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Illustrative Mathematics Unit 6.1, Lesson 17: Squares and Cubes

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Learn more about squares, cubes, and how exponents relate to the area or volume of squares and cubes. After trying the questions, click on the buttons to view answers and explanations in text or video.

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Squares and Cubes
Let’s investigate perfect squares and perfect cubes.

17.1 - Perfect Squares

A 3 by 3 square on a grid.

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?

  • Hints

    1. A number is a perfect square when that number of small squares can be arranged into a large square. For example, 9 is a perfect square because 9 copies of a small square can be arranged into a large square:
    A 3 by 3 square on a grid.

    However, 10 is not a perfect square because 10 copies of a small square cannot be arranged into a large square:
    10 squares arranged in 3 different shapes.

    3. To find the side length, you need to find a number which, when multiplied by itself, equals 64. Answer questions 1 and 2 first. Do you get answers more or less than 64? What does that tell you about the number you need?

  • Answers

    1. Perfect squares: 1, 4, 16, 25
    Non-perfect squares: 5, 24
    A number is a perfect square when that number of small squares can be arranged into a large square:
    Four large squares composed of 1, 4, 16, and 25 small squares arranged into large squares respectively.

    2. 7 km × 7 km = 49 sq km

    3. 8 cm × 8 cm = 64 sq cm

  • See Video 1 for Whole Lesson
  • See Video 2 for Whole Lesson




17.2 - Building with 32 Cubes

Open the applet. Drag on the red point on the cube to move it, and click on the point to switch between vertical and horizontal movement. The gray square will give you 32 cubes. Use as many of these as you need to build the largest single cube you can. Each small cube has a side length of 1 unit.

1. How many snap cubes did you use?
2. What is the side length of the cube you built?
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.

  • Hints

    32 is not a perfect cube. You do not need to use all of the cubes in the applet.

  • See Possible Answers

    Two figures: a cube 3 by 3 by 3, and a stack of 5 cubes.

    1. The built cube consists of 27 snap cubes. There are 5 leftover snap cubes.

    2. The built cube has a side length of 3.

    3. The area of any one face of the built cube is 3 × 3 = 9 sq units.

    4. The volume of the built cube is 3 × 3 × 3 = 27 cubic units. This is equal to the number of snap cubes used.


Open the next applet. The gray square will give you 64 cubes. (There is an error in the applet title.) Use as many cubes as you need to build the largest single cube you can.

1. How many snap cubes did you use?
2. What is the edge length of the new cube you built?
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.

  • Hints

    64 is a perfect cube.

  • See Possible Answers

    A cube 4 by 4 by 4, composed of 64 smaller cubes.

    1. The built cube consists of 64 snap cubes. 64 is a perfect cube, so there are no leftovers.

    2. The built cube has an edge length of 4.

    3. The area of any one face of the built cube is 4 × 4 = 16 sq units.

    4. The volume of the built cube is 4 × 4 × 4 = 64 cubic units. This is equal to the number of snap cubes used.




17.3 - Perfect Cubes

A 3 by 3 by 3 cube.

1. The number 27 is a perfect cube.
Find four other numbers that are perfect cubes and two numbers that are not perfect cubes.

2. A cube has side length 4 cm. What is its volume?

3. A cube has side length 10 inches. What is its volume?

4. A cube has side length s units. What is its volume?

  • Hints

    Like perfect squares, a number is a perfect cube when that number of small cubes can be assembled into a larger cube.

    From the previous questions, a cube with a volume of 27 cubic units measures 3 × 3 × 3, and a cube with a volume of 64 cubic units measures 4 × 4 × 4. How might you find more numbers that are perfect cubes?

  • Answers

    Two cubes composed of 1 and 8 small cubes respectively.

    1. Perfect cubes: 1, 8, 64, 125
    Non-perfect cubes: 9, 16
    A number is a perfect cube when that number of small cubes can be assembled into a larger cube. Some examples are shown in the picture above.

    2. 4 cm × 4 cm × 4 cm = 64 cubic cm

    3. 10 in × 10 in × 10 in = 1000 cubic in

    4. s × s × s


17.4 - Introducing Exponents

When we multiply two of the same number together, such as 5 × 5, we say we are squaring the number. We can write the expression as 52. Because 5 × 5 is 25, we can write 52 = 25, and we say, "5 squared is 25". We can also say that 25 is a perfect square. The raised 2 in 52 is called an exponent.

When we multiply three of the same number together like 4 × 4 × 4, we say we are cubing the number. We can write it as 43. Because 4 × 4 × 4 is 64, we can write 43 = 64, and we say, "4 cubed is 64". We can also say that 64 is a perfect cube. The raised 3 in 43 is also an exponent.

Exponents are also used as shorthand for units of area (square units) and units of volume (cubic units).
A square with side length 5 inches has area of 25 square inches, which we can write as 25 in2.
A cube with edge length 4 centimeters has a volume of 64 cubic centimeters, which we can write as 64 cm3.

For these questions, make sure to include correct units of measure as part of each answer.

  1. A square has side length 10 cm. Use an exponent to express its area.
  2. The area of a square is 72 sq in. What is its side length?
  3. The area of a square is 81 m2. Use an exponent to express this area.
  4. A cube has edge length 5 in. Use an exponent to express its volume.
  5. The volume of a cube is 63 cm3. What is its edge length?
  6. A cube has edge length s units. Use an exponent to write an expression for its volume.
  • Answers
    1. 102 cm2 (which is equal to 100 cm2)
    2. 7 in
    3. 92 m2
    4. 53 in3 (which is equal to 125 in3)
    5. 6 cm
    6. s3

The number 15,625 is both a perfect square and a perfect cube. It is a perfect square because it equals 1252. It is also a perfect cube because it equals 253. Find another number that is both a perfect square and a perfect cube. How many of these can you find?

  • See Possible Answers

    1 equals both 12 and 13.
    64 equals both 82 and 43.

    There are an infinite number of possible answers.



Lesson 17 Summary

When we multiply two of the same numbers together, such as 5 × 5, we say we are squaring the number. We can write it like this: 52

Because 5 × 5 is 25, we can write 52 = 25, and we say, "5 squared is 25".

When we multiply three of the same number together like 4 × 4 × 4, we say we are cubing the number. We can write it as 43.

Because 4 × 4 × 4 is 64, we can write 43 = 64, and we say, "4 cubed is 64".

We also use this notation for square and cubic units.
A square with side length 5 inches has area 25 in2.
A cube with edge length 4 centimeters has a volume 64 cm3.

To read 25 in2, we say "25 square inches", just like before.

The area of a square with side length 7 kilometers is 72 km2. The volume of a cube with edge length 2 millimeters is 23 mm3.

In general, the area of a square with side length s is s2, and the volume of a cube with edge length s is s3.



Glossary Terms

cubed

We use the word cubed to mean "to the third power". This is because a cube with side length s has a volume of s × s × s, or s3.


exponent

In expressions like 53 and 82, the 3 and the 2 are called exponents. They tell you how many factors to multiply. For example, 53 = 5 × 5 × 5, and 82 = 8 × 8.


squared

We use the word squared to mean "to the second power". This is because a square with side length s has an area of s × s, or s2.




Practice Problems

1. What is the volume of this cube?

A cube 2 by 2 by 2 cm, made of 8 smaller cubes.


2. a. Decide if each number on the list is a perfect square.
b. Write a sentence that explains your reasoning.

  1. 16
  2. 20
  3. 25
  4. 100
  5. 125
  6. 144
  7. 225
  8. 10,000
  • Hints

    Use your knowledge of times tables or multiplications of 10: if a number in the list is not the result of multiplying a number by itself, it is not a perfect square.

  • Answers
    1. 16 = 42
    2. 20 = 4 × 5 or 2 × 10, not a perfect square
    3. 25 = 52
    4. 100 = 102
    5. 125 is not a perfect square; the closest perfect square is 121 = 112
    6. 144 = 122
    7. 225 = 152
    8. 10,000 = 1002

3. a. Decide if each number on the list is a perfect cube.
b. Explain what a perfect cube is.

  1. 1
  2. 3
  3. 8
  4. 9
  5. 27
  6. 64
  7. 100
  8. 125
  • Answers
    1. 1 = 13
    2. 3 is a prime number and is not a perfect cube.
    3. 8 = 23
    4. 9 = 32, not a perfect cube
    5. 27 = 33
    6. 64 = 43
    7. 100 = 102, not a perfect cube
    8. 125 = 53

    If a given number of cubes can be arranged into a larger cube, that number is a perfect cube. Another way of expressing this is that a perfect cube is the result of raising any whole number to the third power.


4. a. A square has side length 4 cm. What is its area?
b. The area of a square is 49 m2. What is its side length?
c. A cube has edge length 3 in. What is its volume?

  • Answers

    a. 42 cm2 = 16 cm2

    b. 49 m2 = 72 m2
    Side length = 7 m

    c. 33 in3 = 27 in3


5. Prism A and Prism B are rectangular prisms. Prism A is 3 inches by 2 inches by 1 inch. Prism B is 1 inch by 1 inch by 6 inches.

Select all statements that are true about the two prisms.

A: They have the same volume.
B: They have the same number of faces.
C: More inch cubes can be packed into Prism A than into Prism B.
D: The two prisms have the same surface area.
E: The surface area of Prism B is greater than that of Prism A.

  • Answers

    A: True. Both Prism A and Prism B have a volume of 6 cubic inches.

    B: True. Both are rectangular prisms, which must have 6 faces.

    C: False. Both Prism A and Prism B have the same volume, and therefore can hold the same number of inch cubes.

    D: False. Surface area of Prism A = 2(3 × 2) + 2(3 × 1) + 2(2 × 1) = 22 in2.
    Surface area of Prism B = 4(6) + 2(1) = 26 in2.

    E: True. As seen in the explanation of D above, Prism B has a greater surface area than Prism A.


6. a. What polyhedron can be assembled from this net?

A net consisting of 3 rectangles in a row with 2 triangles above and below the last rectangle.

b. What information would you need to find its surface area? Be specific, and label the diagram as needed.

  • See Possible Answers

    a. This net can be assembled into a triangular prism.

    b. The side lengths of the rectangles (s1, s2, s3, and s4), and the height of the triangles h is needed. s1, s2, and s3 are also equal to the side lengths of the triangles.

    A net consisting of 3 rectangles in a row with 2 triangles above and below the last rectangle. Sides of the rectangles have been labeled S1, S2, S3, and S4. A dashed line labeled h indicates the height of one of the triangles.


7. Find the surface area of this triangular prism. All measurements are in meters.

A triangular prism. The side lengths of the triangular bases are 1, 1, and 1.2. The side lengths of the rectangular faces are 1 and 1.2. The height of the triangular bases is 0.8.

  • Answers

    Area of 2 triangular bases = 2 × ½ × 1.2 m × 0.8 m = 0.96 m2
    Area of the bottom rectangular face = 1.2 m × 1.2 m = 1.44 m2
    Area of two sloping faces = 2 × 1 m × 1.2 m = 2.4 m2
    Surface area = 0.96 m2 + 1.44 m2 + 2.4 m2 = 4.8 m2



The Open Up Resources math curriculum is free to download from the Open Up Resources website and is also available from Illustrative Mathematics.

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