Solving word problems with fractions

By the end of the lesson, students can solve one- and two-step word problems involving addition and subtraction of fractions with unlike denominators, and can explain why a bar model and the written equation agree.

Before this lesson: Students can find a common denominator, add and subtract fractions with unlike denominators, and name the fraction of a whole shown in a diagram.

• One printed problem set per student
• Half sheet of paper for each pair's bar model
• Ruler for drawing equal parts
• Chalkboard for the worked example

50 minutes · printed activities

Student problem set — pair activity For each problem, underline the question, draw a bar, write the equation, solve, and explain the answer with a unit. A. A class uses 1/3 of a roll of paper in the morning and 1/4 of the same roll in the afternoon. What fraction of the roll is used? B. A container holds 5/6 liter of juice. Students pour out 1/4 liter. How much juice remains? C. A footpath is 3/8 kilometer long. Workers repair 1/6 kilometer. How much still needs repair? D. A class plants 2/5 of a garden plot with pechay and 1/3 with tomatoes. What fraction remains unplanted?

Read the same problem twice without changing it. First reading, students listen only. Second reading, they write the numbers they hear and the question being asked. Say: 'The numbers are not the problem. Deciding what is happening to them is the problem.' Show a bar being cut into equal parts for each fraction in the problem, because a bar makes the whole visible and stops students from adding numerators and denominators across the same fraction. Then show the written equation side by side with the finished bar, one step at a time, so students can see that the equation is only a short way of writing the picture. Insist on the two habit questions before any computation: what is the whole here, and are we putting parts together or taking one part away. Point out that an answer such as 5/4 for a problem about one tray of bibingka must be read again: one tray and a quarter is possible, but the teacher must say what that means. Close by naming the three traps in this set: adding the denominators, ignoring the words 'the rest' or 'left', and giving a fraction that answers a question nobody asked.

1. 1Ana sold 2/5 of a tray of bibingka in the morning and 1/3 of the same tray in the afternoon. What fraction of the tray did she sell in all? The whole is one tray and the parts are put together, so add. Common denominator 15: 2/5 = 6/15 and 1/3 = 5/15, so 11/15 of the tray was sold. Show the bar split into fifteenths with 6 and 5 shaded.
2. 2Using the same tray: what fraction was left? The whole is 15/15, so 15/15 - 11/15 = 4/15. Point at the unshaded part of the same bar. Nothing is redrawn.
3. 3A bottle contains 3/4 liter of cooking oil. A family uses 1/8 liter. How much remains? The unit whole is one liter. Rename 3/4 as 6/8, then subtract: 6/8 - 1/8 = 5/8 liter. Label the answer with its unit.
4. 4Two-step non-example to correct together: 'Ruel ate 1/4 of the sandwiches and Jomar ate 1/2. How many are left?' A student writes 1/4 + 1/2 = 3/4 and stops. Ask what the question asked, then complete: 4/4 - 3/4 = 1/4 of the sandwiches.

1. 1Together: a drum is 2/3 full of water. After a class uses some, it is 1/4 full. What fraction was used? Identify the whole first, then the operation, then compute.
2. 2Together: Liza walked 3/8 of the way to school, then rode a tricycle for the rest. What fraction did she ride? Draw the bar before computing.
3. 3Together: correct this student answer out loud, and say which trap it fell into: 1/2 + 1/3 = 2/5.

1. 1A farmer harvested 5/8 of his plot on Saturday and 1/4 on Sunday. What fraction of the plot was harvested in all?
2. 2Mother bought 3/4 kilogram of fish and cooked 1/3 kilogram. How much fish was not cooked?
3. 3In a class of 40, 3/8 joined the clean-up drive. What fraction did not join?
4. 4A tank was 1/2 full. Rain added 2/5 of the tank. What fraction is full now, and is it more than one full tank?
5. 5Mila read 2/9 of a book on Monday and 4/9 on Tuesday. What fraction is left?

• Each student writes the one sentence they would say to a classmate who added the denominators instead of finding a common denominator.
• Ask: 'In independent problem 4, is 9/10 more or less than one full tank? What does that tell you about overflow?'
• Collect the bar drawings only, no answers, and check that the wholes are the same size across a pair's work.

• Adding the denominators because the numbers looked like whole numbers, giving 1/2 + 1/3 = 2/5.

  What to do: Go back to the bar. Ask the student to shade one half, then one third of the same bar, and look at the total. It is clearly more than half. Then rename both fractions in sixths so they can see 3/6 + 2/6 = 5/6.

• Answering the fraction that was used when the question asked for the fraction that is left.

  What to do: Have the student read the underlined question aloud, then point at the unshaded part of their own bar and ask which part the question wants.

• Changing the whole halfway through the problem, for example comparing a fraction of a tray to a fraction of a liter.

  What to do: Require the unit to be written next to every number in the working. A fraction without a unit is not finished in a word problem.