Fractions beyond one whole
Express an improper fraction as whole units and remaining parts, and connect it to a mixed number.
Begin with the activity ↓How much is seven quarter-pieces?
Four quarter-pieces make one whole. Seven make one whole with three quarters left: 7/4 = 1 and 3/4.
Divide two identical bars into quarters. Seven of those pieces exceed one bar. Each piece is 1/4, so seven pieces are 7/4.
Bundle four pieces into a whole bar, leaving three. The amount is one and three quarters, written as the mixed number 1 3/4. The space means addition of the whole and fractional parts, not multiplication.
A fraction with numerator smaller than its denominator is proper; one with numerator at least its denominator is improper. Thus 4/4 is improper and equals 1. Improper does not mean an invalid number.
To convert back, count four quarter-pieces per whole. Two and one quarter has eight pieces in two wholes and one more, giving 9/4. The denominator remains 4 because the piece size stays the same.
Locate 7/4 on a number line and compare it with moving one whole and then three quarters. Both names mark the same point; changing notation does not change the amount.
Listen one paragraph at a time
If the space looked like multiplication
Write 1+3/4 beneath 1 3/4. Put down a whole and add the remaining pieces beside it.
Now change it yourself
Increase the number of equal pieces past one whole. Compare the total piece count with wholes and remaining pieces.
7/4 · One outlined strip = one equal-size whole
Read a mixed number as a sum. All diagrams must use the same-size whole.
Try with paper and objects
Draw two equal-size wholes, divide each into quarters and shade seven pieces.
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2 1/4 · Two groups of four leave one quarter-piece.
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Speak, point or draw on paper. Explain the reason, not just the numerical answer.
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Explain equivalent fractions by subdividing every part equally while preserving the amount.