Multiplication & Division: Counting-based Strategies


Learning Goal

Develop counting-based strategies for multiplication and division

Learning Goal Description

Counting-based strategies for multiplication and division are related to the idea of repeated addition and subtraction. These strategies involve counting in multiples, rather than counting the individual items. The learning progression includes forming equal groups and making equal shares, number word sequences for multiples, and a range of tasks for multiplication and division. A major focus is on developing methods for simultaneously keeping track of the number of groups, the number of items in each group, and the total number. This knowledge creates a strong foundation for developing multiplicative strategies and building fluency with multiplication and division basic facts.

Pre-Requisite Knowledge

For Sub-goals A and B, students should have facility with the number word sequence to 100 and be able to solve addition and subtraction tasks by counting from one, by ones, with materials. For Subgoals C and D, students should be able to solve addition and subtraction tasks using count-up-from and count-down-from strategies and possibly some non-count-by-one strategies. 

(see Ready Set Math modules Early Numeracy: Words, Symbols, and Quantities and Addition & Subtraction: Counting-based Strategies)  

Sub-goal A : Form equal groups and make equal shares

See Instructional Resources
Forming equal groups provides a foundation for early multiplicative thinking, with students attending to the number of groups and number of items in each group. Equal sharing involves organizing a collection of individual items into a specified number of groups (sharing) or a specified group size (grouping). The instructional resources focus on creating, organizing, and describing equal groups and equal shares.
  1. Ask students to organize items into equal groups (e.g., Using 3 small plates and 15 counters representing candies, set up a scenario with 9 counters on the first plate, 2 on the second plate, and 4 on the third plate. Ask students to make it so each plate has the same number of candies).
  2. Ask students to form a specified number of groups, each having a specified number of items (e.g., Provide a collection of 20 counters, ask the student to make 4 groups with 3 in each group). 

If students are not successful, the resources for this sub-goal can support designing instruction for forming equal groups and equal sharing. 

Sub-goal B

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Sub-goal C : Solve multiplication tasks using counting-based strategies

See Instructional Resources
Counting-based strategies for multiplication involve conceiving of a group of individual items as a composite unit (a new unit made up of individual unit items) and iterating (repeating) that unit a specified number of times. The instructional resources promote counting in multiples (involving 2s, 5s, 3s, and 4s) to solve multiplication tasks, simultaneously keeping track of the number of groups, the number of items in each group and the total number of items. Tasks advance from visible materials, to screen materials, to the absence of materials. (Note: The goal across this module is not to automatize multiplication and division for 2s, 5s, 3s, and 4s, but rather, to use these multiples as a means for developing strategies that involve iterating and keeping track. Larger multiples are not emphasized because they involve long counts and can be better handled with other strategies.)
  1. Ask students to work out a range of multiplication tasks (e.g., What is 6 groups of 2?; How much is six 5s?; 6 x 2; 4 x 7). Ask students how they worked out their answers. 

If students are not successful or use counting by ones, check first for knowledge with forming equal groups in Sub-goal A and verbal sequences for multiples in Sub-goal B.  If students have met Sub-goals A and B but are unsuccessful, use counting by ones, or are inconsistent using a multiplicative counting strategy, the resources for this sub-goal can support designing instruction for counting-based strategies for multiplication. 

Sub-goal D

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Math Recovery Purple Book

Wright, R. J., Stanger, G., Stafford, A. K., & Martland, J. (2015). Teaching number in the classroom with 4-8 year olds (2nd ed.). London: Sage Publications, Ltd. (Chapter 10)

The knowledge developed in this module can be extended to learning related to multiplicative strategies and multiplication and division basic facts.

(see Ready Set Math module Multiplication and Division: Strategies and Basic Facts)  Consideration should also be given to early learning involving base-10 thinking. 

(see Ready Set Math module Place Value: Counting involving base-ten units) 

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