The square in the middle has an area of 1 square unit. What is the area of the entire rectangle in square units? Explain your reasoning.
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4 square units. Sample reasoning:
Put together the two small triangles to make a square. Its area is 1 square unit. Decompose each medium triangle into two small triangles that can be arranged as a square. Each of these squares has an area of 1 square unit. Together with the square in the middle, the sum of the areas of these pieces is 4 square units.
A small triangle has an area of 21 square unit, and a medium triangle has an area of 1 square unit. 1+1+1+21+21=4
A maritime flag is shown. What is the area of the shaded part of the flag? Explain or show your reasoning.
An image of a maritime flag, with a shaded portion and a triangular portion removed from the right side. Measurements are indicated around the shaded area indicate 8 inches across the top, two equivalent measurements of 6 inches on the side, and two equivalent measurements of 4 inches on the bottom.
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72 square inches. Sample reasoning: If we draw a line down the middle of the shaded area, we would have a 4 inch-by-12 inch rectangle on the left and two right triangles. The 4-by-12 rectangle has an area of 48 square inches. The two triangles on the right can be composed into a 4 inch-by-6 inch rectangle, so their combined area is 24 square inches. 48+24=72
Section A Check
Section A Checkpoint
Problem 1
Find the area of each shaded region. Explain or show your reasoning.
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The areas is 16 square units. Sample reasoning:
The rectangle can be decomposed into 1 square of 4 square units and 6 identical right triangles. Two right triangles make a square of 4 square units, so 6 right triangles make 3 squares with a combined area of 12 square units. 4+12=16
The rectangle can be enclosed by a 6-by-6 square (36 square units). The square creates 2 larger right triangles and 2 smaller ones. The larger triangles can be rearranged to make a 4-by-4 square (16 square units). The smaller triangles make a 2-by-2 square (4 square units). Subtracting the areas of the triangles from the 6-by-6 square gives 16 square units. 36−16−4=16
30 sq cm. Sample reasoning: The two small triangles can be rearranged into a 3-by-3 square (9 sq cm). The two large triangles can be rearranged into a 3-by-7 rectangle (21 sq cm). The combined area is 9+21, or 30 sq cm.
How would you find the area of this parallelogram? Describe your strategy.
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Sample responses:
Decompose a triangle from one side of the parallelogram and move it to the other side to make a rectangle. Multiply the base and side (height) lengths of the rectangle.
Draw a rectangle that just fits around the parallelogram, multiply the bottom length of that rectangle by its side length to find the area of the rectangle, and then subtract the combined area of the triangles that do not belong to the parallelogram.
Count how many squares are across the bottom of the parallelogram and how many squares tall it is and multiply them.
Find the area of the parallelogram. Explain or show your reasoning.
Was there a length measurement you did not use to find the area? If so, explain why it was not used.
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54 sq cm. Sample reasoning: A base is 9 cm and its corresponding height is 6 cm. 9⋅6=54.
The 7.5 cm length was not used. Sample reasoning:
If the side that is 7.5 cm was used to find area, we would need the length of a perpendicular segment between that side and the opposite side as its corresponding height. We don't have that information.
The parallelogram can be decomposed and rearranged into a rectangle by cutting it along the horizontal line and moving the right triangle to the bottom side. Doing this means the side that is 7.5 cm is no longer relevant. The rectangle is 6 cm by 9 cm; we can use those side lengths to find area.
Section B Check
Section B Checkpoint
Problem 1
Select all parallelograms that show a base and its corresponding height (as a dashed segment).
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A, C, D
Problem 2
Find the area of the parallelogram. Explain or show your reasoning.
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32 sq cm. Sample reasoning:
If the side that is 4 cm is the base, then its corresponding height is 8 cm. 4⋅8=32
If the side that is 10 cm is the base, then its corresponding height is 3.2 cm. 10⋅(3.2)=32
The parallelogram can be decomposed along the dashed line that is 3.2 cm long. Rearranging the pieces makes a rectangle that is 10 cm by 3.2 cm. 10⋅(3.2)=32
Problem 3
A parallelogram has an area of 60 square inches and a base that is 5 inches long. How long is the corresponding height?
Circle all quadrilaterals that you think can be decomposed into two identical triangles using only one line.
What characteristics do the quadrilaterals that you circled have in common?
Here is a right triangle. Show or briefly describe how two copies of it can be composed into a parallelogram.
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Quadrilaterals B, C, D, and F should be circled.
They all have two pairs of parallel sides. They are all parallelograms.
Sample response: Joining two copies of the triangle along a side that is the same length (for instance, the shortest side of one and the shortest side of the other) would make a parallelogram. (Three parallelograms are possible, since there are three sides at which the triangles could be joined. One of the parallelograms is a rectangle.)
Elena, Lin, and Noah all found the area of Triangle Q to be 14 square units but reasoned about it differently, as shown in the diagrams. Explain at least one student’s way of thinking and why his or her answer is correct.
Three images of triangle Q labeled Elena, Lin, and Noah. Elena’s triangle has two additional triangles next to it to compose a rectangle, Lin’s triangle has a copy of the same triangle composed into a parallelogram, and Noah’s triangle shows the top portion of the triangle cut off and moved next to the bottom portion to create a parallelogram.
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Sample responses:
Elena drew two rectangles that decomposed the triangle into two right triangles. She found the area of each right triangle to be half of the area of its enclosing rectangle. This means that the area of the original triangle is the sum of half of the area of the rectangle on the left and half of the rectangle on the right. Half of (4⋅5) plus half of (4⋅2) is 10+4, so the area is 14 square units.
Lin saw it as half of a parallelogram with the base of 7 units and height of 4 units (and thus an area of 28 square units). Half of 28 is 14.
Noah decomposed the triangle by cutting it at half of the triangle’s height, turning the top triangle around, and joining it with the bottom trapezoid to make a parallelogram. He then calculated the area of that parallelogram, which has the same base length but half the height of the triangle. 7⋅2=14, so the area is 14 square units.
For each triangle, identify a base and a corresponding height. Use them to find the area. Show your reasoning.
A
A triangle labeled A. Triangle A has sides of length 7.2, 3, and unknown. The perpendicular length from the side of length 3 to the opposite vertex is 6. The perpendicular length from the side of length 7.2 to the opposite vertex is 2.5. All lengths are in inches.
B
A triangle labeled B. Triangle B has sides of length 5, 6, and 5. The perpendicular length from the side of length 5 to the opposite vertex is 4.8. The perpendicular length from the side of length 6 to the opposite vertex is 4. All lengths are in centimeters.
Here are two five-pointed stars. A student said, “Both figures A and B are polygons. They are both composed of line segments and are two-dimensional. Neither have curves.” Do you agree with the statement? Explain your reasoning.
Here is a five-sided polygon. Describe or show the strategy you would use to find its area. Mark up and label the diagram to show your reasoning so that it can be followed by others. (It is not necessary to actually calculate the area.)
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Disagree. Only Figure B is a polygon. Sample reasoning: Every segment in Figure A meets or crosses more than two segments at its ends, so it is not a polygon. Each segment in Figure B meets only one other segment at each end.
Sample responses:
The polygon can be decomposed into three triangles: one with a base of 6 units and a height of 3, a second one with a base of 7 and a height of 6, and a third with a base of 4 and a height of 6. All areas can be calculated using the area formula.
The polygon can be decomposed into two triangles and a rectangle. One triangle has a base of 6 and a height 3, and the second has a base of 6 and a height of 3. Their areas can be calculated with the area formula. The rectangle is 6 by 4, so its area is the product of 6 and 4.
Section C Check
Section C Checkpoint
Problem 1
Identify a base and a height that you can use to find the area of each triangle. (You don’t have to actually find the areas.)
Label each base with “b.”
Draw a segment for each height and label it with “h.”
Show Solution
Sample response:
Problem 2
Find the area of the triangle. Explain or show your reasoning.
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84 square inches. Sample reasoning: If the side that is 14 inches long is the base, its corresponding height is 12 inches. 21⋅14⋅12=21⋅168=84
Problem 3
Find the area of the shaded polygon in square units. Show your reasoning.
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15 square units. Sample reasoning:
The polygon can be decomposed into two right triangles.
The area of the small triangle is half of a 3-by-2 rectangle, which is 3 square units. The area of the large triangle is half of a 6-by-4 rectangle, which is 12 square units. 3+12=15
Write your best definition or description of a polyhedron. If possible, use the terms you learned in this lesson.
Which of these five polyhedra are prisms? Which are pyramids?
A
B
C
D
E
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Answers might include one or more of these elements: A polyhedron is a three-dimensional figure made from faces that are filled-in polygons. Each face meets one and only one other face along a complete edge. The points where edges meet are called vertices.
net on a grid. 4 adjacent rectangles, from left to right, 4 by 3, 4 by 2, 4 by 3, 4 by 2. Above second rectangle 3 by 2 rectangle. Below fourth rectangle 3 by 2 rectangle.
What kind of polyhedron can be assembled from this net?
Find the surface area (in square units) of the polyhedron. Show your reasoning.
In this net, the two triangles are right triangles. All quadrilaterals are rectangles. What is its surface area in square units? Show your reasoning.
A net of five shapes. Three rectangles in a row with a right triangle above and below the middle rectangle. The left rectangle has sides 5 and 10, the middle rectangle has sides 5 and 8, the right rectangle has sides 5 and 6. Each triangle has sides 8, 10, and 6.
If the net is assembled, which of the following polyhedra would it make?
A
B
C
D
Show Solution
168 square units. Sample reasoning: There are two triangular faces with area of 24 square units each. 21⋅6⋅8=24. There is a rectangular face with area of 50 square units. 10⋅5=50. There is one rectangular face with area of 40 square units. 5⋅8=40. There is one rectangular face with area 5⋅6=30 square units. 2⋅24+50+40+30=168
Choose two figures that have the same surface area but different volumes. Show your reasoning.
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Figures D and E both have a surface area of 26 square units, but D has a volume of 6 cubic units, and E has a volume of 7 cubic units.
Section D Check
Section D Checkpoint
Problem 1
Select all nets that can be assembled into this triangular prism.
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B, D, E
Problem 2
Sketch a net for this square pyramid and label the known lengths.
Find the surface area of the pyramid in square units. Show your reasoning.
Show Solution
Sample response:
72 square units. Sample reasoning: The square base is 4⋅4 or 16 square units. The area of each triangle is 21⋅4⋅7, which is 14 square units. There are 4 triangles, so the surface area is: 16+(4⋅14), which is 16+56 or 72.
