Predict what a multiplier will do

d1. Without calculating, compare 3/4 × 8 with 8. What does 3/4 tell you?

Answer: The product is less than 8 because it takes three quarters of the original positive amount.

Look for:

• Predicts less than 8.
• Explains the factor as part of one copy.

Teacher response: Likely difficulty: Assuming multiplication always makes a number greater. Response: Shade three out of four equal parts of a strip labeled 8.

d2. Without calculating, compare 5/4 × 8 with 8.

Answer: The product is greater than 8 because 5/4 is more than one copy.

Look for:

• Predicts greater than 8.
• Recognizes 5/4 > 1.

Teacher response: Likely difficulty: Assuming every fraction is below 1. Response: Show 5/4 as 1 + 1/4 of the starting strip.

d3. A student says 6/6 × 7 must be larger than 7 because the factor has a 6. Explain.

Answer: 6/6 = 1, so the product equals 7.

Look for:

• States equality with 7.
• Uses 6/6 = 1.

Teacher response: Likely difficulty: Judging a factor by its numerator instead of its value. Response: Group six sixths into exactly one whole copy.

w1. A 12 cm ribbon is scaled by 2/3. Predict and calculate the new length.

1. The factor 2/3 is less than 1, so expect a shorter positive length.
2. Partition the 12 cm ribbon into three equal 4 cm sections.
3. Take two sections: 2 × 4 = 8 cm.
4. Compare 8 with 12 to check the prediction.

Worked example: The new length is 8 cm, less than 12 cm.

Look for:

• Predicts a shorter length.
• Finds 8 cm as two of three equal sections.

Teacher response: Likely difficulty: Interpreting a multiplier as an amount to add. Response: Show two thirds of the original length rather than attaching an extra two-thirds-centimeter piece.

w2. Scale a 6 cm drawing by 3/2. Predict and calculate its new length.

1. The factor 3/2 is 1 + 1/2, so expect more than the original length.
2. Keep one full 6 cm copy.
3. Add half of that copy, which is 3 cm.
4. The result is 6 + 3 = 9 cm, matching the prediction.

Worked example: 9 cm, greater than 6 cm.

Look for:

• Recognizes 3/2 > 1.
• Finds 9 cm using the original as one copy.

Teacher response: Likely difficulty: Treating the denominator as a command to make every product smaller. Response: Represent three halves as one whole copy and one half copy.

p1. Compare 7/8 × 24 with 24 without multiplying. Explain.

Answer: It is less than 24 because 7/8 < 1 and the starting amount is positive.

Look for:

• Predicts less.
• Uses the factor relative to 1.

Teacher response: Likely difficulty: Focusing on the whole number 24 while ignoring the scale factor. Response: Ask whether seven eighths is a full copy of the original.

p2. Compare 9/8 × 24 with 24 without multiplying. Explain.

Answer: It is greater than 24 because 9/8 > 1.

Look for:

• Predicts greater.
• Explains more than one copy.

Teacher response: Likely difficulty: Treating any written fraction as a reduction. Response: Rewrite nine eighths as one whole and one eighth.

p3. Choose the factor that keeps a 10 cm length unchanged: 3/4, 4/4, or 5/4.

Answer: 4/4, because it equals 1.

Look for:

• Selects 4/4.
• Explains the identity factor 1.

Teacher response: Likely difficulty: Choosing a factor by denominator rather than total value. Response: Build the factor from fourth-sized pieces to see which makes exactly one copy.

p4. Scale 15 cm by 2/5. Predict first, then calculate.

Answer: The result is smaller: 6 cm.

Look for:

• Predicts smaller.
• Finds 6 cm with a valid part-of argument.

Teacher response: Likely difficulty: Adding 2/5 to 15 instead of taking part of 15. Response: Divide the original length into five equal 3 cm parts and select two.

p5. Which result is greater: 2/3 × 18 or 4/3 × 18? Explain using copies, then check.

Answer: 4/3 × 18 is greater. It uses twice as many third-sized copies; the results are 12 and 24.

Look for:

• Chooses the 4/3 product.
• Checks 12 and 24 or explains the doubled number of copies.

Teacher response: Likely difficulty: Comparing the common denominator alone. Response: Hold the starting length constant and compare two thirds with four thirds.

p6. A 9 cm strip becomes 6 cm after scaling. Could the factor be 3/2? Give the correct factor.

Answer: No. The factor is 2/3; 3/2 would enlarge the length to 13 1/2 cm.

Look for:

• Rejects an enlargement factor.
• Gives 2/3 as the scale factor.

Teacher response: Likely difficulty: Reversing the factor between new and original lengths. Response: Compare the new length with the original: 6/9 = 2/3.

t1. A machine enlarges a positive length with a factor of 1 1/4. An operator says the result is smaller because the factor contains 1/4. Explain and test the claim using 8 cm.

Answer: The whole factor is 5/4, greater than 1, so it enlarges. An 8 cm length becomes 10 cm.

Look for:

• Uses the value of the entire mixed factor.
• Predicts enlargement and finds 10 cm.

Teacher response: Likely difficulty: Ignoring the whole-number part of a mixed multiplier. Response: Show one entire copy plus a quarter copy of the original.

t2. Write one scale factor that shrinks a positive length, one that preserves it, and one that enlarges it. Does “a factor below 1 makes the result smaller” stay strictly true when the starting amount is 0?

Answer: Examples: 2/3, 1, and 4/3. For a starting amount of 0 the result stays 0, so the strict shrinking statement needs a positive starting amount.

Look for:

• Provides factors below, equal to, and above 1.
• Explains the zero exception; other valid positive factors count.

Teacher response: Likely difficulty: Applying a strict size-change rule to zero. Response: Draw zero length and ask how taking a part could make it shorter.