The Multiplicative Inverse Property

Students discover the Multiplicative Inverse Property, which tells us that multiplying a number and its reciprocal always produces one.

The Commutative and Associative Properties help us re-order expression to make them easier to think about and solve. We also know that these properties apply to multiplication, but not division. But what if there was a way to rewrite division as multiplication? Then we could apply the Commutative and Associative Properties to division expressions, too! Let’s explore this idea with a game.

Draw the following table on the board:

@table{

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I’m going to write a number in the left-hand column. You are going to tell me the reciprocal - or the value I should multiply my starting number by to get a product of 1. I’ll record your response in the right-hand column.

Allow a variety of students to share. Record responses on the table.

These number pairs all represent reciprocals. The reciprocal of a real number 𝑛 is ¹/_𝑛. These number pairs are also illustrative of the Multiplicative Inverse Property: when we multiply them together, we always end up with a product of 1.

Every number, except zero, has a multiplicative inverse.

Students rewrite multiplication expressions as division, and division expressions as multiplication by applying their knowledge of the Multiplicative Inverse Property.

Now that we understand what a reciprocal is, we are ready to think about how we can put it to use. Perhaps it can make some computations simpler?

Hopefully, you discovered two big ideas:

In other words, when you see multiplication or division by a unit fraction (a fraction with a numerator of 1), there is likely a path forward using mental computation only. Understanding how the reciprocal can help us to write equivalent expressions can often make computation much simpler!

Now, students are ready to continue their exploration of multiplication as the inverse of division, while also integrating and applying their knowledge of the Commutative and Associative Properties of Multiplication.

Invite students to share which problems were most challenging, and which ones felt simple. Have students share strategies for determining equivalence.

Students learn an algorithm taught in Kenya, which is used for solving certain types of problems. They then compare and contrast it with an algorithm they have likely seen before. They discover that the Commutative Property and Associative Property are more powerful than they initially thought!

Invite students to share their responses. If your students have spent any time at all studying the order of operations, they will notice both multiplication and division in the expression. From there, they will likely conclude that they must work from left to right to arrive at a correct result.

If there is a brave student who opted to divide before multiplying, invite them to share their method and then ask other students to weigh in. If all students worked left to right, ask students to evaluate the Circle of Evaluation below and then assess if it is equivalent to the Circle of Evaluation, above. (Spoiler alert: It is!)

We’ve learned that the Associative Property applies for expressions with only multiplication…​ not multiplication and division. Many of us have also learned that when an expression includes multiplication and division, we must work from left to right. So… what’s going on!?

In Kenya, students are actually taught that, when confronted with an expression like 100 × 20 ÷ 5, they must divide first…​ and then multiply! But does it actually work, every time? Let’s investigate.

Encourage students to think deeply about why this algorithm works – and if you’d like, invite them to consider and discuss why students all across the country are typically taught just one algorithm when, typically, there are an abundance to choose from!

Students apply their knowledge of examples in WeScheme to think about multiplication and division as inverse operations.

Lead a discussion where students share their thinking and strategies for predicting if the examples would pass or fail.

Debrief with students to ensure that they are looking at the messages that appear in WeScheme. This activity not only provides practice thinking about the multiplicative inverse; it also gives students exposure to tests - bits of code used to verify that code is working as we would expect. Examples and tests are widely used in programming! We explore examples in greater depth in Functions: Contracts, Examples & Definitions.

Let’s revisit our conversation about solving left-to-right…​ or right-to-left.

Students should observe that when multiplication precedes division, they can solve in any order. When division precedes multiplication, however, they must divide first.