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### 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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# Part and wholes

In years 3 and 4, students use two dice, one labelled 1 to 6 (called the whole-number die) and the other labelled with fractions (called the fraction die). The fraction die has fractions appropriate to the year level (e.g. for year 3 the fractions could be $$\frac{1}{2}$$, $$\frac{1}{4}$$, $$\frac{1}{8}$$, $$\frac{1}{3}$$, $$\frac{1}{5}$$).

Pairs of students roll the two dice to generate numbers to insert into the sentence:

(fraction roll) of my collection is (whole-number roll) so altogether I have ____

Example: One quarter of my collection is 5 so altogether I have __

Students model the problem, using counters to make a group of five.

They then make more groups of five until there are four equal groups, and calculate the total.

Teachers should:

• encourage the formation of arrays to make the relationship with factors and multiples more obvious
• discuss strategies for working out the total without having to build the array
• ask what other fractions can be determined from the array, such as
$$\frac{3}{4}$$ of 20 = 15
$$\frac{1}{5}$$ of 20 = 4

In year 6, students solve problems to determine fractions of collections and multiples of those fractions. For example:

I lost $$\frac{1}{8}$$ of my marbles. I have 42 left. How many did I start with?

Student work sample

Discuss with the class:

• possible strategies for solving the problems
• the effectiveness of various strategies.

For students not confident in factors and multiples, creating arrays on grid paper can provide scaffolding for finding strategies.