Section A Practice Problems

Problem 1

Find the area of each rectangle. Explain your reasoning.

  1.  

    Area diagram. 3 rows of 6.

  2.  

    Diagram. Rectangle partially partitioned to show 4 rows of 7 same sized squares.

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Solution
  1. 18 square units. There are 3 rows of 6, so that’s 3× 63 \times 6 or 18 square units.
  2. 28 square units. It takes 4 rows of 7 squares to cover the rectangle without gaps or overlaps, so that’s 4× 74 \times 7 or 28 square units.

Problem 2

On the grid, draw a rectangle whose area is represented by each expression.

  1. 3× 53 \times 5
  2. 4× 84 \times 8
blank grid of 12 by 12.
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Solution
Grid. Two rectangles.

Problem 3

Tyler wants to build a rectangle with an area of 20 square units using square tiles.

  1. Can Tyler build a rectangle with a width of 4 units? Explain or show your reasoning.
  2. Can Tyler build a rectangle with a width of 6 units? Explain or show your reasoning.
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Solution
  1. Yes, a rectangle with a width of 4 units and a length of 5 units has 20 square units in it.
  2. No, if the width is 6 and the length is 3, that means using 18 square tiles. If the length is 4, that means using 24 square tiles.

Problem 4

List the possible whole-number side lengths of rectangles with an area of 32 square units. Explain or show how you know your list is complete.

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Solution
1 unit by 32 units, 2 units by 16 units, 4 units by 8 units. Those are all of the possible rectangles because there are no other factors of 32.

Problem 5

List the factor pairs of each number. Is each number prime or composite? Explain or show your reasoning.

  1. 37
  2. 27
  3. 77
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Solution
  1. Prime, because the only factor pair is 1 and 37.
  2. Composite, because it has the factor pairs 1 and 27 and also 3 and 9.
  3. Composite, because it has the factor pairs 1 and 77 and also 7 and 11.

Problem 6

  1. Use multiplication facts to calculate the area of each rectangle.

    Grid with 4 rectangles labeled A, B, C, D.

  2. How did you use multiplication facts to calculate the areas?
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Solution
  1. Rectangle A is 5 units by 7 units, so its area is 5×75 \times 7 or 35 square units. Rectangle B is 8 units by 4 units, so its area is 32 square units. Rectangle C is 9 units by 7 units, so its area is 63 square units. Rectangle D is 3 units by 7 units, so its area is 21 square units.
  2. I multiplied the side lengths, and that calculation uses multiplication facts. 

Problem 7

  1. You want to arrange all of the students in your class in equal rows.

    1. How many rows can you have? How many students would be in each row?
    2. What if you add the teacher to the arrangement? How would your rows change?

  2. Find some objects at home (such as silverware, stuffed animals, cards from a game) and decide how many equal rows you can arrange them in and how many objects are in each row.

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Solution

Answers vary.

Problem 8

What is the largest prime number you can find? Explain or show why it is a prime number.

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Solution
Sample response: 97 is prime because its only factor pair is 1 and 97. I tried all of the other numbers and could not find another factor pair.