# Number Sense in Decimals – Multiplication and Division

We love this 5th grade question for decimals:

If 8 apples cost \$4.16 and 13 lemons cost as much as 5 apples, what is the cost of 8 lemons?

First, here’s our suggested solution. Why do we like this question? It requires students to break apart the problem into 3 or 4 separate steps, but more importantly, it provides a chance to use two very important skills in problem solving.

## Distributive Property

In the first step of the problem, we need to find out how much one apple cost so that we can find out how much 5 would cost. By splitting up 4.16 into 4.00 + 0.16, and operating on (dividing by 8) them separately, and then adding the results back to get the final answer, students rely on the distributive property of division. This greatly simplifies the calculation involved and builds up on their mental math.

## Number Sense

In the second part of the problem, we need to find out how much 1 lemon costs in order to figure out how much 8 of them would cost. To do this decimal division by standard algorithm is rather tedious and prone to errors, but if we recognize that two 13’s go into 26, i.e. 26 divided by 13 is 2, and relying on our number sense, we can guess the answer is 0.2 without working out the answer explicitly.

## Observation

Hence, a question does not have to be wordy or complex. A short and simple question like this can exercise the three aspects of rigor fully — conceptual understanding , procedure fluency and application.

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