In the curriculum Investigations in Number, Data, and Space, students develop knowledge of the multiplication pairs to 12 x 12 beginning with activities in grade 3 and continuing through grade 5. The goal is for students to become fluent with all these pairs (as well as with many other multiplication pairs). Fluency means that students can use these number pairs quickly and easily as they solve problems.
In Investigations, students encounter multiplication and division together. They are encouraged to use the relationship between these two operations to solve problems. For instance, to find how many packages of 8 balloons you need in order to give a balloon to each of 48 children, you might reason about division relationships ("there are three 8's in 24, and since 24 and 24 is 48, there are six 8's in 48 -- so 6 boxes"), you might draw on relationships between multiplication and division ("8 x 6 is 48, so 48 ? 8 is 6"), or you might reason about halves and doubles ("half of 48 is 24, half of 24 is 12 -- so twelve 4's in 48, and half of 12 is 6 -- six 8's in 48."). As students develop and share strategies for activities such as those listed below, they learn ways to solve multiplication and division pairs, and they practice with these pairs.
Grade 3
Students develop meaning for multiplication pairs through work with a variety of models and materials, including objects, arrays, coins, 100 charts, story problems, pictures, multiples, factors, and equations.
Over the year, students begin to develop a repertoire of multiplication pairs they know. Most students solve unfamiliar pairs with strategies involving counting by one of the numbers in the pair (mentally, on paper, with objects arranged in arrays, or on 100 charts) or repeated addition ("5 + 5 + 5 + 5 is 20, so four 5's is 20" or, "5 + 5 is 10, and 10 and 10 is 20, so 5 x 4 is 20"). For some pairs, students reason about familiar coin relationships ("there are five nickels in a quarter, so 5 x 5 is 25").
Some students begin using pairs they know to solve unfamiliar ones ("4 x 6 is like 2 x 6 twice, so it's 12 and 12, and that's 24"). Students who use such reasoning strategies often do so only for pairs involving small or very familiar numbers: these are the quantities they understand well. They continue to use strategies involving counting by one of the numbers in the pair or repeated addition for pairs involving larger or less familiar numbers.
Selected activities: Cover 50, Arranging Chairs, Multiplication Pairs, Count and Compare, Writing and Solving Story Problems, Many, Many Legs (Things That Come in Groups); Making Counting Charts for 24, Factors of 36 and 48 (Landmarks in the Hundreds).
Grade 4
Students learn all the multiplication pairs up to 10 x 10. They come to learn many of the pairs through repeated use and familiarity. Others they solve with quick and comfortable reasoning strategies that build on their knowledge of multiplication pairs and relationships among them: breaking a difficult pair into smaller, more manageable parts and then combining the parts ("7 x 8 is 4 x 8 and 3 x 8 -- that's 24 and 32, or, 56"), finding relationships among pairs ("5 x 8 is 40, and 6 x 8 is 8 more"), and building on doubles ("6 x 8 is 3 x 8 and 3 x 8, so it's 24 and 24, or, 48"). Students work on many activities that support the development of such strategies. For instance, they compare the size of arrays, combine two or three small arrays to form larger ones, and find multiples of 2, 3, and 10.
Part-way through the year, students identify and work on specific pairs they are having difficulty with. They find and record strategies for learning these difficult pairs, and they share their strategies with their classmates. Students may continue to rely on strategies such as repeated addition or counting with an array to help solve or check difficult pairs.
Selected activities: Finding All Possible Arrays of a Number, Multiplication Clusters, Multiple Bingo, Which Pairs Are Hard for You? (Arrays and Shares); Multiplication Pairs, Count and Compare, Small Array/Big Array, (Arrays and Shares and Packages and Groups); Making a Multiplication Table, Discussing Difficult Multiplication Pairs, Division Bingo (Packages and Groups).
Grade 5
Students review the multiplication pairs, and they list any they are still finding difficult. They develop and share strategies for learning these pairs. They continue to work on these pairs until they know them fluently.
Selected activities: Strategies for Remembering Factor Pairs (Mathematical Thinking at Grade 5).
Marlene Kliman, TERC
1997
This information was reprinted with permission of CESAME, Northeastern Univ., and the Educational Alliance, Brown University.
