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Section 30.1 Making Groups of Things

In this book, we've already seen two different concepts for multiplication. When we were working with the distributive property (Definition 3.1.4), we talked about multiplication representing groups of things and used this example:

\begin{equation*} \begin{aligned} 4 \cdot (2 \text{ apples} + 3 \text{ oranges} ) \amp = 8 \text{ apples} + 12 \text{ oranges} \end{aligned} \end{equation*}

We also talked about multiplication being represented as an area (Section 7.1):

In this section, we're going to explore these multiplication concepts more deeply in order to understand how they relate to each other and to some of our other representations of numbers.

Activity 30.1.1. Multiplication as \(A\) Groups of \(B\).

The concept of using groups to represent multiplication is often referred to as "\(A\) groups of \(B\)" (meaning the number of objects you have when you have \(A\) groups of \(B\) objects each). If we think about groups of we can represent those as individual objects in the following manner:

It's important to recognize that this is not the same picture as \(B\) groups of \(A\) even though the total number of objects is the same.

Try it!

Draw a diagram to represent \(2 \cdot 4\) and \(4 \cdot 2\text{.}\)

Solution.

Activity 30.1.2. Multiplication as Area.

The concept of area as a representation of multiplication comes from the formula for the area of a rectangle: \(A = \ell \cdot w\) (where \(\ell\) is the length of the rectangle and \(w\) is the width). The area of a shape is the number of unit squares that we can fit into it. In the case of rectangles, things fit perfectly (compared with other shapes, like circles, where you have lots of pieces of squares to deal with).

Try it!

Represent \(4 \cdot 7\) using a rectangle and determine the product by determining the area.

Solution.

We can start to see the connection between thinking about groups and thinking about areas if we simply think about the groups being either horizontal or vertical collections of squares.

These diagrams work well for positive numbers, but if we want to use negative numbers, we need a more robust image. We can look at this either with integer chips or with movement diagrams. We will work our way through the different possibilities.

To do positive groups of positive objects, it's not significantly different than what we've already done. Here are groups of integer chips:

And here are 3 movements of 4:

From here, it becomes fairly obvious what 3 groups -4 of must look like:

But this leads us to try to figure out what a negative number of groups might mean. Instead of using the word "negative" it is better to think of this as "opposite." If we have a group of integer chips, it is pretty clear what the corresponding "opposite" group would be:

Similarly, given a particular movement, the opposite is the same total movement but in the opposite direction.

With this in mind, the product \((-3) \cdot 4\) would be three groups of the "opposite" of 4. In other words, three groups of four negative integer chips.

And also three movements that are the "opposite" of 4:

Activity 30.1.3. Working with Opposite Groups.

Try it!

Using the idea of "opposite" groups and "opposite" movements, determine the value of \(-3 \cdot (-4)\) using both an integer chip diagram and a movement diagram.

Solution.

We can condense this all into a familiar set of "rules" (but remember that it's not about the rule, but about the logic that gives us the rule).