Question: How do you teach multiplication and division?
Answer: Questions about the way Investigations teaches multiplication and division are commonplace. Where are the flash cards? What are landmark numbers? I could use some help developing a better understanding of cluster problems. Why do children work on multiplication and division together? In fact, over the past several months we have received questions on many different aspects of teaching multiplication and division in Investigations. We decided to respond to as many of them as we could by laying out the way that Investigations approaches these topics through the grades.
Things that Come in Groups
In grades K through three students begin to work on multiplication and division by looking at contexts in which things come in groups, or in which a group of things needs to be shared. In the primary grades this involves solving problems like how many hands or fingers in our classroom, and counting things by groups such as 2's, 5's, and 10's. By third grade, students more systematically explore things that come in groups, from 2's to 12's, making lists of these things and then using them to compose multiplication and division story problems and equations. A student might be working on 4's multiplication facts, for instance, and choose something from the 4's list to make a multiplication story with. Such a story might look something like this:
This kind of work leads students toward representing these situations with notation. For instance, a student might represent this problem as either:
4 x 6 = 24 or 6 x 4 = 24
This is an opportunity for students to see that changing the order of the numbers does not change the answer.
Another focus in learning multiplication and division in Investigations is on skip counting. This is a practice that is familiar to all of us from elementary school. Indeed, we are familiar still with the chants of "five, ten, fifteen, twenty, twenty-five" that helped us to remember how to count by fives. However, skip counting in Investigations goes deeper than simply chanting numbers, and students work with numbers which have much more complex patterns than those found in counting by 2's or 5's or 10's.
Skip counting is done through a variety of activities in Investigations. In the Ten-Minute Math activity, Counting Around the Classroom each student says one number in the count while the teacher records the list. When complete, students use this list to identify patterns found when counting by a particular number and then use their knowledge of these patterns to become familiar with multiplication and division facts for that number. Students skip count on the hundreds chart (with which they've become familiar in first grade) and again identify patterns in multiples, but extend this further when they create equations to represent how skip counting by a number got them to a multiple. For example, when counting by 4's they will color in 4, 8, 12, 16, 20, 24, etc. and then identify patterns in those multiples. A student might see that the pattern in the ones place goes "4, 8, 2, 6, 0, 4" and then continues to repeat itself. Students will then make multiplication and division equations that represent this counting such as 2 x 4 = 8 or 8 ÷ 2 = 4. In addition to this, students use hundreds charts to look for patterns in multiples, and relationships between the multiples of one number and those of another (3 and 6, for instance) to help them as they learn facts.
Skip counting and the work that accompanies it (identifying patterns, creating equations to represent relationships between factors and multiples) is a significant part of learning multiplication and division in the curriculum. It is a strategy that students often use to find a multiple when they don't yet have it memorized and it is another sound method for practice.
An important model for multiplication and division in the Investigations curriculum is the array model. In working with this geometric, or area, model, students use their knowledge of number relationships, and of the operations of multiplication and division, to further their understanding and to develop efficient strategies for solving problems. In second grade students first encounter arrays as they use tiles to make and explore rectangles. In third grade, they begin by finding as many ways as they can to arrange 18 chairs in equal rows. For example:
These are all of the possible arrays that students can come up with, but they will often do things like create two of the same array, such as the 2 x 9 and the 9 x 2 as well, believing that these are two different ways to make 18. This is an opportunity to talk about congruence as well as the commutative property.
Students use graph paper to create a larger set of arrays for many other numbers. When they create arrays for numbers such as 11, 19, etc. they recognize that these arrays can only be made in one way -- and that the only factors are one and the number itself. This is an opportunity to explore prime numbers. When students recognize that there are some array sets (all of the arrays for one number) that have squares in them (16, 25), as well as an odd number of factors, this is an opportunity for them to become familiar with square numbers.
Students identify arrays in their world (i.e. cans in a six pack, tiles on a floor, panes in a window, etc.) and create number equations (similar to those in the 4's example above) to represent these. They make a set of array cards, identifying the dimensions of these rectangles and the total number of square space (area) in the rectangle. They consider the relationship between the dimensions and the total, and this is a place in which the relationship between multiplication and division becomes particularly explicit. Seeing that a total is created when two dimensions (factors) are in place can help children to understand that a total (multiple) can be broken apart into equal groups using those two dimensions again. For instance, 2 x 9 = 18 and 18 ÷ 2 = 9. The factor pairs are evident in both equations, but the action that is taking place is different. Using multiplication to solve division problems is a strategy that we want children to become familiar with.
Students in grade four play games that support their understanding of the properties of multiplication and division and the relationship between these operations (games such as Small Array/Big Array, Multiplication Pairs, Count and Compare) and that, ultimately, help them to learn facts.
You might think of the array cards as a set of "conceptual" flash cards. Indeed, the array cards are used to practice learning facts at the same time that students are seeing how the arrays (multiples) can be broken up into pieces (factors and factor pairs; focus on the distributive property) and put back together again.
The work with arrays is a central part of learning multiplication and division in Investigations. It is focused on several big ideas that are significant to understanding the operations, as well as how to use understanding of number relationships and operations when solving multiplication and division problems. These big ideas include:
- understanding the properties of multiplication (commutative, distributive)
- learning about the relationship between multiplication and division
- learning number facts, and
- developing strategies for solving multiplication and division problems with fluency (efficiency, accuracy, and flexibility)
A significant part of the grade four and five multiplication/division curriculum is work with Cluster Problems. These are sets of problems that are used to support understanding in two different ways. First, they help students think about how to start solving a problem that s/he might perceive as difficult. Clusters do this by including several smaller, but related, problems that allow students access to a solution for a more complex problem. Second, Clusters offer an opportunity for students to see, use, and make sense of the distributive property. In other words, they allow students to see the way in which a problem can be broken up (products distributed) into parts that are easier to work with [i.e. 23 x 5 = (20 x 5) + (3 x 5)]. Both of these objectives support the end goal which is to help students develop computational fluency --accuracy, efficiency, and flexibility in solving problems. (For more on this definition of fluency, see the article by Susan Jo Russell at http://investigations.terc.edu/library/families/comp_fluency.cfm.)
For example, a cluster for 23 x 5 might include the following:
3 x 5
20 x 5
10 x 5
23 x 5
Students can see how this problem can be broken into 20 x 5 and 3 x 5 and the products added together. Or a student might see that they could start solving the problem with any one of the problems in this set, thus "chunking" the problem into more manageable parts. Students are not expected to always solve every one of the problems in a cluster and/or to use every one to get to an answer. But this is an opportunity for them to consider how the problem can be broken into more manageable parts and to use what they know (10 x 5, for instance, equals 50, double that to get 20 x 5 = 100) to help them arrive at an accurate answer efficiently. Sometimes when working on clusters students find that there are other useful "starts" or problems for the cluster. For instance, in a problem like the one above, a student might find it easier to start with 25 x 5 or with 2 x 5. Students are encouraged to add those problems to the cluster.
Landmark numbers such as ten or multiples of ten, multiples of one hundred, and twenty-five are numbers that students are familiar with through their work in Investigations. As students become familiar with composing and decomposing these numbers they develop a foundation for operating with these numbers and are encouraged to do so when solving multiplication and division problems. Landmark numbers frequently appear in Cluster Problems (as is evident in the 20 x 5 and 10 x 5 problems in this cluster). (For more on the use of Landmark Numbers in Investigations, see the Ask an Author on this topic).
Story Problems/Problem Situations
All students using the Investigations curriculum in grades K-5 are working with story problems and problem situations that help them to better understand the four operations. These problems allow them to see the different situations that can be represented using the same operation and the same numbers.
For instance, consider the problems below that illustrate two different division situations:
How many groups? (partitioning)
There are 21 children in Lisa's party. If the children break into groups of three for a game, how many groups will there be?
This problem could be represented by an equation such as:
21 ÷ 3 = ?
- or -
3 x ? = 21
How many in each group? (sharing)
There are 21 children at Lisa's party. There are three small tables for the children to sit at. How many children will be at each table?
This problem could be represented by either of the above equations as well. However, each answer will represent a different outcome depending on the situation. In the first example, there will be 7 groups of 3 children; in the second, 3 tables of 7 children. (For more on these two different kinds of division, see the following Ask an Author.)
Students are also asked to create equations, similar to those above, from story problems that accurately reflect the situations and actions in the stories.
Throughout the curriculum at grades 3-5 there is a great deal of opportunity for students to practice as they develop an understanding and knowledge of multiplication and division facts. In addition to the games noted above, students write and solve Multiplication Riddles (in Things that Come in Groups) and solve problems in the Froggy Races activities. They play Cover 50, Multiplication BINGO and Division BINGO, games that support practice with multiplication as well as division facts. Through work with many of the Ten-Minute Math activities, students get regular practice with multiplication and division. Activities such as the Estimation Game, Nearest Answer, Counting Around the Class, and Broken Calculator give students practice with these operations and can be modified to meet the needs of individual students as they are learning facts. For instance, in a classroom where students are working on division of three-digit numbers, the Estimation Game might be played using the following equation:
78 ÷ 16 = _______
The students will see the problem for a few moments and then be asked to compute mentally to find an answer that is accurate, or close to accurate, using what they know about multiplication and division. This offers students the opportunity to develop efficient strategies for solving problems since they can't write an algorithm on paper. A student might use their knowledge of multiples of ten to deal with a big chunk of this problem. A strategy such as this one, when described, might sound like this:
"I know that ten times sixteen is 160. I have eighteen left-over. So one more sixteen and then two left-over. My answer to 178 ÷ 16 is eleven with 2 left."
Another student might have the same answer, but represent it differently. For instance, as 11 2/16 or 11 1/8.
Because the student was able to use knowledge of landmark numbers (multiples of ten) and has an understanding of the relationship between multiplication and division, they were able to deal with a big "chunk" of the problem quite rapidly, and to get to an answer accurately and efficiently.
It is important to note that a good deal of student practice with multiplication and division happens just as it does for adults -- when solving problems outside of the classroom or outside of the operations unit. When students in Investigations classrooms are working on data or geometry units, they have to use their knowledge and understanding of multiplication and division to solve problems in these strands as well.
Teaching Multiplication and Division Together
In Investigations, multiplication and division are often taught together. Below is an excerpt from the Teacher Note, "The Relationship Between Multiplication and Division" (p. 15 of Things that Come in Groups and p. 23 of Arrays and Shares) which describes this relationship and the benefits of teaching these two operations together.
Multiplication and division are related operations. Both involve two factors and the multiple created by multiplying those two factors. For example, here is a set of linked multiplication and division relationships:
8 x 3 = 24
24 ÷ 8 = 3
3 x 8 = 24
24 ÷ 3 = 8
Mathematics educators call all of these "multiplicative" situations because they all involve the relationship of factors and multiples. Many problem situations that your students will encounter can be described by either multiplication or division. For example:
"I bought a package of 24 treats for my dog. If I give her 3 treats every day, how many days will this package last?"
The elements of this problem are: 24 treats, 3 treats per day, and a number of days to be determined. This problem could be written in standard notation as either division or multiplication:
24 ÷ 3 = 8 or 3 x ____ = 24
As described above, students work towards fluency with multiplication and division facts and computation in a variety of ways. They develop a meaningful sense of operations and the actions they represent as they think about the context of things that come in groups, and solve story problems. They develop a visual image for multiplication, and for the "size" of various facts through the arrays, which also help them see and understand properties of multiplication and division (such as distributivity). Cluster problems also help students make such connections (12 x 7 = 10 X 7 + 2 x 7) and to use what they know to solve more difficult problems. Practice also comes in many forms -- multiplication and division games, story and cluster problems, bare number problems, Ten Minute Math activities, and regular classroom math activities. The real benefit is that all of these activities support both learning and practice.
Elizabeth Van Cleef and Megan Murray, TERC
Russell, Susan Jo. Relearning to Teach Arithmetic: Multiplication and Division. Dale Seymour Publications, 1999.
Tierney, Cornelia, Berle-Carman, Mary, Akers, Joan. Things That Come in Groups from the third grade Investigations in Number, Data, and Space curriculum. Scott Foresman Publications, 1998.
Economopoulos, Karen, Tierney, Cornelia, Russell, Susan Jo. Arrays and Shares from the fourth grade Investigations in Number, Data, and Space curriculum. Scott Foresman Publications, 1998.