seegongsik
Grade 6 (age 11-12)

grade 6 Counting Cases

Counting cases means finding, without missing any, how many different ways something can turn out. Toss one coin and there are 2 ways, heads or tails; roll one die and there are 6, from 1 to 6. When choices are made one after another, like picking a top and then a bottom, multiplying the number of cases at each step counts every way without missing one. In the first picture, change the number of coins to count cases on a tree diagram or look at the faces of a die; in the second, count outfits by joining tops and bottoms with lines.

What are cases?

🎲 Every way it can turn out
①Toss one coin and you get heads or tails: 2 cases
②Toss 2 coins and each of the first coin's 2 cases pairs with the second coin's 2, so 2 × 2 = 4 cases; count the ends of the branches on the tree diagram
③Each extra coin doubles the number of cases: 8 for 3 coins and 16 for 4
④One die has 6 cases, from 1 to 6, and any one particular face is 1 of them
2

Counting by multiplying

👕 Counting outfits
①With 3 tops and 3 bottoms, each top can be paired with each of the 3 bottoms
②So there are 3 × 3 = 9 outfits, and the picture has 9 lines
③Add one more top and you add 3 more lines: 12 outfits for 4 tops and 15 for 5
④When choices are made one after another, you multiply the number of cases at each step; this is called the multiplication rule
3

Summary

Addition rule
A or B = (cases for A) + (cases for B)
When only one of two things can happen, add their cases
Multiplication rule
A and then B = (cases for A) × (cases for B)
When things happen one after another, multiply their cases
Cases for n coins
2n (multiply 2 by itself n times)
1 coin gives 2 cases, 2 coins 4, 3 coins 8 and 4 coins 16
🎯 At a glance
①Number of cases: how many different ways something can turn out
②Joined by "or": add (addition rule)
③Joined by "and" or "one after another": multiply (multiplication rule)
④A tree diagram helps you count without missing any
⑤n coins give 2 multiplied n times; one die gives 6

Examples & Unit Review

Example 1
You have 3 tops and 3 bottoms (a skirt, trousers and shorts). How many different outfits can you wear?
1
You choose a top and then a bottom, so use the multiplication rule: 3 × 3.
2
3 × 3 = 9, so 9 outfits.
▸ 9 ways
This is the starting setting of the second picture, which has 9 lines joining tops and bottoms.
Example 2
You toss 3 coins at once. How many different results are possible?
1
One coin has 2 cases, heads or tails, and there are 3 coins, so work out 2 × 2 × 2.
2
2 × 2 × 2 = 8, so 8 cases.
▸ 8 outcomes
Set the coins to 3 in the first picture and the tree diagram splits into 8 ends.
Unit Review
When you roll one die, in how many cases is the number even?
①3
②2
③6
④4
▸ ① 3
1
Of the faces 1, 2, 3, 4, 5 and 6, the even ones are 2, 4 and 6.
2
There are 3 even faces, so 3 cases.
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Graph Interpretation

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