Equivalent fractions: new parts, same amount
Explain equivalent fractions by subdividing every part equally while preserving the amount.
Begin with the activity ↓Does changing 2/3 to 4/6 make more?
Split every part into two equal smaller parts. There are twice as many shaded parts, but the shaded amount stays the same.
Shade two of three equal parts of a whole: 2/3. Now split every part in half, including the unshaded one. The whole contains six smaller parts.
Each shaded part also splits in two, giving four shaded parts. The amount is now named 4/6, while the shaded region is unchanged. Therefore 2/3=4/6; these are equivalent fractions.
Doubling only the denominator tells a different story. 2/6 takes just two of the smaller parts, so it is not equal to 2/3. Both total divisions and parts taken must scale together.
You can also bundle small parts equally. Group the pieces in 6/8 by twos: the whole has four groups and three are shaded, giving 3/4. Dividing numerator and denominator by the same common factor is simplifying a fraction.
To work with 1/2 and 1/3 together, choose a common part size. Split halves into three and thirds into two to make sixths. Then 1/2=3/6 and 1/3=2/6. This is finding common denominators.
Listen one paragraph at a time
If you changed only one number
Keep the shaded boundary fixed. Adding subdivision lines alone cannot change the amount. Count both total parts and shaded parts again.
Now change it yourself
Choose a shaded amount, then split all parts in two or three. The shaded boundary does not move.
2/3 · One outlined strip = one equal-size whole
2/3 · One outlined strip = one equal-size whole
Multiply numerator and denominator by the same positive integer, or divide both by a common factor. A denominator cannot be zero.
Try with paper and objects
Shade half a rectangle, then draw equal subdivisions across the whole to read it as 2/4 and 3/6.
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2/3 · Dividing both by three gives 2/3, the same amount.
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Compare fractions of the same whole by common units or by part size when numerators match.