ASSESSMENT TWO

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Curtin University *

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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. 1
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. 2
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
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 4 Wallet being filled ¾ by a $0.50c coin
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