Family III · Equivalence and comparison.
Same Value, Different Forms
How 1/2 and 2/4 can be the same number, why simplifying never shrinks a fraction, and how to compare fractions by reasoning instead of guessing.
- Slice 11
Different Numbers, Same Amount
Is (one half) the same as (two fourths)?
Sounds like “(two fourths) is bigger than (one half). The numbers are bigger.”
The big idea Different numbers can name the same amount. Split every piece into the same number of equal pieces and the amount stays the same: (one half) = (two fourths) = (four eighths).
- Slice 12
Simplifying Doesn’t Shrink
Does simplifying a fraction make it smaller?
Sounds like “If you simplify (six eighths) to (three fourths), you lose some.”
The big idea Simplifying changes the pieces you count with, not the amount. Group six eighths in pairs and you get three fourths: bigger pieces, fewer of them, the same amount.
- Slice 13
Same Bottom, Compare the Top
How do you compare fractions with the same denominator?
Sounds like “(two sevenths) and (five sevenths) are both sevenths, so they’re the same size.”
The big idea When the bottom numbers match, the pieces are the same size, so count them. More of the same pieces is more: (five sevenths) > (two sevenths).
- Slice 14
Same Top, Compare the Pieces
How do you compare fractions with the same numerator?
Sounds like “(three fifths) and (three eighths) both have 3 pieces, so they’re equal.”
The big idea When the top numbers match, both fractions have the same number of pieces, so compare the pieces. Fewer parts in the whole means bigger pieces: (three fifths) > (three eighths).
- Slice 15
Compare Without Guessing
Which is bigger, (three eighths) or (four sevenths)?
Sounds like “The tops and bottoms are all different, so I just guessed.”
The big idea Compare by position, not digits. (three eighths) is less than one half and (four sevenths) is more than one half, so (four sevenths) is bigger.