Equivalent Fractions
4A · Chapter 6 · Lesson 1 · MATH-4A-C06-L01
Learning goal. Construct and justify equivalent positive fractions by refining or coarsening equal partitions.
Before you begin
Recall: Grade 3B: fractions of equal wholes, equivalence, number lines and simplifying. Try these three warm-up problems. If they are difficult, revisit the prerequisite lesson before advancing.
A club has 3,603 tickets. After buying 32 more, they bundle tickets into packs of 8. How many full packs and tickets left?
Show answer & reasoning
Add first: 3,635. Divide by 8: 454 packs, with 3 tickets remaining.
A club has 1,273 tickets. After buying 63 more, they bundle tickets into packs of 7. How many full packs and tickets left?
Show answer & reasoning
Add first: 1,336. Divide by 7: 190 packs, with 6 tickets remaining.
A club has 963 tickets. After buying 9 more, they bundle tickets into packs of 3. How many full packs and tickets left?
Show answer & reasoning
Add first: 972. Divide by 3: 324 packs, with 0 tickets remaining.
Key vocabulary
- equivalent fractions
- different numerators and denominators representing the same quantity
- mixed number
- whole number plus proper fraction
- improper fraction
- fraction with numerator at least its denominator
- denominator
- number of equal parts in one whole
Understand the concept
Fractions are equal-sized parts of a chosen whole. Dividing every fourth into two smaller equal parts turns 1/4 into 2/8 without changing the covered amount. Multiply or divide numerator and denominator by the same nonzero whole number to make an equivalent representation. Always compare parts of the same-sized whole.
Model it: aligned fraction strips with equal wholes. 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
Complete an equivalent fraction: 3/10 = ?/30.
The denominator was multiplied by 3, so the numerator must also be multiplied by 3.
Check your understanding 1
Complete an equivalent fraction: 3/4 = ?/16.
Show answer & reasoning
The denominator was multiplied by 4, so the numerator must also be multiplied by 4.
Worked example 2
Complete an equivalent fraction: 2/8 = ?/16.
The denominator was multiplied by 2, so the numerator must also be multiplied by 2.
Check your understanding 2
Complete an equivalent fraction: 7/10 = ?/20.
Show answer & reasoning
The denominator was multiplied by 2, so the numerator must also be multiplied by 2.
Worked example 3
Complete an equivalent fraction: 5/6 = ?/24.
The denominator was multiplied by 4, so the numerator must also be multiplied by 4.
Check your understanding 3
Complete an equivalent fraction: 5/8 = ?/32.
Show answer & reasoning
The denominator was multiplied by 4, so the numerator must also be multiplied by 4.
Guided practice
Complete an equivalent fraction: 2/4 = ?/16.
Show answer & reasoning
The denominator was multiplied by 4, so the numerator must also be multiplied by 4.
Complete an equivalent fraction: 8/10 = ?/40.
Show answer & reasoning
The denominator was multiplied by 4, so the numerator must also be multiplied by 4.
Complete an equivalent fraction: 9/12 = ?/48.
Show answer & reasoning
The denominator was multiplied by 4, so the numerator must also be multiplied by 4.
Complete an equivalent fraction: 7/12 = ?/60.
Show answer & reasoning
The denominator was multiplied by 5, so the numerator must also be multiplied by 5.
Are 2/8 and 4/16 equivalent? Explain.
Show answer & reasoning
Multiply numerator and denominator of 2/8 by the same factor 2; the represented amount is unchanged.
Independent practice
Complete an equivalent fraction: 3/5 = ?/10.
Show answer & reasoning
The denominator was multiplied by 2, so the numerator must also be multiplied by 2.
Complete an equivalent fraction: 2/5 = ?/10.
Show answer & reasoning
The denominator was multiplied by 2, so the numerator must also be multiplied by 2.
Complete an equivalent fraction: 3/6 = ?/18.
Show answer & reasoning
The denominator was multiplied by 3, so the numerator must also be multiplied by 3.
Are 1/2 and 3/6 equivalent? Explain.
Show answer & reasoning
Multiply numerator and denominator of 1/2 by the same factor 3; the represented amount is unchanged.
Are 4/6 and 16/24 equivalent? Explain.
Show answer & reasoning
Multiply numerator and denominator of 4/6 by the same factor 4; the represented amount is unchanged.
Complete an equivalent fraction: 5/6 = ?/18.
Show answer & reasoning
The denominator was multiplied by 3, so the numerator must also be multiplied by 3.
Complete an equivalent fraction: 3/12 = ?/36.
Show answer & reasoning
The denominator was multiplied by 3, so the numerator must also be multiplied by 3.
Complete an equivalent fraction: 2/4 = ?/20.
Show answer & reasoning
The denominator was multiplied by 5, so the numerator must also be multiplied by 5.
Challenge and transfer
Explain the relationships and justify your method. Multiple correct approaches may be accepted if the reasoning is valid.
A student claims 2/3 equals 3/4 because the numerator and denominator each increase by 1. Explain the error.
Show answer & reasoning
Equivalent fractions multiply numerator and denominator by the same nonzero factor. 2/3 = 8/12 while 3/4 = 9/12.
Are 2/5 and 6/15 equivalent? Explain.
Show answer & reasoning
Multiply numerator and denominator of 2/5 by the same factor 3; the represented amount is unchanged.
Exit check
Complete an equivalent fraction: 1/2 = ?/6.
Show answer & reasoning
The denominator was multiplied by 3, so the numerator must also be multiplied by 3.
Complete an equivalent fraction: 6/12 = ?/60.
Show answer & reasoning
The denominator was multiplied by 5, so the numerator must also be multiplied by 5.
Are 1/6 and 2/12 equivalent? Explain.
Show answer & reasoning
Multiply numerator and denominator of 1/6 by the same factor 2; the represented amount is unchanged.