WEEK FOUR: Division

There are two types of division:

(Week Four Lecture)

PARTITION: If Robert had 10 cookies to share between five friends, how many cookies would each friend get?

(Week Four Lecture)

QUOTITION: Robert has 10 cookies and wants to give 2 to each of his friends. How many friends can he give cookies to?

** QUOTITION IS REPEATED SUBTRACTION - In the above example, cookies were repeatedly taken away until Robert had none left **

Further examples:

Partition: Jessica received 8 ribbons at her sports day. Jessica wanted to share these between her 3 friends who did not receive any places in the events. If Jessica shared all 8 between her and her 3 friends, how many would each of them receive?

Quotition: Quinn had 12 lollies which he wanted to share by giving 3 to each of his friends. How many friends could Quinn share his lollies with?

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Concept, Skill and Strategy:

Concept:

The concept that division is sharing something into equal groups

Skill:

The skill to view a problem and use division correctly to work out the answer.

Division algorithm are the most difficult, because;

- Computation begins left-to-right (unlike addition, subtraction and multiplication)

- The algorithm involves basic division, subtraction and multiplication

- Several patterns

- Involves trial quotients - Estimation may be incorrect initially

(Reys, Lindquist, Lambdin, Smith, Rogers, Falle, Frid & Bennett, 2012)

Simple Division:

(VerbalReasoningTests.com, 2011)

Long Division:

(VerbalReasoningTests.com, 2011)

Strategies:

(Retrieved from Lecture 4 part 3)

1. Think multiplications - Division is the inversion of multiplication

(Burnett & Irons, 2002)

Figure 1. Multiplication Mat - If Sam had 5 plates with 3 peas on each, how many peas would there be altogether?

Figure 2. Division Mat - If Sam had 15 peas to share amongst 5 plates, how many would each plate get? (Partition)

These concrete/visual activities reinforce the relationship between division and multiplication, creating a concrete understanding of turn arounds and fact families.

2. Use the counting strategy - 5's and 10's Facts

- 5, 10, 15, 20, 25: We know that 5 x 5 = 25 - So, 25 / 5 = 5 (Fact family)

- 10, 20, 30, 40: We know that 4 x 10 = 40 - So, 40 / 10 = 4 or 40 / 4 = 10 (fact family)

3. Use double facts - 2's, 4's, 8's

- x2: Double 3 is 6

÷2: Half of 6 is 3

x4: Double, Double 3 is 12

÷4: Halve 12, Halve 6 is 3

x8: Double, Double, Double 3 is 24

÷8: Halve, Halve, Halve 24 is 3

4. Build-up or Build Down

(Week Four Lecture)

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Language of Division:

Thinking Strategy - Use Double facts / Double 5 is 10 / Meaning, 5 goes into 10 twice.

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Misconception: That division is a separate and stand alone operation. But this is not the case and students MUST grasp the concept that division is the inverse of multiplication.

For Example: Students may be able to facilitate multiplication facts, such as 6 x 5= 30. But, can not check their answers using division, such as 30 ÷6 = 5 or 30 ÷ 5 = 6 (Fact Family activities would be a fantastic way for students to create a firm conceptual idea of division)

It is for this reason, division is introduced last (following addition, subtraction and multiplication), as division is the inverse of multiplication (therefor connected to addition) and can require repeated subtraction. Students must have a good understanding of all operations to create strong concepts of division.

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An activity we created in tutorials:

The Doorbell Rang - By Pat Hutchins

This great book lends itself to division, with a storyline that revolves around sharing biscuits between a group of children, yet the group keeps growing as more children visit. We used the above resource to act as the biscuit tray and the green discs to represent the biscuits. We thought this was a great concrete/visual resource to use, as it helps separate the 'biscuits' to create an easy visual, but also looks like an array model helping reinforce the relationship between division and multiplication.

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Teaching strategies and resources:

Picture 1- Sally Buttons. (n.d.). Long division helper house [Photograph]. Retrieved from https://www.teacherspayteachers.com/Product/Long-Division-Helper-House-1645512

Picture 2 - Teach Beside Me. (2013). 10 books about division [Photograph]. Retrieved from http://teachbesideme.com/fun-ways-to-teach-division-to-kids/

Picture 3 - Sims4392. (n.d.). Division man [Photograph]. Retrieved from http://sims4392.blogspot.com.au/search?q=division

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How has my understanding of Division Changed?

For me, this week was about remembering and re-learning the skill of completing division algorithms. Even after watching the lectures and participating in the tutorials, I still struggled with the idea of how to complete the algorithm with a deep understanding, and not by rote. The videos in this weeks portfolio helped me re-establish my understandings of division, once again. As per previous weeks, I found the turn around and fact families most helpful to understand the concept of division. It may be a generational thing (as I am a mature age student), but I was never encouraged to explore the concepts and skills behind operations, we just followed the regimented order without asking, "why?".

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ACARA links to Division in Year 2

http://www.australiancurriculum.edu.au/curriculum/contentdescription/ACMNA032

(Australian Curriculum, n.d.)

Scootle Resource:

Furry little kittens need your help! Can you divide it up?

- The below resource explores division in student language using a visual game that asks for help dividing up toys between kittens.

(http://splash.abc.net.au/home#!/media/33047/divide-it-up-kittens)

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Idea Summaries from Textbook:

Chapter 9: Operations meanings and basic facts - division

- Measurement/Quotition - Is a problem where repeated subtraction is used (could use a number line to visualise to students the notion of repeated subtraction)

- Partition - Dividing an amount between a specified number of groups, to work out how many each group will get.

- Think multiplication - The primary thinking strategy for division.

* If 7 x 5 = 35 Then 35 / 7 = 5 - Because of this relationship, multiplication can be used to solve division problems (use fact family activities to re-iterate this connection with students)

Chapter 11: Solving mathematical problems with standard and alternative written strategies - division

- Using materials - Forms a bridge between the real-life problem situation and the abstract algorithm.

- Division algorithms - Computation begins left-to-right (unlike addition, subtraction and multiplication), the algorithm involves basic division, subtraction and multiplication, Involves trial quotients (estimation may be incorrect initially)

- Use calculators to strengthen understandings of numbers

- Use concrete examples when teaching children about remainders - Eg; Pass out 17 chocolates to 3 children (each child receives 5 chocolates with 2 chocolates left over. Or, if the chocolates can be cut into pieces, each child could have 5 chocolates plus 2/3 of a chocolate).

(Reys, Lindquist, Lambdin, Smith, Rogers, Falle, Frid & Bennett, 2012)