standard_description: Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
problem_activity: Activity
In the previous unit, students studied scale drawings of real-world objects. In grade 8, they will study dilations and similarity. The purpose of this activity is to use students’ recent study of scale drawings as a transition to the study of proportional relationships. Initially, students may describe the difference between Moons A, B, C, and D in qualitative terms, e.g., “D is more squished than the others.” They may also use the term “scaled copies,” which appeared in the work of the previous unit, but struggle to identify measurements to use in these figures that consist only of curved sides. It is important to ask students to articulate what they mean by “squished” in quantitative terms, for example, by talking about the height relative to the width and helping students to define “height” and “width” of a moon in some appropriate way. Once students have that, they can note that the height of the enclosing rectangle is always one and a half times its width for Moons A, B, and C, but not D, or they might note that the distance tip to tip is three times the width of the widest part of the moon for Moons A, B, and C, but not so for D.
[IMG]
As students explore these transformations, ask questions with the goal of having students articulate that for two images to look like scaled copies of each other, the ratios of the side lengths need to be the same.
In the third question, students are asked to represent the situation with tables and double number line diagrams, providing students with an opportunity to recall these representations from their work in grade 6.
Digital
Launch
Give students 3 minutes of quiet think time and tell them to pause after the first two questions. After the quiet think time, ask students to discuss their answers with a partner to describe how Moon D is different. Use MLR 2 (Collect and Display) as students share. Record the explanations that students are using to describe the moons. Ensure that students see some ways to measure lengths associated with the moons, then complete the last question.
If using the digital activity, give students 3 minutes of quiet think time and tell them to pause after the first two questions. After the quiet think time, students can discuss answers with a partner. Based on student conversations, you may want to have a whole-group discussion to ensure they see a way to measure the lengths associated with the moons before they attempt to answer the last question.
Action and Expression: Internalize Executive Functions.
To support development of organizational skills, check in with students within the first 2–3 minutes of work time. Check to make sure students have a method for comparing the measurements of each moon. For example, create a rectangle around each moon and compare the width-height ratios.
Supports accessibility for: Memory; Organization
Student Facing
Here are four different crescent moon shapes.
What do Moons A, B, and C all have in common that Moon D doesn’t?
Use numbers to describe how Moons A, B, and C are different from Moon D.
Use a table or a double number line to show how Moons A, B, and C are different from Moon D.
Student Response
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Student Facing
Are you ready for more?
Can you make one moon cover another by changing its size? What does that tell you about its dimensions?
Student Response
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Print
Launch
Give students 3 minutes of quiet think time and tell them to pause after the first two questions. After the quiet think time, ask students to discuss their answers with a partner to describe how Moon D is different. Use MLR 2 (Collect and Display) as students share. Record the explanations that students are using to describe the moons. Ensure that students see some ways to measure lengths associated with the moons, then complete the last question.
If using the digital activity, give students 3 minutes of quiet think time and tell them to pause after the first two questions. After the quiet think time, students can discuss answers with a partner. Based on student conversations, you may want to have a whole-group discussion to ensure they see a way to measure the lengths associated with the moons before they attempt to answer the last question.
Action and Expression: Internalize Executive Functions.
To support development of organizational skills, check in with students within the first 2–3 minutes of work time. Check to make sure students have a method for comparing the measurements of each moon. For example, create a rectangle around each moon and compare the width-height ratios.
Supports accessibility for: Memory; Organization
Student Facing
Here are four different crescent moon shapes.
[IMG]
What do Moons A, B, and C all have in common that Moon D doesn’t?
Use numbers to describe how Moons A, B, and C are different from Moon D.
Use a table or a double number line to show how Moons A, B, and C are different from Moon D.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
For question 2, students might attempt to find the area of each moon by counting individual square units. Suggest that they create a rectangle around each moon instead and compare the width-height ratios.
For question 3, if students are not sure how to set up these representations, providing a template may be helpful.
Activity Synthesis
As students suggest ways to characterize the difference between Moons A, B, and C and Moon D, ask questions that help them clarify and make their statements more precise. For example,
“What does it mean to be ‘smashed down?’ What measurements might you make to show that this is true?”
“Is there anything else about what A, B, and C have in common that you can identify?”
“What things might we measure about these moons to be able to talk about what makes them different in a more precise way?”
“How can you represent the ratios of the measurements you are comparing using a table and a double number line diagram?”
If some students are still struggling with the tables or diagrams, ask students who were successful to share their representations with the class.
Speaking: MLR8 Discussion Supports.
To aid students in producing statements about comparisons of proportional relationships. Provide sentence frames for students to use when they are comparing and contrasting such as: “All _____ have _____ except _____,” “What makes _____ different from the others is _____.” Ask students to further add to the statement to clarify the comparisons if they used descriptor words such as “wider, narrower, etc.” Improved statements should include mathematical language to describe the measurements (such as length, width, ratio, etc.). This will help students practice and develop language for comparisons.
Design Principle(s): Support sense-making


aligns: no