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Painting the Fence: Sections, Boards, and Equivalent Fractions

See why multiplying the top and bottom by the same number names the same amount.

Sections into boards

Every fence section is split into boards of the same size. Fill in the blanks.

1.
Kai's fence has 5 equal sections. He painted 3 sections. Then each section is split into 2 equal boards.

Sections:
Boards:
The fence has 5 × 2 =   boards in all.
Kai painted 3 × 2 =   boards.
So 3/5 =  / 

2.
Mr. Patel's fence has 2 equal sections. 1 section is painted blue. Each line shows the same fence split into smaller boards. Write the fraction of the boards that are blue.

Each section is 1 board: 1/2 of the boards are blue.
Each section is 2 boards: 2/4 of the boards are blue.
Each section is 3 boards:  /  of the boards are blue.
Each section is 4 boards:  /  of the boards are blue.
Each section is 6 boards:  /  of the boards are blue.

3.
Look at Kai's fence in question 1 again. Circle every sentence that is true.

There are 2 times as many boards in all as sections.More of the fence is painted now than before.There are 2 times as many painted boards as painted sections.Each board is the same size as a section.Each board is half as wide as a section.3/5 and 6/10 name the same amount of fence.

Split or group

Multiply or divide the top and bottom numbers by the same number to find equivalent fractions.

4.
Rosa's fence has 12 equal boards. She painted 9 of them. Every 3 boards side by side make one section.

Boards:
12 boards ÷ 3 =   sections in all
9 painted boards ÷ 3 =   painted sections
So 9/12 =  / 

5.
Each fraction tells how many boards of a fence are painted. Group the boards into bigger sections. Match each fraction to an equivalent fraction.
  1. 10/12
  2. 6/8
  3. 4/10
  4. 2/12
  1. A. 2/5
  2. B. 1/6
  3. C. 5/6
  4. D. 3/4
6.
Find the missing number in each pair of equivalent fractions.

4/5 =  /10
5/6 = 10/ 
3/10 =  /100
8/12 = 2/ 

Fix it and explain

Read each fence story. Decide what is true and explain your thinking.

7.
Jordan painted 2/3 of a fence. Then each section is split into 2 equal boards. Jordan says, “Now 4/3 of the fence is painted.” What mistake did Jordan make?
  • A. He should have added 2 to the top and bottom to get 4/5.
  • B. Nothing is wrong. 4/3 is the same amount as 2/3.
  • C. He multiplied only the top number. There are also twice as many boards in all, so 2/3 = 4/6.
  • D. He should have multiplied only the bottom number to get 2/6.
8.
Each pair of fractions tells how much of the same fence is painted. Sort the pairs.
1/4 and 3/122/5 and 3/63/5 and 6/55/6 and 10/121/2 and 1/47/10 and 70/100
Same amount paintedDifferent amounts painted
9.
Lena painted 1 of the 4 equal sections of her fence. Which fence has the same fraction of its boards painted?
  • A. 2 of 6 boards painted
  • B. 4 of 8 boards painted
  • C. 1 of 8 boards painted
  • D. 3 of 12 boards painted
10.
Mrs. Okafor painted 2/3 of her fence. Her son says the fence is 8/12 painted. Use sections and boards to explain why both fractions name the same amount of fence.