Decompose, add and subtract equal fraction parts

Learning goal: Build fractions from unit fractions, decompose them in several ways, and solve addition and subtraction problems with like denominators. A unit fraction names one equal part of a whole. Five eighths means five copies of one eighth: 5/8 = 1/8 + 1/8 + 1/8 + 1/8 + 1/8. This equation counts the pieces rather than changing their size.

The entire strip in the illustration is one whole divided into eight equal regions. The first three selected regions and next two selected regions together cover five regions. Grouping the selected pieces in different ways gives 5/8 = 3/8 + 2/8 and 5/8 = 1/8 + 4/8. The groups differ, but both cover the same total selected length. The remaining three unselected regions are still part of the whole.

Addition joins amounts measured with the same unit. If a ribbon length is 3/8 meter and another is 2/8 meter, the combined length is 5/8 meter. The numerator counts how many eighth-meter pieces there are altogether. The denominator stays eight because joining pieces does not cut the meter into smaller pieces. Writing 5/16 would describe five sixteenths of a meter, which is too small even to contain the original three eighths.

Subtraction separates or compares amounts in the same unit. Begin with 7/8 meter of ribbon and cut away 3/8 meter. Four eighth-meter pieces remain: 7/8 − 3/8 = 4/8. Check by adding the removed and remaining amounts: 3/8 + 4/8 = 7/8. The result may also be called one half, but 4/8 is already a correct fraction name.

A difference question need not involve physically removing anything. A red ribbon is 7/8 meter long and a blue ribbon is 3/8 meter long. Align their starting points. The red ribbon extends 4/8 meter farther, so the same subtraction finds how much longer it is. Explain whether a story asks for a combined amount, a remaining amount or a difference.

Fractions can describe more than one whole. Nine fourths means nine copies of one fourth. Decompose it as 4/4 + 4/4 + 1/4, or 8/4 + 1/4, or 5/4 + 4/4. Two complete sets of four fourths make two wholes, leaving one fourth, so 9/4 = 2 1/4. A whole number written together with a fraction in this way is called a mixed number. Use multiple equal-sized strips; each strip is one whole, not the entire collection of strips.

A sum can cross a whole-number boundary. Seven eighths plus three eighths equals ten eighths. Eight eighths make one whole and two eighths remain, so 10/8 = 1 2/8. The numerator may exceed the denominator. Forcing the answer to stay below one would lose part of the combined amount.

Always name the common unit in a story. Three eighths of one meter and two eighths of one meter can be joined as five eighths of a meter. Three eighths of a short ribbon and two eighths of a different long ribbon do not necessarily use equal pieces. Establish the same whole before adding the counts.

Worked story: A group walks 4/6 kilometer before a break and 5/6 kilometer afterward. Write d = 4/6 + 5/6 = 9/6 kilometers. Six sixths make one kilometer, leaving three sixths, so d = 1 3/6 kilometers. Check that the total is larger than either part and lies between one and two kilometers.

Worked decomposition: Show 6/8 in two ways: 2/8 + 4/8 and 3/8 + 3/8. Draw six selected eighths and mark a boundary after the second piece, then after the third. The grouping changes; the six selected pieces do not.

Guided practice: Build fraction trains with equal eighth strips. Describe a joining story and a separating story for the same pieces. Draw the unused regions so the whole remains visible. Label each equation with meters, liters or the stated common unit.

Independent evidence: Decompose 7/8 in two different ways, solve a same-unit addition story and a subtraction story, and check subtraction by addition. Show a sum greater than one on two matching whole strips. Require an explanation of why the denominator stays the same, not just a remembered rule.