Fractions

Theoretical Background

Core Concepts/Skills/Strategies

Concept: A concept is a 'mental picture' that is formed in relation to a particular topic. When looking at fractions, one concept involves knowing and understanding the concept of halves. In a sequence of lessons, I would first teach this concept of fractions before incorporating skills or strategies associated with it. To introduce this concept, I would use student's language and provide them with a circular piece of paper which I will call a pizza. Students will then be instructed to fold their piece of paper into two equal pieces, with the teacher demonstrating.

After students understand how to fold their piece of paper, teacher will then explain that their pizza is is now in half. Students will be instructed to cut their piece of paper down the centre line so that their two halves are seperate. They will be asked to exchange half their pizza for someone elses half. This concept will then be emphasised by a teacher having an apple and cutting it in half infront of the students.

Skill: A skill involving this topic, could involve students showing their understanding of the concept. Students should have the skill to be able to halve many different items or pictures. For this stage, the teacher will use a combination of student's and mathematics language to instruct students to halve their items. Students will be given a combination of items on a piece of paper that they will need to be able to cut in half to demonstate the skill. Here is a sample of the objects provided for the students.

Strategy: Once the students have been able to show that they understand the concept of halves and have the skill to demonstrate their understanding of the concept, a strategy may be introduced. A strategy involves the ability to come up with an answer in relation to a concept/skill. This strategy will still be taught using student's language and mathematics language, and will be the last stage before moving on the symbolic language. This strategy involves asking students to determine how they know if their items are halved. They are then questioned how they know if their 2 halves are the same size as each other. This will continue until the answer that they both fold together without any excess paper on either side is obtained. In the lesson sequence, this will then transition to teaching the symbolic language.

Concept: Another concept involving fractions is for a student to understand that the same fractional amount can be represented in multiple ways. For example, 4/8 represents the same value as 1/2. In the lesson sequence, this concept would be introduced using childs language and then mathematics and symbolic language. To introduce this concept, students will be given four circular pieces of paper that look like pizzas. Each pizza will be divided into a different number of equal pizzas, one of 4 pieces, 6 pieces, 8 pieces and 10 pieces. Students will be asked to to cut their pizza slices out and then remove half of them. As they already understand the concept of translating a half into symbolic language, students will then be taken through those steps again with each pizza, with the end concept students knowing that 1/2 can be shown in different ways.

Skill: A skill involving this learning process involves a student being able to find an equivalent fraction for the fraction 2/8. Students should either multiply or divide the numerator AND the denominator by a common multiple or factor. This stage will be taught using symbolic and mathematical language after students have been introduced to the concepts of equivalent fractions, numberator and denominators and the concept of a commone multiple or factor. If students are successful with this process it shows that through this skill, the students understand these concepts.

Strategy: To demonstrate strategy, students should be able to use halving or doubling to generate an equivalent fraction.  In the learning sequence, students should be able to determine the correct answer using this strategy. Both symbolic and mathematical language will be used in this process and students should be able to demonstate that they can translate mathematical language into symbolic language on a worksheet with a number of questions for students to answer.

Concept: A concept that is also concerning the fraction topic is for a student to understand that a fraction is less then a whole and made up of parts. This should be taught in the very early lesson sequence as students are being introduced to mathematics and symbolic language. To help students to understand this concept, the teacher will use a number line on the board. The teacher will explain to students that the fractions are the parts between the numbers and will talk about the pizza examples previously shown, where the pizza was a whole and the slices were part.

Skill: For a student to show understanding of the concept, students will use their pizza cut into 8 equal parts. Using symbolic language, students will be told that Susie ate 2/8 of the pizza and asked to remove the pieces that she ate. They will then be told that James ate 3/8 of the pizza and asked to remove the appropriate amount of slices. This skill will involve adding two fractions together to determine how many pieces are left.

Strategy: A strategy involving this learning process, involves teaching students how to add two fractions together. This learning process will use symbolic language and for the purpose of this strategy, students will be asked to add a set of fractions using the 'strip model' (Van De Walle, 2010). This is a process which involves comparing two different fractions, and using this comparison to determine an answer. Students will be shown examples of this model on a slide show produced on the Interactive Whiteboard using the steps provided in the textbook. After this lesson, students will be shown other strategies they may use to obtain an answer when being asked to add two fractions together. Extension questions may also be offered for students to add more then two together.

Learning and Teaching Resources

1. Click here to view video

This first resource is a clip from the Khan Academy Website. It involves explaining the process of how to add fractions with common denominators. In the classroom, this video could be used to reinforce the strategy of adding fractions together. In a lesson sequence this should be shown after the concept that fractions are less then one is introduced and that adding two fractions together will still be equal to or lesser than one.  This clip is a very effective tool to use in the classroom because it not only uses ICT to integrate learning but it is also demonstrates appropriate language from the language model for mathematics. It mathematical and symbolic language in addition to showing a visual example of symbolic language on the board. It shows an effective method for adding fractions together in a simple yet effective way. This resource should be used in a senior classroom when it is evident students already have a sound understanding of the concept of adding fractions. To improve this resource perhaps a visual model could be used as a physical representation of the sum. The length of this video is effective as it will keep students attention.

2. Colour Sheet Activity

This resource is a sheet that involves different shapes all divided into 4 equal parts. It instructs students, using a colour coded system, to colour 1/4 of a shape in red and 1/2 in blue. This resource would be effective to use to reinforce the introduction of the concept of symbolic language. In the learning sequence, this lesson would be effectively used the day after introduction of the symbolic language to help reinforce the concept and allow students to experiment. This is an enjoyable activity as students are able to use colouring in to enhance their learning experience. To improve this resource I would make is slightly more complex by using a different number of parts in some shapes to help further student's understanding.

3. http://www.primaryresources.co.uk/maths/mathsB6.htm

Lesson Plan

3. Fraction Cards

This resource (obtained from http://www.primaryresources.co.uk/maths/mathsB6.htm) is a set of cards which uses symbolic language and then a visual representation of a fraction. This tool for learning is very useful in many classroom circumstances, but most effectively as a tool to reinforce the concept of fractions being a part of a whole. These cards can be used in a game sense where students could all be given a card. A fraction, such as, 10/20 could be written on the board and students are to decide if their card is equivalent to that fraction. These cards could also be held up with one part covered so students could be asked to write the fraction in symbolic language or draw a picture to represent this fraction. This resource would be most effective in a year 4 classroom as students would understand the symbolic language. These cards could be improved by using different shapes on each to make them a little harder.

Reference List

Bancerek, Grzegorz. (2003) Journal of Formalized Mathematics: Cardinal Numbers: Warsaw: Warsaw University

Jamieson-Proctor, R. (2011). EDX 1280: ppt Retrieved from              
http://usqstudydesk.usq.edu.au/mod/resource/view.php?id=405689

Karp K, Bay-Williams J & Van De Walle J. (2010) Elementary & Middle School Mathematics 7th ed. USA: Pearson Education

Lovell K. (1971) The Growth of Understanding in Mathematics: Kindergarten through grade three. New York: Holt, Rinehart and Winston

Nunes, T., & Bryant. P. (2007). Key Understandings in Mathematics Learning: Paper 2: Understanding   Whole Numbers. (Review commissioned by the Nuffield Foundation).  Retrieved from:    http://89.28.209.149/fileLibrary/pdf/P2.pdf