Add and subtract mixed numbers with justified exchanges

Learning goal: Combine and subtract mixed numbers, including cases requiring an exchange of one whole, while preserving the same whole unit. Explain every renaming and use an estimate to judge whether the result makes sense.

Scope: This lesson completes the mixed-number part of 5.NF.1 and 5.NF.2. The companion lesson on unlike fractions and common units builds the equivalent-fraction method used here.

A mixed number combines a whole-number amount and a fractional amount. 2 3/4 means 2 + 3/4. If each whole is one meter, it represents two full meters and three fourths of another meter. It does not mean two times three fourths. Naming the operation hidden by mixed-number notation prevents errors when combining amounts.

Suppose a project uses 2 3/4 meters of cord in the morning and 1 5/6 meters later. Rename the fractions in twelfths: 3/4 = 9/12 and 5/6 = 10/12. The whole parts total three, while the fractional parts total 19/12. Nineteen twelfths contain one whole and seven twelfths, so the entire sum is 4 7/12 meters.

An alternative is to rename each mixed number as one fraction. 2 3/4 = 11/4 = 33/12, and 1 5/6 = 11/6 = 22/12. Adding gives 55/12, which is 4 7/12 because four wholes use 48 twelfths and seven remain. Both methods count the same units. Choose one that makes the explanation clear and use the other as a check when helpful.

For subtraction, start with 3 1/4 meters and remove 1 5/6 meters. Rename the original amounts as 3 3/12 and 1 10/12. Three twelfths alone cannot supply ten twelfths, but the original amount includes three complete wholes. Exchange one whole for twelve twelfths: 3 3/12 = 2 15/12. This is an equivalent representation, not a change in the available length.

Subtract one whole and ten twelfths from two wholes and fifteen twelfths. The remainder is 1 5/12 meters. Check with addition: 1 5/12 + 1 10/12 = 2 15/12 = 3 3/12 = 3 1/4. The inverse check confirms that the removed and remaining lengths reconstruct the original.

A model can show thirteen quarter units in 3 1/4, then rename each quarter as three twelfths for a total of 39 twelfths. The removed 1 5/6 is 22 twelfths, leaving 17 twelfths, or 1 5/12. If drawing strips, every full strip must represent the same meter and use the same equal partition after renaming.

Estimate with benchmarks. The addition combines a number just below three with a number just below two, so a result just below five is reasonable. The subtraction starts near three and one fourth and removes nearly two, so a result near one and one half fits. A negative result or an amount larger than the starting positive length would contradict this subtraction context.

Do not subtract the smaller numerator from the larger just to avoid exchange. In 3 3/12 − 1 10/12, replacing 3 − 10 with 10 − 3 reverses the fractional subtraction while keeping the whole subtraction unchanged. That describes no consistent removal of the intended amount.

Worked whole-number subtraction: 4 − 1 2/3 = 3 3/3 − 1 2/3 = 2 1/3. Exchanging one whole supplies the thirds needed for the fractional subtraction.

Worked addition: 1 2/5 + 2 3/10 = 1 4/10 + 2 3/10 = 3 7/10. Here the fractional sum stays below one, so no extra whole is composed.

Guided practice: Use equal whole strips to exchange one whole for the selected common denominator. Say the value before and after the exchange aloud. Partners compare a mixed-number method with a single-fraction method.

Independent performance: Solve an addition composing a new whole, a subtraction requiring exchange and a whole-minus-mixed example. Include measurement units, a model or equivalent equations, an estimate and inverse check. Explain why exchange preserves the original amount. Credit equivalent fractions that accurately represent the result.