Unit 2

The Shadow Knows

Scale Drawings, Similarity, and Slope

Lesson 19

Expressions and Equations

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Let’s use shadows to find the heights of an object.

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Unit 2 ● Lesson 19

Learning

Goal

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Long Shadows and Short Shadows

Unit 2 ● Lesson 19 ● Activity 1

What do you notice?

What do you wonder?

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Warm-up: Notice and Wonder

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

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Long Shadows and Short Shadows

Unit 2 ● Lesson 19 ● Activity 1

Warm-up: Notice and Wonder

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

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Objects and Shadows

Unit 2 ● Lesson 19 ● Activity 2

How would you go about measuring the height of each person and the lamppost?

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

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Objects and Shadows

Unit 2 ● Lesson 19 ● Activity 2

Here are some measurements that were taken when the photo was taken. It was impossible to directly measure the height of the lamppost, so that cell is blank.

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1. What relationships do you notice between an object’s height and the length of its shadow?
2. Make a conjecture about the height of the lamppost and explain your thinking.

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

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Objects and Shadows

Unit 2 ● Lesson 19 ● Activity 2

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

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Justifying the Relationship

Unit 2 ● Lesson 19 ● Activity 3

Explain why the relationship between the height of these objects and the length of their shadows is approximately proportional.

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

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The Height of a Tall Object

Unit 2 ● Lesson 19 ● Activity 4

1. Head outside. Make sure that it is a sunny day and you take a measuring device (like a tape measure or meter stick) as well as a pencil and some paper.
2. Choose an object whose height is too large to measure directly. Your teacher may assign you an object.
3. Use what you have learned to figure out the height of the object! Explain or show your reasoning.

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Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

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Using Equations for Lines

Unit 2 ● Lesson 19

• The data is not always perfect
• To explain why the object height to shadow length relationship might be proportional, simplifying assumptions were needed.
• Mathematical models can be used to make accurate guesses or predictions about quantities that are difficult or impossible to measure directly. In this case, a relationship between the height of an object and the length of its shadow was observed from easier-to-get measurements and justified by reasoning about similar triangles. Once we knew this relationship existed and why it existed, we could reasonably expect the same relationship to hold for a very tall object nearby at the same time of day, and use the length of the tall object's shadow to find its height.
• An interesting historical connection: over 2,000 years ago, the ancient Greek mathematician Eratosthenes also studied shadows closely (in a slightly different way) and used this to estimate the circumference of Earth with an error of less than 2%!

Lesson Synthesis

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

7

Unit 2 ● Lesson 19

I can model a real-world context with similar triangles to find the height of an unknown object.

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Learning

Targets

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This slide deck is copyright 2021 by Kendall Hunt Publishing, https://im.kendallhunt.com/, and is licensed under the Creative Commons Attribution-NonCommercial 4.0 International License (CC BY-NC 4.0).

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IM 6–8 Math™ was originally developed by Open Up Resources and authored by Illustrative Mathematics^®, and is copyright 2017-2019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY-NC 4.0). OUR's 6–8 Math Curriculum is available at https://openupresources.org/math-curriculum/.

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Adaptations to add additional English language learner supports are copyright 2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Adaptations and additions to create IM 6–8 Math Accelerated are copyright 2020 by Illustrative Mathematics^®, www.illustrativemathematics.org, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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