nys common core mathematics curriculum 6 student book2.pdfnys common core mathematics curriculum 2...

NYS COMMON CORE MATHEMATICS CURRICULUM 7•6 Lesson 1

Lesson 1: Complementary and Supplementary Angles S.1

This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from G7-M6-TE-1.3.0-10.2015

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Lesson 1: Complementary and Supplementary Angles

Classwork

Opening Exercise

As we begin our study of unknown angles, let us review key definitions.

Term Definition

Two angles, ∠𝐴𝑂𝐶 and ∠𝐶𝑂𝐵, with a common side 𝑂𝐶⃑⃑⃑⃑ ⃑, are angles if 𝐶 is in the interior of ∠𝐴𝑂𝐵.

When two lines intersect, any two non-adjacent angles formed by those lines are called angles, or angles.

Two lines are if they intersect in one point, and any of the angles formed by the intersection of the lines is 90°. Two segments or rays are if the lines containing them are lines.

Complete the missing information in the table below. In the Statement column, use the illustration to write an equation that demonstrates the angle relationship; use all forms of angle notation in your equations.

Angle

Relationship Abbreviation Statement Illustration

Adjacent Angles

The measurements of adjacent angles add.

Vertical Angles

Vertical angles have equal measures.

a°b°

B

A

C

D

c°

b°a°

E

D

F

G

C

http://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_US








60°120°

Angles on a Line

If the vertex of a ray lies on a line but the ray is not contained in that line, then the sum of measurements of the two angles formed is 180°.

Angles at a Point

Suppose three or more rays with the same vertex separate the plane into angles with disjointed interiors. Then, the sum of all the measurements of the angles is 360°.

Angle

Relationship Definition Diagram

Complementary Angles

Supplementary Angles

C

A B D

a° b°

a°b°

B

A

C

D

c°

60°

30°

60°

30°

60°120°









Exercise 1

1. In a complete sentence, describe the relevant anglerelationships in the diagram. Write an equation for theangle relationship shown in the figure and solve for 𝑥.Confirm your answers by measuring the angle with aprotractor.

Example 1

The measures of two supplementary angles are in the ratio of 2: 3. Find the measurements of the two angles.

Exercises 2–4

2. In a pair of complementary angles, the measurement of the larger angle is three times that of thesmaller angle. Find the measurements of the two angles.

22°x°









3. The measure of a supplement of an angle is 6° more than twice the measure of the angle. Find themeasurement of the two angles.

4. The measure of a complement of an angle is 32° more than three times the angle. Find themeasurement of the two angles.

Example 2

Two lines meet at a point that is also the vertex of an angle. Set up and solve an appropriate equation for𝑥 and 𝑦.

16°

x°

y°








Lesson Summary

Supplementary angles are two (or more) angles whose sum is 180°. Complementary angles are two (or more) angles whose sum is 90°. Once an angle relationship is identified, the relationship can be modeled with an equation

that will find an unknown value. The unknown value may be used to find the measure ofthe unknown angle.

1. Two lines meet at a point that is also the endpoint of a ray. Set up and solve the appropriateequations to determine 𝑥 and 𝑦.

Name _______________________________________________ Class ______________

Lesson 1 Problem Set





oacs

Highlight

oacs

Highlight

oacs

Highlight

oacs

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2. Two lines meet at a point that is also the vertex of anangle. Set up and solve an appropriate equation for𝑥 and 𝑦.

28°

x° y°

3. Two lines meet at a point that is also the vertex of an angle.Set up and solve the appropriate equations to determine𝑥 and 𝑦. 32°

x°

y°









5. Two lines meet at a point that is also the endpoint of two rays. Set up and solve theappropriate equations for 𝑚 and 𝑛.

6. The supplement of the measurement of an angle is 16° less than three times the angle. Findthe measurement of the angle and its supplement.

4. Set up and solve the appropriate equations for 𝑠 and 𝑡.





7. The ratio of the measurement of an angle to its complement is 1: 2. Find the measurementof the angle and its complement.

8. The ratio of the measurement of an angle to its supplement is 3: 5. Find the measurementof the angle and its supplement.

9. Let 𝑥 represent the measurement of an acute angle in degrees. The ratio of the complement of𝑥 to the supplement of 𝑥 is 2: 5. Guess and check to determine the value of 𝑥. Explain why youranswer is correct.


Lesson 2: Solving for Unknown Angles Using Equations S.8



Lesson 2: Solving for Unknown Angles Using Equations

Classwork

Opening Exercise

Two lines meet at a point. In a complete sentence, describe the relevant angle relationships in the diagram. Find the values of 𝑟, 𝑠, and 𝑡.

Example 1

Two lines meet at a point that is also the endpoint of a ray. In a complete sentence, describe the relevant angle relationships in the diagram. Set up and solve an equation to find the value of 𝑝 and 𝑟.

25°r°

s° t°

16°

p°

r°

28°









Exercise 1

Three lines meet at a point. In a complete sentence, describe the relevant angle relationship in

the diagram. Set up and solve an equation to find the value of 𝑎.

Example 2

Three lines meet at a point. In a complete sentence, describe the relevant angle relationships in the diagram. Set up and solve an

equation to find the value of 𝑧.

19°

z°









Exercise 2

Three lines meet at a point; ∠𝐴𝑂𝐹 = 144°. In a complete sentence, describe the relevant angle relationships in the diagram. Set up and solve an equation to determine the value of 𝑐.

Example 3

Two lines meet at a point that is also the endpoint of a ray. The ray is perpendicular to one of the lines as shown. In a complete sentence, describe the relevant angle relationships in the diagram. Set up and solve an equation to find the value of 𝑡.

144°

c°

A B

C

D

E

F

O

26°

t°









Exercise 3

Two lines meet at a point that is also the endpoint of a ray. The ray is perpendicular to one of the lines as shown. In a complete sentence, describe the relevant angle relationships in the diagram. You may add labels to the diagram to help with your description of the angle relationship. Set up and solve an equation to find the value of 𝑣.

Example 4

Three lines meet at a point. In a complete sentence, describe the relevant angle relationships in the diagram. Set up and solve an equation to find the value of 𝑥. Is your answer reasonable? Explain how you know.

46°

v°

x°

8°









Exercise 4

Two lines meet at a point that is also the endpoint of two rays. In a complete sentence, describe the relevant angle relationships in the diagram. Set up and solve an equation to find the value of 𝑥. Find the measurements of ∠𝐴𝑂𝐵 and ∠𝐵𝑂𝐶.

Exercise 5

a. In a complete sentence, describe the relevant angle relationships in the diagram. Set up andsolve an equation to find the value of 𝑥. Find the measurements of ∠𝐴𝑂𝐵 and∠𝐵𝑂𝐶.

25°

2x°

3x°

A O

B

C

5x°

(2x+20)°

AO

C

B









1. Two lines meet at a point that is also the endpoint of aray. Set up and solve an equation to find the value of 𝑐.

2. Two lines meet at a point that is also the endpoint of a ray.Set up and solve an equation to find the value of 𝑎.Explain why your answer is reasonable.

c°17°

33°a°

49°

Lesson Summary

Angles on a line and supplementary angles are not the same relationship. Supplementary angles are twoangles whose angle measures sum to 180° whereas angles on a line are two or more adjacent angleswhose angle measures sum to 180°.

Name _______________________________________________ Class ______________









4. Two lines meet at a point that is also the vertex of an angle. Set upand solve an equation to find the value of 𝑚.

68°

m°

24°

3. Two lines meet at a point that is also the endpoint of a ray.Set up and solve an equation to find the value of 𝑤.









5. Three lines meet at a point. Set up and solve an equation to find the value of 𝑟.

6. Three lines meet at a point that is also the endpoint of a ray. Setup and solve an equation to find the value of each variable in thediagram.

7. Set up and solve an equation to find the value of 𝑥. Find the measurement of ∠𝐴𝑂𝐵 and of ∠𝐵𝑂𝐶.

34°

122°r°

51° 43°

x°y°

z°

w°

v°







10. Write an equation to solve for x, then solve for x and the angles.

(4x+5)° (5x+22)°

A O C

B

2x°10x°

(x + 18)°

jumoore

Stamp






Classwork

Opening Exercise

Two lines meet at a point that is also the vertex of an angle; the measurement of ∠𝐴𝑂𝐹 is 134°. Set up and solve an equation to find the values of 𝑥 and 𝑦. Are your answers reasonable? How do you know?

Example 1

Set up and solve an equation to find the value of 𝑥.

x°

14°

134° y°

A

B

C

D

E

F

O

x°

123°









Example 2

Four rays meet at a common endpoint. In a complete sentence, describe the relevant angle relationships in the diagram. Set up and solve an equation to find the value of 𝑥. Find the measurements of ∠𝐵𝐴𝐶 and ∠𝐷𝐴𝐸.

Exercise 2

Four rays meet at a common endpoint. In a complete sentence, describe the relevant angle relationships

in the diagram. Set up and solve an equation to find the value of 𝑥. Find the measurement of ∠𝐶𝐴𝐷.

5x°

x°

204°

A

B

C

D

E

3x°

12x°

60°

AB

C

D

E









Example 3

Two lines meet at a point that is also the endpoint of two rays. In a complete sentence, describe the relevant angle relationships in the diagram. Set up and solve an equation to find the value of 𝑥. Find the measurements of ∠𝐵𝐴𝐶 and ∠𝐵𝐴𝐻.

Exercise 3

Lines 𝐴𝐵 and 𝐸𝐹 meet at a point which is also the endpoint of two rays. In a complete sentence, describe the relevant angle relationships in the diagram. Set up and solve an equation to find the value of 𝑥. Find the measurements of ∠𝐷𝐻𝐹 and ∠𝐴𝐻𝐷.

5x°

3x°

30°

H

BC

D

E

F

A









Example 4

Two lines meet at a point. Set up and solve an equation to find the value of 𝑥. Find the measurement of one of the vertical angles.

Exercise 4

Set up and solve an equation to find the value of 𝑥. Find the measurement of one of the vertical angles.

3x° (x+30)°









1. Two lines meet at a point. Set up and solve an equation to find the value of 𝑥.

2. Three lines meet at a point. Set up and solve an equation to find the value of 𝑎. Is your answerreasonable?Explain how you know.

Lesson Summary

Steps to Solving for Unknown Angles

Identify the angle relationship(s). Set up an equation that will yield the unknown value. Solve the equation for the unknown value. Substitute the answer to determine the angle(s).

72° (x+15)°

Name _______________________________________________ Class ______________









4. Three lines meet at a point that is also the endpoint of a ray. Set up and solve an equation tofind the values of 𝑥 and 𝑦.

5. Two lines meet at a point. Find the measurement of one of the vertical angles. Is your answerreasonable?Explain how you know.

3x°

(x+104)°

39°

x°y°

3. Two lines meet at a point that is also the endpoint of two rays. Set up and solve an equation tofind the values of 𝑎 and 𝑏.

32°

a°

b°





6.

7. Three adjacent angles are at a point. The second angle is 20° more than the first, and thethird angle is 20° more than the second angle.

a. Find the measurements of all three angles.b. Compare the expressions you used for the three angles and their combined expression.

Explain how they are equal and how they reveal different information about this situation.

y°

Three lines meet at a point that is also the endpoint of a ray. Set up and solve an equation to find the valueof 𝑦. 15°

Not drawn to scale





9. Four lines meet at a point. Solve for 𝑥 and 𝑦 in the following diagram.

8. Three adjacent angles are at a point. The ratio of the measurement of the second angle to themeasurement of the first angle is 4: 3. The ratio of the measurement of the third angle to themeasurement of the second angle is 5: 4. Determine the measurements of all three angles.

Not drawn to scale









3x°

(x+64)°


Classwork

Opening Exercise

The complement of an angle is four times the measurement of the angle. Find the measurement of the angle and its complement.

Example 1

Find the measurements of ∠𝐹𝐴𝐸 and ∠𝐶𝐴𝐷.

Two lines meet at a point. List the relevant angle relationship in the diagram. Set up and solve an equation to find the value of 𝑥. Find the measurement of one of the vertical angles.









Exercise 1

Set up and solve an equation to find the value of 𝑥. List the relevant angle relationship in the diagram.

Find the measurement of one of the vertical angles.

Example 2

Three lines meet at a point. List the relevant angle relationships in the diagram. Set up and solve an equation to find the value of 𝑏.

Exercise 2

Two lines meet at a point that is also the endpoint of two rays. List the relevant angle relationships in the

diagram. Set up and solve an equation to find the value of 𝑏.

37°

b°

43°

5x°

(x+132)°

95°

80°b°









Example 3

2

3 the measurement of its supplement. Find the measurements of the angle and its The measurement of an angle is

supplement.

Exercise 3

1

4 the measurement of its complement. Find the measurements of the two The measurement of an angle is

complementary angles.









Example 4

Three lines meet at a point that is also the endpoint of a ray. List the relevant angle relationships in the diagram. Set up and solve an equation to find the value of 𝒛.

Exercise 4

Two lines meet at a point that is also the vertex of an angle. Set up and solve an equation to find the value

of 𝑥. Find the measurements of ∠𝐺𝐴𝐹 and ∠𝐵𝐴𝐶.

z°

29°

5x°

36°

4x°

B

AC D

E

FG









1. Four rays have a common endpoint on a line. Set up and solve an equation to find the value of 𝑐.

2. Lines 𝐵𝐶 and 𝐸𝐹 meet at 𝐴. Set up and solve an equation to find the value of 𝑥. Find the measurements of∠𝐸𝐴𝐻 and ∠𝐻𝐴𝐶.

H

B

C

D

E

F

A

3x°

4x°

57°

Lesson Summary

Steps to Solving for Unknown Angles

Identify the angle relationship(s). Set up an equation that will yield the unknown value. Solve the equation for the unknown value. Substitute the answer to determine the measurement of the angle(s). Check and verify your answer by measuring the angle with a protractor.

59°

140°

c°

d°

Name _______________________________________________ Class ______________









3. Five rays share a common endpoint. Set up and solve an equation to find the value of 𝑥. Find themeasurements of ∠𝐷𝐴𝐺 and ∠𝐺𝐴𝐻.

4. Four lines meet at a point which is also the endpoint of three rays. Set up and solve an equation tofind the values of 𝑥 and 𝑦.





NYS COMMON CORE MATHEMATICS CURRICULUM




Two lines meet at a point. Set up and solve an equation to find the value of 𝑥. Find themeasurement of one of the vertical angles.

4x°

(x+117)°

5.






Lesson 20: Real-World Area Problems S.122



Lesson 20: Real-World Area Problems

Classwork

Opening Exercise

Find the area of each shape based on the provided measurements. Explain how you found each area.






S.123

Example 1

A landscape company wants to plant lawn seed. A 20 lb. bag of lawn seed will cover up to 420 sq. ft. of grass and costs $49.98 plus the 8% sales tax. A scale drawing of a rectangular yard is given. The length of the longest side is 100 ft. The house, driveway, sidewalk, garden areas, and utility pad are shaded. The unshaded area has been prepared for planting grass. How many 20 lb. bags of lawn seed should be ordered, and what is the cost?

Exercise 1

A landscape contractor looks at a scale drawing of a yard and estimates that the area of the home and garage is the same as the area of a rectangle that is 100 ft.× 35 ft. The contractor comes up with 5,500 ft2. How close is this estimate?









Example 2

Ten dartboard targets are being painted as shown in the following figure. The radius of the smallest circle is 3 in., and each successive larger circle is 3 in. more in radius than the circle before it. A can of red paint and a can of white paint is purchased to paint the target. Each 8 oz. can of paint covers 16 ft2. Is there enough paint of each color to create all tentargets?









1. A stop sign is a regular octagon (i.e., a polygon with eight equal sides and eight equal angles).The dimensions of the octagon are given. One side of the stop sign is to be painted red. IfTimmy has enough paint to cover 500 ft2, can he paint 100 stop signs? Explain your answer.

Lesson Summary

One strategy to use when solving area problems with real-world context is to decompose drawings into familiar polygons and circular regions while identifying all relevant measurements.

Since the area problems involve real-world context, it is important to pay attention to the units needed in each response.

Name _______________________________________________ Class ______________





2. A farmer has four pieces of unfenced land as shown to the right in the scale drawing where thedimensions of one side are given. The farmer trades all of the land and $10,000 for 8 acres ofsimilar land that is fenced. If one acre is equal to 43,560 ft2, how much per square foot for theextra land did the farmer pay. (round your answer to the nearest cent)

3. An ordinance was passed that required farmers to put a fence around their property. The leastexpensive fences cost $10 for each foot. Did the farmer save money by moving the farm?





4. The Smith family is renovating a few aspects of their home. The following diagram is of a newkitchen countertop. Approximately how many square feet of counter space is there?





5. In addition to the kitchen renovation, the Smiths are laying down new carpet. Everything butclosets, bathrooms, and the kitchen will have new carpet. How much carpeting must bepurchased for the home?

6. Jamie wants to wrap a rectangular sheet of paper completely around cans that are in. high and 4 in. in diameter. She can buy a roll of paper that is in. wide and 60 ft. long. How many cans will this muchpaper wrap?


Lesson 25: Volume of Right Prisms S.158



Lesson 25: Volume of Right Prisms

Classwork

Opening Exercise

On the next page, cut out the copy of the triangle below. Also cut along the dotted lines.(The vertical line is the altitude, and the horizontal line joins the midpoints of the two sides of the triangle.) Arrange the four pieces so that they fit together to form a rectangle.

If a prism were formed out of each shape, the original triangle, and your newly rearranged rectangle, and both prisms had the same height, would they have the same volume? Discuss with a partner.









𝟖 𝐜𝐦

𝟖 𝐜𝐦

𝟒.𝟓 𝐜𝐦

𝟖 𝐜𝐦

𝟖 𝐜𝐦

𝟗 𝐜𝐦

Exercise 1

a. Show that the following figures have equal volumes.

b. How can it be shown that the prisms will have equal volumes without completing the entire calculation?

Example 1

Calculate the volume of the following prism.









Example 2

A container is shaped like a right pentagonal prism with an open top. When a cubic foot of water is dumped into the container, the depth of the water is 8 inches. Find the area of the pentagonal base.

Example 3

Two containers are shaped like right triangular prisms, each with the same height. The base area of the larger container is 200% more than the base area of the smaller container. How many times must the smaller container be filled with water and poured into the larger container in order to fill the larger container?









Exercise 2

Two aquariums are shaped like right rectangular prisms. The ratio of the dimensions of the larger

aquarium to the dimensions of the smaller aquarium is 3: 2.

Addie says the larger aquarium holds 50% more water than the smaller aquarium.

Berry says that the larger aquarium holds 150% more water.

Cathy says that the larger aquarium holds over 200% more water.

Are any of the girls correct? Explain your reasoning.





S.162


1. The pieces in Figure 1 are rearranged and put together to form Figure 2.

a. Use the information in Figure 1 to determine the volume of the prism.

b. Use the information in Figure 2 to determine the volume of the prism.

c. If we were not told that the pieces of Figure 1 were rearranged to create Figure 2, would it bepossible to determine whether the volumes of the prisms were equal without completing theentire calculation for each?

2. A right prism container with the base area of 4 ft2 and height of 5 ft. is filled with water until it is 3 ft.deep. If a solid cube with edge length 1 ft. is dropped to the bottom of the container, how much willthe water rise?

Lesson Summary

The formula for the volume of a prism is 𝑉 = 𝐵ℎ, where 𝐵 is the area of the base of the prismand ℎ is the height of the prism.

A base that is neither a rectangle nor a triangle must be decomposed into rectangles andtriangles in order to find the area of the base.

Figure 1 Figure 2

Name _______________________________________________ Class ______________









5. A milliliter (mL) has a volume of 1 cm3. A 250 mL measuring cup is filled to 200 mL. A small stone isplaced in the measuring cup. The stone is completely submerged, and the water level rises to 250 mL.

a. What is the volume of the stone in cm3?

b. Describe a right rectangular prism that has the same volume as the stone.

3. A right prism container with a base area of 10 ft2 and height 9 ft. is filled with water until it is 6 ft.deep. A large boulder is dropped to the bottom of the container, and the water rises to the top,completely submerging the boulder without causing overflow. Find the volume of the boulder.

4. A right prism container with a base area of 8 ft2 and height 6 ft. is filled with water until it is 5 ft. deep.A solid cube is dropped to the bottom of the container, and the water rises to the top. Find the lengthof the cube.






Lesson 26: Volume of Composite Three-Dimensional Objects S.164



Lesson 26: Volume of Composite Three-Dimensional Objects

Classwork

Example 1

Find the volume of the following three-dimensional object composed of two right rectangular prisms.

Exercise 1

Find the volume of the following three-dimensional figure composed of two right rectangular prisms.









Exercise 2

The right trapezoidal prism is composed of a right rectangular prism joined with a right triangular

prism. Find the volume of the right trapezoidal prism.









Example 3

A box with a length of 2 ft., a width of 1.5 ft., and a height of 1.25 ft. contains fragile electronic equipment that is packed inside a larger box with three inches of styrofoam cushioning material on each side (above, below, left side, right side, front, and back).

a. Give the dimensions of the larger box.

b. Find the volume of the styrofoam cushioning material by computing the difference betweenthe volume of the larger box and the volume of the smaller box.









1. Find the volume of the three-dimensional objectcomposed of right rectangular prisms.

Lesson Summary

To find the volume of a three-dimensional composite object, two or more distinct volumes must be added together (if they are joined together) or subtracted from each other (if one is a missing section of the other). There are two strategies to find the volume of a prism:

Find the area of the base and then multiply times the prism’s height.

Decompose the prism into two or more smaller prisms of the same height and add the volumesof those smaller prisms.

Name _______________________________________________ Class ______________









3. A plastic die (singular for dice) for a game has an edgelength of1.5 cm. Each face of the cube has the number of cubiccutouts as its marker is supposed to indicate (i.e., the facemarked 3 has 3 cutouts). What is the volume of the die?

2. A smaller cube is stacked on top of a larger cube. An edge of the smaller cube measures1

2cm in length, while the larger cube has an edge length three times as long. What is the

total volume of the object?









4. A wooden cube with an edge length of 6 inches has square holes (holes in the shape of rightrectangular prisms) cut through the centers of each of the three sides as shown in the figure. Findthe volume of the resulting solid if the square for the holes has an edge length of 1 inch.

5. A right rectangular prism has each of its dimensions (length, width, and height) increased by50%. By what percent is its volume increased?





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