Adding fractions with a model

What the model shows

This model adds two fractions with different denominators the way the written method does, one step at a time. A blue bar shows the first fraction and a red bar shows the second. Their parts are different sizes, so they cannot be counted together yet. Split every part of each bar into 2 to 6 pieces until both bars have the same number of equal parts; that number is a common denominator. Then press Combine the pieces and the shaded pieces slide down into one sum. The readout lists each step: matching the parts, renaming each fraction, combining and simplifying.

Why same-size parts matter

The most common mistake in adding fractions is adding the tops and adding the bottoms, so that 1/3 + 1/4 becomes 2/7. A child who has watched the pieces wait until they are the same size before combining knows why that cannot work: thirds and fourths are different units, just as 1 foot plus 1 inch is not 2 of anything. Splitting the parts to make them match is exactly what renaming with equivalent fractions means, so the written steps and the picture tell the same story.

Common mix-ups

• Adding numerators and denominators. 1/3 + 1/4 is not 2/7. In fact 2/7 is less than 1/3 on its own, so it cannot be the sum.
• Thinking only the smallest common denominator is allowed. Any common denominator works; the smallest one just keeps the numbers small. The tool says when a smaller one exists.
• Changing the denominator but not the numerator. When thirds become twelfths, each shaded third becomes 4 shaded twelfths, so 1/3 = 4/12, not 1/12.
• Forgetting a sum can be more than 1. 2/3 + 3/4 = 17/12, which is 1 5/12, so the sum fills one whole bar and part of another.

A 10-minute activity

• Minutes 1 to 2: set 1/2 + 1/4. Ask why the pieces cannot be combined straight away, then split the halves into 2 and combine.
• Minutes 3 to 6: set 1/3 + 1/4. Let the child try splits until the parts match, then read each step aloud.
• Minutes 7 to 8: set 2/3 + 3/4 and ask whether the answer will be more or less than 1 before combining.
• Minutes 9 to 10: press New problem, solve it on paper first, then check with the model.

Where it fits by grade

• Grade 4: adding and subtracting fractions with like denominators, such as 2/5 + 1/5 (Common Core 4.NF.B.3).
• Grade 5: adding and subtracting fractions with unlike denominators by replacing them with equivalent fractions (5.NF.A.1), and solving word problems while estimating with benchmark fractions (5.NF.A.2).
• Grade 6: fluent fraction addition inside ratio, measurement and equation problems.
• England: Year 5 pupils add and subtract fractions with the same denominator and with denominators that are multiples of the same number, and Year 6 pupils add and subtract fractions with different denominators and mixed numbers using equivalent fractions.

Learning more

Adding fractions with unlike denominators is a skill on Eurekami's skill map, along with the equivalent fractions it builds on. Lessons draw fraction bars from each question's numbers, and Daily Practice brings the skill back when it starts to fade.