Practice 1
One tray holds 6 counters. Another holds 4 times as many. How many counters are in the other tray?
Answer and teaching response
24 · Four copies of six give twenty-four counters.
Distinguish times as many from more than
Use equal comparison units to separate multiplicative comparisons from additive differences and identify the unknown quantity.
Get the student and teacher materialsPrepare the group
Learning goal: A learner can choose multiplication or division for a times-as-many comparison and explain why adding the multiplier answers a different question.
You will need: Paper, pencil and optional counters. Use fictional collections; no learner names or personal records are needed.
Start with this question
Lee has 5 counters. Sam has 3 times as many counters as Lee. How many counters does Sam have?
15 counters. One comparison unit is Lee's set of 5; Sam has three copies of that entire amount.
- Identify the reference quantity: Lee's five counters make one comparison unit. The phrase three times as many describes how many copies of that unit to use.
- Draw three equal boxes for Sam, each containing five. Keep the box size consistent so every copy represents the same count.
- Multiply 3 × 5 = 15. Contrast this with three more than five, which would be 5 + 3 = 8 and would describe a different relationship.
- Check the comparison by division: 15 ÷ 5 = 3. The quotient counts copies of Lee's amount, not extra counters beyond Lee's set.
Run the 15-minute workshop
- 3 minutes. Read the story aloud and ask what one comparison unit represents.
- 4 minutes. Draw both three times as many and three more than five, then connect each drawing to an equation.
- 5 minutes. Solve the practice while naming whether the unknown is the larger amount, the copy count or the reference amount.
- 3 minutes. Complete the independent check and use a division equation to verify the comparison.
A useful error to discuss
A learner might say: Three times as many as five means five plus three.
Ask whether the three counts individual extra counters or complete copies of the original group. Draw both situations and compare their totals.
Practise, then check independently
Use the first three tasks for guided practice. Try the final task without help. Record a drawing or equation as well as the answer.
Practice 2
One collection has 21 counters and another has 7. How many times as many counters are in the first collection?
Answer and teaching response
3 · Divide twenty-one into groups of seven to count the comparison copies.
Practice 3
A collection of 32 counters is 4 times as large as a smaller collection. How many counters are in the smaller collection?
Answer and teaching response
8 · The smaller collection is one of four equal comparison groups.
Independent check
A red ribbon is 6 cm long. A blue ribbon is 5 times as long. Find the blue length in cm and explain why 11 cm would answer a different question.
Answer and teaching response
30 · Five copies of six centimetres make thirty; eleven would be five centimetres more than six.
Transfer the idea
For a reference amount of 4, compare three times as many with three more. State both results and describe what the 3 counts.
Discuss a possible response
Three times as many is 12, with the 3 counting copies of four. Three more is 7, with the 3 counting additional individual units.
Choose the next teaching step
Look for:
- Identifies the reference quantity before operating.
- Explains the multiplier as a number of equal copies.
If the idea is not secure: Label each box with the reference amount and count copies before calculating the total.
If the learner is ready: Write three related equations for a comparison of 36 and 9: find the larger amount, the copy count and the smaller amount.
Materials for your group
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