EDMA411-CourtneyWright

Courtney Wright – S00118415

1

EDMA411- Assessment Task 1

Courtney Wright

S00118415

In year 7, students should be able to multiple and divide fractions and decimals using efficient written

strategies and digital technologies (Australian Curriculum, Assessment and Reporting Authority, 2013).

This leads to the first two big ideas of dividing fractions and multiplying fractions. The Australian

curriculum for year 7 also describes comparing, ordering, adding and subtracting integers as something

students should be able to achieve (Australian Curriculum, Assessment and Reporting Authority, 2013).

Addition and subtraction of integers will the the third big idea to be discussed.

For each of these big ideas, I have researched and discussed the difficulties or misconceptions that

students face when learning these ideas to find out how to best challenge these issues with tasks and

activities.

Big Idea 1 – Dividing Fractions When students move into dividing fractions, they bring along their prior experiences of whole number

division and this can often lead to misunderstanding and error (Johanning & Mamer, 2014). Gregg and

Underwood-Gregg (2007) describes that dividing fractions is one of the most mechanical and least

understood areas of mathematics in the middle school, where students performance in these tasks is

typically extremely poor. This is due to the rule as described by Van De Walle (2007) as “Invert the

divisor and multiply”. The inability to understand and describe the inverting fractions rule is not just

limited to the primary or secondary school, Gregg and Underwood-Gregg (2007) state that this is also

continued through to training elementary teachers in college, still unable to explain why the algorithm

works. In accordance, Cramer, Monson, Whitney, Leavitt and Wyberg (2010) explain that the invert and

multiply rule is generally introduced to grades six and seven students when learning division of

fractions and is commonly just a mechanical rule with little understanding.

Using context is a great way to build understanding in mathematical tasks, especially in fractions,

however very few textbooks use context as a way to form meaning of the division of fractions (Cramer,

Monson, Whitley, Leavitt, & Wyberg, 2010). Even when context is used, students connection between

the invert and multiply algorithm and the context is not substantial. Coughlin (2010) describes how it is

difficult to teach fractions as not only is the computation complicated, but it is also challenging to

explain fractions in the contect of word problems.

As previously stated, Johanning and Mamer (2014) discussed how students bring their prior experiences

and understanding of division of whole numbers when they begin to learn division of fractions. In

addition, Johanning and Mamer (2014) describe how this can lead to students not understanding how

dividing can result in getting a larger quotient. When students begin to learn division of fractions, it is

important that their understanding of division is expanded to know that it is possible to divide two

numbers and find a quotient that is larger than either the divisor or the dividend. Coughlin (2010)

states that dividing by fractions is considered to be one of the most complicated procedures in

elementary mathematics, this makes it a big idea and necessary for tasks to develop students’

mathematical understanding.


2

Tasks to address big idea: Using models to represent division of fractions by fractions.

In this task, students will solve contextual worded problems involving division of fractions by fractions

by using models. This task is based on an activity by Smith (2013).

First, students should be introduced to a problem involving a whole number, divided by a fraction, to

familiarize themselves with the concept.

Take 3 divided by ¾.

Teacher should model the drawing of 3 whole units on the board.

Sample 1.

As we are finding how many lots of three quarters there are, we divide our 3 whole pieces into

quarters.

Sample 2.

We then shade ¾ of each whole (as seen by orange in sample 3). Then, we will notice that there is ¼

left in each of the 3 wholes, making another set of ¾ (as seen by blue).

Sample 3.

So there are 4 groups of ¾ in 3.


3

Next, students should be shown a worded problem involving division of two fractions.

Eg. Courtney has ½ a litre of chocolate milk. She decided to divide the chocolate milk into servings

that are ¼ of a litre each. How many servings could she make?

Assist students in this task by rewording it as “How many groups of ¼ are there in 1/2?”

Let students try and attempt this problem themselves using the previous questions explanation before

assisting.

This task can be solved like this.

First, you draw the whole unit. As Courtney only had ½ a litre of milk, you can draw a line to half the

unit and shade the ½.

Sample 4.

Now, as we want to know how many ¼s will fit in that half, we add another whole unit below the other

and divide it into quarters. As Courtney wants to divide her milk into ¼ of a litres, you need to look at

how many ¼’s fit into the half. It is evident that exactly 2/4’s fit into ½. So 2 servings of milk can be

portioned.

Sample 5.


4

Students can judge the reasonableness of this answer by looking at the original fractions. As ¼ is less

than ½, we would expect that we could fit ¼ into ½ more than one time. This addresses the issue

suggested by Johanning and Mamer (2014) of students not understanding why the quotient can be

larger than the dividend.

Discuss with students the relationship between the diagrams and how this compares to solving the

problem with the rule.

Students should continue to use this method to solve worded division of fraction problems.

Impact on student:

As discussed in my research, students merely using the computational approach of invert and multiply

was a significant issue in the teaching and learning of division of fractions. By introducing a task,

teaching students how to address and solve fractional division in another way, students understanding

has been extended beyond what they already knew.

Cramer, Monson, Whitley, Leavitt and Wyberg (2010) stated how using context in mathematical tasks

can be beneficial to students as it creates interest and also enriches students understanding of the task

due to being able to relate to the problem with its real life context.

Giving students a visual representation of the division problem, challenges the misconception for

students that in division the quotient will be a smaller than the dividend. Students can find the answer

using the models and assess the reasonableness of this answer, creating a deeper conceptual

understanding of the task than simply inverting and multiplying.

Big Idea 2 – Multiplying Fractions Similarly to dividing fractions, multiplying fractions challenges students to examine the previous ideas

they have learnt through multiplying whole numbers (Wu, 2001). Multiplying fractions can be a difficult

idea to teach, the computational point of view is rather simple, however, the conceptual point of view

is where the issues lie (Tsankova & Pjanic, 2010). Bezuk and Armstrong (1992) agree by saying that it is


5

easy to multiply fractions symbolically, however to construct meaning and to determine if answers are

reasonable is more difficult. Unlike others, the algorithm for multiplying fractions is much easier to

learn, however it is evident that students often have trouble applying the algorithm flexibly (Tsankova

& Pjanic, 2010). Tsankova and Pjanic (2010) state that students struggle to recognize when the

algorithm should be used, do not use the algorithm to multiply decimals and also have difficulty

creating an appropriate pictorial representation of a problem. Wu (2001) discusses how students who

do not have a conceptual understanding of multiplying fractions and the algorithm will have limited

ability to generalize the information to other situations, especially with more advanced and complex

tasks.

In earlier years, students are introduced to multiplication with whole numbers with the repeated

addition approach (Wu, 2001). Tsankova and Pjanic (2010) believe that the understanding of

multiplication of natural numbers as repeated addition is a prerequisite to the learning of fraction

multiplication. The repeated addition model is a useful link between multiplication and addition

however it is very limited if it is the students only conception of multiplication, especially when

applying multiplication to fractions (Wu, 2001). When applying context into multiplication of fractions

with mixed numbers or common fractions, the repeated addition model can be difficult to interpret

and make sense of (Wu, 2001).

Tsankova and Pjanic (2010) state that to develop new understandings and skills, students need to

recognize how different mathematically ideas connect and how they build on their previously acquired

knowledge. In saying this, Tsankova and Pjanic (2010) suggest that teaching multiplication of fractions

must be constructed on students’ prior knowledge of the multiplication algorithms and their

understanding of the representations. Previous knowledge that students would have attained by this

point in their schooling would include operations with natural numbers, the meaning of fraction as

“part of a whole” and “part of a set” and the concept of measurement (Tsankova & Pjanic, 2010).

Tasks to address big idea: Using area and length models as a method to solve multiplication of fractions.

This activity is based on a class activity completed in EDMA309, involving folding paper squares as a

method for solving multiplication of fractions.

Students will already be familiar with the general rule for multiplying fractions which involves

multiplying the numerators and multiplying the denominators.

Eg.

All students should be given a kinder square.


6

Use the same example as above, and ask the students to fold their squares into quarters and then

unfold it.

Ask students to then shade one quarter of the square.

Sample 1.

Students should then be asked to fold the same piece paper in half the OPPOSITE way to how they

folded previously. Students should open this up and be asked to shade in a different colour, one half of

the square.

Sample 2.

Tell students that this now represents the multiplication problem. Get students to investigate what the

answer is and how this is represented in the square. Also ask students to make connection between the

representation and the rule for solving multiplication of fractions.

The shaded segment of the square represents the numerator of the answer and the denominator

represents all the segments. So the part is 1 and the whole is 8.

Students are able to check their answer through the method of multiplying the numerators and

denominators.


7

Using this method of multiplication also helps students to see that

1 ×

1 is the same as

1 of

1 .

4 2 4 2

Another method for representing this multiplication is on a number line.

First, students need to create a number line, labelling where 0 is and where 1 is.

Secondly, students divide this number into the first fraction like they did with the paper.

Lastly, students need to divide the first labelled fraction into the second.

Sample 3

1 ×

1

4 2

One more example involving fractions where the numerator is not 1.


8

This question also involves the answer being simplified, which Bezuk and Armstrong (1992) state is

something students have difficulty or errors with.

Students should first fold the paper into eighths. Teacher/student discussion can be had to how this is

done, especially relating it to the first example of quarters.

Once paper is successfully folded into eighths, students should shade three eighths.

Sample 4.

Students should then be prompted to fold their square into thirds the other way. Students may have

difficulty as it does not involve halving the paper, so students should be assisted in methods to do this.

Students should then open up their square and shade two of those thirds.

Sample 5.


9

The overlapping shaded areas is represented by 6 fractional pieces and the whole is 24 pieces. (Fold

lines have been drawn in for assistance in viewing individual parts)

Sample 6.

However, 6/24 is not the simplest form of this fraction. To simplify fractions, divide the numerator and

denominator by the highest possible number that can divide into both numbers exactly. In this

example, it is easy as 6 is a multiply of 24, and goes into 24 four times, so the answer is ¼.


10

Sample 7

As seen in sample 7, the number line method did not require simplifying of the fraction as it was in its

lowest form already.

Once students are familiar with both these methods, students should complete a number of different

fraction multiplication problems involving a variety of numerators and denominators, example of

activity in appendix 1.

Impact on students: Students mathematical understanding of multiplying fractions is progressed beyond what they already

knew with this task as, as Tsankova and Pjanic (2010) stated, students are usually well practiced in

using the computational method for multiplication of fractions however lack conceptual understanding

of what the methods or rules entail. By students doing this hands on task where they can visually see

what it is that the multiplication does, they deeper their understanding of the method they have been

using. =

Bezuk and Armstrong’s (1992) research also suggests that this activity for multiplying fractions would

be worthwhile as they believe that it is easy to multiply fractions symbolically, however to construct

meaning and to determine if answers are reasonable is more difficult. This task gives students an

alternate way to determine answers to fraction multiplication questions. This task will take students

from having an instrumental understanding of multiplying fractions to a relational understanding.

Students will be encouraged to think mathematically as they investigate what the squares and shaded

regions mean and represent about the multiplication problem. Students are asked to relate this back to

the original multiplication of fractions and the method they use, to understand it how works.


11

Big Idea 3 – Adding and Subtracting Integers Ponce (2007) describes the adding and subtracting of integers to be one of the first major roadblocks to

student success in the learning of algebra. Similar to both big ideas previously discussed, when working

with integers, students struggle to make the transition from working with whole numbers (Ponce,

2007). Khalid and Badarudin (2008), describe that teachers often find it simpler to teach rules than to

teach for meaning with integers, hoping that students understanding will develop as they operate

successfully. Students find it difficult to establish the rules for themselves through working,

consequently they just remember the rules rather than understanding. Much like the invert and

multiply rule of dividing fractions, students working with integers know how to apply the method

mechanically without any awareness for the significance of the answer (Badarudin & Khalid, 2008).

Students that do not succeed in making the transition from whole numbers to integers will be at a

severe disadvantage when trying to understand following mathematical concepts that rely on their

understanding of addition and subtraction of integers (Ponce, 2007). Usiskin (2005, p10) states that

“algebra is a prerequisite for virtually all other mathematics”, making it important that we find ways

to help students progress through the learning of integers.

Tasks to address big idea: I will describe two tasks on how to address students understanding of adding and subtracting integers.

The first is adapted from ‘Adding and Subtracting Integers on the Number Line’ as described by Çemen

(1993) and the second is ‘A Manipulative Aid for Adding and Subtracting Integers’ as described by Grady

(1978).

Adding and Subtracting Integers on the Number Line

In the activity described by Çemen (1993), students are given two sheets of rules and examples showing

how to add integers on a number and also how to subtract integers on a number line (featured below).


12

For my task, these rules should not be given to students, rather discovered and discussed by class and

teacher while progressing through the activity. Introducing the rules in this way will ensure better

conceptual understanding of the process rather than students just following and memorizing rules,

which is an issue with the learning of integers.

Rather than using printed diagrammatic examples, a large number line should be created across the

floor or wall of the classroom for all students to see. As shown in Çemen’s sheets, students are

directed to start at 0 always and move forwards for positive numbers and backwards for negative

numbers. This should be replicated on the large scale number line in the classroom using the examples

by Çemen. Students should be chosen to stand up and create the path on the number line that the

equation suggests. The rest of the class should map the path on written number lines. The teachers’

role during this activity is to ask questions and promote discussion to find out why students made

different moves and why they went the direction that they did as well as the reasonableness of the

conclusion.

Following this activity, students should practice more integer equations using the number line

technique.

Adding and Subtracting Integers with Counters – A Manipulative Aid

In this task, students should be given students chips, blocks or some form of small manipulative of 2

different colours. For explanation purposes, I will be using yellow squares to represent positive

numbers and red circles to represent negative numbers.

Through this technique, it is important that it is known that +1 + -1 = 0, or when using the chips, one

yellow circle and one red circle is equal to 0.

Example 1 – Positive number plus a negative number.

Given the example +6 + -3. First you get chips out to represent the first part of the equation, the +6.

Next, you add -3.

Lastly, we remove the zeros. These are the pairs of yellow and red circles.

This leaves you with three yellow circles, which represents +3.


13

So, +6 + -3 = 3

Example 2 – Negative number plus a negative number

Another example is -4 + -2.

First you make the -4.

Then, you add -2.

As there are no zero’s in this example, the operation is complete. -4 + -2 = -6

Example 3 – Negative number plus a positive number

Lastly, - 7 + +4.

Firstly, make the -7.

Next, add +4.

Remove the zero’s.

The sum of -7 + +4 is -3.


14

Impact on students: Çemen (1993) believes that the number line method for adding and subtracting integers is valuable as

is represents the adding and subtracting of integers in a way that clearly distinguishes between

subtraction and negative numbers. Çemen’s model also clarifies why subtracting an integer has the

same effect as adding its inverse, which can often be confusing for students.

Using the number line technique helps students to visualize the adding and subtracting of signed

integers. The ability to visualize the mathematics that is occurring helps students to get a more

conceptual understanding of what is happening in the equation. Students will be able to understand

what adding a negative number looks like as well as taking away a positive number, which Çemen

(1993) describes as a confusing area. Implementing Çemen’s activity by challenging the students with

the task and to come up with the rules themselves allows students to engage in thinking

mathematically and the result of the activity is developing students’ mathematical understanding

beyond what they had already possessed.

Grady (1978) believes that the number line method for teaching the adding and subtracting of integers

can often be confusing so has offered an alternative technique involving manipulatives to assist

students in learning this vital concept. Students can use this technique and experience success with the

abstract idea and will have confidence in their results which will help as they learn the later rules for

adding and subtracting integers.


15

References Australian Curriculum, Assessment and Reporting Authority . (2013). Mathematics. Retrieved from

Australian Curriculum: http://www.australiancurriculum.edu.au/mathematics/Curriculum/F-

10#level7

Badarudin, B. R., & Khalid, M. (2008). Using the jar model to improve students' understanding of

operations on integers. Research and Development in the Teaching and Learning of Number

Systems and Arithmetic (pp. 85-94). Mexico: International Congress on Mathematical

Education .

Bezuk, N. S., & Armstrong, B. E. (1992). Understanding fraction multiplication. The Mathematics

Teacher 85(9), 729-739.

Çemen, P. B. (1993). Adding and subtracting integers on the number line. The Arithmetic Teacher

40(7), 388-389.

Coughlin, H. A. (2010). What is the divisor's role? Mathematics Teaching in the Middle School 16(5),

280-287.

Cramer, K., Monson, D., Whitley, S., Leavitt, S., & Wyberg, T. (2010). Dividing fractions and problem

solving. Mathematics Teaching in the Middle School 15(6), 338-346.

Grady, M. B. (1978). A manipulative aid for adding and subtracting integers. The Arithmetic Teacher

26(3), 40.

Gregg, J., & Underwood-Gregg, D. (2007). Measurement and fair-sharing models for dividing fractions.

Mathematics Teaching in the Middle School 12(9), 490-496.

Johanning, D. I., & Mamer, J. D. (2014). How did the answer get bigger? Mathematics Teaching in the

Middle School 19(6), 344-351.

Ponce, G. A. (2007 ). It's all in the card - adding and subtracting integers. Mathematics Teaching in the

Middle School 13(1), 10-17.

Smith, A. (2013). Use models for division of fractions by fractions. Retrieved from LearnZillion:

http://learnzillion.com/student/lessons/204-use-models-for-division-of-fractions-by-fractions

Tsankova, J. K., & Pjanic, K. (2010). The area model of multiplication of fractions. Mathematics

Teaching in the Middle School 15(5), 281-285.

Usiskin, Z. (2005). Should all students learn a significant amount of algebra? In C. Greens, & C. Findell,

Developing Students' Algebraic Reasoning Abilities (pp. 4-16). Lakewood, CO: National Council

of Supervisors of Mathematics and Houghton Mifflen.

van de Walle, J. A. (2007). Elementary and Middle School Mathematics: Teaching Developmentally.

Boston, MA: Allyn & Bacon.

Wu, Z. (2001). Multiplying fractions. Teaching Children Mathematics 8(3), 174-177.


16

Appendix 1.