Section A Practice Problems

Problem 1

  1. Partition the rectangle into 4 equal rows and 5 equal columns.
  2. How many small squares are there in the rectangle?

A rectangle.

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Solution

  1. Students are not expected to draw rows and columns that are exactly equal. 

    Diagram. Rectangle partitioned into 4 rows of 5 of the same size squares.

  2. 20 small squares

Problem 2

Is the number of dots in each image even or odd? Explain how you know.

  1.  
    Group of dots.

  2.  
    9 dots arranged 3 by 3.

  3.  
    A group of dots.

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Solution
  1. Even. Sample response: The columns are pairs of circles.
  2. Odd. Sample response: I can pair up two columns and then make one more pair and there is one circle left over.
  3. Even. Sample response: The rows can be paired up.

Problem 3

How many dots are in each array? Explain or show your reasoning.

  1.  
    Array. 3 rows of 5 dots.

  2.  
    18 dots, arranged in 9 by 2 rectangle.

  3.  
    12 dots, arranged in 3 by 4 rectangle.

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Solution
  1. 15 dots. Sample response: There are 3 rows of 5, so that’s 5+5+55 + 5 + 5 circles.
  2. 18 dots. Sample response: There are 2 rows of 9, so that’s 9+99 + 9 circles.
  3. 12 dots. Sample response: There are 3 columns of 4, so that’s 4+4+44 + 4 + 4 circles.

Problem 4

Use the centimeter ruler to find the lengths of Rectangles A and B. Explain your reasoning.

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Solution
Rectangle A is 7 centimeters long and Rectangle B is 6 centimeters long. Sample response: Rectangle A starts at 0 and goes to 7. Rectangle B starts at 3 and goes to 9, and 9−3=69 - 3 = 6.

Problem 5

Which shape is the largest? Which shape is the smallest? Explain your reasoning. You may trace and cut out the shapes if it is helpful.

3 shapes labeled A, B, and C.

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Solution
Shape A is the largest and Shape C is the smallest. Sample response: Shape C can fit inside Shape A and Shape B can fit inside Shape A if we cut off the top and move it. Shape C fits inside Shape A and can fit inside Shape B if we cut it up and move the pieces around.

Problem 6

Lin, Han, and Elena made letters from squares. Put the letters in order from least area to greatest area. Explain your reasoning.

 

Letters L, H, E. L, 8 squares. E, 11 squares. H, 12 squares.

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Solution
L, E, H. Sample response: L has the least area because it uses 8 squares, E uses 11 squares, and H has the largest area because it uses 12 squares.

Problem 7

  1. Find the area of each rectangle drawn on the grid.

    Grid with 3 rectangles labeled A, B, and C.

  2. Can rectangles with different shapes have the same area? Explain your reasoning.
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Solution
  1. Rectangle A has an area of 12 square units. Rectangle B has an area of 12 square units. Rectangle C has an area of 20 square units.
  2. Yes. Sample response: Rectangles A and B have the same area, but their side lengths are not the same.

Problem 8

Find the area of the rectangle. Explain or show your reasoning.

<span>Diagram. Rectangle partitioned into 5 rows of 8 of the same size squares.</span>

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Solution
40 square units. Sample response: Each column has 5 squares, so there are 5, 10, 15, 20, 25, 30, 35, 40 squares total. 

Problem 9

Which shape has the greater area, the green triangle pattern block or the tan rhombus pattern block? Explain your reasoning

A triangle and a rhombus.

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Solution
The rhombus. Sample response: I cut the picture of the rhombus in half and put the two pieces together, and they completely covered the triangle with some extra. 

Problem 10

Here are 2 rectangles.

Diagram. Rectangle partitioned into 6 rows of 8 of the same size squares.

a square gridded with same size squares.

  1. What is the area of the larger rectangle?
  2. What is the area of 3 of the smaller rectangles combined?
  3. Can you completely cover the first rectangle with 3 of the smaller rectangles without cutting them up? Explain or show your reasoning.
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Solution
  1. 48 square units
  2. 48 square units
  3. No. Sample response: Although the area is the same, I can’t rearrange the smaller rectangles into the larger one, without cutting them.

Problem 11

  1. How many different rectangles can you make with 36 square tiles? Describe or draw the rectangles.
  2. How are the rectangles alike? How are they different?
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Solution

Sample response:

  1. I can make rectangles that are 6 by 6, 9 by 4, 12 by 3, 18 by 2 and 36 by 1. 
  2. The shapes are all different. One is a square, and the others are long and skinny. They all have the same area, 36 square units.