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Fraction Classroom · 2.3.0 · Teacher answers & responses

Distinguish times as many from more than

Use equal comparison units to separate multiplicative comparisons from additive differences and identify the unknown quantity.

Lee has one group of 5 counters. Sam has three equal groups of 5, so the comparison is multiplicative.Reference: 5 counters53 times as many555
Lee has one group of 5 counters. Sam has three equal groups of 5, so the comparison is multiplicative.

Build the idea

model. Lee has 5 counters. Sam has 3 times as many counters as Lee. How many counters does Sam have?

  1. 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.
  2. Draw three equal boxes for Sam, each containing five. Keep the box size consistent so every copy represents the same count.
  3. Multiply 3 × 5 = 15. Contrast this with three more than five, which would be 5 + 3 = 8 and would describe a different relationship.
  4. Check the comparison by division: 15 ÷ 5 = 3. The quotient counts copies of Lee's amount, not extra counters beyond Lee's set.

Worked example: 15 counters. One comparison unit is Lee's set of 5; Sam has three copies of that entire amount.

Practise and explain

Try p1–p3 with a partner. Use a drawing and label your units. Try e1 independently before comparing responses.

p1. One tray holds 6 counters. Another holds 4 times as many. How many counters are in the other tray?

Answer: 24

Teacher response: Four copies of six give twenty-four counters.

p2. One collection has 21 counters and another has 7. How many times as many counters are in the first collection?

Answer: 3

Teacher response: Divide twenty-one into groups of seven to count the comparison copies.

p3. A collection of 32 counters is 4 times as large as a smaller collection. How many counters are in the smaller collection?

Answer: 8

Teacher response: The smaller collection is one of four equal comparison groups.

e1. 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: 30

Look for:

  • Identifies the reference quantity before operating.
  • Explains the multiplier as a number of equal copies.

Teacher response: Five copies of six centimetres make thirty; eleven would be five centimetres more than six.

Transfer the idea

transfer. For a reference amount of 4, compare three times as many with three more. State both results and describe what the 3 counts.

Answer: Three times as many is 12, with the 3 counting copies of four. Three more is 7, with the 3 counting additional individual units.

Run the 15-minute workshop

teach-1. 3 minutes: Read the story aloud and ask what one comparison unit represents.

teach-2. 4 minutes: Draw both three times as many and three more than five, then connect each drawing to an equation.

teach-3. 5 minutes: Solve the practice while naming whether the unknown is the larger amount, the copy count or the reference amount.

teach-4. 3 minutes: Complete the independent check and use a division equation to verify the comparison.

Respond to the learner

misconception. Three times as many as five means five plus three.

Teacher response: Ask whether the three counts individual extra counters or complete copies of the original group. Draw both situations and compare their totals.

support. Label each box with the reference amount and count copies before calculating the total.

extend. Write three related equations for a comparison of 36 and 9: find the larger amount, the copy count and the smaller amount.

Teaching notes