Multiplying a Whole Number by a Fraction — Part 2
4A · Chapter 8 · Lesson 6 · MATH-4A-C08-L06
Learning goal. Compare efficient multiplication/partition orders and verify equivalence.
Before you begin
Recall: 4A Ch6/7: equal partitions, mixed numbers and fraction operations; Multiplying a Whole Number by a Fraction — Part 1. Try these three warm-up problems. If they are difficult, revisit the prerequisite lesson before advancing.
Find 3/5 of a set of 35 objects.
Show answer & reasoning
Divide into 5 equal groups of 7, then choose 3 groups for 21 objects.
Find 2/10 of a set of 80 objects.
Show answer & reasoning
Divide into 10 equal groups of 8, then choose 2 groups for 16 objects.
Find 8/10 of a set of 30 objects.
Show answer & reasoning
Divide into 10 equal groups of 3, then choose 8 groups for 24 objects.
Key vocabulary
- unit fraction
- fraction with numerator one
- fraction of a set
- part of a discrete collection
- reference whole
- quantity to which a fraction applies
- remaining amount
- quantity after some part is removed
Understand the concept
You can either divide the whole set by the denominator first and then multiply by the numerator, or multiply first and divide afterward. With 30×2/5, each approach yields twelve. Choose a computation order that keeps the arithmetic simple, and justify the equivalence rather than treating the shortcut as a magic rule.
Model it: paired partition-first and multiply-first diagrams. Explain what every part of the model represents, then write a matching equation.
Fully worked examples
Read the problem first and predict the answer. Then study how each quantity is represented and calculated.
Worked example 1
Find 1/2 of a set of 14 objects.
Divide into 2 equal groups of 7, then choose 1 groups for 7 objects.
Check your understanding 1
Find 4/6 of a set of 24 objects.
Show answer & reasoning
Divide into 6 equal groups of 4, then choose 4 groups for 16 objects.
Worked example 2
Find 1/4 of a set of 28 objects.
Divide into 4 equal groups of 7, then choose 1 groups for 7 objects.
Check your understanding 2
Find 1/6 of a set of 12 objects.
Show answer & reasoning
Divide into 6 equal groups of 2, then choose 1 groups for 2 objects.
Worked example 3
Find 1/3 of a set of 24 objects.
Divide into 3 equal groups of 8, then choose 1 groups for 8 objects.
Check your understanding 3
Find 1/2 of a set of 10 objects.
Show answer & reasoning
Divide into 2 equal groups of 5, then choose 1 groups for 5 objects.
Guided practice
Find 3/5 of a set of 30 objects.
Show answer & reasoning
Divide into 5 equal groups of 6, then choose 3 groups for 18 objects.
Find 4/6 of a set of 12 objects.
Show answer & reasoning
Divide into 6 equal groups of 2, then choose 4 groups for 8 objects.
Find 1/3 of a set of 9 objects.
Show answer & reasoning
Divide into 3 equal groups of 3, then choose 1 groups for 3 objects.
Find 3/6 of a set of 36 objects.
Show answer & reasoning
Divide into 6 equal groups of 6, then choose 3 groups for 18 objects.
Find 2/6 of a set of 54 objects.
Show answer & reasoning
Divide into 6 equal groups of 9, then choose 2 groups for 18 objects.
Independent practice
Find 1/6 of a set of 18 objects.
Show answer & reasoning
Divide into 6 equal groups of 3, then choose 1 groups for 3 objects.
Find 2/3 of a set of 27 objects.
Show answer & reasoning
Divide into 3 equal groups of 9, then choose 2 groups for 18 objects.
Find 2/3 of a set of 15 objects.
Show answer & reasoning
Divide into 3 equal groups of 5, then choose 2 groups for 10 objects.
Find 3/5 of a set of 20 objects.
Show answer & reasoning
Divide into 5 equal groups of 4, then choose 3 groups for 12 objects.
Find 6/10 of a set of 50 objects.
Show answer & reasoning
Divide into 10 equal groups of 5, then choose 6 groups for 30 objects.
Find 1/5 of a set of 40 objects.
Show answer & reasoning
Divide into 5 equal groups of 8, then choose 1 groups for 8 objects.
Find 1/4 of a set of 12 objects.
Show answer & reasoning
Divide into 4 equal groups of 3, then choose 1 groups for 3 objects.
Find 1/2 of a set of 18 objects.
Show answer & reasoning
Divide into 2 equal groups of 9, then choose 1 groups for 9 objects.
Challenge and transfer
Explain the relationships and justify your method. Multiple correct approaches may be accepted if the reasoning is valid.
Calculate 30 × 2/5 using division before multiplication. Explain why the order works.
Show answer & reasoning
Split 30 into five groups of six, then take two groups: 12.
Find 2/4 of a set of 32 objects.
Show answer & reasoning
Divide into 4 equal groups of 8, then choose 2 groups for 16 objects.
Exit check
Find 3/8 of a set of 48 objects.
Show answer & reasoning
Divide into 8 equal groups of 6, then choose 3 groups for 18 objects.
Find 6/8 of a set of 16 objects.
Show answer & reasoning
Divide into 8 equal groups of 2, then choose 6 groups for 12 objects.
Find 5/8 of a set of 40 objects.
Show answer & reasoning
Divide into 8 equal groups of 5, then choose 5 groups for 25 objects.