Section B Practice Problems

Problem 1

  1. Draw a rectangle on the grid whose area can be represented by 5×75 \times 7.
  2. How does your rectangle represent the expression 5×75 \times 7?

An empty grid.

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Solution

  1. Sample response:

    A grid with a 5 by 7 square.

  2. Sample response: There are 5 rows of square units that cover the rectangle and there are 7 square units in each row.

Problem 2

Here are 2 squares. One is a square centimeter, and the other is a square inch.

Which square is a square centimeter? Which square is a square inch? Explain how you know.

2 squares labeled A and B.

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Solution

Square A is a square centimeter, and Square B is a square inch. Sample response: I know because a centimeter is smaller than an inch.

Problem 3

For each object, decide if you would use square centimeters, square inches, square feet, or square meters to measure area. Explain your reasoning.

  1. a baseball field
  2. a table top
  3. a cell-phone screen
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Solution
  1. Square meters or square feet. Sample response: The baseball field is large.
  2. Square feet. Sample response: It would take a lot of square inches.
  3. Square inches or square centimeters. Sample response: The phone screen is too small to use feet or meters.

Problem 4

The sides of the rectangle are marked in centimeters.

What is the area of the rectangle? Explain your reasoning.

A rectangle marked in centimeters.

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Solution

28 square centimeters. Sample response: The side lengths are 4 centimeters and 7 centimeters, so there are 4 rows of 7 centimeter squares. The area is 4×74 \times 7 or 28 square centimeters.

Problem 5

  1. Use a centimeter ruler.

    1. Find the area of the rectangle in square centimeters.

      A rectangle.

    2. Draw a rectangle the area of which is 18 square centimeters.
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Solution
  1. 24 square centimeters. Sample response: The rectangle is 8 cm wide and 3 cm tall so the area is 3×83 \times 8 or 24 square centimeters.
  2. Sample response: Student uses the ruler to make a rectangle that is 3 cm by 6 cm or 2 cm by 9 cm or 1 cm by 18 cm.

Problem 6

Tyler has 40 carpet squares with sides of 1 foot. He wants to use all the squares to make a rectangle-shaped carpet.

The longest side of the carpet cannot be more than 12 feet. What could be the side lengths of Tyler's carpet?

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Solution

Sample responses: 5 feet and 8 feet, 10 feet and 4 feet

Problem 7

  1. Describe some patterns you see for the numbers in the table.

    Multiplication table with numbers in the 3 column, the 6 column, and the 9 column.

  2. Write some equations that show 1 of the patterns you found. Explain or show why that pattern happens.
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Solution

Sample responses:

  1. The numbers in the 3 column increase by 3 each time I go down one row while the numbers in the 6 column increase by 6 and the numbers in the 9 column increase by 9. If I add the numbers in the 3 and 6 columns I get the number in the 9 column.
  2. 3+3=63 + 3 = 6, 6+3=96 + 3 = 9, 9+3=129 + 3 = 12 Each time I go down the column with '×3\times 3,' I'm adding 1 group of 3 more.

Problem 8

  1. Find in your classroom or at home an object or a space that is shaped like a rectangle. Describe the rectangle.
  2. Would you use square centimeters, square inches, square feet, or square meters to measure the area of the rectangle? Explain your reasoning.
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Solution

Sample response:

  1. The carpet in my room is a rectangle. It’s not very long, and is closer to a square.
  2. I would use square feet to measure it. It’s too large for square inches and I think it’s too small for square meters.

Problem 9

What patterns do you notice in the 3 filled-in columns of the multiplication table?

Multiplication table with 2, 4, and 5 columns filled in.

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Solution

Sample responses: To get 4×64 \times 6, for example, I can subtract 6 from 5×65 \times 6. This same pattern is true for the other numbers in 4 and 5 the columns. Also, I can double 2×62 \times 6 to get 4×64 \times 6, and this pattern is also true for the other numbers in the 2 and 4 columns.

Problem 10

Mai picks a mystery number that is less than 30. She says that she can draw, on this grid, 3 rectangles with different side lengths, whose areas in square units are the same as her mystery number.

Empty grid.

What could be Mai’s mystery number? Explain or show your reasoning.

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Solution

Sample responses:

  • 24. The three rectangles could be 4 by 6, 8 by 3, and 12 by 2.
  • 12. The three rectangles could be 4 by 3, 6 by 2, and 12 by 1.

No other number less than 30 has 3 different rectangles that will fit on the 12-by-12 grid.