verify your solution using a table or a graph. 82 feet must be converted to inches = 984 inches An equation may be used to determine when L is 984. L = n 984 = n = 11.25n n = 85.1, where n represents the number of carts There cannot be a fractional number of carts, so the number of carts that fit into 82 feet is 85 carts; 86 carts are longer than 82 feet. Verify this answer using the table: Linear Functions, Equations, and Inequalities 137

6 Extension Questions If the function rule is L = n for a different set of shopping carts, what is the same about the two sets of carts? The portion that is added for each new cart is the same because the rate, 11.25, has not changed. However, the y-intercept has changed, so the part that is nested into the rest of the carts is not the same. What is a reasonable domain for the function rule representing this different set of carts? The function L = n is a linear function. The domain of the function is all real numbers. What is a reasonable domain for the problem situation? The domain for the problem situation represents the number of carts and must be the set of positive integers. However, the domain is determined by the physical and logistical constraints of the situation, such as the available storage space and customer capacity. What is the rate of change for this situation. The rate of change is inches for every one cart. How does this change affect the graph? The graph has a different y-intercept, but the same slope. The lines are parallel. If the function rule is L = n for a different set of carts, what is the same about the two sets of carts? The portion that is added for each new cart has changed because the rate has changed (from to 14). However, the y-intercept has not changed, so the part that is nested into the rest of the carts remains the same. How does this change affect the graph? The graph has the same y-intercept, but different slopes. The lines are not parallel, but intersect at the point (0, 26.25). 138 Linear Functions, Equations, and Inequalities

Algebra I Assessment. Eligible Texas Essential Knowledge and Skills

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