Division: Using Math to Help with Language
By Rob Madell, Ph.D., and Jane R. Madell, Ph.D., CCC A/SLP, LSLS Cert. AVT
This is the fifth and final article in a series of articles about the word problems of elementary arithmetic. In the first, How Arithmetic Word Problems Help Language Development, we proposed that learning to solve such problems involves language learning as much as it involves arithmetic. In the subsequent articles we focused on addition, subtraction and multiplication, respectively.
This article addresses the language of division word problems. Think about the following three examples. You may recognize the first problem as an example of Easy Multiplication:
- Problem 1: Suppose there are 3 buses and that each of those buses has 4 chickens on it. How many chickens are there altogether?
- Problem 2: There are 12 chickens. They are on 3 buses, with the same number of chickens on each bus. How many chickens are on each bus?
- Problem 3: There are 12 chickens that have to be put on buses. Each bus can only hold 4 chickens. How many buses are needed?
In all three problems there are 3 buses, 4 chickens on each bus and 12 chickens altogether. The differences between the problems have to do with what information you are provided with and what information you have to solve for. Problem 2 is an example of what is called Partitive Division; Problem 3 is an example of Measurement Division:
||4 per bus
||4 per bus
||4 per bus
The Partitive Division Model
To help a child model Problem 2, you could help him or her to:
- Count out 3 paper cups (RED) to represent the buses (Figure 1a).
- Count out 12 marbles (GREEN) to represent the chickens.
- Distribute the marbles so that each cup (bus) gets the same number of marbles (chickens). One way to do that is by:
- First putting one marble in each cup (Figure 1b).
- Then putting a second marble in each cup (Figure 1c).
- Then putting a third marble in each cup (Figure 1d).
- Then putting a fourth marble in each cup (Figure 1e).
The child can then use this model to solve the problem by:
- Counting the number of marbles in any one of the cups.
Of course it is not necessary that children dole out the marbles one at a time. For example, some children may start by guessing that 3 chickens go on each bus. Others may start by guessing that 5 chickens go on each bus. The thing that is important is that in the end, each bus has the same number of chickens.
Other Partitive Division Word Problems
In the article on multiplication we identified, in addition to Problem 1, all of the following as examples of Easy Multiplication. The point was that just one model (in this case, the Easy Multiplication model) can represent various types of word problems.
- Price Problems – Isabella buys 3 candy bars. Each one costs 4 cents. How much do the candy bars cost altogether?
- Rate Problems – Jacob plants 4 sunflower seeds each day. How many sunflower seeds does he plant in 3 days?
- Array Problems – The seats in a small classroom are arranged in 3 rows, with 4 seats in each row. How many seats are there altogether?
So it will come as no surprise that there is more than one type of Partitive Division word problem. In fact, each of the Easy Multiplication problems above can be modified to become a Partitive Division problem. For example, the Price Problem can be edited to become:
- Isabella buys 3 candy bars that each cost the same amount. Altogether they cost 12 cents. How much does each candy bar cost?
Consider how you would model this problem and how, if at all, your model would differ from the model illustrated in Figure 1. We also encourage you to edit both the Rate and Array problems so that your edited versions are examples of Partitive Division. (Hint – you should be able to represent your problems by the equation 12 ÷ 3 = □.)
The Measurement Division Model
To help a child model Problem 3 you will once again need a supply of “buses” and a supply of “chickens.” You will need to help the child to:
- Count out 12 marbles to represent the 12 chickens.
- Put 4 marbles in a cup (representing the first bus).
- Put 4 marbles in a second “bus.”
- Put 4 marbles in a third “bus.”
Now that all 12 “chickens” are on “buses” the problem can be solved by helping the child:
- Count the number of cups.
It is important to see that the process for modeling Problem 3 is different than that for modeling Problem 2.
Just as the Price, Rate and Array problems above can be edited to become examples of Partitive Division, they can be edited in a different way to become examples of Measurement Division. We encourage the interested reader to do so. As above, you should model your examples and think about how those models differ from one another – if you think that they differ at all. (Hint – all of your problems should be represented by the equation 12 ÷ 4 = □.)
Still More Division Problems
As in the case of Easy Multiplication, there are division problems that correspond to each Hard Multiplication problem. For example, you can turn a Hard Multiplication problem like this one:
- Matt has 4 chickens. Oscar has 3 times as many chickens as Matt has. How many chickens does Oscar have?
- Into two related division problems:
- Matt has some chickens. Oscar has 12 chickens and that is 3 times as many as Matt. How many chickens does Matt have?
- Matt has 4 chickens. Oscar has 12 chickens. Oscar has how many times as many chickens as Matt?
Once again we encourage readers to model these problems on their own and to think about how they are related to Partitive and Measurement problems.
Over the course of this series we have introduced to the reader, in some detail, many different kinds of word problems. As the series comes to an end it is worth stepping back to look at the big picture.
- Before children start to memorize “facts” (like 9 – 4 = 5 and 3 x 4 = 12), they should first learn to solve word problems. To do that they should learn what those word problems mean.
- The meaning of every word problem can be represented by a physical model. Every word problem that a child is likely to see in elementary school can be modeled by one or the other of the models described in this series.
- Some children learn how to model and solve some word problem without direct, explicit instruction. But very few children (hearing or not) learn to solve all the different types in this way.
- Therefore, parents, teachers, speech-language pathologists, and listening and spoken language specialists should familiarize themselves with all the different kinds of word problems and with the models that represent them. They should systematically introduce children to those problems. They should help children to understand the meaning of those problems by helping them to solve them with models.
Source: Volta Voices, July/August 2011