Using everything you know about multiplication, solve the problem above without using the long

division algorithm.

What connections can you make to strategies for and models of multiplication.

169' 14

“Guzenta”

“How many times does6 go into 42?”

Suppose that 6 is divided into 12

6 goes into 12 evenly.

There is no remainder.

Often terminology that is used with division is vague, inappropriate, or misleading. It can even interfere with the development of a rich understanding of division. Proper use of terminology and symbols can increase understanding and help to avoid mistakes.

NCTM, Developing Essential Understanding of Multiplication and Division, Grades 3-5

Now, write a story problem.

169' 14

As a group• Share your word problems and select 1 for each expression

(get OK from presenter)

• On chart papera) Write each word problem on a separate piece of paperb) Provide two strategies for each problem. Over the two problems be

sure to include, at least, one example of an additive, transitional, and multiplicative strategy

c) Post word problems with solution on the wall

(These materials were created by the Vermont Mathematics Partnership funded by the US Department of Education (Award Number S366A020002) and the National Science Foundation (Award Number EHR-0227057) © Vermont Institutes 2007)V3

August 2007

Problem sort

Let’s classify the problem situations into groups.

What similarities and differences do you see?

Work with your group to sort the problems in your

bag into two groups

DiscussDiscuss

How did you sort your problems?

What did you notice about the problems?

Two Types of Division Two Types of Division SituationsSituations

Partitive division

Quotative division

(These materials were created by the Vermont Mathematics Partnership funded by the US Department of Education (Award Number S366A020002) and the National Science Foundation (Award Number EHR-0227057) © Vermont Institutes 2007)V3 August 2007

Partitive DivisionPartitive Division

You know: How many you are starting with How many groups you will be making

Trying to find out How many are in each group

The action involved is “dealing out”

What is the meaning of the remainder?

Quotitive DivisionQuotitive Division

You know: How many you are starting with How many are in each group

Trying to find out How many groups there are

The action involved is “measuring out”

What is the meaning of the remainder?

Division Types

Partitive• Gene has 4 tomato plants.

There are the same number of tomatoes on each plant. Altogether there are 20 tomatoes. How many tomatoes are there on each tomato plant?

Quotitive• Gene has some tomato

plants. There are 6 tomatoes on each plant. Altogether there are 24 tomatoes. How many tomato plants does Gene have?

Quotative and Partitive DivisionQuotative and Partitive DivisionInstructional ConsiderationsInstructional Considerations

Students should experience both partitive and quotative division situations, but most students only experience partitive division. (Greer, 1992)

Equations may look the same without labels, but are not the same when you consider the meaning of the quantities. A focus on the meaning of the quantities in division is critical for understanding. (Fosnot, 2005)

The act of solving the two types of division is different. Partitive division can be solved by sharing out and quotative division can be solved using repeated subtraction.

An area model can be used to model both partitive and quotative division.

(These materials were created by the Vermont Mathematics Partnership funded by the US Department of Education (Award Number S366A020002) and the National Science Foundation (Award Number EHR-0227057) © Vermont Institutes 2007)V3 August 2007

Write 2 word problems for each of the Write 2 word problems for each of the expressions below (1 partitive and 1 expressions below (1 partitive and 1 quotitive), and solve each problem quotitive), and solve each problem

using two different strategies using two different strategies (traditional algorithm not allowed).(traditional algorithm not allowed).

29 ÷ 4 =

613 ÷ 25 =

(These materials were created by the Vermont Mathematics Partnership funded by the US Department of Education (Award Number S366A020002) and the National Science Foundation (Award Number EHR-0227057) © Vermont Institutes 2007)V3 August 2007

A Common ModelA Common Model

(These materials were created by the Vermont Mathematics Partnership funded by the US Department of Education (Award Number S366A020002) and the National Science Foundation (Award Number EHR-0227057) © Vermont Institutes 2007)V3 August 2007

100 Jelly Beans

20 students or 20 jelly beans per package

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ArticlesArticles

Two Ways of Thinking about Division

Conceptualizing Division with Remainders