At a glance. One lesson of about 60 minutes. It assumes mean and median from a list and the idea of a frequency table. Students find midpoints, estimate a mean from f × midpoint, find a median class from running totals, and say why the answers are estimates.
Calculator: scientific calculator. A scientific calculator's memory keys help to build the sum of f × midpoint. The by-hand route is the point, because each product is small and an error is easier to see on paper.
What students will learn
Know
- The midpoint of a class is halfway between its boundaries.
- Estimated mean = sum of f × midpoint divided by the total frequency.
- The median class is where the running total passes half of the values.
Understand
- Grouping throws information away, so every answer is an estimate and a band can be given for the mean.
- Frequencies weight the midpoints; an unweighted average of midpoints is a different thing.
- The median class is found by counting, not by position in the table.
Do
- Estimate a mean from a grouped table with equal or unequal widths.
- Name the modal class and the median class.
- Work backwards from a given mean to a missing frequency.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: midpoints, the estimated mean and median class (questions 1 to 8).
- Criterion B. Investigating patterns: the first figure. Ask students to predict how the mean moves as the values slide, and what the biggest and smallest possible means are.
- Criterion C. Communicating: laying out the f × midpoint column clearly (the worked example and question 4).
- Criterion D. Applying mathematics in real-life contexts: commute times, fruit masses, waiting times, rainfall; discussing how far an estimate can be trusted.
The one thing to do with the figure
Make students commit before they slide. In the first figure, ask “What is the smallest the mean could possibly be, and the largest?” and take answers before the slider moves. Then ask where the midpoint estimate sits in that band.
In the second figure, ask students to find the 10th and 11th values before they slide, using the running totals printed under the bars.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 7 min | Hook: the lost notebook. Collect predictions about whether a mean can be found, and how far to trust it. |
| 10 min | First figure and key idea: the band, the midpoint, the formula. |
| 10 min | Second figure, the warning card and the worked example; pairs finish the faded example. |
| 5 min | The check. If many choose A or B, return to the idea of a stand-in value. |
| 18 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 8. |
The answers
| Warm-up | 8; 6; 18. |
|---|---|
| Hook | The mean can be estimated by standing in the midpoint of each class; for the first figure it must lie between 22.5 and 32.5 and the midpoint estimate is 27.5. It cannot be trusted exactly, because the exact times are not in the table. |
| Worked example | Mean 149.5 g; median class 140 to under 160. |
| Faded example | 9.5 minutes. |
| 1. What a class adds to the sum | D. A uses the start of the class, B the end, C the width. |
| 2. Midpoint of a class | 30. (24 + 36) ÷ 2. |
| 3. Sum of frequency times midpoint | 140. 15 + 75 + 50. |
| 4. Estimated mean | 42. 1260 ÷ 30. |
| 5. Median class | B. A is the modal class; C has the second-highest frequency; D is where the running total is only 11. |
| 6. A missing frequency from a mean | 30. 900 + 30x = 720 + 36x. |
| 7. Mean from classes of unequal width | 24.3. 730 ÷ 30 = 24.33… |
| 8. Estimated mean from a small table | A. B averages the midpoints without frequencies; C uses class starts; D uses class ends. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or C in question 1 | Uses the start or the end of the class, or the class width. | Draw a class as a bar and mark the middle with a finger. |
| 12, 60 or 36 in question 2 | Used the class width, the sum of the boundaries, or the upper boundary. | Ask: where is halfway between 24 and 36 on a number line? |
| 90, 190 or 45 in question 3 | Used the start or the end of the class, or forgot to multiply by frequency. | Write the three products in a column before adding. |
| 40, 32 or 1260 in question 4 | Averaged the midpoints, used class starts, or did not divide. | Ask: how many people are there, and has the sum been shared among them? |
| A, C or D in question 5 | Picked the modal class, the second-highest bar, or a class that is too early. | Make a running-total row under the table and circle where it passes 25. |
| 16, 150 or −6 in question 6 | Treated the mean as the count, dropped the 20 values, or left x out of the total. | Write the sum and the total number of values as two separate lines before setting up the equation. |
| 20.7, 23.3 or 14 in question 7 | Used the class widths, averaged the midpoints, or used the class starts. | Ask students to say, in words, what each term in the sum stands for. |
| B, C or D in question 8 | Averaged the midpoints, or used the class starts or ends. | Compare the answer with the band the table allows. |
Questions worth asking
- Why is the mean from a grouped table only an estimate? Listen for “we do not know where in the class the values are, so we use the middle”.
- Could the true mean be outside the band from the first figure? No: even with every value at the start or the end of its class, the mean stays inside the band.
- Why can the modal class not be named from the median class? They answer different questions: the highest frequency, and the position of the middle value.
- What makes an estimate better? Narrower classes, so each stand-in is closer to the values it replaces.
Differentiation
A way in for anyone stuck
- Give a table with the midpoint column and the f × midpoint column already set out, with the numbers blank.
- Use a checklist: midpoints? times frequencies? total? divided by the number of values?
- Use a number line under each class to find the midpoint by sight before using the formula.
Stretch
- Find the smallest and largest means a table could have, using the class starts and the class ends. Prove that the midpoint estimate is always exactly halfway between them, whatever the class widths.
- A table has an open-ended last class, 50 and over. What can and cannot be said about the mean?
Students learning in English as an additional language
“Class” here means a group of values, not a group of students. “Boundary” is the edge of a class. “Estimate” means a careful answer that may not be exact. “Midpoint” means halfway between two ends.
| Word | Plain meaning | Example |
|---|---|---|
| class | a group of values in a grouped table, such as 20 to under 30 | the interval 60 to under 64 |
| class boundary | the lower or upper end of a class; a value on the upper boundary belongs to the next class | 64 is the start of the next class |
| midpoint | halfway between the boundaries; the stand-in value for a class | 62 for the class 60 to under 64 |
| estimated mean | the sum of frequency times midpoint divided by the total frequency | the average found from a table, not from the raw data |
| modal class | the class with the highest frequency, when the classes have equal width | the tallest bar |
| median class | the class that holds the middle value of the ordered data | where the running total passes half the values |
Sentence starters for explaining aloud or in writing:
- The midpoint of this class is … because ….
- This is an estimate because ….
- The median class is … because the running total reaches ….
Two things not to say
“The median class is the middle class.” It is the class where the running total passes half the values.
“The mean is the average of the midpoints.” Each midpoint has to be counted as many times as its frequency.
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.
- MidpointWhat is the midpoint of the class 42 to under 50?
46. - TotalTwo values in 0 to under 4 and four in 4 to under 8. Total of f × midpoint?
2 × 2 + 4 × 6 = 28.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextHeights of 20 sunflowers, in cm: 100 to under 120 (3), 120 to under 140 (9), 140 to under 160 (8). Estimate the mean.
(110 × 3 + 130 × 9 + 150 × 8) ÷ 20 = 2700 ÷ 20 = 135 cm. - Earlier work (Year 8)Find the median of 11, 4, 9, 16, 7.
9. The ordered list is 4, 7, 9, 11, 16.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningA table has 8 values in 0 to under 10 and 12 in 10 to under 20. Between which two numbers must the true mean lie?
Between (0 × 8 + 10 × 12) ÷ 20 = 6 and (10 × 8 + 20 × 12) ÷ 20 = 16. - Working backwardsA table has 10 values in 0 to under 10 and y values in 10 to under 20. The estimated mean is 11. Find y.
(50 + 15y) ÷ (10 + y) = 11 gives 50 + 15y = 110 + 11y, so y = 15.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider, then the arrow keys) and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Give pairs a strip of 20 numbered cards, grouped into four piles, and have them find the 10th and 11th by counting.