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LearnZillion Illustrative Mathematics
Grade 6
Unit 1
Adapted from Open Up Resources under Creative Commons Attribution 4.0 International License. https://creativecommons.org/licenses/by/4.0
All adaptations copyright LearnZillion, 2018
This book includes public domain images or openly licensed images that are copyrighted by their respective owners. Openly licensed images remain under the terms of their respective licenses. See the image attribution section for more information.
Some images have been created with GeoGebra (www.geogebra.org).
ContributorsWriting Team
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Mike Nakamaye
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Supports for Students with Special Needs
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Table of Contents
Unit 1: Area and Surface Area
Lesson 1: Tiling the Plane………………………………………………………………..... 9
Lesson 2: Finding Area by Decomposing and Rearranging…………….... 14
Lesson 3: Reasoning to Find Area…………………………………………………….. 21
Lesson 4: Parallelograms……………………………………………………………….... 26
Lesson 5: Bases and Heights of Parallelograms……………………………..... 31
Lesson 6: Area of Parallelograms………………………………………………….….. 39
Lesson 7: From Parallelograms to Triangles…………………………………..... 45
Lesson 8: Area of Triangles…………………………………………………………….... 51
Lesson 9: Formula for the of a Triangle………………………………………….... 59
Lesson 10: Bases and Heights of Triangles…………………………………….... 66
Lesson 11: Polygons…………………………………………………………………….….. 73
Lesson 12: What is Surface Area? ……………….……………………………….….. 80
Lesson 13: Polyhedra……………………………………………………………….……... 85
Lesson 14: Nets and Surface Area………………………………………...….…..... 92
Lesson 15: More Nets, More Surface Area…………………………………….... 98
Lesson 16: Distinguishing Between Surface Area and Volume…….....103
Lesson 17: Squares and Cubes……………………………………………………....109
Lesson 18: Surface Area of a Cube………………………………………………....114
Lesson 19: Designing a Tent…………………………………………………………...118
Unit 1 9 Lesson 1
1.2: More Rhombuses, Trapezoids, or Triangles?
Your teacher will assign you to look at Pattern A or Pattern B.
In your pattern, which shape covers more of the plane: rhombuses,
trapezoids, or triangles? Explain how you know.
Pattern A
Pattern B
Unit 1 10 Lesson 1
Unit 1 11 Lesson 1
Unit 1 12 Lesson 1
Unit 1 13 Lesson 1
Unit 1 14 Lesson 2
2.2: Composing Shapes
Your teacher will give you one square and some small, medium, and
large right triangles. The area of the square is 1 square unit.
1. Notice that you can put together two small triangles to make a square. What is the
area of the square composed of two small triangles? Be prepared to explain your
reasoning.
2. Use your shapes to create a new shape with an area of 1 square unit that is not a
square. Trace your shape.
3. Use your shapes to create a new shape with an area of 2 square units. Trace your
shape.
Unit 1 15 Lesson 2
Unit 1 16 Lesson 2
2.3: Tangram Triangles
Recall that the area of the square you saw earlier is 1 square unit.
Complete each statement and explain your reasoning.
1. The area of the small triangle is ___ square units. I know this because ...
2. The area of the medium triangle is ___ square units. I know this because ...
3. The area of the large triangle is ___ square units. I know this because ...
Unit 1 17 Lesson 2
Unit 1 18 Lesson 2
Unit 1 19 Lesson 2
Unit 1 20 Lesson 2
Unit 1 21 Lesson 3
Unit 1 22 Lesson 3
Unit 1 23 Lesson 3
Unit 1 24 Lesson 3
Unit 1 25 Lesson 3
Lesson 4: Parallelograms
Let's investigate the features and area of parallelograms.
4.1: Features of a Parallelogram
Figures A, B, and Care parallelograms. Figures D, E, and F are not parallelograms.
A F· (
I I, , /J J
V V "
r F
I -....
r--..... I \ ,.__ J
I I \ I I \ I I
I ' ,
Study the examples and non-examples. What do you notice about
1. the number of sides that a parallelogram has?
2. opposite sides of a parallelogram?
3. opposite angles of a parallelogram?
4.2: Area of a Parallelogram
Find the area of each parallelogram. Show your reasoning.
\ \
-
- -
\
\
- - \ \ \
_,_ _,_
\
·\ -
/ " " /
" /
"
"'
Unit 1 26 Lesson 4
Unit 1 27 Lesson 4
Unit 1 28 Lesson 4
Unit 1 29 Lesson 4
Unit 1 30 Lesson 4
Lesson 5: Bases and Heights of Parallelograms
Let's investigate the area of parallelograms some more.
5.1: A Parallelogram and Its Rectangles
Elena and Tyler were finding the area of this parallelogram:
I I f I
j j
I I I I
Here is how Elena did it
' j j ' . j
I I I I . I I , I . I
j ' ' J . J
I I I I . I I ,r
I f I .
I •
.
r
C 0
Here is how Tyler did it
j . j
I I I . I f , .
f'
j .
I I I .
I .f , . f
C b
How are the two strategies for finding the area of a parallelogram the same? How they
are different?
Unit 1 31 Lesson 5
5.2: The Right Height?
1. Each parallelogram has a side that is labeled "base."
Study the examples and non-examples of bases and heights of parallelograms.
Then, answer the questions that follow.
Examples: The dashed segment in each
drawing represents the corresponding
height for the given base.
Non-examples: The dashed segment in
each drawing does not represent the
corresponding height for the given
base.
base
base
'
'
'
'
'
'
'
'
'
'
'
base
base
base
base
Select all statements that are true about bases and heights in a parallelogram.
a. Only a horizontal side of a parallelogram can be a base.
b. Any side of a parallelogram can be a base.
c. A height can be drawn at any angle to the side chosen as the base.
d. A base and its corresponding height must be perpendicular to each other.
e. A height can only be drawn inside a parallelogram.
f. A height can be drawn outside of the parallelogram, as long as it is drawn at a
90-degree angle to the base.
g. A base cannot be extended to meet a height.
Unit 1 32 Lesson 5
Unit 1 33 Lesson 5
Unit 1 34 Lesson 5
Unit 1 35 Lesson 5
Unit 1 36 Lesson 5
Unit 1 37 Lesson 5
Unit 1 38 Lesson 5
Lesson 6: Area of Parallelograms
Let's practice finding the area of parallelograms.
6.1: Missing Dots
••••••
••••••
••••••
•• •
•• •
••••••
How many dots are in the image?
How do you see them?
6.2: More Areas of Parallelograms
1. Find the area of each parallelogram. Show your reasoning.
6cm Bern ···
--10cm--
10 cm
D
I I I I
I I I I
1cm
1 J, m
Unit 1 39 Lesson 6
Unit 1 40 Lesson 6
Unit 1 41 Lesson 6
Unit 1 42 Lesson 6
Unit 1 43 Lesson 6
Unit 1 44 Lesson 6
Unit 1 45 Lesson 7
7.2: A Tale of Two Triangles (Part 1)
Two polygons are identical if they match up exactly when placed one
on top of the other.
1. Draw one line to decompose each of the following polygons into two identical
triangles, if possible. Use a straightedge to draw your line.
A R C" i.,..,-
,,. ...... r-....
........ i--.. I
......_ ,,,,,. .....
) _..,.... --I--
I_ - -......
i--... _,,, -� --
n -
F ':; ...
l l ) J
I I I '
I ) ) \ , I I I '
J ) I I
\ '
'
2. Which quadrilaterals can be decomposed into two identical triangles?
Pause here for a small-group discussion.
-
3. Study the quadrilaterals that can, in fact, be decomposed into two identical triangles.
What do you notice about them? Write a couple of observations about what these
quadrilaterals have in common.
Unit 1 46 Lesson 7
Are you ready for more?
On the grid, draw some other types of quadrilaterals that are not already shown. Try to
decompose them into two identical triangles. Can you do it?
Come up with a rule about what must be true about a quadrilateral for it to be
decomposed into two identical triangles.
7.3: A Tale of Two Triangles (Part 2)
Your teacher will give your group several pairs of triangles. Each
group member should take 1-2 pairs.
1. a. Which pair(s) of triangles do you have?
b. Can each pair be composed into a rectangle? A parallelogram?
2. Discuss with your group your responses to the first question. Then, complete each of
the following statements with all, some, or none. Sketch 1-2 examples to illustrate
each completed statement.
a. _____ of these pairs of
identical triangles can be composed into
a rectangle.
b. _____ of these pairs of
identical triangles can be composed into
a parallelogram.
Unit 1 47 Lesson 7
Unit 1 48 Lesson 7
Unit 1 49 Lesson 7
Unit 1 50 Lesson 7
Unit 1 51 Lesson 8
Unit 1 52 Lesson 8
Unit 1 53 Lesson 8
Unit 1 54 Lesson 8
Unit 1 55 Lesson 8
Unit 1 56 Lesson 8
Unit 1 57 Lesson 8
Unit 1 58 Lesson 8
Unit 1 59 Lesson 9
Unit 1 60 Lesson 9
Unit 1 61 Lesson 9
Unit 1 62 Lesson 9
Unit 1 63 Lesson 9
Unit 1 64 Lesson 9
Unit 1 65 Lesson 9
Lesson 10: Bases and Heights of Triangles
Let's use different base-height pairs to find the area of a triangle.
10.1: An Area of 12
On the grid, draw a triangle with an area of 12 square units. Try to
draw a non-right triangle. Be prepared to explain how you know
the area of your triangle is 12 square units.
Unit 1 66 Lesson 10
Unit 1 67 Lesson 10
Unit 1 68 Lesson 10
Unit 1 69 Lesson 10
Unit 1 70 Lesson 10
Unit 1 71 Lesson 10
Unit 1 72 Lesson 10
Unit 1 73 Lesson 11
Unit 1 74 Lesson 11
Unit 1 75 Lesson 11
Unit 1 76 Lesson 11
Unit 1 77 Lesson 11
Unit 1 78 Lesson 11
Unit 1 79 Lesson 11
Unit 1 80 Lesson 12
12.3: Building with Snap Cubes
Here is a sketch of a rectangular prism built from 12 cubes:
•
•
•
•
•
It has six faces, but you can only see three of them in the sketch.
It has a surface area of 32 square units.
You have 12 snap cubes from your teacher. Use all of your snap
cubes to build a different rectangular prism (with different edge
lengths than shown in the prism here).
1. How many faces does your figure have?
2. What is the surface area of your figure in square units?
3. Draw your figure on isometric dot paper. Color each face a different color.
Unit 1 81 Lesson 12
Unit 1 82 Lesson 12
Unit 1 83 Lesson 12
Unit 1 84 Lesson 12
Unit 1 85 Lesson 13
Unit 1 86 Lesson 13
Unit 1 87 Lesson 13
Unit 1 88 Lesson 13
Unit 1 89 Lesson 13
Unit 1 90 Lesson 13
Unit 1 91 Lesson 13
Unit 1 92 Lesson 14
Unit 1 93 Lesson 14
Unit 1 94 Lesson 14
Unit 1 95 Lesson 14
Unit 1 96 Lesson 14
Unit 1 97 Lesson 14
Unit 1 98 Lesson 15
Unit 1 99 Lesson 15
Unit 1 100 Lesson 15
Unit 1 101 Lesson 15
Unit 1 102 Lesson 15
Unit 1 103 Lesson 16
16.2: Building with 8 Cubes
Your teacher will give you 16 cubes. Build two different shapes using
8 cubes for each. For each shape:
• Give a name or a label (e.g., Mae's First Shape or Eric's Steps).
• Determine the volume.
• Determine the surface area.
• Record the name, volume, and surface area on a sticky note.
Pause for further instructions.
16.3: Comparing Prisms Without Building Them
Three rectangular prisms each have a height of 1 cm.
• Prism A has a base that is 1 cm by 11 cm.
• Prism B has a base that is 2 cm by 7 cm.
• Prism C has a base that is 3 cm by 5 cm.
1. Find the surface area and volume of each prism. Use the dot paper to draw the
prisms, if needed.
Unit 1 104 Lesson 16
Unit 1 105 Lesson 16
Unit 1 106 Lesson 16
Unit 1 107 Lesson 16
Unit 1 108 Lesson 16
Lesson 17: Squares and Cubes
Let's investigate perfect squares and perfect cubes.
17.1: Perfect Squares
1. The number 9 is a perfect square.
Find four numbers that are perfect squares and two numbers
that are not perfect squares.
2. A square has side length 7 km. What
is its area?
3. The area of a square is 64 sq cm.
What is its side length?
17.2: Building with 32 Cubes
Your teacher will give you 32 snap cubes. Use them to build the
largest single cube you can. Each small cube has an edge length of 1
unit.
1. How many snap cubes did you use?
2. What is the edge length of the cube
you built?
17.3: Perfect Cubes
1. The number 27 is a perfect cube.
3. What is the area of each face of the
built cube? Show your reasoning.
4. What is the volume of the built cube?
Show your reasoning.
Find four other numbers that are perfect cubes and two
numbers that are not perfect cubes.
Unit 1 109 Lesson 17
Unit 1 110 Lesson 17
Unit 1 111 Lesson 17
Unit 1 112 Lesson 17
Unit 1 113 Lesson 17
Unit 1 114 Lesson 18
Unit 1 115 Lesson 18
Unit 1 116 Lesson 18
Unit 1 117 Lesson 18
Unit 1 118 Lesson 19
Unit 1 119 Lesson 19
Unit 1 120 Lesson 19
Image Attribution
Licensing and attribution for images appearing in this unit appears below.
Additional Attribution: ‘Notice and Wonder’ and ‘I Notice/I Wonder’ are trademarks of the National Council of Teachers of Mathematics, reflecting approaches developed by the Math Forum (http://mathforum.org), and used here with permission.
Lesson 6: Practice Problem 4, “Architecture Port E lbe Hamburg Building” via Max Pixel. Public Domain. http://maxpixel.freegreatpicture.com/Architecture-Port-E lbe-Hamburg-Building-416779.
Lesson 12: Practice Problem 2, “Trunk 1” via Wikimedia Commons. Public Domain. https://commons.wikimedia.org/wiki/File:Trunk_1.jpg
