Explain equivalent fractions and fair comparisons

Learning goal: Use visual models and number-line positions to explain equivalent fractions, whole-number fractions and comparisons with a shared numerator or denominator. Fractions are equivalent when they name the same number. Different-looking numerators and denominators can represent the same amount when the whole stays fixed.

Begin with a strip split into two equal halves. Shade one half. Now divide each half into two equal pieces without changing the whole or the shaded amount. The strip has four equal pieces and two are shaded. Therefore 1/2 = 2/4. The denominator increased because the unit pieces became smaller; the numerator increased because more of those smaller pieces describe the same shaded area.

Similarly, divide each third of a strip into two equal pieces. Two shaded thirds become four shaded sixths, so 2/3 = 4/6. Verify the shaded boundary rather than relying only on a numerical pattern. On a number line with a common zero-to-one unit, equivalent fractions occupy the same point. The point halfway to one can be labeled both 1/2 and 2/4.

A whole number can be written as a fraction. Three thirds make one, so 3/3 = 1, and on a number line 3/3 and 1 label the same point, just as 4/4 and 1 do. Two whole strips divided into thirds contain six thirds altogether, so 6/3 = 2. Three wholes each considered as one full-size unit give 3/1 = 3. The denominator one means that each part is an entire whole, not that the number must be less than one.

When denominators match, compare the number of same-size parts. For the same whole, 3/8 < 5/8 because three eighth-size parts are less than five eighth-size parts. The open side of a comparison symbol faces the larger number. You can check by placing the fractions on the same eighths number line; five eighths lies farther right.

When numerators match, compare the size of the parts. For equal wholes, 2/3 > 2/6 because a third is larger than a sixth and both amounts use two parts. A larger denominator does not automatically make a larger fraction. It indicates more equal parts within the whole, which makes each part smaller. Use matching strips to make this visible.

A fair fraction comparison requires the same whole when comparing actual portions. Half of a large sheet might have more area than three fourths of a small sheet. Without information about the sheet sizes, those portion labels alone do not settle the area comparison. Comparing the numbers 1/2 and 3/4 on a common unit line is different: the unit is fixed, and one half is less than three fourths.

Worked equivalence: Shade two thirds of a rectangle and split each third into two equal areas. Count four shaded pieces among six total. The colored area has not moved or grown, so 2/3 and 4/6 are equivalent. Place both labels at the same point on a line to verify the number relationship.

Worked comparisons: Compare 5/6 and 1/6: five sixths is larger because both use sixth-size parts of the same whole. Compare 3/4 and 3/8: three fourths is larger because a fourth is twice the size of an eighth and both count three parts. Neither example requires a general cross-multiplication rule.

Guided practice: Use equal-size paper strips to construct equivalences and keep the shaded boundary unchanged while subdividing. Represent one and two as fractions with different denominators. Then alternate same-denominator and same-numerator comparisons so learners must explain which feature stays fixed.

Independent evidence: Draw a model proving an equivalence, label a whole number with a fraction, and justify one comparison of each kind using <, > or =. Include an example with unequal physical wholes and explain why the portion sizes cannot be inferred from fraction labels alone. If a learner compares only numerators, return to the size of the parts and the unit whole.