standard_description: Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.
problem_activity: Activity
Now that students have some experience drawing and interpreting histograms, they use histograms to compare distributions of two populations. In a previous activity, students compared the two dot plots of students in a keyboarding class—one for the typing speeds at the beginning of the course and the other showing the speeds at the end of the course. In this activity, they recognize that we can compare distributions displayed in histograms in a similar way—by studying shapes, centers, and spreads.
Launch
Arrange students in groups of 2. Give students 4–5 minutes of quiet work time and 1–2 minutes to share their responses with a partner.
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer interactions. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their reasoning. For example, “Histogram ___ represents the heights of _______ players because . . .”, and “I agree/disagree because . . .”
Supports accessibility for: Language; Social-emotional skills
Student Facing
Professional basketball players tend to be taller than professional baseball players.
Here are two histograms that show height distributions of 50 male professional baseball players and 50 male professional basketball players.
Decide which histogram shows the heights of baseball players and which shows the heights of basketball players. Be prepared to explain your reasoning.
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Write 2–3 sentences that describe the distribution of the heights of the basketball players. Comment on the center and spread of the data.
Write 2–3 sentences that describe the distribution of the heights of the baseball players. Comment on the center and spread of the data.
Student Response
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Activity Synthesis
Select a few students to share their descriptions about basketball players and baseball players. After each student shares, ask others if they agree with the descriptions and, if not, how they might revise or elaborate on them. In general, students should recognize that the distributions of the two groups of athletes are quite different and be able to describe how they are different.
Highlight the fact that students are using approximations of center and different adjectives to characterize a distribution or a typical height, and that, as a result, there are variations in our descriptions. In some situations, these variations might make it challenging to compare groups more precisely. We will study specific ways to measure center and spread in upcoming lessons.
Representing, Writing: MLR1 Stronger and Clearer Each Time.
Use this routine to give students a structured opportunity to revise and refine their written responses to the questions about describing the distribution of heights of basketball and baseball players. Ask students to meet with 2–3 partners, to share and get feedback on their responses. Provide listeners with prompts for feedback that will clarify the descriptions about the distributions shown in the histograms (e.g., “What details about the histograms are important are important to describe?”). Finally, give students time to revise their initial written responses to reflect input they received. This will help students write descriptions of the center and spread of distributions displayed in histograms using precise language.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness


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