ASSESSMENT TWO
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Mathematics
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Jun 18, 2024
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Miroslawa Kirkby
Student ID: 21082393
Part A Fractions: What to teach and how to teach it?
Fractions are one of the most crucial topics for students to learn in order to succeed in later mathematical
studies, such as algebra, but it is an area in which Australian students struggle. According to research,
students have a poor understanding of fraction concepts (Pearn, 2007; Sowder & Wearne, 2006; Wearne &
Kouba, 2000 as cited in (Van de Walle, et al., 2019, p. 254). This lack of comprehension manifests itself in
difficulties with fraction computation, decimal and percentage concepts, and the use of fractions in other
content areas (Pearn & Stephens, 2015, 2018; Bailey, Hoard, Nugent, & Geary, 2012; Brown & Quinn, 2007;
National Mathematics Advisory Panel, 2008 as cited in (Van de Walle, et al., 2019, p. 254). As a result, it is
crucial that to teach fractions effectively, show fractions as exciting and significant, and commit to assisting
students in understanding the big principles.
Understanding fractions is a developing process. Fraction experiences should start in Year 1. Students are
required to identify and explain one-half as one of two equal halves of a whole in the Australian
Curriculum: Mathematics (ACMNA016; (ACARA, n.d.). In Year 2, students will be required to identify and
explain typical uses of forms and collections in halves, quarters, and eighths (ACMNA033). The emphasis in
Year 3 is on modelling and portraying unit fractions and their multiples to the whole (ACMNA058). Year 4
focuses on context-based fraction equivalency (ACMNA077) and counting, locating, and portraying
quarters, half, and thirds on a number line (ACMNA078). Students continue to compare and rank common
unit fractions, as well as find and depict them on a number line (ACMNA102) in Year 5. They also look into
ways for solving problems requiring fraction addition and subtraction with the same denominator
(ACMNA103). Students in Year 6 are expected to compare fractions with related denominators, as well as
locate and depict them on a number line (ACMNA125). They are expected to answer problems involving
the addition and subtraction of fractions with the same or related denominators (ACMNA126) and the
determination of a simple fraction of a quantity when the result is a whole number, both with and without
the use of digital technology (ACMNA127).
According to Van de Walle et al., (2019, p. 272) “fractions are the gateway to algebra and beyond”. They
emphasise that comprehending fractions is vital for establishing the foundations for recognising rational
numbers, decimals, percentages and other complex mathematics.
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Miroslawa Kirkby
Student ID: 21082393
Understanding fractions entails comprehending all of the concepts that fractions can convey. Part-whole is
one of the most prevalent definitions of fraction. According to Clarke, Roche, & Mitchell, (2008); Lamon,
(2012); Siebert & Gaskin, (2006), many fraction understanding researchers believe that emphasising
additional fraction meanings might help students grasp fractions better (as cited in Van de Walle et al.,
(2019, p. 255).
Van de Walle et al., (2019) highlight five fraction constructions employed in mathematics instruction: part-
whole, division, ratio, measurement, and operator. These constructions can be used to assist students
comprehend fractions and how they relate to other mathematical ideas. The following are examples of
teaching methodologies and materials that use each fraction component in the context of Australian Coins:
Part-Whole:
This concept entails viewing fractions as components of a whole. This model is excellent for teaching
fractions because it shows students how a fraction can represent a portion of a larger object or amount.
Yet, it may not be as successful for comprehending more difficult operations such as multiplication and
division, which necessitate a more in-depth grasp of the connection between the numerator and
denominator. (Van de Walle, et al., 2019). Students could, for example, use Australian coins to investigate
how different coin combinations can represent different fractions of a dollar. A teaching technique that
utilises this concept could include instructing students to put coins together to form a specified fraction,
such as 1/4 of a dollar. Division: This concept entails comprehending fractions obtained by dividing one value by another. This construct can
assist students in understanding the connection involving division and fractions. This additionally has the
potential to assist students in solving practical problems involving fractional parts of a quantity. However,
because it doesn't concentrate on the link between the numerator and denominator, it may be less useful
for establishing conceptual comprehension. (Van de Walle, et al., 2019). Students could, for example, use
Australian coins to investigate how many of one coin are required to make a given fraction of another coin.
A teaching technique that utilises this concept could include asking students to determine the number of 5
cent coins required to make 1/4 of a dollar. Coin worksheets and interactive online activities are examples
of resources that could be used for this.
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Miroslawa Kirkby
Student ID: 21082393
Ratio:
Understanding fractions as a comparison of two quantities is necessary for the ratio construct. This
construct is excellent for showing students how fractions may be used to compare values. It is also useful
for solving issues that require proportional thinking, such as determining the corresponding value of a
specific fraction. However, it may be unsuccessful in building a knowledge of fractions as components of a
whole or as operators (Van de Walle, et al., 2019). For example, students could compare the ratio of two
distinct sorts of currencies, such as the ratio of 10-cent coins to 20-cent coins, using Australian coins.
Incorporating this concept into a teaching technique could entail asking students to find the ratio of the
number of 50-cent coins to the number of 20-cent coins in a particular set of coins. Coin charts and graphs
are examples of resources that could be used for this.
Measurement:
Understanding fractions as a technique to measure amounts is required for this concept. This construct is
great for showing students how to use fractions to measure length, weight, or volume. It can also be used
to solve difficulties with fractional parts of measurements. It may, however, be less successful in building a
knowledge of fractions as elements of a whole or as operators (Van de Walle, et al., 2019). Students could,
for example, use Australian coins to investigate how fractions can be used to measure length, weight, or
volume. A teaching technique that utilises this concept could include encouraging students to measure the
length of a line or the weight of an object using coins. Measuring tapes and scales are examples of
resources that could be utilised for this.
Operator:
This notion entails viewing fractions as operations that can be done on numbers. This construct is beneficial
to establish calculation and problem-solving skills since it shows students ways fractions can be added,
subtracted, multiplied, or divided. It can also be used to solve difficulties involving determining a fractional
component of a quantity. However, because it doesn't focus on the link between the numerator and
denominator, it may be less useful for establishing conceptual comprehension (Van de Walle, et al., 2019).
Students could, for example, use Australian coins to investigate how to add, subtract, multiply, and divide
fractions. This concept could be used in a teaching technique that asks students to answer problems
3
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Miroslawa Kirkby
Student ID: 21082393
involving fractions of money, such as calculating change following a purchase. Coin puzzles and games are
examples of resources that could be used for this.
In summary, every fraction construct has advantages and disadvantages, and an array of constructions can
be employed to assist students build an in-depth knowledge of fractions and their connections to other
concepts in mathematics. Teachers can incorporate these components into their lesson using a number of
teaching tactics and tools, according to the requirements and preferences of their students.
Van de Walle et al., (2019) examine three fraction models for representing fractions: area models, linear
models, and set models.
Area Models:
Fractions are represented as sections of a larger area, such as a square or rectangle, in area models. The
area is divided into equal pieces, each of which represents a fraction of the total. A rectangle, for example,
can be divided into halves, thirds, fourths, and so on (Van de Walle et al., 2019, p. 256). Use a rectangle to
symbolise a wallet, and different colours or types of coins to represent the different sections of the fraction,
such as 3/4 of the wallet being filled with 50-cent coins.
Figure 1
Illustration of area models inspired from
(Van de Walle, et al., 2019)
Linear Models:
Linear models describe fractions as segments or portions of lines. The line is divided into equal sections,
each representing a fraction of the total. A line segment, for example, can be divided into halves, thirds,
fourths, and so on (Van de Walle, et al., 2019, p. 257). A number line could be used to represent the
amounts of different coins, with each segment representing a fraction of the overall in an image including
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Wallet being filled ¾ by a $0.50c
coin
Miroslawa Kirkby
Student ID: 21082393
the theme of Australian Coins. A segment from 0 to 50 cents, for example, represents 1/2 of a dollar, while
a segment from 0 to 25 cents represents 1/4 of a dollar.
Figure 2
Illustration of linear models inspired from (Van de Walle, et al., 2019)
Set Models:
Fractions are represented as subsets of a bigger set in set models. The set is divided into equal pieces, each
of which represents a fraction of the total. A set of objects, for example, can be divided into halves, thirds,
fourths, and so on (Van de Walle et al., 2019, p. 257). To illustrate the theme of Australian Coins, use a
collection of coins to represent the entire set, and different containers or groupings to represent the
various portions of the fraction. A jar of 20 one-dollar coins, for example, can be divided into four equal
groups, with each group representing 1/4 of the total amount.
Figure 3
Illustration of set models inspired from (Van de Walle, et al., 2019)
Fraction concepts can be difficult for students, and they frequently encounter challenges and
misconceptions.
5
¼ ½ ¾ 1
$0.25c $0.50c $0.75c $1.00
20 $1.00
coins
make up
a whole
jar
4 jars have 5 $1.00 coins divided equally make each jar ¼ of the whole jar.
Miroslawa Kirkby
Student ID: 21082393
Issues understanding fractions as pieces of a whole:
Students believe that the numerator and denominator are distinct values and have trouble recognising
them as one. To address this, teachers can employ tangible manipulatives such as fraction circles, fraction
strips, or fraction bars, which allow students to see and touch the parts of a whole (Cramer & Wyberg,
2016). These manipulatives can assist children in visualising fractions and understanding their links.
Difficulty regarding equivalent fractions: Students might find it difficult with comprehending equivalent fractions, or fractions having the same value.
They may be unaware that 2/3 refers to two equal-sized sections (albeit not necessarily equal-shaped
things) (Van de Walle, et al., 2019, p. 255). Teachers might use illustrations such as fraction bars or fraction
circles to highlight how different fractions can be comparable to assist children understand this idea
(Cramer & Wyberg, 2016). Teachers can also urge students to compare fractions using objectives such as
1/2 or 1/4 to help them understand how fractions relate to one another.
Common misconceptions about fraction operations include: Students may have misunderstandings regarding fractions whether it comes to adding, subtracting,
multiplying, or dividing. Since 5 is less than 10, students believe that a fraction such as 1/5 is smaller than a
fraction such as 1/10. Students may be informed the opposite - the larger the denominator, the smaller the
fraction. If such rules are taught without explanation, students may conclude that 1/5 is greater than 7/10
(Van de Walle, et al., 2019, p. 255). For example, they may believe it is impossible to add fractions with
distinct denominators. Teachers can address this by using real-world situations and problem-solving
assignments that require students to employ fraction operations (Empson & Levi, 2011). These activities
can assist students in understanding the importance of fraction procedures and how they can be utilised to
resolve issues.
Feelings of negativity about fractions:
Students may have an adverse disposition regarding fractions because they believe they are excessively
difficult or complicated. Students make the error of applying whole-number operation 'rules' to fractions;
for example, they add the numerators and denominators individually, as shown in 1/2 + 1/2 = 2/4 (Van de
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Walle, et al., 2019, p. 255). Students that make these mistakes do not comprehend fractions. They will
continue to make mistakes by overapplying whole-number concepts until they understand fractions
meaningfully (Cramer 2010). To address this, teachers should foster a positive classroom culture centred on
fractions by incorporating interesting and pertinent tasks, activities, and narratives that show students the
enjoyment and appeal of fractions (Lamon, 2012). Embracing student's different cultural, ethnic, or
religious backgrounds might benefit fractions learning. Teachers may utilise fractions in traditional meals or
crafts from different countries, such as breaking a cake into equal parts or weaving a basket using fraction
measures (Cai & Lester, 2010). This may assist students comprehend how fractions are used in different
cultural settings and how they apply to their life. Teachers can also invite students to discuss their own
cultural customs utilising fractions, which can foster a sense of affiliation in the classroom.
Part B
Decimals: Mathematics Content Knowledge Test
Question 1:
Year and CD
Year 6 - Investigate and calculate percentage discounts of 10%, 25% and 50% on
sale items, with and without digital technologies (ACMNA132)
Multi-choice
question
During the Boxing Day sale, a pair of black denim jeans is discounted by 25%. The original price of the jeans was $40.00. What is the discounted price of the jeans?
Options
A $10.00 B $20.00 C $30.00
D $50.00 Significance
This question measures students' comprehension of computing percentage discounts, which is an important part of numeracy. It also assesses their ability to apply their information in a real-world situation, which is essential for problem-solving.
The correct solution “key” is B.
A The mistaken belief that the discount is deducted from the original price rather than computed as a percentage of the original price.
C The mistaken belief that the sale price and the reduced price are the same.
D The mistaken belief that the sale price is determined by adding the discount to the original price rather than removing it.
Diagnostic
A How would you calculate a 20% discount on a $75.00 item? B If a shirt is on sale with a 25% discount and its original price is $60, what is the
sale price?
C If a dress is on sale with a 10% discount and its sale price is $55, what was the original price of the shirt?
D If Kmart offers a 50% discount on all clothing items and a skirt originally costs $60, what is the sale price of the skirt?
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Miroslawa Kirkby
Student ID: 21082393
Question 2:
Year and CD
Year 5 -Compare, order and represent decimals (ACMNA105) Multi-choice
question
Which of the following is the equivalent of 7.5?
Options
A 0.75 B 75
C 750
D 7.05 Significance
This assesses Year 5 students abilities to evaluate decimals and comprehend their relative values. It asks them to choose the proper equivalent of 7.5 from a list of possibilities, which will test their knowledge of place value and decimal locations.
The correct solution “key” is D .
A Students have confused the place value of the digits
B&C Are much larger than 7.5 and therefore doesn’t make sense to the be the equivalents.
Diagnostic
A What is the decimal form of seventy-five hundredths?
B What is the decimal form of seventy-five units?
C What is the decimal form of seven hundred and fifty units?
D What is the decimal form of seven and five tenths?
I used GENAI for some parts of this assessment. It helped me gain further insight into expanding my
resources and cross-checking the reliability of what I read. I made sure that what used was all paraphrased
in my own words and referenced accordingly.
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Miroslawa Kirkby
Student ID: 21082393
References
ACARA. (n.d.). Australian Curriculum, Assessment and Reporting Authority [ACARA]
, 8.4. Retrieved from Australian Curriculum, Assessment and Reporting Authority: https://www.australiancurriculum.edu.au/f-10-curriculum/mathematics
Cai, J., & Lester, F. K. (2010). Teaching and learning of fractions. In J. Cai, & E. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives
(pp. 309-326). Springer.
Cramer, K., & Wyberg, T. (2016). Developing fractions concepts using concrete manipulatives. Teaching Children Mathematics, 22
(6), 336-345.
Curtin University. (n.d.). Children As Mathematical Learners. Topic 8: Fraction Constructs and Models
. Retrieved from learn-ap-southeast-2-prod-fleet01-xythos.content.blackboardcdn.com/
5dc3e34515a0e/20718020?X-Blackboard-S3-Bucket=learn-ap-southeast-2-prod-fleet01-xythos&X-
Blackboard-Expiration=1683439200000&X-Blackboard-Signature=53AO5cZIIaU9iv6rz9crPy2Otc
%2FZLGGa8EtHDk
Empson, S. B., & Levi, L. (2011). Extending children's mathematics: Fractions and decimals: Innovations in Cognitively Guided Instruction.
Heinemann.
Lamson, S. J. (2012). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers.
New York: Routledge.
Van de Walle, J. A., Karp, K. S., Bay-Williams, J. M., Brass, A., Bentley, B., Ferguson, S., . . . Wilkie, K. (2019). Primary and middle years mathematics - Teaching Developmentally
(1st Australian ed.). Melbourne:
Pearson Australia.
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