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# Division

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Tweet## Chloe Page

on 15 March 2013#### Transcript of Division

The Big Ideas Why is understanding division so important? What division Looks like in our classrooms: How you can help your child learn Some students are more vulnerable than others when it comes to mathematics and numeracy (Gervasoni, 2005) . Our job is to recognize when a student is struggling or when they have misconceptions about certain ideas and help them to change their views. Some misconceptions students have about division include:

Some believe that division is only used to make the numbers smaller (Graeber, 1993). This can lead to difficulties in later years especially when dividing decimals and percentages. We can help students eliminate this misconception by using arrays and helping them fully understand division as a concept.

Some students believe that it is simply impossible to divide by zero however have little knowledge as to why dividing by 0 will always equal 0 (Van De Walle, 2006). Multiplicative thinking Learning division from an early age is vital in developing mathematical understanding and strong multiplicative thinking. If an individual does not have a strong foundational understanding of these concepts learning even more complex ideas such as algebra will be very challenging (Young-Loveridge & Mills, 2012). Students may also struggle with the Learning Division We aim to create a classroom environment that is accessible to all students no matter what their abilities. In order to do this for division we provide lots of open ended tasks where the students can create their own methods and come to their own conclusions about the division tasks (Van De Walle, Karp & Bay-Williams, 2006). Open ended tasks allow students to use a wide range of materials to develop and support their theories and are accessible to students of all skill levels (Ferguson, 2009 & Gervasoni, 2005).

The use of concrete materials can help build up a mental image of problems and solutions and also provides support to those who are struggling with a concept.

Most importantly we don't just want the correct answers, we want to know HOW they worked it out, Each pair takes a collection of tiles (20-30) and records the quantity (anything around the house can be used instead of tiles such as buttons, lima beans ext)

Take turns to roll a die and organise your collection into an array with that many rows or columns.

If there are leftovers’ they record this in some way.

Winner is the person with most leftovers’.

In pairs, list the numbers that divided equally.

Discuss what other numbers might be divisible by these numbers

Is there a pattern?

(Adapted from Downton, 2013)

Note: This game can easily be extended by using a 10 sided die or even two 6 sided die and a larger collection of tiles. An overview of division

in our classrooms Division is more than the traditional algorithm that you use in class, it is a concept and a tool for every day use, and we want our students to understand this. In our classroom we will provide open learning experiences where the students are encouraged to create their own methods of solving these problems. In order for children to become flexible problem solvers they need to be shown multiple representations of a solution to a problem (Young-Loveridge, 2005).

We also want children to relationally understand what division means and how it relates to not only other maths concepts, but how it can be used in real life situations. order to become multiplicative thinkers. They need to understand that a group of numbers they are dividing is a unit. (Gervarom, 2004)

We want our students to understand the relationship between numbers, multiplication and division, for example, if 3 X 4 = 12 then 12 ÷ 3 = 4. We also want students to use the knowledge we taught them to eventually be able to check their weekly pay against their rates and hours they worked. Misconceptions children can have about multiplication and division: When children learn division it must be integrated into their existing knowledge bank and by the time that most students are formally introduced to division in schools many will have prior knowledge (Anghileri, 1995). Therefore our main aim when teaching division and multiplication to our students is to build on their knowledge and work towards them becoming multiplicative thinkers (Clarke and Cheeseman, 2000).

Multiplicative thinkers are "able to think about multiplication in a number of different ways to recognise when multiplication is required and how it relates to division, support efficient mental and written computation, and solve a wider range of problems involving equal groups, simple proportion, combinations, and rate" (DEECD, 2012)

For example:

If i invested $2.00 and got back $8.00 and then invested $6.00 and got back $12.00. What is the better deal?

An additive thinker would say that they are equally good because in both cases, you gained $6.00 .A multiplicative thinker could appreciate that the 1st deal quadrupled their investment whilst the 2nd deal merely doubled it. (Young-Loveridge, 2005) Anghileri, J. (2005) Negotiating Meanings for Division. Mathematics in

School. Cambridge.

Clarke, D., Cheeseman, J. (2000). Some insights from the first year of

early numeracy research project. ACER research conference

2000 improving numeracy learning: what does the research tell

us?, p 6 -10.

DEECD. (2012) Common Misunderstandings - Level 3 Multiplicative

Thinking. State Government of Victoria. Retrieved from

www.education.vic.gov.au/school/teachers/teachingresources

/discipline/maths/assessment/Pages/lvl3multi.aspx

Downton, A. (2013) Day 5 Session Notes. Retrieved from

http://leo.acu.edu.au/mod/resource/view.php?id=374137

Graeber, A. (1993). Misconceptions about multiplication and division.

The arithmetic teacher, March 1993 (40), 408.

Gervarom, A. (2004) Extending Mathematical Understanding Specialist

Teaching Manual. Ballarat.

Ferguson, S. (2009). Same task, different paths: catering for student

diversity in the mathematical classroom. APMC, 13 (2), P 32 – 36.

Young-Loveridge, J. & Mills, J. (2012) Deepening students'

understanding of multiplication and division by exploring

divisibility by nine. Australian Mathematics Teacher. 68 (3), P 15-

20.

Young-Loveridge, J. (2005) Fostering Multiplicative Thinking Using

Array-Based Materials. The Australian Mathematics Teacher.

61(3) p.34-40

Van De Walle, J. A., Karp, K. S., & Bay-Williams, J. M (7th ed,). (2010).

Elementary & MiddleSchool Mathematics: Teaching

Developmentally. Boston, United States of America: Pearson

Education Inc. First and foremost, talk about division during everyday experiences like going shopping and when cooking. Children have been known to understand division from as early as kindergarten and most of that knowledge and language is learned at home (Taber & Canonic, 2008). By asking simple questions such as, "If we are making 12 cupcakes, how many does everybody get?" and "If we want $6 worth of canned soup, how many cans can we get?" children are likely to encounter and probably use more phrases associated with division" then they would in formal classroom experiences. (Anghileri, J). Division is more than a formula Dividing by zero is not allowed Multiplication makes larger and division makes smaller. References Questions to ask your children: Some questions you can ask to get them to think deeper about the maths they are doing in the left overs game include:

Can you prove that that is the right answer? Convince me.

Is there a faster way?

What number would you like to roll to gain the most left overs? Why?

What numbers divided equally?

If I had (x) amount of tiles, how many ways could I equally divide it? Show me.

How could we record our game so we can know what numbers had remainders and what numbers divided equally? Left overs' Game concept of proportional reasoning and advanced problem solving skills if their foundational knowledge is weak. Division is also all around us. We unconsciously use division when shopping, cooking, playing sports, working out averages and so much more. We want to be able to give your children the best tools so that they can face any challenge that faces them in their lives. By Chloe Page & Sharni Spriggs

EDMA 201 - Assessment 2

2013 S00090442 & S00084077

Full transcriptSome believe that division is only used to make the numbers smaller (Graeber, 1993). This can lead to difficulties in later years especially when dividing decimals and percentages. We can help students eliminate this misconception by using arrays and helping them fully understand division as a concept.

Some students believe that it is simply impossible to divide by zero however have little knowledge as to why dividing by 0 will always equal 0 (Van De Walle, 2006). Multiplicative thinking Learning division from an early age is vital in developing mathematical understanding and strong multiplicative thinking. If an individual does not have a strong foundational understanding of these concepts learning even more complex ideas such as algebra will be very challenging (Young-Loveridge & Mills, 2012). Students may also struggle with the Learning Division We aim to create a classroom environment that is accessible to all students no matter what their abilities. In order to do this for division we provide lots of open ended tasks where the students can create their own methods and come to their own conclusions about the division tasks (Van De Walle, Karp & Bay-Williams, 2006). Open ended tasks allow students to use a wide range of materials to develop and support their theories and are accessible to students of all skill levels (Ferguson, 2009 & Gervasoni, 2005).

The use of concrete materials can help build up a mental image of problems and solutions and also provides support to those who are struggling with a concept.

Most importantly we don't just want the correct answers, we want to know HOW they worked it out, Each pair takes a collection of tiles (20-30) and records the quantity (anything around the house can be used instead of tiles such as buttons, lima beans ext)

Take turns to roll a die and organise your collection into an array with that many rows or columns.

If there are leftovers’ they record this in some way.

Winner is the person with most leftovers’.

In pairs, list the numbers that divided equally.

Discuss what other numbers might be divisible by these numbers

Is there a pattern?

(Adapted from Downton, 2013)

Note: This game can easily be extended by using a 10 sided die or even two 6 sided die and a larger collection of tiles. An overview of division

in our classrooms Division is more than the traditional algorithm that you use in class, it is a concept and a tool for every day use, and we want our students to understand this. In our classroom we will provide open learning experiences where the students are encouraged to create their own methods of solving these problems. In order for children to become flexible problem solvers they need to be shown multiple representations of a solution to a problem (Young-Loveridge, 2005).

We also want children to relationally understand what division means and how it relates to not only other maths concepts, but how it can be used in real life situations. order to become multiplicative thinkers. They need to understand that a group of numbers they are dividing is a unit. (Gervarom, 2004)

We want our students to understand the relationship between numbers, multiplication and division, for example, if 3 X 4 = 12 then 12 ÷ 3 = 4. We also want students to use the knowledge we taught them to eventually be able to check their weekly pay against their rates and hours they worked. Misconceptions children can have about multiplication and division: When children learn division it must be integrated into their existing knowledge bank and by the time that most students are formally introduced to division in schools many will have prior knowledge (Anghileri, 1995). Therefore our main aim when teaching division and multiplication to our students is to build on their knowledge and work towards them becoming multiplicative thinkers (Clarke and Cheeseman, 2000).

Multiplicative thinkers are "able to think about multiplication in a number of different ways to recognise when multiplication is required and how it relates to division, support efficient mental and written computation, and solve a wider range of problems involving equal groups, simple proportion, combinations, and rate" (DEECD, 2012)

For example:

If i invested $2.00 and got back $8.00 and then invested $6.00 and got back $12.00. What is the better deal?

An additive thinker would say that they are equally good because in both cases, you gained $6.00 .A multiplicative thinker could appreciate that the 1st deal quadrupled their investment whilst the 2nd deal merely doubled it. (Young-Loveridge, 2005) Anghileri, J. (2005) Negotiating Meanings for Division. Mathematics in

School. Cambridge.

Clarke, D., Cheeseman, J. (2000). Some insights from the first year of

early numeracy research project. ACER research conference

2000 improving numeracy learning: what does the research tell

us?, p 6 -10.

DEECD. (2012) Common Misunderstandings - Level 3 Multiplicative

Thinking. State Government of Victoria. Retrieved from

www.education.vic.gov.au/school/teachers/teachingresources

/discipline/maths/assessment/Pages/lvl3multi.aspx

Downton, A. (2013) Day 5 Session Notes. Retrieved from

http://leo.acu.edu.au/mod/resource/view.php?id=374137

Graeber, A. (1993). Misconceptions about multiplication and division.

The arithmetic teacher, March 1993 (40), 408.

Gervarom, A. (2004) Extending Mathematical Understanding Specialist

Teaching Manual. Ballarat.

Ferguson, S. (2009). Same task, different paths: catering for student

diversity in the mathematical classroom. APMC, 13 (2), P 32 – 36.

Young-Loveridge, J. & Mills, J. (2012) Deepening students'

understanding of multiplication and division by exploring

divisibility by nine. Australian Mathematics Teacher. 68 (3), P 15-

20.

Young-Loveridge, J. (2005) Fostering Multiplicative Thinking Using

Array-Based Materials. The Australian Mathematics Teacher.

61(3) p.34-40

Van De Walle, J. A., Karp, K. S., & Bay-Williams, J. M (7th ed,). (2010).

Elementary & MiddleSchool Mathematics: Teaching

Developmentally. Boston, United States of America: Pearson

Education Inc. First and foremost, talk about division during everyday experiences like going shopping and when cooking. Children have been known to understand division from as early as kindergarten and most of that knowledge and language is learned at home (Taber & Canonic, 2008). By asking simple questions such as, "If we are making 12 cupcakes, how many does everybody get?" and "If we want $6 worth of canned soup, how many cans can we get?" children are likely to encounter and probably use more phrases associated with division" then they would in formal classroom experiences. (Anghileri, J). Division is more than a formula Dividing by zero is not allowed Multiplication makes larger and division makes smaller. References Questions to ask your children: Some questions you can ask to get them to think deeper about the maths they are doing in the left overs game include:

Can you prove that that is the right answer? Convince me.

Is there a faster way?

What number would you like to roll to gain the most left overs? Why?

What numbers divided equally?

If I had (x) amount of tiles, how many ways could I equally divide it? Show me.

How could we record our game so we can know what numbers had remainders and what numbers divided equally? Left overs' Game concept of proportional reasoning and advanced problem solving skills if their foundational knowledge is weak. Division is also all around us. We unconsciously use division when shopping, cooking, playing sports, working out averages and so much more. We want to be able to give your children the best tools so that they can face any challenge that faces them in their lives. By Chloe Page & Sharni Spriggs

EDMA 201 - Assessment 2

2013 S00090442 & S00084077