Rename fractions without changing their size

Learning goal: Explain and generate equivalent fractions by subdividing equal parts. A fraction describes an amount relative to a defined whole. The denominator tells how many equal parts make that whole; the numerator counts how many of those parts are selected. The whole includes both selected and unselected regions. Changing the partition can change the written numbers without changing the selected amount.

Start with a rectangular strip divided into three equal parts. Shade two. The entire strip is one whole, and the shaded amount is two thirds. Now split every third into two equal pieces. There are six equal pieces in the same whole. Each original shaded third contains two shaded sixths, so four sixths are shaded. No paper or shading was added: 2/3 = 4/6.

Both counts double because every original part becomes two parts. The individual pieces become smaller: each sixth is half as large as each third. More selected pieces do not automatically mean a larger amount. Here four smaller pieces cover exactly the same region as two larger pieces. Compare the outer boundaries and the shaded endpoints in the illustration.

Split each original third into four equal pieces instead. The whole now contains twelve pieces and the shaded region contains eight: 2/3 = 8/12. The subdivision factor is four. For a fraction a/b, splitting every part into n equal pieces produces (n × a)/(n × b), where n is a positive whole number. Multiplying both counts preserves the fraction's value because the unit pieces shrink as their count grows.

Multiplying only one number does something different. Changing 2/3 to 4/3 doubles the selected amount while keeping thirds as the pieces. Changing 2/3 to 2/6 keeps the selected count but halves the size of each piece. Neither is the same as subdividing both shaded and unshaded thirds. Check a model instead of relying on similar-looking digits.

Equivalent names can also be found by grouping small equal pieces. Six eighths can be grouped into three pairs. Each pair of eighths is one fourth, so 6/8 = 3/4. The denominator and numerator are both divided by two. Group the entire whole consistently, including the unshaded pieces. You cannot group only the shaded pieces and keep an unrelated denominator.

The same reasoning works beyond one whole. Six fourths occupy one whole strip and half of a second matching strip. Group pairs of fourths to get three halves, so 6/4 = 3/2. Each strip still defines one whole. The fraction bar does not require the numerator to be smaller than the denominator.

Use denominators such as 2, 3, 4, 5, 6, 8, 10, 12 and 100 in this sequence. An equivalent fraction need not be reduced to the smallest possible numbers to be correct. A useful name depends on the purpose: twelfths might help compare amounts, while halves might make the size easier to recognize.

Worked construction: Rename 3/4 in eighths. Split all four quarter regions into two equal pieces. The whole has eight pieces, and the three shaded quarters contain six. Therefore 3/4 = 6/8. In the other direction, pair the six shaded eighths and pair the two unshaded eighths to recover three shaded fourths out of four.

Worked missing number: Find a in 2/5 = a/10. Each fifth is split into two tenths, so two selected fifths become four tenths: a = 4. Adding five to both numbers would give 7/10, which shades a different amount.

Guided practice: Fold equal-length paper strips into halves, fourths and eighths. Mark the same half-strip region on each. State how the piece count, piece size and shaded amount change. The adult should check that the folds make equal regions before accepting the fraction names.

Independent evidence: Draw and explain two equivalent names for 1/2 and for 2/3. Then show why 2/3 and 2/6 are unequal. Record the subdivision factor and describe what happens to each original piece. A correct multiplication sentence without a model or explanation is only part of the evidence.