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5 Practices

5-practices

Productive mathematical discussions lead to students who can think, reason and engage effectively in quantitative problem solving, characteristics which are much needed but may not routinely emerge from our classrooms. So just how can teachers create such discussions?

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Number lines

This activity uses eighths, but can be adjusted to use other fractions and diagrams. Examples of same denominator problems are available.

Together the class constructs a number line from 0 to 2, labelled with eighths. The line should be labelled with both improper fractions and mixed numbers.

Present a contextual problem.

The family bought some pizzas. I ate \(\frac{4}{8}\) of the pepperoni pizza and \(\frac{3}{8}\) of the ham and pineapple pizza. How much pizza did I eat?

  • Ask students to predict whether the answer will be less than, equal to, or more than one, and explain their reasons.
  • Use questioning to scaffold the modelling of the problem with circles partitioned into eighths to represent the pizzas.
  • On the number line, mark the two jumps to \(\frac{7}{8}\). Display the equation \(\frac{4}{8}\) + \(\frac{3}{8}\) = \(\frac{7}{8}\).
Number line between 0 and 1,divided into eighths, showing ‘jumps’ from 0 to 4/8 then a further 3/8 to 7/8.

Student work sample showing addition of fractions with related denominators.

Extend the problem.

Later I ate \(\frac{2}{8}\) of the vegetarian pizza. How much pizza had I eaten altogether?

  • Ask students to predict the answer, then model with the circle diagrams and the extra jump on the number line. Discuss the equivalence of \(\frac{9}{8}\) and 1\(\frac{1}{8}\).
  • Display the equation: \(\frac{4}{8}\) + \(\frac{3}{8}\) + \(\frac{2}{8}\) = \(\frac{9}{8}\) = 1\(\frac{1}{8}\).
  • In pairs, students work on a few similar problems, recording their diagrams and solutions. You might find the recording template useful.

As a class, review the solutions and discuss strategies for completing similar additions without the use of diagrams or number lines.

A similar approach can be used with subtraction problems.