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What Division Means · Set 5 · Grade 3
Combinations: The Matching Machine
Shirts and pants make outfits. Cones and flavors make treats.
When you know the matches, division finds a missing choice.
When you know the matches, division finds a missing choice.
pants A
pants B
red shirt
outfit 1
outfit 2
blue shirt
outfit 3
outfit 4
green shirt
outfit 5
outfit 6
every shirt meets every pants
3 × 2 = 6
shirts × pants = outfits
6 ÷ 3 = 2
outfits ÷ shirts = pants
6 ÷ 2 = 3
outfits ÷ pants = shirts
The chart is just an array in disguise: shirts are the rows, pants are the columns. Set 2's tricks all work here.
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Combinations · Set 5
Name
Date
Build the Matching Chart
The ice cream stand has 2 cones and 3 flavors.
How many different treats can it make?
How many different treats can it make?
1.Fill every empty box with its treat. Write it like "cake + mint". The first one is done for you.
mint
berry
mango
cake cone
cake + mint
sugar cone
The stand can make different treats. 2 × 3 =
2.A new flavor arrives! Now there are 4 flavors and still 2 cones. Do NOT draw the new chart. Just picture one more column. How many treats now?
2 × 4 = treats
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Combinations · Set 5
Name
Date
The Missing Choice
Now division gets its turn. You know the TOTAL of matches.
You know ONE of the choices. Find the other.
You know ONE of the choices. Find the other.
1.A sandwich shop brags: "12 different sandwiches!" It has 3 breads. How many fillings must it have? Finish the chart: label the bread rows, then split the 12 sandwiches evenly among them and count the columns you need.
filling ___
filling ___
filling ___
filling ___
bread ___
bread ___
bread ___
12 sandwiches ÷ 3 breads = fillings
2.A game has 10 characters made from hats and faces. There are 5 faces. How many hats? No chart needed if you can see it.
10 ÷ 5 = hats
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Combinations · Set 5
Name
Date
Branch It Out
Another way to see matches: a tree.
Each choice branches into all the next choices.
Each choice branches into all the next choices.
1.Trace the tree with your finger: 2 cones, each branching to 3 flavors. Count the ends.
cake cone
mint
berry
mango
sugar cone
mint
berry
mango
The tree has ends. Same 6 as the chart!
2.Draw the tree for this one: 8 outfits come from 2 jackets, each branching into some shirts. Start with the 2 jackets and figure out how many branches each one needs.
8 ÷ 2 = shirts
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Combinations · Set 5
Name
Date
Design Your Own Menu
You own a food truck. Your menu must offer
exactly 12 different meals. You decide how.
exactly 12 different meals. You decide how.
1.Pick your two categories (like tacos and salsas, or noodles and toppings). Decide how many of each so the matches make exactly 12. Draw your menu chart with real names for everything.
2.Prove your menu works:
I chose and .
× = 12, and 12 ÷ =
Hint: there are three ways to hit 12 with two categories. 2 and 6, 3 and 4, or 1 and 12 (a boring truck).
Roots of Reason
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What Division Means · Set 5 · Grade 3
Answer Key and Teacher Notes
Build the Matching Chart (page 2)
1. 6 treats (cake/sugar × mint/berry/mango) 2. 2 × 4 = 8
The Missing Choice (page 3)
1. 12 ÷ 3 = 4 fillings (all four columns used) 2. 10 ÷ 5 = 2 hats
Branch It Out (page 4)
1. 6 ends 2. Each jacket branches into 4 shirts; 8 ÷ 2 = 4
Design Your Own Menu (page 5)
Answers vary: 2 × 6, 3 × 4, 4 × 3, 6 × 2, or the boring 1 × 12. Check the chart really shows every match and the division sentence uses their own numbers.
This closes the loop on the whole series: the combination chart IS the array from Set 2, the "how many treats now?" question is Set 3's stretch, and the menu design forces a student to run multiplication and division as one machine. A student who designs a working 3-by-4 menu and can say why 12 ÷ 3 = 4 has all five meanings in hand.