6th grade - center for teaching &...

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6th Grade

Geometry

2015-12-01

www.njctl.org

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Table of ContentsClick on a topic to go to that section

Area of Rectangles

Area of Irregular Figures Area of Shaded Regions

Area of ParallelogramsArea of Right Triangles

Area of TrapezoidsMixed Review

3-Dimensional Solids

Surface AreaVolume

More Polygons in the Coordinate PlaneGlossary & Standards

Surface Area and Volume Application Problems

Area of Acute and Obtuse Triangles

Nets

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Table of ContentsClick on a topic to go to that section

Area of Rectangles

Area of Irregular Figures Area of Shaded Regions

Area of ParallelogramsArea of Right Triangles

Area of TrapezoidsMixed Review


Surface AreaVolume

More Polygons in the Coordinate PlaneGlossary & Standards

Surface Area and Volume Application Problems


Nets

[This object is a pull tab]

Teac

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in the presentation. The text box the word is in is then linked to the page at the end of the presentation with the word defined on it.

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Area of Rectangles

Return to Table of Contents

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Area is:

10 ft

5 ft

Areathe number of square units (units2) it takes to cover the surface of a figure.

ALWAYS label units2!!!

http://www.njctl.org

http://www.njctl.org

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How many 1 ft2 tiles does it take to cover the rectangle?

Use the squares to find out!

Look for a faster way than covering the whole figure.

10 ft

5 ft

Area Practice

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A = length(width)A = lw

A = side(side)A = s2

The Area (A) of a rectangle is found by using the formula:

The Area (A) of a square is found by using the formula:

Area

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1 What is the Area (A) of the figure?

13 ft

7 ft


1 What is the Area (A) of the figure?

13 ft

7 ft


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2 Find the area of the figure below.

8


2 Find the area of the figure below.

8


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3 Michelle needs new carpeting for her bedroom that is 12 feet by 9 feet. Does Michelle need to find the area or perimeter of her bedroom in order to figure out how much carpet to order?

A Area

B Perimeter


3 Michelle needs new carpeting for her bedroom that is 12 feet by 9 feet. Does Michelle need to find the area or perimeter of her bedroom in order to figure out how much carpet to order?

A Area

B Perimeter[This object is a pull tab]

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*Note - perimeter is a linked vocabulary word. Click on the text box to go to the vocab page.

Since the carpeting will cover the floor in her room, it is area, A.

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4 Now solve the problem....

Michelle needs new carpeting for her bedroom that is 12 feet by 9 feet. How many square feet of carpet does Michelle need to order?


4 Now solve the problem....

Michelle needs new carpeting for her bedroom that is 12 feet by 9 feet. How many square feet of carpet does Michelle need to order?


Ans

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5 A rectangle measures 3 in by 4 in. If the lengths of each side double, what is the effect on the area?

A The area doubles

B The area quadruples

C The area is cut in half

D There is no effect


5 A rectangle measures 3 in by 4 in. If the lengths of each side double, what is the effect on the area?

A The area doubles


C The area is cut in half

D There is no effect[This object is a pull tab]

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6 The area of a desktop is 24 sq. units. The length of the desktop is 6 units. What is the width of the desktop?


6 The area of a desktop is 24 sq. units. The length of the desktop is 6 units. What is the width of the desktop?


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7 The 6th grade class at Immersion Middle School is building a giant I for their school. The I will be 10 ft. tall and 2 ft. wide. How large will the I be if measured in square inches?


7 The 6th grade class at Immersion Middle School is building a giant I for their school. The I will be 10 ft. tall and 2 ft. wide. How large will the I be if measured in square inches?


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8 The lumber that will be used to make the Immersion School I is 6 in by 1 ft. How many pieces of wood are needed to complete the project?


8 The lumber that will be used to make the Immersion School I is 6 in by 1 ft. How many pieces of wood are needed to complete the project?


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Area of Parallelograms


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11 units

How can we find the area of this parallelogram? Cut out your parallelogram and work with your table to come up with a way to determine the area.

10 units

15 units

click

Area of a Parallelogram


11 units

How can we find the area of this parallelogram? Cut out your parallelogram and work with your table to come up with a way to determine the area.

10 units

15 units

click



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Print this rectangle out full page so that each student has one. After reviewing their predictions, ask how can we change the parallelogram into a rectangle? Click on the "click box" to reveal the height drawn and labeled. Have students cut this triangle off their parallelogram. You can also trace over the triangle yourself on the board and then move it to the other side.

This activity addresses MP1. Additional questions to ask:- How should you start the problem? - Why would you choose do perform a certain step? - What plan can you make to solve the problem?

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9 What is the area of the parallelogram?

10 units

15 units

click

11 units

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Area of a ParallelogramLet's use the same process as we did for the rectangle. How many 1 ft2 tiles fit across the bottom of the parallelogram?

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Area of a ParallelogramLet's use the same process as we did for the rectangle. If we build the parallelogram with rows of ten 1 ft2 tiles, what happens?

How tall is the parallelogram?How can you tell?

10 ft

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How does this help us find the area of the parallelogram?

10 ft

How do you find the area of a parallelogram?

4 ft



How does this help us find the area of the parallelogram?

10 ft

How do you find the area of a parallelogram?

4 ft



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Green trianglecan be moved.

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A = base(height)A = bh

The Area (A) of a parallelogram is found by using the formula:

Note: The base & height always form a right angle!


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Example.

Find the area of the figure.

6 cm

6 cm

2 cm 2 cm1.7 cm

Parallelogram Area Practice


Example.


6 cm

6 cm

2 cm 2 cm1.7 cm



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Try These.

Find the area of the figures.

10.4 in

8.7 in6.2 in

13 m

15 m

13 m

16 m


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Try These.


10.4 in

8.7 in6.2 in

13 m

15 m

13 m

16 m



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10 Find the area.

10 ft 9 ft

11 ft


10 Find the area.

10 ft 9 ft

11 ft


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11 Find the area.

15 in

15 in

10 in 11 in11 in


11 Find the area.

15 in

15 in

10 in 11 in11 in


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12 Find the area.

8.4 m

13.1 m

8.4 m

12.2 m

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12 Find the area.

8.4 m

13.1 m

8.4 m

12.2 m


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13 Find the area.

13 cm

12 cm

7 cm


13 Find the area.

13 cm

12 cm

7 cm


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14 A box with a square opening is squashed into the rhombus shown below. What is the area of the opening?

7 in.

14 in


14 A box with a square opening is squashed into the rhombus shown below. What is the area of the opening?

7 in.

14 in[This object is a pull tab]

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Solving for Missing InformationA parallelogram has an area of 137.7 cm2 and a base of 9 cm. Write an equation that relates the area to the base and height, h. Solve the equation to determine the length of the height.

Step 1: Plug in known information.

A = bh

=

Step 2: Use inverse operations to solve

( (

137.7 cm2 9 cm

Abh

information

(Answer) / 219

Solving for Missing InformationA parallelogram has an area of 137.7 cm2 and a base of 9 cm. Write an equation that relates the area to the base and height, h. Solve the equation to determine the length of the height.


A = bh

=


( (

137.7 cm2 9 cm

Abh

information

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This question addresses MP2.

Additional Questions to Ask:

- What variables are used to represent the variables? - Why are those variables chosen?- Are there different variables we could have chosen? - Why is it important to have variables instead of the original values?

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15 The height of a parallelogram is 12.6 feet and the area is 88.2 square feet. Write an equation that relates the area to the height and the base, b. Solve the equation to determine the length of the base.

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16 The height of a parallelogram is 54 inches and the area is 972 square inches. Write an equation that relates the area to the height and the base, b. Solve the equation to determine the length of the base.


Area of Right Triangles


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Area of Right Triangles



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Students will first complete the Area of Right Triangles Exploratory Challenge Lab prior to starting the slides in this section. This is found in the lab section at:

https://njctl.org/courses/math/6th-grade-math/geometry/

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Area of a TriangleLet's use the same process as we did for the rectangle & parallelogram. How many 1 ft2 tiles fit across the bottom of the triangle?

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Area of a TriangleIf we continue to build the triangle with rows of thirteen 1 ft2 tiles what happens?

How tall is the triangle? How can you tell?

13 ft

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How does this help us find the area of the triangle?

Find the area of the rectangle, then divide by 232.5 ft2

See that the rectangle we built is twice as large as the triangle. How do you find the area of a triangle?

13 ft

5 ft

Area of a Triangle

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The Area (A) of a triangle is found by using the formula:

Area of a Triangle

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Try this. What is the area of the right triangle below?

4 units

14 units

14.7 units

Area of a Triangle Practice

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Try this. What is the area of the right triangle below?

4 units

14 units

14.7 units

Area of a Triangle Practice


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17 Calculate the area.

7 cm.

8 cm.10.5 cm



7 cm.

8 cm.10.5 cm


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9.9 m9 m

4.1 m



9.9 m9 m

4.1 m[This object is a pull tab]

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7 in7 in

9.9 in

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7 in7 in

9.9 in


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Solving for Missing Information

A triangle has an area of 70.8 cm2 and a base of 6 cm. Write an equation that relates the area to the base and height, h. Solve the equation to determine the length of the height.


A = bh

=


( (

70.8 cm2 6 cm

Abh

information

( (


Solving for Missing Information

A triangle has an area of 70.8 cm2 and a base of 6 cm. Write an equation that relates the area to the base and height, h. Solve the equation to determine the length of the height.


A = bh

=


( (

70.8 cm2 6 cm

Abh

information

( ([This object is a pull tab]

Ans

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20 If the area of a triangle is 117 square cm and its base is 20 cm, write an equation that relates the area to the height, h, and the base. Solve the equation to determine the height.


21 Fran is surveying a plot of land in the shape of a right triangle. The area of the land is 45,000 sq. meters. If the base of the triangular plot is 180 m long, what is the height, in meters, of the triangle? Write and solve an equation.

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21 Fran is surveying a plot of land in the shape of a right triangle. The area of the land is 45,000 sq. meters. If the base of the triangular plot is 180 m long, what is the height, in meters, of the triangle? Write and solve an equation.


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What is the difference between these three triangles?

Triangles


What is the difference between these three triangles?

Triangles


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Label the triangles as right, acute and obtuse.

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The height of the right triangle is easy to find, it is a side. It does not always need to be a side of the triangle. The height of a triangle is also called the altitude, which is a line segment from a vertex of the triangle and perpendicular to the opposite side.

hh

b

b

b

h

Triangle Altitudes


The height of the right triangle is easy to find, it is a side. It does not always need to be a side of the triangle. The height of a triangle is also called the altitude, which is a line segment from a vertex of the triangle and perpendicular to the opposite side.

hh

b

b

b

h

Triangle Altitudes


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Label the triangles as right, acute and obtuse. Then students will complete the Area of Acute and Obtuse Triangles Exploratory Challenge Lab prior to starting the rest of the slides in this section. This is found in the lab section at:

https://njctl.org/courses/math/6th-grade-math/geometry/

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Is the formula for the area of a right triangle true for all triangles?Let's see!

Triangle AreaSlide 49 / 219

Example.


8 cm

11 cm 11 cm

11 cm

Triangle Area


Example.


8 cm

11 cm 11 cm

11 cm

Triangle Area


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Try These.


13 ft

11 ft

10 ft 12 ft 1420

16

16

Triangle Area Practice


Try These.


13 ft

11 ft

10 ft 12 ft 1420

16

16

Triangle Area Practice


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22 Find the area.

8 in

6 in

10 in9 in

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22 Find the area.

8 in

6 in

10 in9 in


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23 Find the area.

14 m

9 m10 m 12 m


23 Find the area.

14 m

9 m10 m 12 m


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24 Find the area.

10 in.

6 in.14 in.

5 in.


24 Find the area.

10 in.

6 in.14 in.

5 in.


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(Answer) / 219 Slide 55 / 219

26 Chauncey is building a storage bench for his son's playroom The storage bench will fit into the corner and then go along the wall to form a triangle. Chancey wants to buy a cover for the bench. If the storage bench is ft. along one wall and ft. along the other wall, how big will the cover have to be to cover the entire bench?

(Problem derived from )


26 Chauncey is building a storage bench for his son's playroom The storage bench will fit into the corner and then go along the wall to form a triangle. Chancey wants to buy a cover for the bench. If the storage bench is ft. along one wall and ft. along the other wall, how big will the cover have to be to cover the entire bench?



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Use what you know to try and figure out how can we calculate the area of this triangle.

Hint

Triangle Area


Use what you know to try and figure out how can we calculate the area of this triangle.

Hint

Triangle Area


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Calculate and subtract the area of the surrounding triangles from the area of the whole rectangle.

Students should complete each step on the next slide, and click on the reveal box to check answers as they go.

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11.5 7+ 10 28.5

49- 28.5 20.5

Triangle Sum

Difference

Trianglea = 1/2bha = 1/2(2)(7)a = 7 u2

Triangle a = 1/2bha = 1/2(3)(7)a = 11.5 u2

Trianglea = 1/2bha = 1/2(5)(4)a = 10 u2

a = lwa = 7(7)a = 49 u2

The shaded triangle is 20.5 u2

Square

Step 1: Calculate the area of the square Step 2: Calculate the area of the triangles.Step 3: Find the sum of the areas of the triangles.

Step 4: Subtract the sum of the triangle areas from the rectangle area.

http://www.engageny.org/sites/default/files/resource/attachments/math-g6-m5-teacher-materials.pdf


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27 What is the area of the shaded figure? Students type their answers here




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1 A=1/2bh A=1/2(1)(4) A=2 u2

2 A=1/2bh A=1/2(4)(3) A=6 u2

3 A=1/2bh A=1/2(1)(5) A=2.5 in2

Area of rectangle 20 u2 - Sum of triangles - 10.5 u2

= Area of Shaded Triangle 9.5 u2

Rectangle A=lw A=5(4) A=20 u2

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1 A=1/2bh A=1/2(1)(4) A=2 u2

2 A=1/2bh A=1/2(4)(3) A=6 u2

3 A=1/2bh A=1/2(1)(5) A=2.5 in2

Area of rectangle 20 u2 - Sum of triangles - 10.5 u2

= Area of Shaded Triangle 9.5 u2

Rectangle A=lw A=5(4) A=20 u2

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1 A=1/2bh A=1/2(4)(4) A=8 in2

2 A=1/2bh A=1/2(2)(7) A=14 in2

3 A=1/2bh A=1/2(2)(3) A=3 in2

Area of rectangle 28 in2 - Sum of triangles - 25 in2

= Area of Shaded Triangle 3 in2

Rectangle A=lw A=4(7) A=28 in2

1

2

3

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This method can be used with any shape, as long as you can find the base and height of the

triangles that form the surrounding rectangle.

Area of Any Shape


This method can be used with any shape, as long as you can find the base and height of the

triangles that form the surrounding rectangle.

Area of Any Shape


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Area of square 25 u2 - Sum of triangles - 13 u2

= Area of Shaded Quadrilateral 12 u2

Square A=s2

A=52

A=25 in2

1 A=1/2(3)(3) A=4.5 u2

2 A=1/2(2)(3) A=3 u2

3 A=1/2(2)(4) A=4 u2

4 A=1/2(2)(1) A=1.5 u2

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= Area of Shaded Rhombus 11 in2

Rectangle A=lw A=4(5) A=20 in2

1 A=1/2(2)(3) A=3 u2

2 A=1/2(1)(3) A=1.5 u2

3 A=1/2(3)(2) A=3 u2

4 A=1/2(1)(3) A=1.5 u2

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= Area of Shaded Rhombus 17 in2

Square A=s2

A=52

A=25 in2

1 A=1/2(1)(3) A=1.5 u2

2 A=1/2(1)(2) A=1 u2

3 A=1/2(1)(3) A=1.5 u2

4 A=1/2(4)(2) A=4 u2

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Area of Trapezoids


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Area of a Trapezoid· Draw a diagonal line to break the trapezoid into two triangles.· Find the area of each triangle· Add the area of each triangle together

See the diagram below. 10 in

12 in

5 in

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The Area (A) of a trapezoid is also found by using the formula:

Note: The base & height always form a right angle!

10 in

12 in

5 in

Area of a Trapezoid

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Example.

Find the area of the figure by drawing a diagonal and splitting it into two triangles.

12 cm

10 cm 11 cm

9 cm

Trapezoid Area Practice


Example.

Find the area of the figure by drawing a diagonal and splitting it into two triangles.

12 cm

10 cm 11 cm

9 cm



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12 cm

10 cm 11 cm

9 cm

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Try These.

Find the area of the figures using the formula.

12 ft

9 ft

7 ft 8 ft

13

10

8 ft 7 86


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page45svg

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(Answer) / 219

Try These.

Find the area of the figures using the formula.

12 ft

9 ft

7 ft 8 ft

13

10

8 ft 7 86



Ans

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32 Find the area of the trapezoid by drawing a diagonal.

9 m

11 m

8.5 m


33 Find the area of the trapezoid using the formula.

20 cm

13 cm

12 cm


33 Find the area of the trapezoid using the formula.

20 cm

13 cm

12 cm


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34 The shape of the state of Arkansas resembles a trapezoid. The population density of Arkansas is 54.8 people per square mi. What is the approximate total population of this state?

280 mi

210 mi

235 mi

(Answer) / 219

34 The shape of the state of Arkansas resembles a trapezoid. The population density of Arkansas is 54.8 people per square mi. What is the approximate total population of this state?

280 mi

210 mi

235 mi [This object is a pull tab]

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Additional questions to ask:

- Are there any other states that we can measure with our area formulas?

- What are some of the benefits/disadvantages of using an area model in real life?

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35 Each of the four sides of this tent are congruent. How much fabric was used to make all four sides of this tent?

23 in.

36.5 in.

32 in.


35 Each of the four sides of this tent are congruent. How much fabric was used to make all four sides of this tent?

23 in.

36.5 in.

32 in.


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Mixed Review:Area


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36 Find the area of the figure.

5 cm

4 cm 3 cm 4 cm

11 cm



5 cm

4 cm 3 cm 4 cm

11 cm


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8 yd

10.5 yd

10.5 yd 10.5 yd



8 yd

10.5 yd

10.5 yd 10.5 yd


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4.7 m

7.2 m



4.7 m

7.2 m


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9 in 7 in

15 in



9 in 7 in

15 in


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40 Find the area of the figure by drawing a diagonal and creating triangles.

17 cm

16 cm 15 cm 16 cm

22 cm


40 Find the area of the figure by drawing a diagonal and creating triangles.

17 cm

16 cm 15 cm 16 cm

22 cm


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7 in 5.2 in

12.4 in



7 in 5.2 in

12.4 in


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11 yd12 yd

13 yd

12 yd



11 yd12 yd

13 yd

12 yd


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4.6 m

8.7 m



4.6 m

8.7 m


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44 The Andersons were going on a long sailing trip during the summer. However, one of the sails on their sailboat ripped, and they have to replace it. The sail is pictured below.

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

Derived from


44 The Andersons were going on a long sailing trip during the summer. However, one of the sails on their sailboat ripped, and they have to replace it. The sail is pictured below.

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

Derived from


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$96

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45 A wall is 56" wide. You want to center a picture frame that is 20" wide on the wall. How much space will there be between the edge of the wall and the frame?


45 A wall is 56" wide. You want to center a picture frame that is 20" wide on the wall. How much space will there be between the edge of the wall and the frame?


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18 inches on each side

http://www.engageny.org/resource/grade-6-mathematics-module-5




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46 Daniel decided to walk the perimeter of his triangular backyard. He walked 26.2 feet north and 19.5 feet west and back to his starting point. What is the area of Daniel's backyard?


46 Daniel decided to walk the perimeter of his triangular backyard. He walked 26.2 feet north and 19.5 feet west and back to his starting point. What is the area of Daniel's backyard?


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47 If the area of a parallelogram is sq. km. and the base is km., write an equation that relates the area to the base, b, and the height. Solve the equation to determine the height.


47 If the area of a parallelogram is sq. km. and the base is km., write an equation that relates the area to the base, b, and the height. Solve the equation to determine the height.

[This object is a pull tab]A

nsw

er

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48 If the area of a right triangle is sq. ft. and the height is ft., write an equation that relates the area to the base, b, and the height. Solve the equation to determine the base.


48 If the area of a right triangle is sq. ft. and the height is ft., write an equation that relates the area to the base, b, and the height. Solve the equation to determine the base.


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/ 21949 Below is a drawing of a wall that is to be covered with either

wallpaper or paint. It is 8 ft. high and 16 ft. long. The window, mirror and fireplace will not be painted or papered. The window measures 18 in. by 14 ft. The fireplace is 5 ft. wide and 3 ft. high, while the mirror above the fireplace is 4 ft. by 2 ft.

Part A: How many square feet of wallpaper are needed to cover the wall?

Derived from continued


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50 Part B: The wallpaper is sold in rolls that are 18 in. wide and 33 ft. long. Rolls of solid color wallpaper will be used so patterns do not have to match up. What is the area of one roll of wallpaper?

Continued from previous page.


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51 Part C: How many rolls would be needed to cover the wall?






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52 Part D: This week the rolls of wallpaper are on sale for $11.99/ roll. Find the cost of covering the wall with wallpaper.


52 Part D: This week the rolls of wallpaper are on sale for $11.99/ roll. Find the cost of covering the wall with wallpaper.


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$11.99 x 2 = $23.98

Slide 91 / 219

53 Part E: A gallon of special textured paint covers 200 ft2 and is on sale for $22.99/ gallon. The wall needs to be painted twice (the wall needs two coats of paint). Find the cost of using paint to cover the wall.


53 Part E: A gallon of special textured paint covers 200 ft2 and is on sale for $22.99/ gallon. The wall needs to be painted twice (the wall needs two coats of paint). Find the cost of using paint to cover the wall.


Ans

wer

If the wall needs to be painted twice, we need to paint a total area of 84 ft2 x 2 = 168 ft2.

One gallon is enough paint for this wall, so the cost will be $22.99.

Slide 92 / 219

54 The area of a rectangular patio is square yards, and its length is yards. What is the patio's width in yards?

A

B

C

D

From PARCC PBA sample test non-calculator #3


54 The area of a rectangular patio is square yards, and its length is yards. What is the patio's width in yards?

A

B

C

D

From PARCC PBA sample test non-calculator #3


Ans

wer

A

http://practice.parcc.testnav.com/#


/ 219

55 Joanne buys a rectangular rug with an area of 35/4 square meters. The length of the rug is 7/2 meters. What is the width, in meters, of the rug?

From PARCC EOY sample test non-calculator #1


55 Joanne buys a rectangular rug with an area of 35/4 square meters. The length of the rug is 7/2 meters. What is the width, in meters, of the rug?



Ans

wer

Slide 94 / 219

Area ofIrregular Figures


Slide 95 / 219

Area of Irregular FiguresMethod #1

1. Divide the irregular figure into smaller figures (that you know how to find the area of)

2. Label each small figure and label the new lengths and widths of each shape

3. Find the area of each shape

4. Add the areas

5. Label your answer



1. Divide the irregular figure into smaller figures (that you know how to find the area of)

2. Label each small figure and label the new lengths and widths of each shape

3. Find the area of each shape

4. Add the areas

5. Label your answer[This object is a pull tab]

Mat

h Pr

actic

e

This slide addresses MP8.


- How can you chose which method to use?

- How are the methods similar? different?

- Why is it possible to use previous formulas in this new situation?

Slide 96 / 219

Example:Find the area of the figure.

12 m

8 m

4 m2 m

12 m6 m

4 m2 m #1

#2

2 m

Irregular Figure Area



page1svg

page46svg

page46svg

/ 219


1. Create one large, closed figure

2. Label the small added figure and label the new lengths and widths of each shape

3. Find the area of the new, large figure

4. Subtract the areas

5. Label your answer

Slide 98 / 219

Example:Find the area of the figure.

12 m

8 m

4 m2 m

8 m Whole Rectangle

Extra Rectangle

12 m

8 m

4 m2 m

Irregular Figure Area

Slide 99 / 219

Try This:Find the area of the figure.

3m5m

3m8m

Irregular Figure Area Practice



3m5m

3m8m



Ans

wer

Whole Square

Extra Square

3m

5m

3m8m

5m

Slide 100 / 219


18 ft

12 ft

6 ft

10 ft




18 ft

12 ft

6 ft

10 ft



Ans

wer

18 ft

12 ft

6 ft

10 ft

Whole Triangle

BottomTrapezoid

/ 219

56 Find the area.

4'

3'

1'

2'

10'

8'

5'


56 Find the area.

4'

3'

1'

2'

10'

8'

5'


Ans

wer 4'

3'

1'

2'

5'

8'

5'2'

3'

Top Rectangle

Bottom Rectangle

Vertical Rectangle

Total Area

Slide 102 / 219

57 Find the area.

12

101320

25

10


57 Find the area.

12

101320

25

10


Ans

wer

Whole New Figure New Rectangle

Total Area12

101320

25

10

Slide 103 / 219

58 Find the area.

8 cm 18 cm

9 cm


58 Find the area.

8 cm 18 cm

9 cm


Ans

wer

8 cm 18 cm

9 cm9 cm

Triangle Rectangle

Total Area

/ 219

59 Find the area.

4 ft 9 ft

6 ft

7 ft


59 Find the area.

4 ft 9 ft

6 ft

7 ft


Ans

wer

Triangle Trapezoid

Total Area

4 ft9 ft

6 ft

7 ft

Slide 105 / 219

60 Find the area.

8 mm

8 mm

8 mm10 mm

14 mm

14 mm

6 mm


60 Find the area.

8 mm

8 mm

8 mm10 mm

14 mm

14 mm

6 mm


Ans

wer

8 mm

8 mm

8 mm

10 mm

14 mm

14 mm

6 mm

Area of Triangle 1

Area of Triangle 2

Area of Rectangle

Total Area

Slide 106 / 219

61 Cara wants to put new carpet in both of her bedrooms. How much carpet will she need?


61 Cara wants to put new carpet in both of her bedrooms. How much carpet will she need?


Ans

wer

Area of Bedroom 1 Area of Bedroom 2

Total Area

/ 219

62 How many rectangular tiles are needed to cover this floor?

2 m1 m

Tiles

(Drag and drop to check.)


62 How many rectangular tiles are needed to cover this floor?

2 m1 m

Tiles

(Drag and drop to check.)


Ans

wer

Total Area

Area of Tile

Slide 108 / 219

Area ofShaded Regions


Slide 109 / 219

Area of a Shaded Region

1. Find area of whole figure.

2. Find area of unshaded figure(s).

3. Subtract unshaded area from whole figure.

4. Label answer with units2.

Slide 110 / 219

Example

Find the area of the shaded region.

8 ft

10 ft

3 ft3 ft

Area Whole Rectangle

Area Unshaded Square

Area Shaded Region

Shaded Region AreaSlide 111 / 219

Try ThisFind the area of the shaded region.

14 cm

12 cm

Shaded Region Area

page1svg

(Answer) / 219

Try ThisFind the area of the shaded region.

14 cm

12 cm

Shaded Region Area


Ans

wer

Area Whole Square

Area Triangle

Area Shaded Region

Slide 112 / 219

Try This


16 m

12 m6 m

8 m2 m

Shaded Region Area


Try This


16 m

12 m6 m

8 m2 m

Shaded Region Area


Ans

wer

Area Trapezoid

Area Rectangle

Area Shaded Region

Slide 113 / 219

63 Find the area of the shaded region.

11'

8'

3'4'



11'

8'

3'4'


Ans

wer

Area Whole Rectangle

Area Unshaded

Area Shaded Region

Slide 114 / 219


16"

17"

15"7"

5"

(Answer) / 219


16"

17"

15"7"

5"


Ans

wer

Area Parallelogram

Area Triangle

Area Shaded Region

Slide 115 / 219

13"

14"

8"

5"

9"

4"



13"

14"

8"

5"

9"

4"



Ans

wer

Area Whole

Area Rectangle

Area Shaded Region

13"

14"

8"

5"

9"

4"5"

Slide 116 / 219

4 yd

4 yd

3 yd

8 yd

4 yd

4 yd



4 yd

4 yd

3 yd

8 yd

4 yd

4 yd



Ans

wer Area of 2 Triangles

Area Rectangle

Area Shaded Region

Slide 117 / 219

67 A cement path 2 feet wide is poured around a rectangular pool. If the pool is 13 feet by 9 feet, how much cement was needed to create the path?

(Answer) / 219

67 A cement path 2 feet wide is poured around a rectangular pool. If the pool is 13 feet by 9 feet, how much cement was needed to create the path?


Ans

wer

Area Path & Pool

Area Pool

Area Path

Slide 118 / 219

68 Logan wants to paint his trapezoid-shaped wall shown below. He of course will not be painting over his window. One gallon of paint will cover 50 sq. feet. How many gallons of paint will he need?

23 ft

18 ft

13 ft5 ft

4 ft


68 Logan wants to paint his trapezoid-shaped wall shown below. He of course will not be painting over his window. One gallon of paint will cover 50 sq. feet. How many gallons of paint will he need?

23 ft

18 ft

13 ft5 ft

4 ft


Ans

wer

Area of Window

Area of Trapezoid

Slide 119 / 219

69 An advertising company is designing a new logo that consists of a shaded triangle inside a parallelogram.

Part A

What is the area, in square units, of parallelogram ABCD?

square units

From PARCC EOY sample test calculator #7


69 An advertising company is designing a new logo that consists of a shaded triangle inside a parallelogram.

Part A

What is the area, in square units, of parallelogram ABCD?

square units



Ans

wer

24 square units

Slide 120 / 219

70 Part B

In the new logo, what fraction of the parallelogram is shaded?

square units





(Answer) / 219

70 Part B

In the new logo, what fraction of the parallelogram is shaded?

square units



Ans

wer

1/4

Slide 121 / 219


Return toTable ofContents

Slide 122 / 219

3-Dimensional SolidsCategories & Characteristics of 3-D Solids:

Prisms1. Have 2 congruent, polygon bases which are parallel to one another2. Sides are rectangular (parallelograms)3. Named by the shape of their base

Pyramids1. Have 1 polygon base with a vertex opposite it2. Sides are triangular3. Named by the shape of their base

click to reveal

click to reveal

Slide 123 / 219

Sort the figures. If you are incorrect, the figure will be sent back.3-Dimensional Figures

Slide 124 / 219

3-Dimensional SolidsCategories & Characteristics of 3-D Solids:

Cylinders1. Have 2 congruent, circular bases which are parallel to one another2. Sides are curved

Cones1. Have 1 circular bases with a vertex opposite it2. Sides are curved

click to reveal

click to reveal

Slide 125 / 219

3-Dimensional SolidsVocabulary Words for 3-D Solids:

Face Flat surface of a Polyhedron

Edge Line segment formed where 2 faces meet

Vertex (Vertices) Point where 3 or more faces/edges meet

Face

edge


page1svg

page77svg

page83svg

page86svg

page87svg

page69svg

page74svg

page80svg

/ 219

A polyhedron is a 3-D figure whose faces are all polygons.

Polyhedron Not Polyhedron

Sort the figures to the appropriate side.

Polyhedron

Slide 127 / 219

71 Name the figure.

A rectangular prism

B triangular prism

C triangular pyramid

D cylinder

E square cone

F square pyramid


71 Name the figure.

A rectangular prism

B triangular prism


D cylinder

E square cone

F square pyramid [This object is a pull tab]

Ans

wer

C

Slide 128 / 219

72 Name the figure.

A rectangular prism

B triangular prism


D cylinder E coneF square pyramid


72 Name the figure.

A rectangular prism

B triangular prism


D cylinder E coneF square pyramid


Ans

wer

A

Slide 129 / 219

73 Name the figure.

A rectangular prism

B triangular prism


D pentagonal prism

E cone

F square pyramid

page68svg

(Answer) / 219

73 Name the figure.

A rectangular prism

B triangular prism


D pentagonal prism

E cone

F square pyramid[This object is a pull tab]

Ans

wer

D

Slide 130 / 219

74 Name the figure.

A rectangular prism

B triangular prism


D pentagonal prism

E cone

F square pyramid


74 Name the figure.

A rectangular prism

B triangular prism


D pentagonal prism

E cone

F square pyramid


Ans

wer

E

Slide 131 / 219

75 Name the figure.

A rectangular prism

B cylinder


D pentagonal prism

E coneF square pyramid


75 Name the figure.

A rectangular prism

B cylinder


D pentagonal prism



Ans

wer

B

Slide 132 / 219

Nets


page1svg

/ 219

NetsNets are two-dimensional drawings that represent the surface area of three-dimensional shapes.

There is more than one way to draw a net for a cube, however not all nets can be folded into a cube...

Slide 134 / 219Nets

Exploratory Challenge LabClick for Link to Lab

There are some six square arranglements on your page. Sort each of the six arrangements into one of two piles, those that are nets of a cube and those that are not.

click to reveal answersDerived from

Slide 134 (Answer) / 219Nets

Exploratory Challenge LabClick for Link to Lab

There are some six square arranglements on your page. Sort each of the six arrangements into one of two piles, those that are nets of a cube and those that are not.

click to reveal answersDerived from


Teac

her N

otes

& M

ath

Prac

tice

Each group of students will need a set of 20 (nets A-T). They are sized to wrap around a cube with side lengths of 4 cm, which can be made from 8 Unifix cubes. Each group needs one of these cubes.

The slides that cover nets cover MP6. Additional Questions to Ask:- Why are nets useful?- What are the advanatages/disadvantages of a net?

Slide 135 / 219

Click for a web site with interactive 3-D figures and nets.

Interactive 3-D Figures and Nets

Slide 136 / 219

Nets for prisms will have rectangular faces and two bases for which the shape is named.

Notice the two triangles areopposite from one another (bases).

Prism Nets

Slide 137 / 219

page57svg

page89svg

https://njctl.org/courses/math/6th-grade-math/geometry/nets-exploratory-challenge-lab/



https://njctl.org/courses/math/6th-grade-math/geometry/nets-exploratory-challenge-lab/



page67svg

http://www.learner.org/interactives/geometry/3d_prisms.html

/ 219

76 Name the figure represented by the net.

A rectangular prism

B cylinder

C triangular prism

D pentagonal prism




A rectangular prism

B cylinder

C triangular prism

D pentagonal prism



Ans

wer

C

Slide 139 / 219


A rectangular prism B cylinder

C triangular prism

D pentagonal prism




A rectangular prism B cylinder

C triangular prism

D pentagonal prism



Ans

wer

F

Slide 140 / 219

Use the packaging explorer to view more examples of nets.Interactive Nets

Slide 141 / 219

For each figure, find the number of faces, vertices and edges. What is the relationship between the number of faces, vertices and edges of 3D Figures?

Name Faces Vertices Edges Cube 6 8 12

Rectangular Prism 6 8 12

Triangular Prism 5 6 9

Triangular Pyramid 4 4 6

Square Pyramid 5 5 8

Pentagonal Pyramid 6 6 10

Octagonal Prism 10 16 24

3D Figure Patterns

Mat

h Pr

actic

e

/ 219

Euler's Formula

F + V - 2 = E

The number of edges is 2 less than the sum of the faces and vertices.

click to reveal

Slide 143 / 219

78 How many faces does a cube have?


78 How many faces does a cube have?


Ans

wer

6

Slide 144 / 219

79 How many vertices does a triangular prism have?


79 How many vertices does a triangular prism have?


Ans

wer

6

Slide 145 / 219

80 How many edges does a square pyramid have?

page91svg

(Answer) / 219

80 How many edges does a square pyramid have?


Ans

wer

8

Slide 146 / 219

81 Paige has a figure whose faces are all congruent, and it has 4 vertices. Which shape does Paige have?

A triangular pyramid

B triangular prism

C cube

D square


81 Paige has a figure whose faces are all congruent, and it has 4 vertices. Which shape does Paige have?

A triangular pyramid

B triangular prism

C cube

D square[This object is a pull tab]

Ans

wer

A

Slide 147 / 219

82 Jonathan has 2 cubes. Henry has a square pyramid. How many edges do they have all together?


82 Jonathan has 2 cubes. Henry has a square pyramid. How many edges do they have all together?


Ans

wer

32

Slide 148 / 219

83 Which of these nets can be folded to form a cube?

A

B

C

D

(Answer) / 219

83 Which of these nets can be folded to form a cube?

A

B

C

D[This object is a pull tab]

Ans

wer

All of them.

Slide 149 / 219

Surface Area


Slide 150 / 219

Surface AreaSurface area is the sum of the areas of all outside faces of a 3-D figure.

To find surface area, you must find the area of each face of the figure then add them together.

6 in

2 in7 in

What type of figure is pictured?

How many surfaces are there?

How do you find the area of each surface?

Slide 151 / 219

Surface Area

6 in

2 in7 in

7 in2 in

2 in6 in

A net is helpful in calculating surface area.

Simply label each section and find the area of each.

#2 #4

6 in

#1

#3

#5

#6

Slide 152 / 219

7 in2 in

2 in6 in

#2 #4

6 in

#1

#3

#5

#6

#1 #2 #3 #4 #5 #6

Example

Surface Area ExampleSlide 153 / 219

Try This

Find the surface area of figure using the given net.

#1

#2 #3 #4

#5

12 cm

Surface Area Practice

page1svg

page92svg

(Answer) / 219

Try This

Find the surface area of figure using the given net.

#1

#2 #3 #4

#5

12 cm

Surface Area Practice


Ans

wer

#1 #2 #3

#4 #5

Slide 154 / 219

84 Find the surface area of the figure given its net.

7 yd

7 yd

7 yd

7 ydSince all of the faces are the same, you can find the area of one face and multiply it by 6 to calculate the surface area of a cube.

What pattern did you notice while finding the surface area of a cube?



7 yd

7 yd

7 yd

7 ydSince all of the faces are the same, you can find the area of one face and multiply it by 6 to calculate the surface area of a cube.

What pattern did you notice while finding the surface area of a cube?


Ans

wer

#1 - #6

Slide 155 / 219


9 cm

25 cm

12 cm



9 cm

25 cm

12 cm


Ans

wer

#1 #3

#5

#2

#49 cm

25 cm

12 cm #1

#2 #3 #4

#5

Slide 156 / 219

86 The figure below represents a present you want to wrap for your friend's birthday. How many square centimeters of wrapping paper will you need? On the grid on the next slide, the distance between grid lines represents one centimeter. Use the grid to draw the net for the given figure. Then, calculate its surface area.

4 cm

4 cm8 cm

(Answer) / 219

86 The figure below represents a present you want to wrap for your friend's birthday. How many square centimeters of wrapping paper will you need? On the grid on the next slide, the distance between grid lines represents one centimeter. Use the grid to draw the net for the given figure. Then, calculate its surface area.

4 cm

4 cm8 cm


Ans

wer

#1,#3, #5 and #6

#2, #4

A = 4(8)

A = 32 cm2

A = 4(4)

A = 16 cm2

A = #1 + #2 + #3 + #4 + #5 + #6

A = 32 + 16 + 32 + 16 + 32 + 32

A = 160 cm2

8 cm

4 cm

4 cm

#5

#2 #4#3

#6

#1

4 cm

Slide 157 / 219

Slide 158 / 219

87 Draw the net for the given figure, and calculate its surface area.

7 ft7 ft

11 ft

4 ft

12 ft


87 Draw the net for the given figure, and calculate its surface area.

7 ft7 ft

11 ft

4 ft

12 ft


nsw

er

11 ft

7 ft 7 ft

12 ft#1

#3

#2 #4

4 ft

#1 #2 #3

#4

Slide 159 / 219

88 This is a net of a right rectangular prism.

Part A

Which prism can be made using the net?

A

B

C

D

Slide 160 / 219

89 Part B

What is the surface area, in square feet, of the prism?



(Answer) / 219

89 Part B

What is the surface area, in square feet, of the prism?



Ans

wer

1300 square feet

Slide 161 / 219

Volume


Slide 162 / 219

Volume Activity

Take unit cubes and create a rectangular prism with dimensions of 4 x 2 x 1.

What happens to the volume if you add another layer and make it 4 x 2 x 2?

What happens to the volume is you add another layer and make it 4 x 2 x 3?


Volume Activity

Take unit cubes and create a rectangular prism with dimensions of 4 x 2 x 1.

What happens to the volume if you add another layer and make it 4 x 2 x 2?

What happens to the volume is you add another layer and make it 4 x 2 x 3?

Teac

her N

otes

& M

ath

Prac

tice


Lead students to discover that the area of the base times the height equals the volume.



- Why is it important to be precise when adding cubes?- Can you provide an explanation for your formula? - Does the unit matter for the formula?

Slide 163 / 219

Volume

- Volume is the amount of space occupied by or inside a 3-D Figure.- The number of cubic units needed to fill a 3-D Figure (layering).

Label:Units3 or cubic units

Slide 164 / 219

Volume Formulas

Formula 1

V= lwh, where l = length, w = width, h = height

Multiply the length, width, and height of the rectangular prism.

Formula 2

V=Bh, where B = area of base, h = height

Find the area of the rectangular prism's base and multiply it by the height.


page1svg

page94svg

page94svg

page93svg

/ 219

Find the Volume.

5 m

8 m

2 m

Volume Practice


Find the Volume.

5 m

8 m

2 m

Volume Practice


Ans

wer

VOLUME: 2x 5 10 (Area of Base)x 8 (Height)80 m3

VOLUME:V = B hV = l w hV = 5 2 8V = 10 8V = 80 m3

Slide 166 / 219

Example

Each of the small cubes in the prism shown have a length, width and height of 1/4 inch.

The formula for volume is lwh.

Therefore the volume of one of the small cubes is:

Multiply the numerators together, then multiply the denominators. In other words, multiply across.

Forget how to multiply fractions?

Volume ExampleSlide 167 / 219

To calculate the volume of the whole prism, count the number of cubes, and multiply it by the volume of one cube.

The top layer of this prism has 4 rows of 4 cubes, making a total of 16 cubes per layer.

The prism has 4 layers, 16 cubes per layer, so has 64 small cubes total.

Therefore the total volume of the prism is:

Example

Volume Example

Slide 168 / 219

Example You can also use the formula to find the volume of the same prism.

The length, width, and the height of this prism is four small cubes.

Remember each small cube has a length, width, and height of 1/4 inch.

Therefore, you can find the total volume finding the total length, width, and height of the prism and multiplying them together.

Volume ExampleSlide 169 / 219

ExampleHow would you find the volume of the rectangular prism with side lengths of 1/2 cm, 1/8 cm, and 1/4 cm?

Volume Practice

(Answer) / 219

ExampleHow would you find the volume of the rectangular prism with side lengths of 1/2 cm, 1/8 cm, and 1/4 cm?

Volume Practice


Ans

wer

Since it already tells you the side lengths, you can simply plug it into the volume formula.

Slide 170 / 219

Try This Every cube in the rectangular prism has a length, width and height of 1/5 inch.

Find the total volume of the rectangular prism.

Method 1: Find volume of one small cube and multiply it by the number of cubes.

One cube: Total Volume:

Method 2: Find the length, width, and height of the rectangular prism and use the formula.

Click to Reveal

Volume Example

Click to Reveal

Slide 171 / 219

90 Find the volume of the given figure.




Ans

wer

Slide 172 / 219





Ans

wer

/ 219





Ans

wer

Slide 174 / 219

93 Find the volume of the given figure.The length,

width, and height of one small cube is .


93 Find the volume of the given figure.The length,

width, and height of one small cube is .


nsw

er

Method 2Method 1

Slide 175 / 219

94 Find the volume of the given figure. The length, width,

and height of one cube is .


94 Find the volume of the given figure. The length, width,

and height of one cube is .


Ans

wer

Method 1 Method 2

/ 219

95 A student filled a right rectangular prism-shaped box with one inch cubes to find the volume, in cubic inches. The student's work is shown.

Part A

Explain why the student's reasoning is incorrect. Provide the correct volume, in cubic inches, of the box.

From PARCC PBA sample test calculator #9


95 A student filled a right rectangular prism-shaped box with one inch cubes to find the volume, in cubic inches. The student's work is shown.

Part A

Explain why the student's reasoning is incorrect. Provide the correct volume, in cubic inches, of the box.



Ans

wer

&

Mat

h Pr

actic

e

The student's reasoning is incorrect because they did not count the top layer as part of the height. The calculation should have been 63 x 10, which

equals a total of 630 cubes. Therefore, the volume is 630 cubic inches.

This slide addresses MP3.

Additional questions to Ask:

- What evidence can you use to support the error analysis with?

- What are other common student errors?

- What advice can you tell other students to avoid making the error?

Slide 177 / 219

96 Part B

A second box also has a base of 63 square inches, but it has a volume of 756 cubic inches. What is the height, in inches, of the second box? Explain or show how you determined the height.



96 Part B

A second box also has a base of 63 square inches, but it has a volume of 756 cubic inches. What is the height, in inches, of the second box? Explain or show how you determined the height.



Ans

wer

Volume is equal to the area of the base times the height. V=bh

756 = 63 x height of the cubes

756/63 = height of the cubes

12 = height of the cubes

So, the height of the box is 12 inches since there are 12 1-inch cubes stacked on top of each other.

Slide 178 / 219

Surface Area and Volume

Application Problems


Slide 179 / 219

97 A rectangular storage box is 12 1/4 in wide, 15 3/5 in long and 9 in high. How many square inches of colored paper are needed to cover the surface area of the box?





page1svg

(Answer) / 219

97 A rectangular storage box is 12 1/4 in wide, 15 3/5 in long and 9 in high. How many square inches of colored paper are needed to cover the surface area of the box?


Ans

wer

12.25in

15.6in

9in

Slide 180 / 219

98 A teacher made 2 pair of foam dice to use in math games. Each cube measured 10 2/3 in on each side. How many square inches of fabric were needed to cover the 2 cubes?


98 A teacher made 2 pair of foam dice to use in math games. Each cube measured 10 2/3 in on each side. How many square inches of fabric were needed to cover the 2 cubes?


Ans

wer

Area of 1 cube Area of 2 cubes

Slide 181 / 219

99 A company is packaging their cereal in two rectangular-shaped containers. Container A is 5.5in x 7.25in x 10 3/4 in. Container B is 8 1/2in x 3 1/4 in x 12in. Which container will hold more cereal? Input your answer, then explain your answer in a sentence on your paper.

A Container A

B Container B


99 A company is packaging their cereal in two rectangular-shaped containers. Container A is 5.5in x 7.25in x 10 3/4 in. Container B is 8 1/2in x 3 1/4 in x 12in. Which container will hold more cereal? Input your answer, then explain your answer in a sentence on your paper.

A Container A

B Container B


Ans

wer

Volume Container A

Volume Container B

Container A will hold more cereal becuase the

container has a greater volume.

Slide 182 / 219

100 A company is packaging their cereal in two rectangular-shaped containers. Container A is 5.5in x 7.25in x 10 3/4 in. Container B is 8 1/2in x 3 1/4 in x 12in. Which container will require more cardboard to make the box? Input your answer, then explain your answer in a sentence on your paper.

A Container A

B Container B

(Answer) / 219

100 A company is packaging their cereal in two rectangular-shaped containers. Container A is 5.5in x 7.25in x 10 3/4 in. Container B is 8 1/2in x 3 1/4 in x 12in. Which container will require more cardboard to make the box? Input your answer, then explain your answer in a sentence on your paper.

A Container A

B Container B


Ans

wer

5.5in 7.25in

10.75in

8.5in

12in

3.25in

Container A

Container B

Container B will require more cardboard because it has a

larger surface area.

Slide 183 / 219

101 A toy company manufactured a new set of toy blocks. The packaging manager insists that the cubes be arranged to form a rectangular prism and that the package be designed to hold the blocks exactly, with no leftover packaging. Each block measures 1 in. x 1 in. x 1 in. There are 24 toy blocks to be sold in a box. What are all of the possible box dimensions in inches? (Select all that apply.)

A 1 x 1 x 24

B 1 x 2 x 12

C 1 x 3 x 8

D 2 x 2 x 8

E 2 x 3 x 6

F 1 x 3 x 6

G 1 x 4 x 6

H 2 x 2 x 6

I 2 x 4 x 8


J 2 x 3 x 4


101 A toy company manufactured a new set of toy blocks. The packaging manager insists that the cubes be arranged to form a rectangular prism and that the package be designed to hold the blocks exactly, with no leftover packaging. Each block measures 1 in. x 1 in. x 1 in. There are 24 toy blocks to be sold in a box. What are all of the possible box dimensions in inches? (Select all that apply.)

A 1 x 1 x 24

B 1 x 2 x 12

C 1 x 3 x 8

D 2 x 2 x 8

E 2 x 3 x 6

F 1 x 3 x 6

G 1 x 4 x 6

H 2 x 2 x 6

I 2 x 4 x 8


J 2 x 3 x 4


Ans

wer

A, B, C, G, H, J

Slide 184 / 219

102 (Cont. from previous slide) Which toy block box design will use the least amount of cardboard for packaging? Select one measurement (in inches) for each dimension of the box.

A 1

B 2

C 1

D 2

E 3

F 4

G 4

H 6

I 8

Height Width Length


J 12


102 (Cont. from previous slide) Which toy block box design will use the least amount of cardboard for packaging? Select one measurement (in inches) for each dimension of the box.

A 1

B 2

C 1

D 2

E 3

F 4

G 4

H 6

I 8

Height Width Length


J 12


Ans

wer

B, E, G

Slide 185 / 219

103 A 250 in3 box needs to be packaged for shipment. One shipping container has a length of 7 inches, a height of 5 inches, and a width of 6 inches. The other container has a length of 8 in, a height of 4 inches, and a width of 9 inches. Which container can the package be shipped in? Explain.

A Container A: 7 in x 6 in x 5 in

B Container B: 8 in x 4 in x 9 in





/ 219

104 Small cubes with edge lengths of 1/4 inch will be packed into the right rectangular prism shown.

How many small cubes are needed to completely fill the right rectangular prism?

cubes


Slide 187 / 219

105 The right rectangular prism is built with small cubes.

Part A

What is the volume, in cubic inch(es), of the right rectangular prism? Enter your fraction.



105 The right rectangular prism is built with small cubes.

Part A

What is the volume, in cubic inch(es), of the right rectangular prism? Enter your fraction.



Ans

wer

3/8

Slide 188 / 219

106 Part B

What is the volume, in cubic inch(es), of 1 of the small cubes? Enter your fraction.



106 Part B

What is the volume, in cubic inch(es), of 1 of the small cubes? Enter your fraction.



Ans

wer

1/64

Slide 189 / 219

More Polygons in the Coordinate Plane







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107 Draw a polygon in the coordinate plane using the given coordinates.

(4, -4)

(6, -2)

(8, -6)

What is the area of

the polygon?

Students type their answers here

(Problem from )


107 Draw a polygon in the coordinate plane using the given coordinates.

(4, -4)

(6, -2)

(8, -6)

What is the area of

the polygon?

Students type their answers here

(Problem from )


Ans

wer

Slide 191 / 219

108 A surveyor is mapping a city block on a coordinate grid. The square-shaped block has vertices at (-4,1), (-4, -4), and (1, -4). What are the coordinates of the remaining vertex?


108 A surveyor is mapping a city block on a coordinate grid. The square-shaped block has vertices at (-4,1), (-4, -4), and (1, -4). What are the coordinates of the remaining vertex?


Ans

wer

Vertex is (1,1)

Slide 192 / 219

109 What is the area of the square block described in the previous problem?


109 What is the area of the square block described in the previous problem?


Ans

wer



/ 219

Glossary & Standards



Glossary & Standards



Teac

her N

otes Vocabulary Words are bolded

in the presentation. The text box the word is in is then linked to the page at the end of the presentation with the word defined on it.

Slide 194 / 219

Back to

Instruction

3-D FiguresAn object with three different dimensions: length, width (or depth or breadth), and

height. Also called a solid figure.

One-Dimensional Three-DimensionalTwo-Dimensional

lengthlength

lengthwidthwidth

height

Slide 195 / 219

Back to

Instruction

AltitudeA line segment from a vertex of the triangle and perpendicular to the opposite side. The

height.

h

bb

h

b

h

This is not the height. It is not perpendicular to

the base.

Slide 196 / 219 Slide 197 / 219

Back to

Instruction

Base & HeightBase- the surface that a solid object

stands on

Height- the distance from the base to the top of a solid object.

base

height

base

height

The base and height

always form a right angle.

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/ 219

Back to

Instruction

ConeA 3-dimensional figure with one circular

base, a vertex at the top, and one curved surface connecting the two.

1 Circular Base

1 Curved Surface 1 Vertex

Slide 199 / 219

Back to

Instruction

CubeA 3-dimensional figure with 3 pairs

of parallel, congruent, square bases.

12 Edges8 vertices 6 faces

Slide 200 / 219

Back to

Instruction

CylinderA 3-dimensional figure with two-

congruent, circular bases, and one curved surface connecting them.

2 congruent, parallel, circular

bases

1 curved surface

No vertices

Slide 201 / 219

Back to

Instruction

DiagonalA line that goes from one non-

adjacent vertex to another.

Cannot draw a diagonal, because

all vertices are adjacent.

Slide 202 / 219

Back to

Instruction

DimensionsA measurement of

length in one direction.

1 dimension

2 dimensions

3 dimensions length

Slide 203 / 219

Back to

Instruction

EdgeThe line segment

where two faces meet.

10 edges

edge

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/ 219

Back to

Instruction

Euler's FormulaF + V - 2 = E

For any polyhedron that doesn't intersect itself, the number of edges is 2 less than

the sum of the faces and vertices.

Faces: 6Vertices: 8

6+8-2=12 Edges: 12

Slide 205 / 219

Back to

Instruction

Face

A flat surface of a 3-d figure.

There is still debate over

whether curved surfaces are

faces.Face6 faces

Slide 206 / 219 Slide 207 / 219

Slide 208 / 219

Back to

Instruction

Net

= =

A 2-dimensional pattern which can be folded into a 3-

dimensional figure.

Slide 209 / 219

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/ 219

Back to

Instruction

Perimeter

The distance around an object.

side 1

side 2side 3

P= side 1 + side 2 + side 3

To fence in this rectangular yard,

you would measure the perimeter.

l wP=2l+2w

Slide 211 / 219

Back to

Instruction

Polyhedron

"Polyhedra" is the

singular form of

polyhedron

A three dimensional figure with all flat faces.

non-polyhedraPolyhedra

Slide 212 / 219

Back to

Instruction

PrismA 3-dimensional figure with two congruent, parallel bases, and all other faces are rectangles.

Prisms are named by the

Pentagonal Prism

shape of their bases. 2 triangular bases

Triangular Prism

3 rectangular faces

Slide 213 / 219

Back to

Instruction

PyramidA 3-dimensional figure with one base, a vertex at the top, and all other faces are

triangles.

1 baseAll other faces are triangles

A vertex at the top

Slide 214 / 219

Back to

Instruction

Square Units

1 un

it

1 unit

1 unit x 1 unit = 1 square unit

A measurement in the shape of a square with side lengths

that are one unit long.

Notation:sq unit

unit2

u2

3 units

3 un

its

3 units x 3 units = 9 units2

Slide 215 / 219

6 u2

6 u2

12 u2

8 u2

8 u2

12 u2

Back to

Instruction

Surface AreaThe total area of the surface

of a 3-dimensional figure.

2lw+2lh+2wh2 12+2 6+2 8SA=24+12+16

SA=52u2

Surface Area=

+ + +

+ +

=SA

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/ 219

Back to

Instruction

TrapezoidA quadrilateral with one

pair of parallel sides.

There are no // sides.

Slide 217 / 219

Back to

Instruction

VertexA point where two or more

straight lines meet.

The plural of vertex is "vertices"

APoint A or

vertex A

Slide 218 / 219

Back to

Instruction

Volume

4

3

3

v=lwhv= 4 3 3v= 36 u3

The amount of space within a 3-dimensional object. Measured in cubic units.

11

1

V=1 1 1V= 1 cubic unit

Slide 219 / 219

Throughout this unit, the Standards for Mathematical Practice are used.

MP1: Making sense of problems & persevere in solving them.MP2: Reason abstractly & quantitatively.MP3: Construct viable arguments and critique the reasoning of others. MP4: Model with mathematics.MP5: Use appropriate tools strategically.MP6: Attend to precision.MP7: Look for & make use of structure.MP8: Look for & express regularity in repeated reasoning.

Additional questions are included on the slides using the "Math Practice" Pull-tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used.

If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull-tab.


Throughout this unit, the Standards for Mathematical Practice are used.

MP1: Making sense of problems & persevere in solving them.MP2: Reason abstractly & quantitatively.MP3: Construct viable arguments and critique the reasoning of others. MP4: Model with mathematics.MP5: Use appropriate tools strategically.MP6: Attend to precision.MP7: Look for & make use of structure.MP8: Look for & express regularity in repeated reasoning.

Additional questions are included on the slides using the "Math Practice" Pull-tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used.

If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull-tab.


Mat

h Pr

actic

e

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6th grade - center for teaching &...

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