Multiplication and division Unpacked

Generalization / standard - Multiplication and division are ways to describe how two (or more) values can be operated on to create another value that is equivalent to the initial values in some way.

9th – 12th

Create equal subgroups within a collection.

Associate multiplication and division with repeated groups of equal size.

Division and multiplication as involving the distribution of equal shares or groups.

Multiplication and division as repeated groups of equal amounts or shares.

The meanings of, and relationship between, multiplication and division.

What each number in a multiplication or division expression represents as well as the units each represents. Factors in multiplication and division can refer to different units. (29 cats with each having 4 legs. 116 is the total number of legs on 29 cats.

Multiplication and division problems can be modeled with pictures, diagrams, or concrete materials to illustrate what the factors and the product represent in various contexts. (if there are 112 people traveling by bus and each bus can hold 28 people, how many buses are needed? 112 people put into groups of 28 people, for each bus, will need one bus for each group.) or (If 112 people need to fit among four buses, how many people will need to be on each bus? In this case, 112 people distributed across 4 buses will indicate the number of people on each bus.

Ways to determine what to do with remainders can be modeled when dividing.

How to know what values of remainders can be expected from different divisions. (putting counters into sets of 4 can result in what values of remainders for different amounts of counters?

Inverse relationship between the two operations.

Rates (3 candy bars for 59 cents each).

Comparisons (the book weighs 4 times as much as the tablet).

Combinations (the number of outfits possible from 3 shirts and 2 pairs of shorts).

For example, dividing 28 by 14 and comparing the result to dividing 28 by 7 can lead to the conjecture that the smaller the divisor, the larger the quotient.

Dividing numbers between 0 and 1, such as 1/2, and find a quotient larger than the original number. Explorations such as these help dispel common, but incorrect, generalizations such as "division always makes things smaller."

Area models to show and develop-
the relationship of a product to its factors. An understanding of multiplication properties (Graeber and Campbell 1993). Properties of operations such as the commutativity.
Other relationships by composing and decomposing area models. (The distributive property, 20 * 6 can be split in half, then rearranged to form a 10 x 12 rectangle, and show the equivalence of 10 * 12 and 20 * 6. is particularly powerful as the basis of many efficient multiplication algorithms.

Area models are useful to show multiplication properties (Graeber and Campbell 1993).

Area models can be extended to operations and properties with fractions, decimals, and percentages as well as their combinations with whole numbers and each other.

Operations can be used in general ways, rather than only in particular computations.

Reasoning about the properties of the numbers involved rather than for following procedures to arrive at exact answers.

Judging the reasonableness of results.

Solve problems by balancing equations [5+7-2=2(5)] or [100/25 = 347-343]. 

Create and interprete algebraic equations (8x+2=50).

Understand parenthetical order of  operations in equations such as (5x4)-15+2 or (12-2)(3+4).

Dr. Robert Sweetland's Notes ©