At a glance. One lesson of about 60 minutes. Students find the mean from a list and from a frequency table, understand it as a fair share, and use the total to find a missing value.
Calculator: no calculator. Totals are chosen to divide cleanly, so the arithmetic stays on the idea. A calculator would not add anything here.
What students will learn
Know
- mean = total ÷ number of values.
- From a frequency table: total of (value × frequency), divided by total frequency.
- The mean lies between the smallest and largest value and need not be one of the values.
Understand
- The mean is the fair share: what every value would be if the total were shared equally.
- It is the balance point: the amount above it equals the amount missing below it.
- The total is mean × number of values, so a missing value can be found.
Do
- Find the mean of a list and of a frequency table.
- Use the total to find a missing value.
- Combine two groups by their totals, not by averaging their means.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: using the mean on lists and tables (questions 1 to 5).
- Criterion B. Investigating patterns: the level figure. Ask students to find the rule linking the two totals as the level moves one step.
- Criterion C. Communicating: showing the total and the division as separate steps, and explaining why a combined mean is not the mean of the means (question 7).
- Criterion D. Applying mathematics in real-life contexts: goals, tea and books as fair shares, and what a mean of 1.25 siblings does and does not mean.
The one thing to do with the figure
Ask for a prediction first: if 36 books were shared out equally among 6 students, roughly how many each? Many will say 6. Then move the level and ask why 4 is too low: what is there too much of above it?
Ask them to say what happens to the books above and the gaps below as the level rises by one: the books above fall by the number of bars still above it, and the gaps below rise by the number of bars at or below it. Balance is where those trade evenly.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook and level figure on the board; predictions first. |
| 10 min | Key card and the test-scores worked example; students find the total before dividing. |
| 10 min | Frequency table example; point out the common-mistake card with the 6.25. |
| 5 min | Question 1, the check. Rows or frequencies? Hands up for each letter. |
| 15 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 9. |
The answers
| Warm-up | 8; 6.75; 15. |
|---|---|
| Worked example 1 | Total 75, mean 15; median 17 and mode 17. |
| Worked example 2 | Total 25 siblings among 20 students: 1.25. |
| Faded example | Total 120 km, 88 km already, day 4 is 32 km. |
| 1. Mean from a frequency table | B, 2.1. A ignores the frequencies; C divides by the rows; D stops at the total. |
| 2. Mean of a list | 9. 45 ÷ 5. |
| 3. Mean in context | 2. 8 ÷ 4. |
| 4. Mean from a frequency table | 2.55. 51 ÷ 20. |
| 5. A missing value from a mean | 10. 70 − 60. |
| 6. Which list has the mean | B. A has median 10, C has median and mode 10, D mean 13.75. |
| 7. Combining two groups | 5.4. (120 + 150) ÷ 50. |
| 8. How a new value moves the mean | 16. 72 − 56. |
| 9. What a mean tells you | A. B and D treat the mean as a value in the list; C confuses it with the median. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| 2 or 7 in question 1 | Divided by the number of rows, or ignored the frequencies. | Ask: how many families are in the table? How many pets are there altogether? |
| 45 in question 2 | Stopped at the total. | Ask what the mean of one value would be, then of two. |
| 2.5 in question 3 | Gave the median. | Ask which measure shares the total equally. |
| 2.5 or 12.75 in question 4 | Averaged the values, or divided by the rows. | Draw the table as a list of 20 cups, one per person. |
| 14 or 60 in question 5 | Gave the mean or the total of the four. | Ask what total the mean says five numbers must have. |
| 5.5 in question 7 | Averaged the two means. | Ask which class has more students, and so should count for more. |
| 8 or 72 in question 8 | Gave the new mean, or the new total. | Ask for the total before and after, then subtract. |
Questions worth asking
- What does a mean of 1.25 siblings mean when nobody has 1.25 siblings? It is a fair share, a summary of the whole group, not a real student's value.
- Can the mean be bigger than every value? No: the mean lies between the smallest and largest value.
- Why is the mean of the means not the mean of everyone? The groups can be different sizes, so each mean must count in proportion to the number in its group. Find each group's total (mean × size), add the totals, then divide by everyone.
- What happens to the mean when one more value equal to the mean is added? It stays the same: the new value takes exactly its fair share.
Differentiation
A way in for anyone stuck
- Use counters or linking cubes for the six bars so students can physically move cubes from tall bars to short ones.
- Give a blank frequency table with the third column ready, so the structure is not a barrier.
- Keep totals small (under 60) for the first two practice questions.
Stretch
- Question 8, then: if the mean of 8 numbers is 7, and a new number makes the mean 7 again, what is the new number? (7: it takes exactly its fair share.)
- Ask students to make a list of 5 numbers with mean 8, median 6 and mode 4, then check it.
Students learning in English as an additional language
Say each word while pointing to the figure or the table. “Share equally” is the key phrase, so act it out with real objects.
| Word | Plain meaning | Example |
|---|---|---|
| mean | the total shared out equally, found by total ÷ number of values | mean of 2, 4, 9 is 15 ÷ 3 = 5 |
| total | all the values added together | 3 + 8 + 4 = 15 |
| frequency | how many times a value appears | 5 students have 2 siblings |
| fair share | what each would get if everything were shared equally | 12 sweets among 4 friends: 3 each |
| balance | the level where the amount above equals the gaps below | for 2, 4 and 9 the level 5 balances: 4 above, and gaps of 3 and 1 below |
Sentence starters for explaining aloud or in writing:
- The total is … and there are … values, so the mean is …
- I know the mean is …, so the total must be … × … = …
- This answer is too big because the mean cannot be bigger than the …
Two things not to say
“The average is the mean.” True in everyday talk, but it hides the other two. Say “the mean” when you mean it.
“Add them up and divide.” as a chant with no picture. Say why: sharing the total equally.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- ListFind the mean of 4, 8, 6, 10, 12.
8. The total is 40 and 40 ÷ 5 = 8. - FrequencyValues 1, 2, 3 have frequencies 1, 1, 2. Find the mean.
2.25. The total is 1 + 2 + 6 = 9 among 4 values: 9 ÷ 4.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- Missing valueThe mean of 6 numbers is 9. Five of them add to 47. What is the sixth?
7. The total is 54 and 54 − 47 = 7. - TableA table shows scores of 0 (3 times), 1 (7 times) and 2 (10 times). Find the mean.
1.35. The total is 0 + 7 + 20 = 27 among 20 values.
3Challenge
Reasoning, generalizing and working backwards.
- CombiningTen boys have a mean height of 140 cm and fifteen girls have a mean of 150 cm. What is the mean for all twenty-five?
146 cm. The total is 1400 + 2250 = 3650 and 3650 ÷ 25 = 146. - From lesson 1Find the median and the mean of 2, 3, 3, 4, 18. Which is closer to most of the values?
The median is 3 and the mean is 6. The median is closer, because the 18 pulls the mean up.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The level slider works with the keyboard (tab, then the arrow keys) and by touch. Under the figure, “Show me the steps” gives the same states in words.
No device? Use six towers of linking cubes and move cubes from the tall ones to the short ones until all are level.