Unit 7 Plan - Grade 7

Title	Takeaways	Student Summary	Assessment
Lesson 1 Relationships of Angles	—	When two lines intersect and form four equal angles, we call each one a right angle. A right angle measures $90^\circ$ . You can think of a right angle as a quarter turn in one direction or the other. An angle in which the two sides form a straight line is called a straight angle. A straight angle measures $180^\circ$ . A straight angle can be made by putting right angles together. You can think of a straight angle as a half turn, so that you are facing in the opposite direction after you are done. If you put two straight angles together, you get an angle that is $360^\circ$ . You can think of this angle as turning all the way around so that you are facing the same direction as when you started the turn. When two angles share a side and a vertex, and they don't overlap, we call them adjacent angles.	Identical Octagons (1 problem) This pattern is composed of a square and some regular octagons. In this pattern, all of the angles inside the octagons have the same measure. The shape in the center is a square. Find the measure of one of the angles inside one of the octagons. Show Solution $135^\circ$ . Sample reasoning: The angles in the square are $90^\circ$ . Since the angles around a point add up to $360^\circ$ , then 2 octagon angles must be $360-90$ , or $270^\circ$ . Since all of the octagon angles are the same, each angle is $270\div2$ or $135^\circ$ .
Lesson 2 Adjacent Angles	—	If two angle measures add up to $90^\circ$ , then we say the angles are complementary. Here are three examples of pairs of complementary angles. If two angle measures add up to $180^\circ$ , then we say the angles are supplementary. Here are three examples of pairs of supplementary angles.	Finding Measurements (1 problem) Point $F$ is on line $CD$ . Find the measure of angle $CFE$ . Angle $SPR$ and angle $RPQ$ are complementary. Find the measure of angle $RPQ$ . Show Solution $28^\circ$ $53^\circ$
Lesson 3 Nonadjacent Angles	—	When two lines cross, they form two pairs of vertical angles. Vertical angles are across the intersection point from each other. Vertical angles always have equal measure. We can see this because they are always supplementary with the same angle. For example: This is always true! $a+b = 180$ so $a = 180-b$ . $c+b = 180$ so $c = 180-b$ . That means $a = c$ .	Finding Angle Pairs (1 problem) Name a pair of complementary angles in the diagram. Name a pair of supplementary angles in the diagram. Name a pair of vertical angles in the diagram. Show Solution $ABC$ with one of $BAC$ , $FAD$ , or $ADG$ One of these pairs: $CAD$ with one of $ABC$ , $DAC$ , or $DAG$ $BAF$ with one of $ABC$ , $DAC$ , or $DAG$ Any 2 of $BCA$ , $ACG$ , and $CGA$ One of these pairs: $DAF$ and $BAC$ $BAF$ and $CAD$
Lesson 4 Solving for Unknown Angles	—	We can write equations that represent relationships between angles. The first pair of angles are supplementary, so $x+42 = 180$ . The second pair of angles are vertical angles, so $y = 28$ . Assuming the third pair of angles form a right angle, they are complementary, so $z + 64 = 90$ .	Missing Circle Angles (1 problem) $AD$ , $BE$ , and $CF$ are all diameters of the circle. The measure of angle $AOB$ is 40 degrees. The measure of angle $DOF$ is 120 degrees. Find the measures of the angles: $BOC$ $COD$ Show Solution Angle $BOC=80^\circ$ . Sample reasoning: Given angle $DOF=120^\circ$ , angle $AOC=120^\circ$ because they are congruent vertical angles. Consequently, angles $AOB+BOC=120^\circ$ because they are adjacent. Angle $COD=60^\circ$ . Sample reasoning: Angle $COD$ and angle $DOF$ are supplementary angles, so the sum of their measurements has to be $180^\circ$ .
Lesson 5 Using Equations to Solve for Unknown Angles	—	To find an unknown angle measure, sometimes it is helpful to write and solve an equation that represents the situation. For example, suppose we want to know the value of $x$ in this diagram. Using what we know about vertical angles, we can write the equation $3x + 90 = 144$ to represent this situation. Then we can solve the equation. $\begin{aligned} 3x + 90 &= 144 \\ 3x + 90 - 90 &= 144 - 90 \\ 3x &= 54 \\ 3x \boldcdot \frac13 &= 54 \boldcdot \frac13 \\ x &= 18 \end{aligned}$	In Words (1 problem) Here are three intersecting lines. Write an equation that represents a relationship between these angles. Describe, in words, the process you would use to find $w$ . Show Solution Samples responses: $2w + 76 = 180$ or $4w + 152 = 360$ . Sample responses: Subtract 76 from 180 and then divide by 2 (or multiply by $\frac12$ ). Subtract 152 from 360 and then divide by 4 (or multiply by $\frac14$ ).
Section A Check Section A Checkpoint

Lesson 12 Volume of Right Prisms	—	Any cross-section of a prism that is parallel to the base will be identical to the base. This means we can slice prisms up to help find their volume. For example, if we have a rectangular prism that is 3 units tall and has a base that is 4 units by 5 units, we can think of this as 3 layers, where each layer has $4\boldcdot 5$ cubic units. The volume of the figure is the number of cubic units that fill a three-dimensional region without any gaps or overlaps. That means the volume of the original rectangular prism is $3(4\boldcdot 5)$ , or 60, cubic units. This works with any prism! If we have a prism with a height of 3 cm that has a base with an area of 20 cm², then the volume is $3\boldcdot 20$ cm³ regardless of the shape of the base. In general, the volume of a prism with height $h$ and area $B$ is $\displaystyle V = B \boldcdot h$ For example, these two prisms both have a volume of 100 cm³.	Octagonal Box (1 problem) A box is shaped like an octagonal prism. Here is what the base of the prism looks like. For each question, make sure to include the unit with your answer and explain or show your reasoning. If the height of the box is 7 inches, what is the volume of the box? If the volume of the box is 123 in³, what is the height of the box? Show Solution 287 in³, because the base has an area of 41 in^2, and $41\boldcdot 7=287$ . 3 in, because $41 \boldcdot 3 = 123$ .
Lesson 13 Decomposing Bases for Area	—	To find the area of any polygon, you can decompose it into rectangles and triangles. There are always many ways to decompose a polygon. Sometimes it is easier to enclose a polygon in a rectangle and subtract the area of the extra pieces. To find the volume of a prism with a polygon for a base, you find the area of the base, $B$ , and multiply that by the height, $h$ . $\displaystyle V = Bh$	Volume of a Pentagonal Prism (1 problem) Here is a prism with a pentagonal base. The height is 8 cm. What is the volume of the prism? Show your thinking. Organize it so it can be followed by others. Show Solution The volume is 232 cm³. The area of the base is 29 cm² and can be found in multiple ways, but one way is to consider a 5 by 7 rectangle with a right triangle cut off, then $5 \boldcdot 7 - \frac{1}{2} \boldcdot 4 \boldcdot 3 = 29$ . Since the height is 8 cm, the volume is calculated by $29 \boldcdot 8 = 232$ .
Lesson 14 Surface Area of Right Prisms	—	To find the surface area of a three-dimensional figure whose faces are made up of polygons, we can find the area of each face, and add them up! Sometimes there are ways to simplify our work. For example, all of the faces of a cube with side length $s$ are the same. We can find the area of one face, and multiply by 6. Since the area of one face of a cube is $s^2$ , the surface area of a cube is $6s^2$ . We can use this technique to make it faster to find the surface area of any figure that has faces that are the same. For prisms, there is another way. We can treat the prism as having three parts: two identical bases, and one long rectangle that has been taped along the edges of the bases. The rectangle has the same height as the prism, and its length is the perimeter of the base. To find the surface area, add the area of this rectangle to the areas of the two bases.	Surface Area of a Hexagonal Prism (1 problem) Find the surface area of this prism. Show your reasoning. Organize your explanation so it can be followed by others. Show Solution The surface area is 270 cm². Possible strategy: The area of the base is 27 cm². The perimeter of the base is 24 cm, so the combined area of the sides is 216 cm², because $24 \boldcdot 9=216$ . Therefore the total surface area is 270 cm², because $27 \boldcdot 2 + 216=270$ .
Lesson 15 Distinguishing Volume and Surface Area	—	Sometimes we need to find the volume of a prism, and sometimes we need to find the surface area. Here are some examples of quantities related to volume: How much water a container can hold How much material it took to build a solid object Volume is measured in cubic units, like in³ or m³. Here are some examples of quantities related to surface area: How much fabric is needed to cover a surface How much of an object needs to be painted Surface area is measured in square units, like in² or m².	Surface Area Differences (1 problem) Describe some similarities and differences between a situation that involves calculating surface area and a situation that involves calculating volume. Show Solution Sample response: Volume refers to how much of something fits inside an object. Surface area refers to how much of something is needed to cover the outside of an object.
Lesson 16 Applying Volume and Surface Area	—	Suppose we wanted to make a concrete bench like the one shown in this picture. If we know that the finished bench has a volume of 10 ft³ and a surface area of 44 ft², we can use this information to solve problems about the bench. For example, How much does the bench weigh? How long does it take to wipe the whole bench clean? How much will the materials cost to build the bench and to paint it? To figure out how much the bench weighs, we can use its volume, 10 ft³. Concrete weighs about 150 pounds per cubic foot, so this bench weighs about 1,500 pounds, because $10 \boldcdot 150 = 1,500$ . To figure out how long it takes to wipe the bench clean, we can use its surface area, 44 ft². If it takes a person about 2 seconds per square foot to wipe a surface clean, then it would take about 88 seconds to clean this bench, because $44 \boldcdot 2 = 88$ . It may take a little less than 88 seconds, since the surfaces where the bench is touching the ground do not need to be wiped. Would you use the volume or the surface area of the bench to calculate the cost of the concrete needed to build this bench? And for the cost of the paint?	Preparing for the Play (1 problem) Andre is preparing for the school play. He needs to paint a cardboard box to look like a dresser. The box is a rectangular prism that measures 5 feet tall, 4 feet long, and $2\frac12$ feet wide. Andre does not need to paint the bottom of the box. How much cardboard does Andre need to paint? If one bottle of paint covers an area of 40 square feet, how many bottles of paint does Andre need to buy for this project? Show Solution 75 square feet. $(2.5 \boldcdot4)+2(5 \boldcdot4)+2(2.5 \boldcdot5)=75$ 2 bottles of paint. $\frac {75}{40}=1.875$
Section C Check Section C Checkpoint
Unit 7 Assessment End-of-Unit Assessment

—

When two lines intersect and form four equal angles, we call each one a right angle. A right angle measures $90^\circ$ . You can think of a right angle as a quarter turn in one direction or the other.

Two lines meet to form four angles. Each angle is marked 90 degrees, and with the symbol for perpendicular lines.

An angle in which the two sides form a straight line is called a straight angle. A straight angle measures $180^\circ$ . A straight angle can be made by putting right angles together. You can think of a straight angle as a half turn, so that you are facing in the opposite direction after you are done.

A straight line is indicated to be an angle with measure 180 degrees.

If you put two straight angles together, you get an angle that is $360^\circ$ . You can think of this angle as turning all the way around so that you are facing the same direction as when you started the turn.

A circle from a segment, around one endpoint, indicates that the measure is 360 degrees.

When two angles share a side and a vertex, and they don't overlap, we call them adjacent angles.

Identical Octagons (1 problem)

This pattern is composed of a square and some regular octagons.

In this pattern, all of the angles inside the octagons have the same measure. The shape in the center is a square. Find the measure of one of the angles inside one of the octagons.

A diagram composed of 4 octagons with the same measure. They are arranged so that they each touch two others and form a center in the shape of a square.

Show Solution

135^\circ

. Sample reasoning: The angles in the square are

90^\circ

. Since the angles around a point add up to

360^\circ

, then 2 octagon angles must be

360-90

, or

270^\circ

. Since all of the octagon angles are the same, each angle is

270\div2

or

135^\circ

.

Lesson 2

Adjacent Angles

—

If two angle measures add up to $90^\circ$ , then we say the angles are complementary. Here are three examples of pairs of complementary angles.

Three images. First, adjacent angles, 30 degrees, 60 degrees. Second, non-adjacent angels formed by two lines, 45 degrees. Third, a triangle, angles 90 degrees, 38 degrees, 52 degrees.

If two angle measures add up to $180^\circ$ , then we say the angles are supplementary. Here are three examples of pairs of supplementary angles.

Three images. First, adjacent angles, 55 degrees, 125 degrees. Second, perpendicular lines, non-adjacent angles marked. Third, distinct angles, 152 degrees, 28 degrees.

Finding Measurements (1 problem)

Point $F$ is on line $CD$ . Find the measure of angle $CFE$ .
Angle $SPR$ and angle $RPQ$ are complementary. Find the measure of angle $RPQ$ .

Show Solution

$28^\circ$
$53^\circ$

Lesson 3

Nonadjacent Angles

—

When two lines cross, they form two pairs of vertical angles. Vertical angles are across the intersection point from each other.

Two lines cross, with the 4 angles formed marked.

Vertical angles always have equal measure. We can see this because they are always supplementary with the same angle. For example:

Two images. Both images intersecting lines, an obtuse angle, 150 degrees. One image, the angle adjacent counter-clockwise is 30 degrees, the other image, the angle adjacent clockwise is 30 degrees.

This is always true!

Two images. Both images intersecting lines, an obtuse angle, b. One image, the angle adjacent counter-clockwise is a, degrees, the other image, the angle adjacent clockwise is c degrees.

$a+b = 180$ so $a = 180-b$ .

$c+b = 180$ so $c = 180-b$ .

That means $a = c$ .

Finding Angle Pairs (1 problem)

Name a pair of complementary angles in the diagram.
Name a pair of supplementary angles in the diagram.
Name a pair of vertical angles in the diagram.

Show Solution

$ABC$ with one of $BAC$ , $FAD$ , or $ADG$
One of these pairs:
1. $CAD$ with one of $ABC$ , $DAC$ , or $DAG$
2. $BAF$ with one of $ABC$ , $DAC$ , or $DAG$
3. Any 2 of $BCA$ , $ACG$ , and $CGA$
One of these pairs:
1. $DAF$ and $BAC$
2. $BAF$ and $CAD$

Lesson 4

Solving for Unknown Angles

—

We can write equations that represent relationships between angles.

Three images. First, adjacent angles, x degrees, 42 degrees, form a straight line. Second, vertical angles, 28 degrees, y degrees. Third, complementary angles, z degrees, 64 degrees.

The first pair of angles are supplementary, so $x+42 = 180$ .
The second pair of angles are vertical angles, so $y = 28$ .
Assuming the third pair of angles form a right angle, they are complementary, so $z + 64 = 90$ .

Missing Circle Angles (1 problem)

$AD$ , $BE$ , and $CF$ are all diameters of the circle. The measure of angle $AOB$ is 40 degrees. The measure of angle $DOF$ is 120 degrees.

A circle, points on the circumference labeled A, B, C, D, E, F in that order, center O. Angle AOB measures 40 degrees.

Find the measures of the angles:

$BOC$
$COD$

Show Solution

Angle $BOC=80^\circ$ . Sample reasoning: Given angle $DOF=120^\circ$ , angle $AOC=120^\circ$ because they are congruent vertical angles. Consequently, angles $AOB+BOC=120^\circ$ because they are adjacent.
Angle $COD=60^\circ$ . Sample reasoning: Angle $COD$ and angle $DOF$ are supplementary angles, so the sum of their measurements has to be $180^\circ$ .

Lesson 5

Using Equations to Solve for Unknown Angles

—

To find an unknown angle measure, sometimes it is helpful to write and solve an equation that represents the situation. For example, suppose we want to know the value of $x$ in this diagram.

Two lines meet to form 4 angles. An angle is labeled 144 degrees. It's vertical angle is split into 4 smaller angles, x degrees, x degrees, x degrees, 90 degrees.

Using what we know about vertical angles, we can write the equation $3x + 90 = 144$ to represent this situation. Then we can solve the equation.

$\begin{aligned} 3x + 90 &= 144 \\ 3x + 90 - 90 &= 144 - 90 \\ 3x &= 54 \\ 3x \boldcdot \frac13 &= 54 \boldcdot \frac13 \\ x &= 18 \end{aligned}$

In Words (1 problem)

Here are three intersecting lines.

Three lines intersect at a single point. The angles measures, moving clockwise, are labeled w, blank, 76 degrees, blank, w blank.

Write an equation that represents a relationship between these angles.
Describe, in words, the process you would use to find $w$ .

Show Solution

Samples responses: $2w + 76 = 180$ or $4w + 152 = 360$ .
Sample responses:
- Subtract 76 from 180 and then divide by 2 (or multiply by $\frac12$ ).
- Subtract 152 from 360 and then divide by 4 (or multiply by $\frac14$ ).

Section A Check

Section A Checkpoint

Lesson 12

Volume of Right Prisms

—

Any cross-section of a prism that is parallel to the base will be identical to the base. This means we can slice prisms up to help find their volume. For example, if we have a rectangular prism that is 3 units tall and has a base that is 4 units by 5 units, we can think of this as 3 layers, where each layer has $4\boldcdot 5$ cubic units. The volume of the figure is the number of cubic units that fill a three-dimensional region without any gaps or overlaps.

Two images. First, a prism made of cubes stacked 5 wide, 4 deep, 3 tall. Second, each of the layers of the prism is separated to show 3 prisms 5 wide, 4 deep, 1 tall.

That means the volume of the original rectangular prism is $3(4\boldcdot 5)$ , or 60, cubic units.

This works with any prism! If we have a prism with a height of 3 cm that has a base with an area of 20 cm², then the volume is $3\boldcdot 20$ cm³ regardless of the shape of the base. In general, the volume of a prism with height $h$ and area $B$ is

$\displaystyle V = B \boldcdot h$

For example, these two prisms both have a volume of 100 cm³.

Prism with triangular base, area 20 centimeters squared, and height 5 centimeters.

Prism with irregular base, area 25 centimeters squared, and height 4 centimeters.

Octagonal Box (1 problem)

A box is shaped like an octagonal prism. Here is what the base of the prism looks like.

An octagon, inch grid. From the first vertex, move right 3 to the next vertex, then down 2, right 2, then down 3, then down 2, left 2, then left 3, then up 2, left 2, then up 3, then up 2, right 2.

For each question, make sure to include the unit with your answer and explain or show your reasoning.

If the height of the box is 7 inches, what is the volume of the box?
If the volume of the box is 123 in³, what is the height of the box?

Show Solution

287 in³, because the base has an area of 41 in^2, and $41\boldcdot 7=287$ .
3 in, because $41 \boldcdot 3 = 123$ .

Lesson 13

Decomposing Bases for Area

—

To find the area of any polygon, you can decompose it into rectangles and triangles. There are always many ways to decompose a polygon.

Four images of the same irregular polygon. In two images, the polygon is cut into different triangles and rectangles. In the fourth image, a triangle is added to make the polygon a rectangle.

Sometimes it is easier to enclose a polygon in a rectangle and subtract the area of the extra pieces.

To find the volume of a prism with a polygon for a base, you find the area of the base, $B$ , and multiply that by the height, $h$ .

A prism. The base of the prism is the irregular polygon from the previous images, area B, and the prism has height h.

$\displaystyle V = Bh$

Volume of a Pentagonal Prism (1 problem)

Here is a prism with a pentagonal base. The height is 8 cm.

What is the volume of the prism? Show your thinking. Organize it so it can be followed by others.

A prism with a pentagon base, all lengths centimeters. The pentagon is a rectangle 5 tall by 7 wide, with a triangle 3 tall by 4 wide removed from the corner. The prism has height 8.

Show Solution

The volume is 232 cm³. The area of the base is 29 cm² and can be found in multiple ways, but one way is to consider a 5 by 7 rectangle with a right triangle cut off, then $5 \boldcdot 7 - \frac{1}{2} \boldcdot 4 \boldcdot 3 = 29$ . Since the height is 8 cm, the volume is calculated by $29 \boldcdot 8 = 232$ .

Lesson 14

Surface Area of Right Prisms

—

To find the surface area of a three-dimensional figure whose faces are made up of polygons, we can find the area of each face, and add them up!

Sometimes there are ways to simplify our work. For example, all of the faces of a cube with side length $s$ are the same. We can find the area of one face, and multiply by 6. Since the area of one face of a cube is $s^2$ , the surface area of a cube is $6s^2$ .

We can use this technique to make it faster to find the surface area of any figure that has faces that are the same.

For prisms, there is another way. We can treat the prism as having three parts: two identical bases, and one long rectangle that has been taped along the edges of the bases. The rectangle has the same height as the prism, and its length is the perimeter of the base. To find the surface area, add the area of this rectangle to the areas of the two bases.

Surface Area of a Hexagonal Prism (1 problem)

Find the surface area of this prism. Show your reasoning. Organize your explanation so it can be followed by others.

A prism, all dimensions centimeters. The base of the prism is a 6 by 6 rectangle with a 3 by 3 square removed from the corner. The prism is 9 centimeters high.

Show Solution

The surface area is 270 cm². Possible strategy: The area of the base is 27 cm². The perimeter of the base is 24 cm, so the combined area of the sides is 216 cm², because $24 \boldcdot 9=216$ . Therefore the total surface area is 270 cm², because $27 \boldcdot 2 + 216=270$ .

Lesson 15

Distinguishing Volume and Surface Area

—

Sometimes we need to find the volume of a prism, and sometimes we need to find the surface area.

Here are some examples of quantities related to volume:

How much water a container can hold
How much material it took to build a solid object

Volume is measured in cubic units, like in³ or m³.

Here are some examples of quantities related to surface area:

How much fabric is needed to cover a surface
How much of an object needs to be painted

Surface area is measured in square units, like in² or m².

Surface Area Differences (1 problem)

Describe some similarities and differences between a situation that involves calculating surface area and a situation that involves calculating volume.

Show Solution

Sample response: Volume refers to how much of something fits inside an object. Surface area refers to how much of something is needed to cover the outside of an object.

Lesson 16

Applying Volume and Surface Area

—

Suppose we wanted to make a concrete bench like the one shown in this picture. If we know that the finished bench has a volume of 10 ft³ and a surface area of 44 ft², we can use this information to solve problems about the bench.

<p>A concrete bench formed by two rectangular prism supports, each the same width as the rectangular prism bench seat.</p>

For example,

How much does the bench weigh?
How long does it take to wipe the whole bench clean?
How much will the materials cost to build the bench and to paint it?

To figure out how much the bench weighs, we can use its volume, 10 ft³. Concrete weighs about 150 pounds per cubic foot, so this bench weighs about 1,500 pounds, because $10 \boldcdot 150 = 1,500$ .

To figure out how long it takes to wipe the bench clean, we can use its surface area, 44 ft². If it takes a person about 2 seconds per square foot to wipe a surface clean, then it would take about 88 seconds to clean this bench, because $44 \boldcdot 2 = 88$ . It may take a little less than 88 seconds, since the surfaces where the bench is touching the ground do not need to be wiped.

Would you use the volume or the surface area of the bench to calculate the cost of the concrete needed to build this bench? And for the cost of the paint?

Preparing for the Play (1 problem)

Andre is preparing for the school play. He needs to paint a cardboard box to look like a dresser. The box is a rectangular prism that measures 5 feet tall, 4 feet long, and $2\frac12$ feet wide. Andre does not need to paint the bottom of the box.

How much cardboard does Andre need to paint?
If one bottle of paint covers an area of 40 square feet, how many bottles of paint does Andre need to buy for this project?

Show Solution

75 square feet. $(2.5 \boldcdot4)+2(5 \boldcdot4)+2(2.5 \boldcdot5)=75$
2 bottles of paint. $\frac {75}{40}=1.875$

Section C Check

Section C Checkpoint

Unit 7 Angles Triangles And Prisms — Unit Plan