In week three the following big ideas were covered:
- There are four different ways to model multiplication (Jamieson-Proctor, 2016a):
1. Set model - there are 2 acorns on each of the 3 plates. How many acorns are there all together?
2. The array/area model - the length of a rectangle is 5 centimetres and the width is 3 centimetres.
What is the area of the rectangle?
3. The measurement/length model - I bought 4 hair ribbons each 2 metres long. How many metres
of ribbon did I buy?
4. The combinations/cross products model - I have 3 different coloured shirts and 2 different
coloured pants. How many different outfits can I make?
- There are five different strategies for multiplication (Jamieson-Proctor, 2016a):
1. Use counting strategy for x5 and x10
2. Think real world for x1 and x0
3. Use doubles for x2, x4 and x8
4. Build up for x3, x5 and x6 or build down for x9 and x10
5. The turn-around 7's
- There are five different properties for multiplication (Reys et al., 2012):
1. Null factor property - when a number is multiplied by 0 the product will be 0
2. Identity property - when a number is multiplied by 1 it stays the same
3. Commutative property/turnarounds - 3 x 4 is the same as 4 x 3
4. Associative property - 3 x (4 x 5) is the same as 3 x (4 x 5)
5. Distributive property - 4 x 7 = ?
= 4 x (5 + 2)
= (4 x 5) + (4 x 2)
= 20 + 8
= 28
- "A factor is a whole number that can be divided into another whole number to given a whole
number quotient" (Jamieson-Proctor, 2016b, p. 8). i.e. 2 x 3 = 6, the quotient is 2 and 3
- "A multiple is the quantity achieved by multiplying 2 or more factors together" (Jamieson-Proctor,
2016b, p. 8). i.e. 2 x 3 = 6, the multiple is 6.
- Prime factorisation - all composite numbers are the product of a prime numbers if the order is
ignored (Reys et al., 2012) i.e. 12 is the product of 2 x 2 x 3.
- Lowest Common Multiple (LCM) - the smallest number that a given group of numbers will
divide into exactly (Jamieson-Proctor, 2016b) i.e. 2, 3, 5. Pick the highest number in the group
and list the multiples to find the LCM. e.g. 5, 10, 15, 20, 25, 30, 35. LCM of 2, 3 and 5 is 30
- Greatest Common Factor (GCF) - the largest number that will divide into a group of numbers
evenly. i.e. 6, 12, 18. Pick the lowest number in the group and list the multiples. Find the number
that the numbers all divide equally into. e.g. 6 = 1, 2, 3 and 6. GCF of 6, 12 and 18 is 6.
These big ideas have changed my understanding of the weekly topic in the following ways:
- I now have an understanding that there are four different ways to model multiplication
- I also have a greater understanding of prime factorisation, lowest common multiples and greatest
common factors, allowing me to better teach these to students. Although I always knew what
these concepts were I struggled with understanding them. I now understand them more.
- I also didn't know that there were five different properties for multiplication, I only knew about
null factor, identity property and commutative property.
Demonstrate your understanding of the mathematical concept and related skills and strategies children need to assimilate and be able to use, that are related to the concept of multiplication
- The concept this week was multiplication. Multiplication is repeated addition of equal groups
or sets. This concept of addition also involves the concepts of five different multiplication
properties:
1. Null factor property
2. Identity property
3. Commutative property
4. Associative property
5. Distributive property.
- The concept of multiplication is applied using the skill of multiplication algorithms. A
multiplication algorithm can be seen in figure 1.8.
- There are five different strategies for the concept of multiplication;
1. Use counting strategy
2. Think real world
3. Use doubles
4. Build up
5. The turn-around
 |
| Figure 1.8: Jamieson-Proctor, R. (2016a). EDMA202/262 Mathematics Learning and Teaching 1: Week 3 Part 2. Brisbane, Australia: Australian Catholic University. |
 |
| Figure 1.9: Language Model for Multiplication |
- A specific teaching strategy that could be used for children to understand the concept of
multiplication is the skip counting strategy. Skip counting is something that occurs in everyday life
for children and adults therefore it is an essential strategy to learn (Reys et al., 2012).
- An example of skip counting is 5, 10, 15, 20, 25, 30. These numbers are part of the 5 times
tables.
- When teaching skip counting a hundred chart would be helpful for students to visualise the normal
number pattern and then figure out what the next number in their skip sequence would be. An
example of a hundred chart can be seen in figure 1.10.
 |
| Figure 1.10: Smart About Mathematics (2016). Gallery of 100 chart for math. Retrieved from http://www.smart.dynu.net/100-chart-for-math.html |
- A common misconception that children might have in relation to multiplication is that
multiplication always makes a number bigger. This is a misconception that can affect children's
mathematics ability when they are learning to multiply with rational numbers less than one
(Graeber & Campbell, 1993).
- This misconception can be avoided by ensuring that when teaching the basics of multiplication
children are not told or taught that multiplication always makes numbers bigger, to avoid confusion
in the later years.
Provide appropriate URL links to the ACARA year, strand, sub-strand, content description, elaborations and Scootle resources for the earliest mention of multiplication
- Multiplication can be first seen in Year 2 of the Australian Curriculum, number and algebra strand,
and in the number and place value sub-strand.
- The content description for ACMNA031 is "recognise and represent multiplication as repeated
addition, groups and arrays" (ACARA, 2016).
- The elaborations are "representing array problems with available materials and explaining
reasoning" and visualising a group of objects as a unit and using this to calculate the number of
objects in several identical groups" (ACARA, 2016).
- Scootle resources for multiplication
Provide appropriate links to resources and ideas you have sourced personally to assist students to develop concepts, skills and/or strategies related to multiplication
- Resource/s for children to learn the concept of multiplication
*concept of multiplication
- Resource/s for children to learn the skill of multiplication
* multiplication algorithm
- Resource/s for children to learn the strategies of multiplication
* Skip counting strategy
* Use doubles strategy
* Build up strategy
* Build down strategy
Provide a concise synthesis of the textbook chapter/s related to the weekly topic
- There are four types of multiplication problems:
1. Equal group problems;
2. Comparison problems;
3. Combination problems;
4. Area and array problems (Reys et al., 2012)
- Different skills for multiplication:
* Multiplication with 1 digit multipliers (Reys et al., 2012)
14
x 2
------
8 2 x 4 = 8
+ 20 2 x 10 = 20
-------
28
* Partial-products multiplication algorithm (Reys et al.., 2012)
372
x 28
------
16 (8 x2)
560 (8 x 70)
2400 (8 x 300)
40 (20 x 2)
1400 (20 x 70)
6000 (20 x 300)
----------
10416
* Lattice multiplication algorithm
 |
| Figure 1.11: Reys, R., Lindquist, M., Lambdin, D., Smith, N., Rogers, A., Falle, J., Frid, S., & Bennett, S. (2012). Helping children learn mathematics. Queensland, Australia: John Wiley & Sons Australia, Ltd. |
move 1 place to the left i.e. 10 x 863 = 8630 (Reys et al., 2012).
* Multiplication with zeros (Reys et al., 2012)
306
x 9
--------
2754
9 x 306 = 9 x (300 + 6)
= (9 x 300) + (9 x 6)
= 2700 + 54
= 2754
* Multiplication with 2 digit multipliers (Reys et al., 2012)
54
x 23
--------
12
150
80
1000
-------
1242
* Multiplication with large numbers - when multiplying with large numbers children should be
encouraged to estimate before using the calculator (Reys et al., 2012).
- There are six different thinking strategies for multiplication facts:
1. Commutativity;
2. Skip counting;
3. Repeated addition;
4. Splitting the product into known parts;
5. Patterns;
6. Multiplying by 1 and 0 (Reys et al., 2012).
References
Australian Curriculum and Assessment, Reporting Authority [ACARA].
(2016). Mathematics.
Retrieved from http://www.australiancurriculum.edu.au/mathematics/curriculum/f-10?layout=1
Education Services Australia. (2016). Scootle: Mathematics.
Retrieved from http://www.scootle.edu.au/ec/search?accContentId=ACMNA031&userlevel=(2)
Graeber, A. & Campbell, P. (1993). Misconceptions about multiplication
and division. The Arithmetic Teacher, 40(7),
408-411.
Jamieson-Proctor, R. (2016a). EDMA202/262 Mathematics
learning and teaching 1: Week 3 part 2. Brisbane, Australia: Australian
Catholic University.
Jamieson-Proctor, R. (2016b). EDMA202/262 Mathematics
learning and teaching 1: Week 3 part 3. Brisbane, Australia: Australian
Catholic University.
Kisi Kids Math TV. (2013). Learn
the basics of multiplication: Math lesson for 2nd graders. Retrieved
from https://www.youtube.com/watch?v=uacFH2oLj9M
Math Antics. (2012). Maths
Antics: Multi-digit multiplication pt 1. Retrieved from https://www.youtube.com/watch?v=FJ5qLWP3Fqo
Origio One. (2016). Teaching
the build-up strategy for multiplication. Retrieved from https://www.youtube.com/watch?v=NPC1mMKOl5I
Price, S. (2014). Build-down
strategy folder card. Retrieved from https://www.youtube.com/watch?v=_sYUMQ4j5qY
Reys, R., Lindquist, M., Lambdin, D., Smith, N., Rogers, A., Falle, J., Frid,
S., & Bennett, S. (2012). Helping children learn
mathematics. Queensland, Australia: John Wiley & Sons Australia,
Ltd.
Smart About Mathematics (2016). Gallery of 100 chart for math.
Retrieved from http://www.smart.dynu.net/100-chart-for-math.html
Tenframe. (2011). Times
tables: Multiply with 4, use the doubles strategy. Retrieved from https://www.youtube.com/watch?v=Vnv4-MMOlas
Tenframe. (2011). Times
tables: Multiply with 5, skip counting strategy. Retrieved from https://www.youtube.com/watch?v=R_Y8setGf5k
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