# Subtraction Here are 3 different subtraction problems. Each can be represented by the equation 5 – 3 = □ even though the models of these problems are different from one another. Easy Subtraction: Trixie has 5 lollipops. Some are red and some are green. If she has 3 red lollipops then how many of her lollipops are green? Hard Subtraction 1: Rose has 5 lollipops. Eva has 3 lollipops. Rose has how many more lollipops than Eva? Hard Subtraction 2: Rose has 5 lollipops. Rose has 3 more lollipops than Eva has. How many lollipops does Eva have?

The names of these different problem types are not important - I made them up. (For children it is not even important that they call this "subtraction.") What is important is the fact that your child is going to see all 3 types in school and that the first step in learning to solve such problems is learning to model them in the simplest possible way.

## Easy Subtraction

### 1. What Is The Model For Easy Subtraction?

Here is the easiest way to model the Easy Subtraction problem above:

• Count out 5 objects to represent Trixie's 5 lollipops,
• separate out 3 of those 5 objects to represent the 3 red lollipops and move those 3 objects to the side,
• then count the lollipops that remain.

### 2. How Do You Teach Easy Subtraction?

I have argued elsewhere that the first step in teaching a child to solve any particular type of word problem is direct instruction. So in the case of Easy Subtraction the child should be presented with simple examples and shown how to make the corresponding Easy Subtraction model. Here is Trixie solving a very simple problem.

## Hard Subtraction 1

### 1. What Is The Model For Hard Subtraction 1?

Here is how to model this problem: Rose has 5 lollipops. Eva has 3 lollipops. Rose has how many more lollipops than Eva?

• Count out 5 objects to represent Rose's 5 lollipops.
• Count out 3 objects to represent Eva's lollipops.
• Match up Eva's 3 lollipops with 3 of Rose's lollipops.
• Count the "extra" lollipops that Rose has.

### 2. How Do You Teach Hard Subtraction 1?

• In this video Trixie is only 2 years and 11 months old. It is clear that she needs my help with a Hard Subtraction 1 problem.
• Here she is 5 months later. In some ways she seems much more capable. But she still needs help.

## Hard Subtraction 2

### 1. What Is The Model For Hard Subtraction 2?

Here is how to model this problem: Rose has 5 lollipops. Rose has 3 more lollipops than Eva has. How many lollipops does Eva have?

• Count out 5 objects to represent Rose's 5 lollipops.
• Separate out the 3 lollipops that Rose has that are more than the lollipops that Eva has.
• Give Eva some lollipops - enough to match those of Rose that were not separated out.
• Count the lollipops that Eva has.

### 2. How Do You Teach Hard Subtraction 2?

In this example I help Trixie solve a Hard Subtraction 2 problem. You will see that at 3 years and 4 months she has little understanding of what it is that she is supposed to do.

Easy Subtraction, Hard Subtraction 1, and Hard Subtraction 2 are all different from one another. It is important that children see all 3 types. You have seen above that the child who knows how to model one type does not necessarily know how to model the other two. Click here to download a printable PDF summary, in a new window, of the models of the 3 types of subtraction word problems.

Within any one category of subtraction word problems, some examples may pose more difficulties than others. Consider these two examples of Easy Subtraction.

Trixie had 5 toy cars. She gave 3 of her cars to Rose. How many toy cars does Trixie have left?

Trixie's has a ribbon that is 5 feet long. She cut off the first 3 feet and gave it to Rose. How many feet of ribbon does Trixie have left?

These two problems are modeled in exactly the same way as one another. In each case Trixie has 5 of something and gives 3 of those things away. Nevertheless, the problems are different enough from one another that children should have experiences solving both.