Mathmates Group 1

1. Week 2 Objectives Mathmates Group 1 By Courtney Nayda Sarah Nottingham Keegan Rebbeck Karen Relph Kristie Roberts Fiona Wareham Deborah Whiteley

2. Give a rationale for the inclusion of fractions in early childhood and primary mathematics learning; that is, explain why it is vital that children develop fraction concepts and skills beginning at an early age. • Fractions are a critical foundation for students as they are used in measurement across various professions and they are essential to the study of algebra and more advanced mathematics (Van DerWalle, 2010, p. 287). • Common fractions are important in daily life and are necessary for further study in mathematics (Reys, 2009, p. 265). • Building fraction ideas develops important higher level cognitive processes (Booker, 2010, p. 146). • Understanding fractions underpins many of the uses of mathematics in everyday life (Booker, 2010, p. 78). • Fractions are student’s first introduction to abstraction in mathematics and provide the best introduction to algebra in primary and middle school (Booker, 2010, p. 146). • Today measurement is expressed through decimal fractions and per cent, a convenient way to discuss proportion and change in things such as finance and sporting achievements (Booker, 2010, p. 146). • We need to understand and use decimal fractions, per cents and common fractions to deal with situations and problems in the real world (Booker, 2010, p. 146). • The world we live in requires that we can interpret and use decimal fractions in our work and everyday life (Booker, 2010, p. 165). Objective One

3. Objective Two Use the First Steps in Maths Key Understandings in Fractions as a base for planning appropriate fraction learning experiences for primary aged children.

4. Objective Three Explain and give examples of how fractions relate to part-whole relationships

5. Explain and give examples of the different contexts in which fractions need to be understood (e.g. part of a set/group, region/area, and length). Fractions operate in a variety of contexts not only part-whole (Van DerWalle, 2010, p. 287). Part – whole is one meaning of fractions that usually involves shading a region, another is Measure which identifies a length and then using that length as a measurement piece, determines the length of an object (Van DerWalle, 2010, p.287). This concept focuses on how much rather than how many parts. Another context is division which is not a part-whole scenario. For example sharing $10 with four people still means each person will receive one fourth of the money (Van DerWalle, 2010, p. 287). Fractions can also be used to indicate an operation e.g. two thirds of the audience held up banners (Van DerWalle, 2010, p. 287). Ratiois another context in which fractions are used e.g. ¼ can mean the probability of an event as in one in four. Ration ¾ could be the ratio of people wearing jackets to those not wearing jackets (Van DerWalle, 2010, p. 287). Using a range of different representation models can help students to make sense of fractions and clarify ideas (Van DerWalle, 2010, p. 288). The Region/Area model is the most commonly used model because it is the most concrete and easily handled by children (Reys 2009, p. 267). It is a good place to begin teaching fractions and is essential when doing sharing tasks. It is based on parts of an area or region (Van DerWalle, 2010, p. 289). Region can be any shape, however rectangles re the easiest region for children to draw. Activities using geoboards, pie pieces, and rectangular regions, paper folding and drawing on grids can be used. Lengths or measurements are compared in the Length Model, instead of areas. Length can be partitioned into fractional parts of equal length (Reys, 2009, p. 267). Children can begin to fold (partition) a long thin strip of paper in halves, fourths and so on (Reys, 2009, p. 267). Children need plenty of prior experience with a number line before they can understand it as a model for fractions (Reys, 2009, p. 267). Cuisenaire rods, tape, number lines and fractions bars can be used. The Set Model uses a set of objects as a whole. This model may cause children difficulties because they don’t often consider a set of objects as a unit and children have usually not had experience with physically partitioning the objects in a sent into thirds, fourths and fifths (Reys, 2009, 268). Colour counters, bingo chips and foam shapes are useful materials. Objective Four

6. Manipulatives, such as pattern blocks and fraction bars Physical representations Objective Five Pictorial representations Outline teaching strategies for the effective teaching of fractions. Conceptual Understanding Language development Visual representations Listen to student reasoning Number line Relevant applications Mathematics literature Teaching Strategies Fractions Use different types of models, ie region, area, length, set. Multiembodiment of concepts Develop all three meanings of fractions: part-whole, quotient and ratio Encourage words and symbols

7. Objective Six Relate a sequence for teaching fractions to First Steps and the National Curriculum. Paper folding fractions: Supplying children with paper that can be easily folded into halves. Talk about whole objects and halves of objects. Demonstrate how to fold the paper in half to make one half. Allow the students to try this, have them colour or decorate one half of their whole. Talk must lead to two halves are one whole, this could be shown by using students work e.g. Johns half along with Sarah's half can make one whole. Has a sequence for teaching fractions. Demonstrates wholes and halves whilst explaining two halves are one whole. Can be linked to First Steps in Mathematics: KU 1 : Two equal sized parts e.g. come up with ways to divide into two equal halves (Willis, Devlin, Jacob, Powell, Tomazos, & Treacy, 2004) . Meets The Australian Curriculum Year 1 Fractions and Decimals description: "Recognise and describe one-half as one of two equal parts of a whole. (ACMNA016) " (Australian Curriculum Assessment and Reporting Authority, 2011, p. 15)

8. Objective 7 Demonstrate ways to assess children’s knowledge of fractions.

9. Demonstrate ways to use materials, for example Pattern Blocks and Cuisenaire Rods, to teach fraction knowledge Objective Eight • Designed to make students think about the concept of a fractions rather than just following an algorithmic process or a set of rules for adding or subtracting • Explore geometric models of fractions • Discover relationships among fraction/shapes (NSW Department of Education and Training 1999 – 2011) • Pattern blocks are versatile manipulative used in mathematics classrooms • They assist students understanding of fraction • Assist in other areas on mathematics; geometric concepts, spatial relationships & pattern awareness (NSW Department of Education and Training 1999 – 2011) EXAMPLES How many are in 2 How many are in 2 How many are in 2 Based on these relationships, (not to scale) If = 1, = ___. 1/6 PATTERNBLOCKS • Perfectly proportioned blocks or folded paper used to teach a variety of math concepts • Fractions/basic operation/decimals/percentages/algebraic reasoning geometry (Van de Walle & Folk 2010)

10. Objective 9

11. Australian Curriculum Assessment and Reporting Authority. (2011). The Australian curriculum: mathematics. Retrieved from: http://www.australiancurriculum.edu.au/Mathematics/Curriculum/F-10  IMG00013. [Image].(2006).Retrieved from: www.mathleague.com/help/fractions/fractions.htm NSW Department of Education and Training. (1999 – 2011). Retrieved from: www.curriculumsupport.education.nsw.gov.au/secondary/mathematics/years7_10/teaching/frac.htm Reys, R., Lindquist, M., Lambdin, D., & Smith, N. (2009). Helping childrenlearn mathematics (9th ed.). United States of America: John Wiley & Sons, Inc. Sector of Circle. [Image]. (2011). Retrieved from: http://eversongeometryblog.blogspot.com/2011/05/113-area-of-sectors-and-arc-length.html Van De Walle, J. A., Karp, K. S. & Bay-Williams, J. M. (2010). Elementary and middle school mathematics: Teaching developmentally (7th edition). Boston: Allyn and Bacon/Pearson Education. Ch. 15 pp. 286-308. Willis, S., Devlin, W., Jacob, L., Powell, B., Tomazos, D., & Treacy, K. (2004). First steps in mathematics: Number. Understand whole and decimal numbers/understand fractional numbers. Victoria: Rigby Heinemann. References