Add and subtract matching fraction units
Add or subtract numerators when units match; first find common units when denominators differ.
Begin with the activity ↓Why is 1/2 plus 1/3 not 2/5?
The denominator sets the piece size. Change to equal-size pieces before adding or removing their counts.
Adding three eighth-pieces to two gives five eighth-pieces: 2/8+3/8=5/8. The part count changes, but each part remains 1/8. This is why denominators are not added to make 16.
The pieces in 1/2 and 1/3 differ in size. Calling them two pieces alone loses that information. Express both using sixth-pieces of the same whole.
Since 1/2=3/6 and 1/3=2/6, combine three and two matching pieces to obtain 5/6. Adding top and bottom separately to get 2/5 does not match the amount combined.
Subtraction uses the same idea. Rewrite 3/4−1/6 as 9/12−2/12, leaving seven pieces, or 7/12. Match the part size before subtracting the counts.
A sum may exceed one whole. 3/4+2/3=9/12+8/12=17/12, or 1 and 5/12. This is more than one whole’s worth of matching pieces, not a change of piece size.
Listen one paragraph at a time
If you added or subtracted the denominators
Draw three sixth-pieces and then two more beside them. There are five pieces, but no new subdivision line changed their size.
Now change it yourself
Choose addition or subtraction and compare common pieces. If the removed amount is larger, swap the amounts first to explore a nonnegative difference.
1/2 = 3/6 · One outlined strip = one equal-size whole
1/3 = 2/6 · One outlined strip = one equal-size whole
Both strips use the same whole; one common piece is 1/6.
Use a common whole and measurement unit. Subtraction in this introductory activity is nonnegative; negative differences connect to the later negative-number course.
Try with paper and objects
Shade fractions on equal-size strips and add common subdivisions. Join pieces for addition or mark the pieces removed for subtraction.
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5/8 · 1/4=2/8, so 7/8−2/8=5/8.
Does your explanation include these ideas?
Speak, point or draw on paper. Explain the reason, not just the numerical answer.
Continue, or pause here
Use an overlap diagram to find a fraction of a fraction of the original whole.