Generate equivalent fractions and simplify

What you will learn

• Generate equivalent fractions and simplify

Models used: number sentences, stories, fraction strips.

Before you start

• Represent fractions of a whole or set

Fewer parts, the same share

Four of eight equal parts are shaded. Can we name that same share using fewer, larger equal parts?

1. Four shaded parts out of eight equal parts.

   Start with four shaded eighths. Count the unshaded parts too: all eight belong to our one whole.

   4/8

   A simpler fraction must cover exactly this same length. We may regroup the pieces, but must not change the whole or its shaded share.

2. The lower strip groups the upper strip’s pieces in pairs. The blue lengths still match.

   Pair neighbouring eighths. Two small parts now make one larger part. Four shaded pieces become two shaded pairs; eight pieces become four pairs altogether.

   4/8 = (4 ÷ 2)/(8 ÷ 2) = 2/4

   The size of a fraction depends on both numbers: how many equal parts make the whole, and how many of those parts we take.

   4/8 = 2/4: the same length, so the same amount.

3. The top has two shaded quarters. The bottom has one shaded half.

   Pair the quarters once more. Two shaded quarters become one half, and the four quarters become two halves.

   2/4 = (2 ÷ 2)/(4 ÷ 2) = 1/2

   Divide both counts by the same number because we regroup the shaded pieces and all the pieces in exactly the same way.

   2/4 = 1/2: the same length, so the same amount.

4. Different counts of parts; one unchanged shaded amount.

   We can stop at one half: 1 and 2 have no common factor greater than 1. Check the first strip against the last. Their shaded lengths have not changed.

   4/8 = 1/2

   Simplifying changes the fraction’s name, not its size. Dividing only the bottom number would change the share.

   4/8 = 1/2: the same length, so the same amount.

Worked example: A new example, together

Write 4/8 in simplest form.

1. Each strip is one whole of the same size. Count all the equal parts, then count the shaded parts.

   Match the shaded lengths before and after regrouping. Count both the shaded groups and all the groups.

   Multiplying or dividing both numerator and denominator by the same non-zero number preserves the fraction.

2. Each strip is one whole of the same size. Count all the equal parts, then count the shaded parts.

   Splitting each part into 2 smaller parts changes the count and total by the same factor.

   The size of a fraction depends on both numbers: how many equal parts make the whole, and how many of those parts we take.

   Write 4/8 in simplest form.

   1/2

3. Each strip is one whole of the same size. Count all the equal parts, then count the shaded parts.

   Splitting each part into 2 smaller parts changes the count and total by the same factor.

   The size of a fraction depends on both numbers: how many equal parts make the whole, and how many of those parts we take.

4. Each strip is one whole of the same size. Count all the equal parts, then count the shaded parts.

   Divide numerator and denominator by their greatest common factor 4.

   The size of a fraction depends on both numbers: how many equal parts make the whole, and how many of those parts we take.

5. Each strip is one whole of the same size. Count all the equal parts, then count the shaded parts.

   4/8 and 2/4 cover the same amount. Answer: 1/2.

   The size of a fraction depends on both numbers: how many equal parts make the whole, and how many of those parts we take.

6. Each strip is one whole of the same size. Count all the equal parts, then count the shaded parts.

   Check the result against the question we started with.

   Cross-products agree: 4 × 4 = 2 × 8.

   We found 1/2. Multiplying or dividing both numerator and denominator by the same non-zero number preserves the fraction.

Common mistakes to discuss

• Adding instead of multiplying. Both numbers are multiplied or divided by the same amount. 1/2 = 4/8 because both were multiplied by 4; adding 3 to each would give 4/5, a different amount.

Check your understanding

• Generate equivalent fractions and simplify
• simplify a fraction by regrouping equal parts without changing its value
• check the answer against the original quantities and units

Explain your reasoning to an adult, then try a fresh question in the interactive lesson. A result from one attempt is not proof of lasting mastery.