Unit 1, Lesson 17: Squares and Cubes
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1 Unit 1, Lesson 17: Squares and Cubes Lesson Goals Use units with exponents for area and volume. Understand and explain the meaning of the exponent in the formula. Calculate the volume of a cube given its edge length. Required Materials snap cubes 17.1: Perfect Squares (5 minutes) Setup: Introduce "perfect squares." 2 minutes of quiet think time, followed by a whole-class discussion. Unit 1: Area and Surface Area, Lesson 17: Squares and Cubes 1
2 Student task statement 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? Possible responses 1. Answers vary. For example, here are some squares: 9, 25, 4, 49, 100 and non-squares: 1/2, 2, 3, square inches 3. 8 centimeters Anticipated misconceptions If students do not recall what the abbreviations km, cm, and sq stand for, provide that information. Students may divide 64 by 2 for the third question. If students are having trouble with this, ask them to check by working backwards, i.e. by multiplying the side lengths to see if the product yields the given area measure. Unit 1: Area and Surface Area, Lesson 17: Squares and Cubes 2
3 17.2: Building with 32 Cubes (Optional, 15 minutes) Setup: Students in groups of snap cubes for each group minutes to build a cube and answer questions. Unit 1: Area and Surface Area, Lesson 17: Squares and Cubes 3
4 Student task statement 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. Possible responses How many snap cubes did you use? 2. What is the edge 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 units 3. 9 square units cubic units Anticipated misconceptions Students may neglect to write units for length or area and may need a reminder to do so. When determining area, students may multiply a side by two instead of squaring it. When determining volume, they may multiply a side by three instead of cubing it. If this happens, ask them to count individual squares so that they can see that there is an error in their reasoning. Unit 1: Area and Surface Area, Lesson 17: Squares and Cubes 4
5 17.3: Perfect Cubes (10 minutes) Setup: Students in groups of 2. Introduce "perfect cubes." A few minutes of quiet think time, followed by partner and whole-class discussions. Unit 1: Area and Surface Area, Lesson 17: Squares and Cubes 5
6 Student task statement 1. The number 27 is a perfect cube. Find four other numbers that are perfect cubes and two numbers that are not perfect cubes. Possible responses 1. Answers vary: 1, 8, 64, 125, 216, 1,000. Non-cubes: 2, 3, or 64 cubic cm 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 units. What is its volume? 3. or 1,000 cubic inches 4. cubic units Anticipated misconceptions Watch for students using square units instead of cubic units. Remind them that volume is a measure of the space inside the cube and is measured in cubic units. Students may multiply by 3 when finding the volume of a cube instead of multiplying three edge lengths (which happen to be the same number). Likewise, they may think a perfect cube is a number times 3. Suggest that they sketch or build a cube with that edge length and count the number of unit Unit 1: Area and Surface Area, Lesson 17: Squares and Cubes 6
7 cubes. Or ask them to think about how to find the volume of a prism when the edge lengths are different (e.g., a prism that is 1 unit by 2 units by 3 units). Unit 1: Area and Surface Area, Lesson 17: Squares and Cubes 7
8 17.4: Introducing Exponents (15 minutes) Setup: Ask students whether they have heard of the term "exponent." Introduce the term. Explain use of exponents of 2 and 3 with numbers and in units of measurement. Students in groups of minutes of quiet think time, followed by partner and whole-class discussions. Unit 1: Area and Surface Area, Lesson 17: Squares and Cubes 8
9 Student task statement Make sure to include correct units of measure as part of each answer. Possible responses A square has side length 10 cm. Use an exponent to express its area. 2. The area of a square is sq in. What is its side length? 3. The area of a square is 81 m 2. 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 cm 3. What is its edge length? 6. A cube has edge length units. Use an exponent to write an expression for its volume inches cm 6. units or cubic units Anticipated misconceptions Upon seeing in the fourth question, some students may neglect to interpret the question, automatically calculate, and conclude that the edge length is 216 cm. Ask them to check their answer by finding the volume of a cube with edge length 216 cm. Unit 1: Area and Surface Area, Lesson 17: Squares and Cubes 9
10 Are you ready for more? Find some numbers that are both perfect squares and perfect cubes, and write them using the notation. For example, 1 is both a perfect square because and a perfect cube because, and we can write it like this: Possible Responses The smallest examples are 0, 1, 64, and 729. Lesson Synthesis (5 minutes) Review the language and notation for squaring and cubing a number. Remind students we use this notation for square and cubic units, too. 17.5: Exponent Expressions (Cool-down, 5 minutes) Setup: None. Unit 1: Area and Surface Area, Lesson 17: Squares and Cubes 10
11 Student task statement 1. Which is larger, 5 2 or 3 3? 2. A cube has an edge length of 21 cm. Use an exponent to express its volume. Possible responses 1. and, so 3 3 is larger than cm 3 or 21 3 cubic centimeters Anticipated misconceptions Students may perform calculations on the second question, which is not necessary since the target is an expression with an exponent. Unit 1: Area and Surface Area, Lesson 17: Squares and Cubes 11
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Student Probe Measurement Using Standard Units 6 7 8 9 List the length of each colored line segment: blue, red, green. Explain how you found your answers. Lesson Description The lesson is intended to help