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What Division Means · Grade 3
Division Is a Question
12 ÷ 3 is not just a problem to solve.
It is a question you can ask five different ways.
It is a question you can ask five different ways.
1
Equal groups
If 12 cookies go on 3 plates, how many on each plate?
2
Rows and columns
If 12 chairs make 3 rows, how many chairs in each row?
3
Times as much
If 12 is 3 times as big as my number, what is my number?
4
Fair rates
If 3 pencils cost 12 dollars, how much is one pencil?
$ → 1
5
Combinations
If 3 shirts make 12 outfits, how many pairs of pants do I have?
One more secret: every division fact is a multiplication fact turned around.
3 × 4 = 12, so 12 ÷ 3 = 4.
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What Division Means · 1 of 5
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Equal Groups: Sharing
12 ÷ 3 can ask: "12 cookies, 3 plates. How many cookies on each plate?"
1.Here are 12 cookies and 3 plates. Draw a line from each cookie to a plate. Deal them out like cards: one for you, one for you, one for you. Every plate must get the same.
Each plate gets cookies. So 12 ÷ 3 =
2.Your turn to be the picture-maker. Draw 10 strawberries shared by 2 friends. Then finish the sentence.
10 ÷ 2 =
Sharing happens at every dinner table.
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What Division Means · 1 of 5, part two
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Equal Groups: Making Bags
The SAME problem, 12 ÷ 3, can also ask:
"12 apples, 3 in each bag. How many bags can I fill?"
"12 apples, 3 in each bag. How many bags can I fill?"
1.Here are 12 apples. Draw a loop around every group of 3. Each loop is one full bag. Then count your loops.
I can fill bags. So 12 ÷ 3 = again!
2.Think about it. On the last page, 12 ÷ 3 = 4 meant 4 cookies on each plate. On this page, 12 ÷ 3 = 4 means 4 bags. Same answer, different question. Circle the one where you already know how many groups:
cookies on 3 plates
apples, 3 to a bag
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What Division Means · 2 of 5
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Rows and Columns
15 ÷ 3 can ask: "15 chairs in 3 equal rows. How many chairs in each row?"
1.The chairs are set up. Count how many are in one row, then check: does every row match?
Each row has chairs. 15 ÷ 3 =
2.Now build your own. Draw 8 dots in 2 equal rows in the space below. Big dots! Use the whole box.
8 ÷ 2 =
The magic of an array: turn it sideways and it still works. 3 rows of 5 or 5 rows of 3, the chairs never change.
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What Division Means · 3 of 5
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Times as Much
12 ÷ 3 can ask: "Mia has 12 marbles. That is 3 times as many as Leo. How many does Leo have?"
1.Mia's bar shows 12. It is cut into 3 equal pieces because she has 3 times Leo's amount. One piece is exactly Leo's share. Write the number that goes in each piece, then fill in Leo's bar.
Mia · 12 marbles
Leo · ? marbles
Leo has marbles. 12 ÷ 3 =
2.Draw your own bars: Dad is 2 times as old as Jo, and Dad is 16. Draw Dad's bar cut into 2 pieces, then Jo's bar under it.
Jo is years old.
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What Division Means · 4 of 5
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Fair Rates: How Much for One?
12 ÷ 3 can ask: "3 pencils cost 12 dollars. How much for ONE pencil?"
1.Here are 12 dollar coins. Share them fairly under the 3 pencils: draw each coin's path, or just draw the coins under each pencil.
$
$
$
$
$
$
$
$
$
$
$
$
pencil 1
pencil 2
pencil 3
One pencil costs dollars. 12 ÷ 3 =
2.A snail crawls 8 feet in 2 hours, moving steadily the whole time. How far in ONE hour?
8 ÷ 2 = feet each hour
"How much for one?" is division at the store.
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What Division Means · 5 of 5
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Combinations: Outfits
6 ÷ 3 can ask: "I can make 6 outfits with 3 shirts. How many pairs of pants do I own?"
1.Every box in this chart is one outfit: a shirt matched with pants. Fill in the missing pants labels, then count.
6 outfits ÷ 3 shirts = pairs of pants
2.Ice cream shop: 2 cones and some flavors make 8 different treats. Draw a chart like the one above (cones down the side, flavors across the top).
8 ÷ 2 = flavors
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rootsofreason.org
What Division Means · Grade 3
Answer Key and Teacher Notes
Sharing (page 2)
1. Each plate gets 4; 12 ÷ 3 = 4 2. 10 ÷ 2 = 5
Making bags (page 3)
1. 4 bags 2. Circle cookies on 3 plates (the number of groups is known; the bag problem knows the group SIZE instead)
Rows and columns (page 4)
1. 5 chairs per row 2. 8 ÷ 2 = 4 (2 rows of 4)
Times as much (page 5)
1. Each piece is 4; Leo has 4 2. Jo is 8 (16 ÷ 2)
Fair rates (page 6)
1. 4 dollars each 2. 4 feet each hour
Combinations (page 7)
1. 2 pairs of pants 2. 4 flavors (2 cones × 4 flavors = 8 treats)
The set works as five mini anchor charts. The pair to slow down on is pages 2 and 3: sharing (how many in each group) versus measuring out (how many groups). A student who can say which question a word problem is asking, before computing anything, has the concept. The drawing boxes are the assessment; the number boxes are just the receipt.