Different pieces, the same fraction
Objective: Recognize equivalent fractions by splitting each equal part of an unchanged whole into the same number of smaller parts.
Before you start
- Count equal parts of a whole.
- Read a fraction such as 1/3 as one out of three equal parts.
- Multiply small positive whole numbers.
In 1/3, the bottom number says there are 3 equal parts in the whole. The top number says 1 of those parts is selected.
Explanation
Imagine a strip divided into three equal parts. One part is shaded, so one third of the strip is shaded. We write that as 1/3.
Now split every third into two equal smaller pieces. Do not change the strip or the shaded region. The strip now has six equal pieces, and two are shaded. The same region is 2/6 of the strip. That is why 1/3 = 2/6.
Fractions with the same value are called equivalent fractions. Splitting each old piece into two makes both counts twice as large: 1 × 2 = 2 shaded pieces and 3 × 2 = 6 pieces altogether. The pieces get smaller, but the shaded share stays the same.
You can split each old piece into three smaller pieces instead. Then 1/3 = 3/9. For this activity, multiply the top and bottom numbers by the same positive whole number. Changing only the bottom number changes the fraction.
Keep the whole in mind when comparing physical amounts. Half of a small sheet and half of a large sheet are both the fraction 1/2, but the shaded areas can differ. Equivalent fractions describe the same share; equal actual amounts need equal wholes.
Worked example
Split each of the 3 equal parts into 4 smaller equal parts.
- The whole still has the same size.
- There are 3 × 4 = 12 small parts altogether.
- The selected third contains 1 × 4 = 4 small parts.
- So 1/3 = 4/12 (spoken: one third equals four twelfths).
Split the same whole
Choose how many smaller pieces to make inside every third. Then fill in the missing top number.
Split each third into
The whole has 3 equal original parts. One original part is selected. Each original part is split into 2 smaller equal pieces. There are 6 smaller pieces in the whole. The strip and the selected region have not changed.
Hint 1
Keep the whole and shaded region unchanged. Only the dividing lines change.
Hint 2
The original top number is 1. Multiply 1 by the number of smaller pieces inside each third.
Answer with explanation
1 × 2 = 2; 3 × 2 = 6. So 1/3 = 2/6.
Practice: three questions
A common mix-up
Claim to check: “A bigger denominator always means a bigger fraction.”
The denominator counts equal pieces in one whole. More pieces means each piece is smaller. Compare both numbers: 1/3 and 2/6 have the same value.
Try it on paper (optional, not graded)
Draw a rectangle, divide it into 4 equal parts and shade 3. Add one line through the middle of every part to make 8 equal parts. Explain why the shaded region can be called both 3/4 and 6/8.
No drawing? Say or write what happens to the count of selected parts and the total count when each of 4 equal parts is split in two.
Model answer
Each of the 3 selected parts becomes 2 parts, so 6 are selected. The 4 total parts become 8. The whole and shaded amount are unchanged.
This free sample works on its own. Review the explanation or try the activity again.