Overview

Session Summary

In this session you will consider the number and operation knowledge students need to be able to develop efficient computation strategies for multiplication and division. You will examine your own starting place for mental computation, contexts and visual models for understanding multiplication, and the relationships of multiples of numbers. The session concludes with a look at student work and a discussion of strategies that students use.

In Investigations, the goal for this topic is that, by the end of fifth grade, students have both a solid understanding of multiplication and division situations, and several efficient strategies for solving problems. You will consider how students build understanding of number and operations, and how this understanding leads to strategies for solving multiplication and division problems.

Before you begin to look at what may be more familiar to you, that is, how students think about multiplication and division in the upper grades, consider the foundations they bring from their work with number in the lower grades. What mathematical experiences do young children have that can support them when they encounter ideas about multiplication? What about division?

Mathematics in this session

• Examine strategies for solving multiplication and division problems
• Understand the relationship between multiplication and division
• Find patterns in a sequence of multiples in order to solve problems
• Use arrays to model multiplication
• Consider how students develop computational fluency multiplying and dividing whole numbers

Common Core Alignment

The content in this session is aligned to the following Common Core State Standards and Math Practices:

MATH STANDARDS

Operations and Algebraic Thinking 3.OA
3.OA.1, 2, 3, 4, 5, 6, 7
Represent and solve problems involving multiplication and division.

1. Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 × 7.
2. Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.
3. Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.
4. Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = ? ÷ 3, 6 × 6 = ?.

Understand properties of multiplication and the relationship between multiplication and division.

5. Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)
6. Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.

Multiply and divide within 100.

7. Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.

Number and Operations in Base Ten 3.NBT
3.NBT.3
Use place value understanding and properties of operations to perform multi-digit arithmetic.

3. Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations.

Number and Operations in Base Ten 4.NBT
4.NBT.3, 5, 6
Generalize place value understanding for multi-digit whole numbers.

3. Use place value understanding to round multi-digit whole numbers to any place.

Use place value understanding and properties of operations to perform multi-digit arithmetic.

5. Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
6. Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Number and Operations in Base Ten 5.NBT
5.NBT6
Understand the place value system.

6. Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

MATH PRACTICES

MP1 Make sense of problems and persevere in solving them.

MP2 Reason abstractly and quantitatively.

MP5 Use appropriate tools strategically.

MP6 Attend to precision.

MP8 Look for and express regularity in repeated reasoning.

You may also wish to review the full set of Common Core State Standards in Math and/or videos of the Math Practices enacted in Investigations.

Mental Math: Multiplication

Overview

In this activity you will solve two multiplication problems mentally. As you consider your strategies you will name the first steps that you use in your mental strategies. Both adults and children tend to use more flexible thinking when doing mental computation rather than using learned pencil and paper procedures. We want to take a look at some of the knowledge we use when we compute mentally.

Mental Math Problems

Take a look at the following problem, and estimate whether it is more or less than 150:

29 x 6
More or less than 150?

Write down any steps that will help you keep track of your strategies. When we use the term “mental computation,” we don’t necessarily mean that no recording takes place.

Now consider another problem:

15 x 14
More or less than 200?

Strategy 1

Strategy 2

Strategy 3

When students solve multiplication problems, they most often use strategies that involve breaking numbers apart to create problems that are more manageable and that make use of familiar number relationships. Next you will look at how the use of contexts and representations help students to develop these strategies.

Contexts and Representations

Overview

Now you will look at activities from the curriculum that support students in building an understanding of multiplication and division through the use of contexts and representations. You will consider Things That Come in 3s and solve a pair of related multiplication problems. You will find all the ways to arrange 12 chairs in equal rows and see how this context is used to introduce the representation of arrays. Finally you will work on breaking a 6 x 9 array into smaller pieces and see an example of how unmarked arrays can be used to record strategies.

Things that Come in 3s

Visual mathematical models and contexts are often helpful for understanding multiplication.

In third grade the work on multiplication begins with a discussion about naming things that come in equal-sized groups.

After brainstorming contexts for things that come in groups of 2 to 12, students write and solve many multiplication questions such as this one:

Watch the screencast below to see a representation of this story problem.

A next problem for students to solve might be:

Array Model of Multiplication

One of the models that Investigations uses is an array, which some of you have been exploring in your grade level groups. Students begin working with arrays within a geometry context in second grade, when they examine the properties of rectangles that they construct using color tiles.

In third grade students solve “Arranging Chairs problems” as a context to construct the array model for multiplication.

Find all the ways to arrange 12 chairs, and then click the Show link to check your work.

As students work on the Arranging Chairs problem for totals up to 30, they learn about factors and are introduced to number concepts such as square numbers and prime numbers.

Using Array Cards

Notice that the front of each array card is marked with a grid and is preprinted with dimensions of the card, such as “5 x 7 and 7 x 5.” The back of each card is plain (no grid) and is preprinted with the product in the center and with one of the two dimensions, such as “35” and “5.” The grade 3 array cards include combinations up to a product of 50. The grade 4 cards include combinations up to 12 x 12.

Using your own array cards, find two arrays you can put together to make a rectangle that matches the 6 x 9 array exactly. Try to find several combinations of two arrays. When you are done, watch the slideshow below to see possible solutions.

NOTE: Since the Array Card sets include only one of each array, you will have to visualize a combination that involves doubling an array, as in (3 x 9) + (3 x 9).

NOTE: As students learn how to break apart multiplication problems, they are applying an important property of multiplication, the distributive property. Although it is not important for elementary students to identify the distributive property by name, it is important for teachers to understand how students’ strategies relate to this property in order to teach students how to keep track of which numbers must be multiplied when breaking numbers apart in a multi-digit multiplication problem.

Here is an example of a multiplication card that third and fourth grade students make as they are learning their multiplication combinations. For a “hard” combination, like 9 x 6 can be for many students, they record a “Start with” clue to help them practice that combination.

Based on the ways that you saw in the slideshow to break apart the 6 x 9 array, what are some possible “start with” clues that a student might use to help remember this combination? Click the Show link to see some possible answers.

You may wish to refer to the Teacher Note, Learning and Assessing the Multiplication Combinations for more information about how students learn and practice multiplication “facts” in Investigations.

OPTIONAL ACTIVITY

If time permits, you may want to explore one or more of the following games that students play with the array cards to become familiar with the multiplication combinations.

• In the Factor Pairs game students lay out all of the array cards with the grid-side up. They choose a card and say the product and how they know it. They check their answer by looking at the back of the card. Students create two lists “Combinations I Know” and “Combinations I’m Working on” as they play Factor Pairs.

• The Missing Factors Game is played the opposite way as Factor Pairs. Students lay out all of the array cards with the product side up. They choose a card and say the missing dimension and how they know it. They check their answers by turning the card over. Use the Missing Factors recording sheet for this game.

• Count and Compare is a game played with a partner. Each player gets half of the set of array cards. In each round players play the top array card in their pile, dimension side up and determine which array is larger. Some rounds of this game can be trivial, such as 1 x 3 vs. 10 x 5, but in a round such as the one shown here, students have to reason—without counting by ones—who has the larger product. How might student figure out which is larger, 4 x 7 or 5 x 6? [Possible strategies include overlapping the cards, or subtracting 4 x 5 from each, and comparing 4 x 2 and 5 x 2.]

An example of the use of contexts and representations in grades 4 and 5

Here is one of the problems that we solved earlier in this session expressed in a story problem context that is used in grades four and five as students solve multiplication and division problem--the idea of player and teams. In grades four and five, however, students would be working with larger numbers.

Also in grades four and five students use arrays as a representation for recording their strategies. Unmarked arrays, like this one, do not have grid lines like the Array Cards, but are drawn approximately to scale of the given dimensions.

Patterns of Multiples

Overview

In this activity you will be introduced to the Ten-Minute Math activity, Counting Around the Class. You will use a multiple tower to solve two problems and relate this skip counting work to how students build their understanding of multiplication and division.

Throughout Investigations, students engage in many different counting activities. These include counting orally and reasoning about the numbers they say, counting things in groups, and exploring counting sequences on the 100 Chart. For young children, counting in groups is the basis of their foundation for understanding multiplication and division. A Ten-Minute Math activity that continues to support this in grades three and four is Counting Around the Class.

Watch the video below and pay particular attention to the questions the teachers ask as they count.

Multiple Towers

Take a look at the Partial Multiple Tower for 15 on the right hand side of the screen. Note that some of the multiples of 15 are recorded on this strip, starting at the bottom with 15 (which isn’t visible) and working upwards.

Identifying these landmarks can be helpful for using the tower to solve multiplication and division problems.

Click the Show link to see how other participants might solve this problem.

Click the Show link for a possible solution.

Click the Show link for a possible solution.

Click the Show link for a possible solution.

Throughout Investigations, students count around the class and reason about the counting numbers they say. In the early grades, students count by 1s, and they explore questions like, if we count in a different order will we reach the same number? If two people are absent today, what number will we get to when we count around the class?

By second grade, they count by small, familiar numbers such as 2 and 5. In third grade they move on to numbers such as 12, 15, and 25, and in later grades, they count by less familiar numbers such as 45 and 72, larger numbers such as 250 and 500, and numbers such as .1 and 1/2. Sometimes they start at numbers other than 0 or they count backwards (for instance, if we start at 1,000 and count back by 75’s, will we end up on 0?).

Student Work: Division

In Investigations, students have a great deal of opportunity to build a strong sense of number as they work through number units throughout the grades. In the youngest grades, they spend time counting, looking for patterns, and learning about the base ten number system.

In the middle grades, they continue to work with the number system, develop good strategies for addition and subtraction, and build understanding of multiplication and division through models and contexts.

In the upper grades, we want them to consolidate their knowledge and understanding of numbers and operations as they develop and use efficient and effective strategies for computation.

Here is a division problem from fifth grade for you to solve:
374 ÷ 12 =

There is a wide variety in the strategies that students use to solve a division problem. Solve this problem for yourself in any way you want to. You may want to solve it in more than one way.

Then, look at the following examples of student work. You may wish to review the DIET protocol for looking at student work to guide your analysis.

• Student A
• Student B
• Student C
• Student D
• Student E

As you look at the student work, consider these questions:

• Does the student break the problem into easily manageable parts?
• Does the student solve all the sub-problems (using either division or multiplication) correctly?
• Does the student keep track of what has been solved and what remains to be solved?
• Does the student combine the products or quotients of their partial solutions to find the solution to the original problem?
• Does the student present his/her solution strategy clearly and concisely?
• Does the student correctly identify the remainder?

In many ways, mastering division seems to be the culmination of work that elementary students do on whole numbers and their operations. For many students, it is also the hardest, and for good reason. In order to solve division problems accurately and efficiently, students need to be able to hold on to a variety of ideas about both numbers and operations, and not just the operation of division but the other three as well.

Discussion

What do students need to know and understand to develop efficient strategies for solving multiplication and division problems?

How does the use of contexts and representations help students make sense of and solve multiplication and division problems?

Go to the Forum

Key Learning

• When students solve multiplication problems, they most often use strategies that involve breaking numbers apart to create problems that are more manageable and that make use of familiar number relationships.
• Students use representations, models, such as arrays and contexts to help them make sense of and solve multiplication and division problems.

This session is aligned with the following K-2 Units in Investigations.