At a glance. One lesson of about 60 minutes. Students tally a list in fives, read and check frequencies, and group measurements into equal classes with no gaps or overlaps.
Calculator: no calculator. Nothing here needs one. The counts are small and the checks are sums.
What students will learn
Know
- A tally counts in groups of five; the frequency is how often a value appears.
- The frequencies add to the number of values.
- In this unit classes have equal widths, with no gaps and no overlaps, so frequencies compare directly.
Understand
- A table makes a pattern visible that a list hides.
- Grouping loses exact values but shows the shape.
- A value on a boundary goes in exactly one class.
Do
- Tally a list and write the frequencies.
- Check a table by its total.
- Group measurements into classes and place boundary values.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: tallies, frequencies and classes (questions 1 to 6).
- Criterion B. Investigating patterns: the grouped figure. Ask students to say what stays the same and what changes as the class width changes, and to explain why the total stays 20.
- Criterion C. Communicating: explaining why a set of classes is or is not suitable, using the words width, gap and overlap (questions 4 and 9).
- Criterion D. Applying mathematics in real-life contexts: library loans, plant heights and parcel masses (the hook, the figures and question 9).
The one thing to do with the figure
Ask for a prediction before the first slide. “Which number of books will be most common?” Let students write a number, then slide to 24 and check.
In the second figure ask where a plant of exactly 12.0 cm goes in each of the three groupings before showing the step that says so. Keep the total 20 in view and ask why it never changes.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook and the first figure. Predict the most common value, then tally. |
| 10 min | Key idea and the common mistake; the second figure on classes. |
| 10 min | Worked example on goals; students finish the faded bus-journey example in pairs. |
| 5 min | Question 1, the check. If many choose B, go back to the first figure. |
| 15 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Exit ticket, question 9, on paper or on screen. |
The answers
| Warm-up | 20; 1/4; continuous. |
|---|---|
| Worked example | 0: 3, 1: 5, 2: 5, 3: 1, 4: 1; total 15. |
| Faded example | 10 to under 20: 4; 20 to under 30: 4; 30 to under 40: 2. |
| 1. What a frequency means | A. B confuses frequency with pets; C reads it as a position; D confuses it with the number of rows. |
| 2. Read a tally | 13. 2 × 5 + 3. |
| 3. Total from a frequency table | 12. 3 + 5 + 4. |
| 4. A value on a class boundary | C. B includes the upper boundary; A and D are the wrong end of the scale. |
| 5. Percentage from a frequency table | 44. 11 out of 25. |
| 6. A missing frequency | 6. 40 − 34. |
| 7. Number of classes needed | 5. 160 to under 165 is the fifth class. |
| 8. Build a frequency table | C. A swaps the 1s and 2s; B drops a 3 (total 7); D counts the 0 as a 1. |
| 9. Choose classes | C. A has an overlap at 5; B leaves out values from 9 up; D has unequal widths. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 | Read the frequency as pets in total, a position or the number of rows. | Ask: in the row for 2 pets, what are we counting? |
| 5, 23 or 15 in question 2 | Counted groups as 1, joined the digits, or made the loose strokes a group. | Ask students to write the tally and count in fives aloud. |
| 13 or 3 in question 3 | Totalled the stickers, or counted rows. | Ask whether the question is about stickers or students. |
| A, B or D in question 4 | Put the boundary value in the class that ends there. | Write 'under 20' beside the class and say it means not 20 itself. |
| 11, 56 or 28 in question 5 | Gave the count, the complement or only the row for 2. | Put 'out of 25' and 'which rows?' on the board before the percentage. |
| 34 or 40 in question 6 | Gave the sum so far, or the total. | Write the four frequencies with a blank and the total on the right. |
| 4 or 23 in question 7 | Rounded the class number down, or used the distance. | List the classes one at a time until one holds 163.2. |
| A, B or D in question 8 | Miscounted one value in the list. | Cross out each value as it is tallied and check the total is 8. |
| A, B or D in question 9 | Missed an overlap, a gap or unequal widths. | Check the three rules in turn: equal width, no gap, no overlap. |
Questions worth asking
- Why must the frequencies add to the number of values? Every value is in exactly one row. Listen for “each value is counted once”.
- What can you see in the grouped table that you could not in the list? The shape: where most values sit. Listen for “most are between …”.
- What can you no longer see after grouping? The exact values inside a class.
- Would you tally the heights of 20 plants one value per row? No, almost every height would be different. Listen for “nearly every row would have 1”.
Differentiation
A way in for anyone stuck
- Give a list on a strip of paper and let students cross out each value as it goes into the tally.
- Start with values 0 to 3 and fewer than 15 items; add the grouped table last.
- Provide classes already drawn with the headings “class”, “tally” and “frequency”.
Stretch
- Question 7, then: classes 5 wide starting at 140 for heights up to 187.9. How many classes are needed? (10: 140 to under 145 up to 185 to under 190.)
- Make up a list of 12 values that gives a frequency table with a most common value of 3 and a value that never appears.
Students learning in English as an additional language
A tally is a way of counting with strokes. “Frequency” here means how many times, not how often in time. Say “under 20” with “not including 20”.
| Word | Plain meaning | Example |
|---|---|---|
| tally | a stroke for each value, in groups of five | four strokes and a diagonal across them |
| frequency | how many times a value appears | 2 books has frequency 8 |
| frequency table | a table of each value with its frequency | books borrowed and frequency |
| total frequency | the sum of all the frequencies, the number of values | 3 + 5 + 4 = 12 |
| class | a range of values, used to group measurements | 10 to under 20, written 10 ≤ h < 20 |
| class width | the size of each class | 10 to under 20 has width 10 |
Sentence starters for explaining aloud or in writing:
- The frequency of … is …, which means …
- The frequencies add to …, which is the number of …
- This value goes in the class … because the class starts at …
Two things not to say
“Count the strokes.” Count in fives and then ones; counting strokes one by one invites slips.
“Just put it in the nearest class.” Boundaries are not about nearest: every value has one class, set by the rule.
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.
- TallyA tally shows three full groups of five and four more strokes. What is the frequency?
19. 3 × 5 + 4. - TotalFrequencies 6, 9, 4 and 1. How many values were tallied?
20.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA café counts cups of tea ordered per table: 1 cup, 14 tables; 2 cups, 9 tables; 3 cups, 5 tables; 4 cups, 2 tables. How many tables are there, and what percentage ordered 2 cups?
30 tables; 9 ÷ 30 = 0.3, which is 30%. - Earlier work (Unit 2, lesson 3)Find 20% of 35 values and say what it means for a frequency table.
7; seven of the values.
3Challenge
Reasoning, generalising and working backwards.
- ReasoningA grouped table has frequencies 4, 9, 12 and 5 in classes 0 to under 10, 10 to under 20, 20 to under 30 and 30 to under 40. How many values are 20 or more?
17. 12 + 5. - BoundaryRavi says a value of exactly 30 is in “20 to under 30”. Is he right? Write the class that holds 30 as an inequality for a value h.
No. It is in “30 to under 40”, the class that starts at 30: 30 ≤ h < 40.
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? Write the 24 values on cards and let students tally on paper in turn.