Engage NY Eureka Math Grade 6 Module 5 Lesson 3 Answer Key

Exercises:

Question 1.
Work with a partner on the exercises below. Determine if the area formula A = $$\frac{1}{2}$$ bh is always correct. You may use a calculator, but be sure to record your work on your paper as well. Figures are not drawn to scale.

Question 2.
Can we use the formula A = $$\frac{1}{2}$$ × base × height to calculate the area of triangles that are not right triangles? Explain your thinking.
Yes, the formula A = $$\frac{1}{2}$$ × base × height can be used for more than just right triangles. We just need to be able to determine the height when it is not necessarily the length of one of the sides.

Question 3.
Examine the given triangle and expression.
$$\frac{1}{2}$$ (11 ft.) (4ft.)
Explain what each part of the expression represents according to the triangle.
11 ft. represents the base of the triangle because 8 ft. + 3 ft. = 11 ft.
4 ft. represents the altitude of the triangle because this length is perpendicular to the base.

Question 4.
Joe found the area of a triangle by writing A = $$\frac{1}{2}$$ (11 in.)(4 in.), while Kaitlyn found the area by writing A = $$\frac{1}{2}$$ (3 in.)(4 in.) + $$\frac{1}{2}$$ (8 in.)(4 in.). Explain how each student approached the problem.
Joe combined the two bases of the triangle first and then calculated the area of the entire triangle, whereas Kaitlyn calculated the area of two smaller right triangles and then added these areas together.

Question 5.
The triangle below has an area of 4.76 sq. in. If the base is 3.4 in., let h be the height in inches.

a. Explain how the equation 4.76 in2 = (3.4 in. )h represents the situation.
The equation shows the area, 4. 76 in2, is one-half the base, 3.4 in times the height, in inches, h.

b. Solve the equation.
4.76 in2 = $$\frac{1}{2}$$ (3.4 in.)h
4.76 in2 = (1.7 in.)h
4.76 in2 + 1.7 in. = (1.7 in.)h + 1.7 in.
2.8 in. = h

Eureka Math Grade 6 Module 5 Lesson 3 Problem Set Answer Key

Calculate the area of each shape below. Figures are not drawn to scale.

Question 1.

A = $$\frac{1}{2}$$ (3.3 in.)(4.4 in.) = 7.26 in2
A = $$\frac{1}{2}$$ (6. 1 in.)(4.4 in.) 13.42 in2
A = 7.26 in2 + 13.42 in2 = 20.68 in2
or
A = $$\frac{1}{2}$$ (9.4 in. )(4. 4 in.) = 20.68 in2

Question 2.

A = $$\frac{1}{2}$$ (8m)(14m)= 56m2
A = $$\frac{1}{2}$$ (16m)(14m)= 112m2
A = 56m2 ÷ 112 m2 = 168m2
or
A = $$\frac{1}{2}$$ (24 m)(14 m) = 168m2

Question 3.

A = $$\frac{1}{2}$$ (5 ft) (12 ft) = 30 ft2
A = (12 ft.)(12 ft.) = 144 ft2
A = $$\frac{1}{2}$$ (5 ft.)(12 ft.) = 30 ft2
A = 30ft2 + 144ft2 + 30ft2 = 204 ft2

Question 4.

A = $$\frac{1}{2}$$ (48 km) (7 km) = 168km2
A = (35 km)(48 km) = 1,680 km2
A = $$\frac{1}{2}$$ (48 km) (7 km) = 168km2
A = 168 km2 + 1,680 km2 + 168 km2 = 2,016km2

Question 5.
Immanuel is building a fence to make an enclosed play area for his dog. The enclosed area will be in the shape of a triangle with a base of 48 m and an altitude of 32 m. How much space does the dog have to play?
A = $$\frac{1}{2}$$ bh = $$\frac{1}{2}$$ (48 m) (32 m) = 768 m2
The dog has 768 m2 in which to play.

Question 6.
Chauncey is building a storage bench for his son’s playroom. The storage bench will fit into the comer and against two walls to form a triangle. Chauncey wants to buy a triangular shaped cover for the bench.
If the storage bench is 2$$\frac{1}{2}$$ ft. along one wall and 4$$\frac{1}{4}$$ ft. along the other wall, how big will the cover have to be to cover the entire bench?

A = $$\frac{1}{2}$$ (2$$\frac{1}{2}$$ ft.) (4$$\frac{1}{4}$$ ft.)

= $$\frac{1}{2}$$ ($$\frac{5}{2}$$ ft) ($$\frac{17}{4}$$ ft.)

= $$\frac{85}{16}$$ft2 = 5$$\frac{5}{16}$$ ft2

Chauncey would have to buy a cover that has an area of 5$$\frac{5}{16}$$ ft2 to cover the entire scale. bench.

Question 7.
Examine the triangle to the right.

a. Write an expression to show how you would calculate the area.
$$\frac{1}{2}$$ (7 in.) (4 in.) + $$\frac{1}{2}$$ (3 in.) (4in.) or $$\frac{1}{2}$$ (10 in.) (4in.)

b. Identify each part of your expression as it relates to the triangle.
If students wrote the first expression, then 7 in. and 3 in. represent the two parts of the base, and 4 in. is the height, or the altitude, of the triangle.

If students wrote the second expression, then 10 in. represents the base because 7 in. + 3 in. = 10 in., and 4 in. represents the height, or the altitude, of the triangle.

Question 8.
The floor of a triangular room has an area of 32$$\frac{1}{2}$$ sq. m. If the triangle’s altitude is 7$$\frac{1}{2}$$ m, write an equation to determine the length of the base, b, in meters. Then solve the equation.

Therefore, the base is 8$$\frac{2}{3}$$ m.

Eureka Math Grade 6 Module 5 Lesson 3 Exit Ticket Answer Key

Calculate the area of each triangle using two different methods. Figures are not drawn to scale.

Question 1.

1
A = $$\frac{1}{2}$$ (3ft.) (7 ft.)= 10.5 ft2
A = $$\frac{1}{2}$$ (12 ft.) (7 ft.) = 42 ft2
A= 10.5 ft2 + 42 ft2 = 525 ft2
OR
A = $$\frac{1}{2}$$ (15 ft.) (7 ft.) = 52.5 ft2

Question 2.

A = $$\frac{1}{2}$$ (9 in.)(18 in.) = 81 in2
A = $$\frac{1}{2}$$ (32 in. )(18 in) = 288 in2
A = 81 in2 + 288 in2 = 369 in2
OR
A = $$\frac{1}{2}$$ (41 in.)(18 in.)= 369 in2

Eureka Math Grade 6 Module 5 Lesson 3 Multiplication of Decimals Answer Key

Multiplication of Decimals – Round 1

Directions: Evaluate each expression.

Question 1.
5 × 1 =
5

Question 2.
5 × 0.1 =
0.5

Question 3.
5 × 0.01 =
0.05

Question 4.
5 × 0.001 =
0.005

Question 5.
2 × 4 =
8

Question 6.
0.2 × 4 =
0.8

Question 7.
0.02 × 4 =
0.08

Question 8.
0.002 × 4 =
0.008

Question 9.
3 × 3 =
9

Question 10.
3 × 0.3 =
0.9

Question 11.
3 × 0.03 =
0.09

Question 12.
0.1 × 0.8 =
0.08

Question 13.
0.1 × 0.08 =
0.008

Question 14.
0.01 × 0.8 =
0.008

Question 15.
0.01 × 0.08 =
0.0008

Question 16.
0.3 × 0.2 =
0.6

Question 17.
0.03 × 0.2 =
0.006

Question 18.
0.02 × 0.3 =
0.006

Question 19.
0.02 × 0.03 =
0.0006

Question 20.
0.2 × 0.2 =
0.04

Question 21.
0.02 × 0.2 =
0.004

Question 22.
0.2 × 0.02 =
0.004

Question 23.
5 × 3 =
15

Question 24.
5 × 0.3 =
1.5

Question 25.
0.5 × 3 =
1.5

Question 26.
0.3 × 0.5 =
0.15

Question 27.
9 × 2 =
18

Question 28.
0.2 × 9 =
1.8

Question 29.
0.9 × 2 =
1.8

Question 30.
0.2 × 0.9 =
0.18

Question 31.
4 × 0.4 =
1.6

Question 32.
0.4 × 0.4 =
0.16

Question 33.
0.04 × 0.4 =
0.016

Question 34.
0.8 × 0.6 =
0.48

Question 35.
0.8 × 0.06 =
0.048

Question 36.
0.006 × 0.8 =
0.0048

Question 37.
0.006 × 0.08 =
0.00048

Question 38.
0.7 × 0.9 =
0.63

Question 39.
0.07 × 0.9 =
0.063

Question 40.
0.9 × 0.007 =
0.0063

Question 41.
0.09 × 0.007 =
0.00063

Question 42.
1.2 × 0.7 =
0.84

Question 43.
1.2 × 0.07 =
0.084

Question 44.
0.007 × 0.12 =
0.00084

Multiplication of Decimals – Round 1

Directions: Evaluate each expression

Question 1.
9 × 1 =
9

Question 2.
0.9 × 1 =
0.9

Question 3.
0.09 × 1 =
0.09

Question 4.
0.009 × 1 =
0.009

Question 5.
2 × 2 =
4

Question 6.
2 × 0.2 =
0.4

Question 7.
2 × 0.02 =
0.04

Question 8.
2 × 0.002 =
0.004

Question 9.
3 × 2 =
6

Question 10.
0.3 × 2=
0.6

Question 11.
2 × 0.03 =
0.06

Question 12.
0.7 × 0.1 =
0.07

Question 13.
0.07 × 0.1 =
0.007

Question 14.
0.01 × 0.7 =
0.007

Question 15.
0.01 × 0.07 =
0.0007

Question 16.
0.2 × 0.4 =
0.08

Question 17.
0.02 × 0.4 =
0.008

Question 18.
0.4 × 0.02 =
0.008

Question 19.
0.04 × 0.02 =
0.0008

Question 20.
0.1 × 0.1 =
0.01

Question 21.
0.01 × 0.1 =
0.0001

Question 22.
0.1 × 0.01 =
0.001

Question 23.
3 × 4 =
12

Question 24.
3 × 0.4 =
1.2

Question 25.
0.3 × 4 =
1.2

Question 26.
0.4 × 0.3 =
0.12

Question 27.
7 × 7 =
49

Question 28.
7 × 0.7 =
4.9

Question 29.
0.7 × 7 =
4.9

Question 30.
0.7 × 0.7 =
0.49

Question 31.
2 × 0.8 =
1.6

Question 32.
0.2 × 0.8 =
0.16

Question 33.
0.02 × 0.8 =
0.016

Question 34.
0.6 × 0.5 =
0.3

Question 35.
0.6 × 0.05 =
0.03

Question 36.
0.005 × 0.6 =
0.003

Question 37.
0.005 × 0.06 =
0.0003

Question 38.
0.9 × 0.9 =
0.81

Question 39.
0.09 × 0.9 =
0.081

Question 40.
0.009 × 0.9 =
0.0081

Question 41.
0.009 × 0.09 =
0.00081

Question 42.
1.3 × 0.6 =