Lesson 7 of 8 · Grades 3 to 5
Mean as fair share
The mean is the fair share. Picture towers of different heights: move blocks from the tall towers to the short ones until every tower is the same height.
Sharing fairly
Imagine three children have 15 stickers between them, but not equally. One has 5, one has 7 and one has 3. To share the stickers fairly, you gather them all together and deal them out again, one at a time, until everyone has the same. The amount everyone ends up with is the mean, or the fair share.
The same idea works for days, teams or stacks. Add every amount to get the total, then divide the total by the number of groups. The answer is the value that would make every group equal.
Towers that need levelling
A tower chart makes the fair share easy to see. Each tower is a group, and the height of the tower is its value. These towers show snack boxes packed on four days.
Show the data as a table
| Tower | Count |
|---|---|
| Mon | 6 |
| Tue | 2 |
| Wed | 4 |
| Thu | 4 |
The fair share is 4.
The total is 6 + 2 + 4 + 4 = 16 boxes. Picture moving one box from Monday down to Tuesday, then another: both towers become 4. The four days each end up with 4 boxes, so the fair share is 16 ÷ 4 = 4. The dashed line in the chart sits at that height.
Adding first, dividing second
Always find the total before you divide. For another example, three teams scored 5, 7 and 3 points. The total is 5 + 7 + 3 = 15 points. There are three teams, so the fair share is 15 ÷ 3 = 5 points. Moving the two points above the line from the middle team to the lowest team levels all three teams to 5.
Division only gives a whole number answer when the total is shared exactly. In this course the totals always divide evenly, so every fair share is a whole number. Later you will meet data where the fair share is a decimal, but the method stays the same: add, then divide.
Checking the fair share
The fair share is the number that makes every group equal, so you can check it by multiplying. If four days each pack 4 boxes, the total is 4 × 4 = 16 boxes, which matches the numbers you started with. If the check gives a different total, the fair share is wrong, or the addition was.
The fair share is also called the mean. The mean can sit between the values in the data, so it does not have to be one of them.
Show the data as a table
| Tower | Count |
|---|---|
| Red | 5 |
| Blue | 7 |
| Green | 3 |
The fair share is 5.
In this chart the fair line sits level with the top of the Red tower. Blue is two above the line and Green is two below, so moving one point from Blue to Green would level all three at 5. That two-step move is the same arithmetic as 15 ÷ 3 = 5.
Fair share or total?
Read the question carefully. "How many altogether?" asks for the total, and the answer is 15. "What is the fair share?" asks for the value per group, and the answer is 5. The total is always bigger than the fair share when there is more than one group, so the size of the answer is a useful clue.
Worked example
Four friends have 6, 2, 4 and 4 stickers. What is the fair share for each friend?
Step 1: add the four amounts. 6 + 2 + 4 + 4 = 16 stickers.
Step 2: count the friends. There are 4 friends, so there are 4 shares.
Step 3: divide the total by the number of shares. 16 ÷ 4 = 4 stickers each.
Step 4: check by multiplying back. 4 × 4 = 16, which is the total you started with.
Answer: 4 stickers each.
Watch out for this
Do not divide before you add. The fair share is the total divided by the number of groups, not the first number divided by anything. Also count the groups carefully: three stacks need a division by 3, and four stacks need a division by 4.
Remember
- The fair share is the total shared equally between the groups.
- Add every value, then divide by the number of groups.
- The mean is another name for the fair share.
- Check by multiplying the fair share by the number of groups.
Try it
Answer the five questions below. Work them out first, then open each answer to check.
-
Four days of snack boxes: Monday 6, Tuesday 2, Wednesday 4 and Thursday 4. How many boxes were packed in the four days?
Show answer
16 boxes. 6 + 2 + 4 + 4 = 16.
-
The four days packed 6, 2, 4 and 4 boxes. Share the 16 boxes equally among the four days. What is the fair share?
Show answer
4 boxes. 16 ÷ 4 = 4.
-
Three teams scored 5, 7 and 3 points. How many points were scored altogether?
Show answer
15 points. 5 + 7 + 3 = 15.
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The three teams scored 5, 7 and 3 points. What is the fair share for each team?
Show answer
5 points. 15 ÷ 3 = 5.
-
Five stacks each have 4 cubes. What is the fair share for each stack?
Show answer
4 cubes. Every stack is already level, so the fair share is 4.
Puzzle · Level the towers
Puzzle time
Three stacks have 5, 7 and 3 cubes. If the cubes were shared equally, how many cubes would each stack have?
Show a hint
Add the three stacks first, then divide by the number of stacks.
Show the answer
Answer: 5. 5 + 7 + 3 = 15 cubes, and 15 ÷ 3 = 5 cubes for each stack.