Prioritized Common Core Mathematical Practice Standard(s) Supporting Common Core Mathematical Practice Standards

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1 Math Solutions Lesson from the Classroom Stacking Cups A Lesson for Grades 9 12 Math Goal In this lesson, students use real-life experiences to build the concepts of y-intercept as the starting point and slope as a rate of change. Students investigate slope as a rate of change. They compare, contrast, and make conjectures based on cups-height graphs for different types of stacking containers. Time Two class periods of 45 minutes each Common Core State Standard Content Correlation A-CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. F-IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Prioritized Common Core Mathematical Practice Standard(s) MP4 Model with mathematics. MP7 Look for and make use of structure. Supporting Common Core Mathematical Practice Standards MP6 Attend to precision. MP3 Construct viable arguments and critique the reasoning of others. Language Goal Students will make inferences orally and in writing about the relationships they discover between the number of cups and the height of the cups. They will describe their findings in their small groups and with the whole class using words and equations. Vocabulary constant, function, linear, slope, variable, y-intercept Materials Chart paper (1 sheet per pair of students) Newsprint (2 sheets per pair of students) Gridded chart paper (1 sheet per group of four, plus extras) Yellow construction paper (1 sheet per group of four, plus extras) Adhesive color dots, ¾ inch, red (10 dots per group of four; optional) 1

2 Easel pad markers (one 4-pack per group of four) Meter sticks (1 per group of four) Rulers (1 per group of four) Painter s tape (blue, 1 roll per group of four) Stacking cup collection:»» Have five to eight different kinds of cups or containers for a class of thirty-two students. You will need five to seven of each kind of cup or container. The cups or containers need to stack/nest inside one another. Vary the height of the cups as well as the width of the rims. Sentence Frames Beginning If the number of cups is, then the height of the stack is. If the height of the stack is, then the number of cups is. I think the relationship between the number of cups and the height of the stack is a function. Intermediate/Advanced I know that for n cups, the height of the stack is. I can conclude that the function is, because. Anticipating Confusion It is common that students observe the constant difference in a table of data and assume that value is added to the variable. Help students understand what the constant difference represents in the context of the problem in order to determine that it is not added to the variable (the number of cups) but rather it is to be multiplied by the variable. Students can also be confused as they make measurements because the differences are not exact. It is important to have them reflect on the reasons that measurements are not exact so they can deal with the messiness of the data. Planning Questions These questions will engage students in talking about what they understand or will reveal misconceptions. What is the height of one cup? Two cups? Three cups? What stays the same as you stack cups? What changes? How could you use that information to figure out the height of a stack of cups beyond what you were given to measure? 2

3 Lesson Implementation Minilesson 1. This lesson begins with a review of the important vocabulary that the students will encounter when looking for the relationship between the number of cups in a collection and the height of the stack. Taking the time to explore key vocabulary is essential in order to ensure that students have access to the lesson s math content by tapping prior knowledge and clarifying any confusion. Begin by drawing a table with two columns for the whole class to see. One column is headed with the words Number of Cups; the other column headed with Height of Stack. Number of Cups Height of Stack (cm) Ask students if they know what a table is and how it is used. Listen to responses and clarify its use by saying something like, You can use a table like this to find a pattern of differences between the number of cups and the height of the stack. We can identify a function from a table like this. 2. Check for understanding of the word function. This lets you tap into students prior knowledge in order to connect their prior knowledge with the new knowledge that will be presented in the lesson. Record students ideas on a chart with the prompt A function is: written on the top. 3. After gathering ideas from students, add the following definition to the chart. (You can substitute this one for a definition that comes from your math resource.) A function is a rule that establishes a relationship between two quantities, called the input (or independent variable) and the output (or dependent variable). Ask students to read along as you read the definition. The words that are bold represent key academic terms that you expect students to use during the lesson. 4. Ask, What is a rule? It is important that students are familiar with the meaning of rule in mathematics rather than the everyday meaning of rule. 5. Ask the same question about the other bolded terms. If students are unable to give an adequate explanation of any of them, offer clear examples and check for student understanding again. Introduction 6. Introduce the task by showing students the collection of cups you have assembled. Then, give them the following scenario: You have been hired by a company that makes all kinds of cups and containers of different sizes. They also make holders for these cups and need to know how tall to makes the holders so they hold 50 cups. 3

4 7. Explain that you will distribute a set of cups, rulers, a sheet of large gridded chart paper, a sheet of construction paper, and markers to each group. Once they get their set of cups, students are to do the following: a. Make a table and record in it the measurement data (number of cups and height [cm] of the stack). b. Make a coordinate graph of the data on the large gridded paper. (Do not identify your collection of cups.) Height of cups should be on the y-axis and number of cups on the x-axis. Use intervals of 1 on both axes. c. On a separate sheet of construction paper, express the relationship by writing it in equation form. d. Post your graph and equation in the appropriate places. Exploration 8. Distribute a set of containers and a sheet of gridded chart paper to each group. Designate an area of the room where the equations can be posted, and a separate area where the graphs can be posted. 9. As students work, observe how they tackle the task. Ask the questions below, as appropriate. Q What is the height of one cup? Two cups? Three cups? Q What stays the same as you stack cups? What changes? Q How could you use that information to figure out the height of a stack of cups beyond what you were given to measure? 10. As groups finish, ask students to examine and discuss the graphs of other groups. Tell them to compare the results for different kinds of containers, noting similarities and differences. 11. When the graphs and equations have been posted, have groups put their cups on a table where everyone can see them. Prominently label each of the graphs A, B, C, and so on. By doing it this way, the graphs can be easily identified. The equations should be labeled 1, 2, 3, and so on. The labeling should be done in random order, so it is not obvious which graph matches which equation. Summary 12. Call the class together for a whole-class discussion. Begin by analyzing the graphs that are posted. Ask the following questions to focus students attention on important aspects of the graphs: Q Which graphs represent a collection of cups with wide rims? How do you know? Q Which graphs represent tall cups? How do you know? Q If you were going to match a collection of cups to this graph, what would you look for and why? 13. Next, have students work in pairs to match the equations with the graphs. Ask them to write down the pairing and be ready for a discussion in a few minutes. Remind them that they must have a reason or viable argument to support the pairings they make. 14. Once students have had time to match the equations and graphs, open up the discussion regarding the matches. This is a good opportunity to encourage and support students as they make viable arguments and for others to critique their reasoning by offering counterideas or asking questions. 15. After the matching is completed, ask the following questions to assess students understanding. You may do these orally or select a few and ask students to respond in writing. Q How is the y-intercept on the graph represented in the equation? Q What does the y-intercept represent in the context of our problem? 4

5 Q How can you tell from the graphs whether the height of the stacked cups increased slowly or rapidly? Q Where do you see that increase reflected in the equation? Q How can you use the table of values to determine the equation of the line? Q How can you use the graph of the data to determine the equation of the line? Teacher Notes There are two common ways to see the equation: y = (height of lip) * x + (height of cup without lip) y = (height of lip) * (x 1) + (height of cup) Below are some points to keep in mind: In the context of the problem, the domain is only whole numbers because the data is discrete (there are only a finite number of values possible) yet the line represented by the equation is a useful model to help us predict how tall the stack would be with fifty cups. You might hear students say that the y-intercept of the line represents the height of the original cup minus the lip. However, the y-intercept of the line represented by the equation has no meaning in the context of the problem; if there are zero cups, there would be zero height. Find more classroom lessons online at mathsolutions.com. Visit the Educator Tools section and click Free Classroom Lessons. 5