Download a Sequence of Instruction Graphic Sequence of Instruction
Although it is not necessary to use every unit in the suggested order, they are designed with specific topics and grade levels in mind. For a graphic that provides a visual representation of the sequence and relationships of the components in Investigating Multiplication and Division click on the attached pdf.
Groceries, Stamps, and Measuring Strips: Early Multiplication
by Frans van Galen and Catherine Twomey Fosnot
Using baker's trays, postage stamps, patio tiles, and other realistic contexts, Groceries, Stamps, and Measuring Strips was developed to introduce multiplication in grades 2-3. Initially, formal multiplication notation is not the focus; efficient grouping is, as students are encouraged to make groups (and groups of groups) as an efficient way to deal with repeated addition and determine totals. Carefully crafted arrangements of real-world resources invite strategies such as the use of doubles, doubling and halving, and partial products. As the unit progresses, formal notation and use of the language of "times" is introduced within the context of measuring buildings and other objects in a city in relation to the height of eight-year-old Antonio, who is four feet tall. For example, a tree is determined to be twelve feet high if it is three times the height of Antonio. Subsequently, making measuring strips invites children to discuss relationships that help them automatize the basic multiplication facts.
The Big Dinner: Multiplication with the Ratio Table
by Catherine Twomey Fosnot
In The Big Dinner, the context of preparing a turkey dinner initially engenders a discussion on students' early multiplication strategies. As the unit progresses, the ratio table is introduced and students are supported to use the distributive property with large numbers. Strings of related problems guide learners toward computational fluency with whole-number multiplication and build automaticity with multiplication facts by focusing on relationships. This unit was designed to be used in grade 3.
Muffles' Truffles: Multiplication and Division with the Array
by Antonia Cameron and Catherine Twomey Fosnot
A chocolatier's efforts to cope with the operational challenges of running a truffle shop (counting, pricing, and labeling assorted boxes of chocolates) in Muffles' Truffles introduce students to the array as a model for multiplication and division. A series of investigations explores and invites students to make use of place value—the multiplicative structure of our baseten system and quotative division—and big ideas in multiplication, including the distributive, associative, and commutative properties. As the unit progresses, students design boxes for Muffles and build blueprints of their designs with graph paper arrays, and the array of chocolates arranged in rows and columns is progressively transformed into the open array model. This unit was developed for grade 3.
The Teachers' Lounge: Place Value and Division
by Chris Natale and Catherine Twomey Fosnot
The stocking of water and juice vending machines in The Teachers' Lounge introduces big ideas related to division. In considering different ways to inventory the contents of each machine, students employ a repertoire of strategies, including the use of the ten-times strategy, partial products and partial quotients, and the distributive property of multiplication over addition—the basis for the long division algorithm. They also examine the relationship between partitive and quotative models of division and explore what to do with remainders in various contexts. This unit was designed to be used in grade 4.
Minilessons for Early Multiplication and Division
by Willem Uittenbogaard and Catherine Twomey Fosnot
Minilessons for Early Multiplication and Division is a resource of 75 minilessons. Some of the minilessons use pictures of realistic situations, others make use of quick images with ten-frames and arrays while other minilessons are crafted as a tightly structured series of computation problems. These minilessons will generate discussion on certain strategies or big ideas underlying an understanding of early multiplication and division.
The Box Factory: Extending Multiplication with the Array
by Miki Jensen and Catherine Twomey Fosnot
The Box Factory was designed to be used in grade 4 or 5. Its focus is the deepening and extending of students' understanding of multiplication, specifically the associative and commutative properties and their use with computation, systematic factoring, and the extension of an understanding of two-dimensional rectangular arrays to three-dimensional arrays within rectangular prisms. As the unit progresses, formulas for area, surface area of rectangular prisms, and volume are generalized and the relationship between surface area and volume is explored within the context of a box factory, with students designing boxes to meet specific size and space requirements.
Minilessons for Extending Multiplication and Division
by Catherine Twomey Fosnot and Willem Uittenbogaard
Minilessons for Extending Multiplication and Division contains 77 minilessons. structured as strings of related computation problems. They are likely to generate discussion of certain strategies or big ideas that are landmarks on the landscape of learning for multiplication and division, particularly using numbers with two and three digits.