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Videos and solutions to help Grade 6 students construct the altitude for three different cases and de-construct triangles to justify that the area of a triangle is exactly one half the area of a parallelogram.

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Lesson 4 Student Outcomes
• Students construct the altitude for three different cases: an altitude that is a side of a right angle, an altitude that lies over the base, and an altitude that is outside the triangle.

• Students deconstruct triangles to justify that the area of a triangle is exactly one half the area of a parallelogram.

Opening Exercise

Draw and label the height of each triangle below.

Exploratory Challenge

- Use rectangle “x” and the triangle with the altitude inside (triangle “x”) to show the area formula for the triangle is A = 1/2 x base x height.

a. Step One: Find the area of rectangle x.

b. Step Two: What is half the area of rectangle x?

c. Step Three: Prove, by decomposing triangle x, that it is the same as half of rectangle x. Please glue your decomposed triangle onto a separate sheet of paper. Glue it next to rectangle x. - Use rectangle “y” and the triangle with a side that is the altitude (triangle “y”) to show the area formula for the triangle is A = 1/2 x base x height.

a. Step One: Find the area of rectangle y.

b. Step Two: What is half the area of rectangle y?

c. Step Three: Prove, by decomposing triangle y, that it is the same as half of rectangle y. Please glue your decomposed triangle onto a separate sheet of paper. Glue it next to rectangle y. - Use rectangle “z” and the triangle with a side that is the altitude (triangle “z” to show the area formula for the triangle is A = 1/2 x base x height.

a. Step One: Find the area of rectangle z.

b. Step Two: What is half the area of rectangle z?

c. Step Three: Prove, by decomposing triangle z, that it is the same as half of rectangle z. Please glue your decomposed triangle onto a separate sheet of paper. Glue it next to rectangle z. - When finding the area of a triangle, does it matter where the altitude is located?
- How can you determine which part of the triangle is the base and the height? 6 - 7. Calculate the area of each triangle. Figures are not drawn to scale.
- Draw three triangles (acute, right, and obtuse) that have the same area. Explain how you know they have the same area.

**Lessons 2 - 4**

**Problem Set**

- to 4. Calculate the area of each figure below. Figures are not drawn to scale.

If the sailboat sails are on sale for $2 per square foot, how much will the new sail cost?

6. Darnell and Donovan are both trying to calculate the area of an obtuse triangle. Examine their calculations below.

Which student calculated the area correctly? Explain why the other student is not correct.

7. Russell calculated the area of the triangle below. His work is shown.

Although Russell was told his work is correct, he had a hard time explaining why it is correct. Help Russell explain why his calculations are correct.

8. The larger triangle below has a base of 10.14 m; the gray triangle has an area of 40.325 m

Determine the area of the larger triangle if it has a height of 12.2 m.

Let A be the area of the unshaded (white) triangle in square meters. Write and solve an equation to determine the value of A, using the areas of the larger triangle and the gray triangle.

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