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Lessons

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    (Also available in WeScheme)

    Students develop a more nuanced interpretation of the Commutative and Associative Properties as a result of their exploration of the inverse relationship between multiplication and division.

    Lesson Goals

    Students will be able to…​

    • Recognize that dividing by 𝑥 is the same as multiplying by 1/𝑥.

    • Acknowledge flexibility in the Order of Operations.

    Student-facing Lesson Goals

    • Let’s explore the inverse relationship between multiplication and division.

    Materials

    Key Points For The Facilitator

    • This lesson is all about mathematical flexibility, and may challenge preexisting ideas about arithmetic. For students who’ve had a mostly procedural introduction to mathematics - or who have high math anxiety - the flexibility introduced by this lesson can be intimidating!

    • Embrace an open mind and expect the same of your students, and be attentive to their emotional response as the lesson progresses. If students are getting anxious, they will not be able to concentrate and focus on the content.

    • This lesson digs into multiplication and division of unit fractions and whole numbers, emphasizing structure over computation.

    • This lesson has a similar structure as The Additive Inverse Property. Students may benefit from looking back at the pages they completed during that lesson.
    Glossary
    Associative Property

    When adding three numbers or multiplying three numbers, it does not matter whether you start with the first pair or the last. The same is true when either adding or multiplying four numbers, five numbers, etc.

    Commutative Property

    For any expression involving only addition or only multiplication, changing the order of the numbers will not change the result.

    Multiplicative Inverse Property

    Multiplying a number and its reciprocal always produces 1

    reciprocal

    the reciprocal of any real number n is 1/n.

    🔗The Multiplicative Inverse Property 10 minutes

    Overview

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

    Launch

    We know that the Commutative and Associative 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.

    But first, let’s play a game. Draw a two-column table on the board to record response. Label the left-hand side "Starting Value," and label the right-hand side "?" (see example below).

    Starting Values ?

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

    • The first number is 1/2. What do I need to multiply 1/2 to get 1?

    • The next number is 1/3. What do I need to multiply 1/3 to get 1?

    • How about 1/100?

    • How about 10?

    • Can someone offer me another pair of numbers - a fraction and a whole number - that multiply together to produce 1?

      • 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.

    The Multiplicative Inverse Property tells us that multiplying a number by its reciprocal always produces one.

    Every number, except zero, has a multiplicative inverse.

    Investigate

    • Turn to The Multiplicative Inverse Property.

    • In the first section, practice finding reciprocals and write them in the space provided.

    • Then, fill in the missing number to complete the equations. Some equations use mathematical notation and some use Circles of Evaluation.

    Synthesize

    • Can you think of a way to visually represent that a number multiplied by its reciprocal produces one?

    • How would you explain the Multiplicative Inverse Property to another student?

    🔗Multiplication and Division: Inverse Operations 20 minutes

    Overview

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

    Launch

    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?

    Two main ideas emerged during the previous exploration:

    • Dividing by 𝑥 produces the same result as multiplying by its reciprocal, 1/𝑥.

    • Dividing by 1/𝑥 produces the same result as multiplying by its reciprocal, 𝑥.

    In other words, when students 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.

    Investigate

    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.

    Complete Which One Doesn’t Belong?.

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

    Synthesize

    • Claire and Soraya want to write an equivalent expression for 45 ÷ 9. Claire studies the expression and announces that, because it involves division, the Commutative Property cannot be applied. Is she correct?

    • Soraya grabs a pencil and writes the following: 45 × 1/9. She says, "There! I fixed it. Now we can apply the Commutative Property." Explain what Soraya did. Is she correct?

      • Sample response: Instead of dividing by 9, Soraya is multiplying by the reciprocal. Yes, Soraya has written an equivalent expression and can apply the Commutative Property - but the computation will not be any simpler.

    🔗Is the Order of Operations Universal? 25 minutes

    Overview

    Students examine an algorithm taught in Kenya and consider how and why it differs from what they might have learned previously. They discover that the Commutative Property and Associative Property are more powerful than they initially thought!

    Launch

    • Consider this expression: 100 × 20 ÷ 5

    • Rewrite the expression - either by adding parentheses or drawing a Circle of Evaluation - to show your process for solving.

    • What do we get when we simplify the expression to a single value?

      • 400

    • How did you arrive at your answer?

    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. This solving strategy can be represented by the Circle of Evaluation, below.

    (/ (​* 100 20) 5)

    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!)

    (​* 100 (/ 20 5))

    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!?

    Investigate

    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.

    • Turn to Divide First…​ or Solve Left-to-Right?.

    • There, you will test out the "Kenya algorithm" on several different expressions to see if dividing and then multiplying produces the correct result every time.

    • What do you Notice? What do you Wonder?

    • Why are we able to change the groupings for an expression like 100 × 20 ÷ 5 …​ but not for an expression like 100 ÷ 20 ÷ 5?

    • Describe why the "Kenya algorithm" works. (Hint: Think about the Multiplicative Inverse Property!)

      • We can rewrite any division expression as multiplication by the reciprocal. Once we transform a division expression into a multiplication expression, we can apply the Commutative and Associative Properties freely!

    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!

    Now, let’s put our new knowledge to use! Project the problems below one at a time, and invite students to solve using mental math.

    Scan each expression to determine the simplest solving strategy, then compute mentally.

    • 114 × 17 ÷ 17

      • Solution: 114

    • 15 × 3 ÷ 15

      • Solution: 3 **

    • 2 × 16 × 1/27 × 27

      • Solution: 105

    Synthesize

    • How did it feel to scan the problem, choose your strategy, and then solve mentally?

    • Did you like this new approach - or do you prefer solving from left to right?

    • Knowledge of inverse operations creates more opportunities to apply the Commutative Property and the Associative Property? Explain why this is the case.

    • Do you think the Order of Operations is universal? Why or why not? Yes, there is a basic agreed upon order across countries, but numerous differences exist within tiers and how they are described.

    • Can you think of any other examples - they can be math-related or not! - of when you thought there was just one way to do something…​ and then learned that you were wrong?

    These materials were developed partly through support of the National Science Foundation, (awards 1042210, 1535276, 1648684, 1738598, 2031479, and 1501927). CCbadge Bootstrap by the Bootstrap Community is licensed under a Creative Commons 4.0 Unported License. This license does not grant permission to run training or professional development. Offering training or professional development with materials substantially derived from Bootstrap must be approved in writing by a Bootstrap Director. Permissions beyond the scope of this license, such as to run training, may be available by contacting contact@BootstrapWorld.org.