MATHEMATICS UNIT PLANNER OVERVIEW

Topic/Mathematics Focus: Equivalent Fractions Year Level: 4 Term: 2 Week: 3 Date: 18/09/2011

Introduction: This unit of work aims to teach students about Equivalent fractions using a range of resources, technology and teaching strategies. By the end of this unit, students should be able to identify and draw/model equivalent fractions.

Key Mathematical Understandings being constructed:

Fractions can be represented in different forms, but hold the same equality. Equivalent fractions can be identified by simplifying fractions into their simplest form. Equivalent fractions can be represented as a proper fraction, an improper fraction or as a mixed number. Throughout this unit students will learn how to identify equivalent fractions

Catering for Diversity: There are some considerations which need to be addressed during this unit which will meet the needs of ESL students. Thomas (2007) states that ESL students generally need more time to grasp mathematical concepts as their speaking, writing and listening skills may still be developing. One implication that Thomas suggests teachers implement in their classroom is to use a lot of models, photographs and demonstrations as it is a lot easier for students to develop the concept before they develop the corresponding language. This strategy may also be helpful for visual and “hands on” learners.

When teaching a unit of work, it is important to cater for all learning needs and styles so all students have the opportunity to reach their potential in this unit. According to Blecher et al.,(1997) it is important to “accommodate for a variety of student learning styles through both teaching materials and methods,

interests and develop effective ways to communicate their understanding.” (p. 197)

When teaching this unit of work, it is also important to consider the needs of students with learning and intellectual disabilities in the mathematics classroom. Van de Walle et al., (2010) suggests that teachers only give one instruction at a time, especially when catering for students with learning difficulties. A strategy as suggested in Van de Walle et al., (2010) that can be implemented to support students with intellectual disabilities is to provide opportunities for repetition and concrete learning tools as they often need more time to learn a new concept

Purposes of Assessment: The purpose of assessment in this unit of work is for teachers to gather an understanding of whether students have been successful in these tasks. By assessing students work during this unit plan, it is allowing the teacher to see if a student understands the content, or whether some students need some extra support in this topic. There are a wide range of assessment tools that can be used to mark a wide range of activities. Assessment should ideally relate directly to what students will be learning. Blecher et al. (1997) suggest that assessment goals are made clear to students, teachers and parents at the beginning of planning process. By doing this it is allowing students to be aware of the expectations of assessment and teachers can design lesson plans based around these expectations.

Assumptions about Assumed prior knowledge: By grade 4, it is assumed that students will have had some prior experience with fractions in their previous years of primary school. According to VELS (2007) it states that students can use fractions with numerators other than one eg. 3/4. However this doesn’t necessarily mean that all year 4 students would have this knowledge.

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and allowing students to investigate their own

Rationale

Related VELS Focus / Outcomes: The Australian Curriculum and VELS have both influenced this unit plan to ensure that the activities are aimed at a suitable level which is achievable by grade 4 students.

From VELS level 3 progression points, it states that by progressing towards level 3 students should develop the following key understandings:

Number:

use of fractions with numerators other than one; for example, 3/4 of a block of chocolate (VELS, 2007)

development and use of fraction notation and recognition of equivalent fractions such as1/2 = 4/8 , including the ordering of fractions using physical models (VELS, 2007)

Working Mathematically: (VELS, 2007)

use of materials and models to solve problems and explain answers

explanation and comparison of alternative computation methods

use of materials and models to illustrate and test generalisations

From the Australian Curriculum:

Number and Algebra:

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Year 4 Standard: Students recognise and locate familiar fractions on a number line and make connections between fraction and decimal notations.

What Teachers Need to Know

Before beginning this unit of work, teachers need to be aware that there are misconceptions that students can develop when learning about fractions which need to be addressed and eliminated.

Naiser et al., (2004) state that teachers should find out what their students already know when introducing a new topic as it allows them to identify any misconceptions that students may have already developed. By identifying these misconceptions at the beginning of the unit, it can allow them to be eliminated before they are developed further.

A common misconception associated with equivalent fractions is that some students can view the second fraction as having twice the amount of parts as the first fraction. Afamasaga-Fuata’i & Jigyel, (2007) states that to address this misconception, students need to understand the relationship between the numerator and the denominator which is also known as the part-whole concept. Another strategy to eliminate this misconception and other fraction related misconceptions is to involve a wide range of models and multiple contexts which can cater for the diverse learning needs of the students. (Afamasaga-Fuata’i & Jigyel, 2007)

Investigate Equivalent Fractions used in contexts.

Grouping Strategies: Session 1 (below) has been designed to discover students’ prior knowledge on fractions by applying what they already know for the task. This task can determine the learning groups that will be used for Sessions 2 & 3 which are both small group work. It is preferable that students be placed in learning ability groups which can allow the tasks to specifically cater for their needs, especially the teacher focus group.

Session 2 &3 has been designed so that each group does each activity for half a session, then rotates. This is to ensure that they have enough time to explore the activity provided..

Vocabulary to develop:

Numerator, denominator, equivalent, simplifying, comparison, partitioned, equal and similar.

Teaching strategies: Modeling/demonstrating to students, using authentic tasks that link to real life contexts, using open ended questions/activities, whole class investigations, using virtual and concrete manipulatives, small group activities, teacher focus group and peer teaching.

Links to other contexts: ICT, real life contexts.

Lea

rnin

g st

rate

gies

AnalysingCheckingClassifyingCo-operatingConsidering optionsDesigningElaborating

EstimatingExplainingGeneralisingHypothesisingInferringInterpretingJustifying

ListeningLocating informationMaking choicesNote takingObservingOrdering eventsOrganising

PerformingPersuadingPlanningPredictingPresentingProviding feedbackQuestioning

ReadingRecognising biasReflectingReportingRespondingRestatingRevising

Seeing patternsSelecting informationSelf-assessingSharing ideasSummarisingSynthesising

TestingViewingVisually representingWorking independentlyWorking to a timetable

MATHEMATICALOBJECTIVE

(what you want the children to come to understand or

appreciate)

TUNING IN(sets the scene/ context for what students do in the independent

aspect. e.g., it may be a problem posed, an open-ended question, modelling the use of material or

reading a story.)

INDEPENDENT LEARNING

(extended opportunity for students to work in pairs, small groups or individually. Time for teacher to rove, listen, probe and challenge children’s thinking.)

REFLECTION ANDMAKING CONNECTIONS

(focused planned time for students to reflect on learning by explaining, showing, justifying their thinking; teacher questions gather evidence for general finding, assists children to make link/s, raises possibilities for further thinking /investigation)

ADAPTATIONS DURING LESSON- Enabling prompt(to allow those experiencing difficulty to engage in active experiences related to the initial goal task)- Extending prompt(questions that extend students’ thinking on the initial task)

ASSESSMENTSTRATEGIES

(should relate to objective. Includes what the teacher will listen for, observe, note or analyse)

Session 1

Understanding the relationships between equivalent

Whole class investigation to introduce concept of equivalent fractions.

Students work individually on set task exploring everything they know about their

Teacher asks selected students to share their favourite fraction with their class by sharing all their knowledge on a

Those experiencing difficulty can be posed with a simpler fraction to work with eg. 1/2 or ¼ and focus on writing

Observe which students were able to list at least 1 equivalent fraction of their chosen

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fractions and if a fraction can be represented in a different way.

Introduced using pizza diagram. (See appendix 1 for pizza diagram) Teacher asks, if I have 2 pieces of pizza out of the 8 pieces, what are 2 ways of representing this?

Then pizza is divided into only 4 pieces and students are asked “If I had 1 piece out of 4, would I be having the same amount as the 2 pieces out of 8?”

favourite fraction. Can use concrete materials or physically draw their fraction. Asked to list any equivalent fractions that there may be from their favourite fraction. See Appendix 2 (Activity Adapted from Downton et al., 2006)

fraction, giving students the opportunity to learn off one another. Ask students to justify the equivalent fractions that they have listed.

at least 5 facts about it.Extending students by asking them to represent their fraction in 3 or more ways by using concrete materials or by drawing it using set/discrete models.

fraction and also state 4 other key points about their fraction. Note down those that were unable to by using a class list to record these details.

Session 2Exploring and understanding equivalent fractions in a range of contexts and materials. Focussing on halves, quarters and eighths.

Teacher models to students how to use Cuisenaire rods to make equivalent fractions.Asks students: “What is 2/4 equivalent to?”“How can you show this using Cuisenaire rods.”

Explore equivalent fractions with concrete or virtual manipulatives. One group uses Cuisenaire rods to answer questions eg. “How many quarters in 2/8?” Students record their answers and draw a model. The other group explore the ICT activities on equivalent

Students who explored the Cuisenaire rods can share their findings to the class by stating equivalent fractions that they have found and can draw on the board or show students using the rods. Teacher can ask “How did you find this equivalent fraction?”“Are there anymore equivalent fractions for the ones that you have found?” “Explain how do

Students experiencing difficulty with the rods or ICT tasks can be limited to using halves, or quarters to find equivalent fractions.Students who finish the ICT activities early can use another virtual manipulative which makes the connection to percentages and decimals. (See Appendix 5) Students can also be asked to list

Observe strategies for finding equivalent fractions eg. Were they using the rods to compare fractions to find equivalents?

Anecdotal notes are written on one group of students listing their strategies used to find equivalent fractions and how

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fractions. (See appendices 3 & 4)

you know this?” at least 10 equivalent fractions that they have found.

many they found.

Session 3

Understanding the use of equivalent fractions in addition equations and how we can identify equivalent fractions on a fraction wall.

Whole class investigation, using equivalent fractions when adding. Eg. “If I ate ½ of a chocolate bar and my friend ate a ¼ of the same chocolate bar, how much was eaten altogether?”Introduce using a fraction wall to compare equivalent fractions by labelling all sections then colouring fractions as required.

One group plays fraction game in pairs (See Appendix 6 & 7) looking at equivalent fractions using a fraction wall.Teacher focus group works on fraction equations using equivalent fractions on a fraction wall. Exploring “If Bob ate 2/8 of a liquorice stick and Sue ate ½ of a liquorice stick, how much was eaten all together? (Adapted from: Van de Walle et al., 2010 p. 311 used in tutorial 7)

Students can reflect in pairs/small groups on the game, who won, why do you think they won? Is there a strategy?Teacher Focus Group shares their answers within the group, justifying how they got their answer. Teacher can ask “Is there another way of representing the fractions that each Bob and Sue ate which still gives us the same answer?” Why or why not?

Can extend the fractions game by modifying it, using different dice/different fractions including sixteenths and twentieths. Can simplify the game by limiting the fractions to halves, quarters and eighths.Can extend Teacher Focus Group by changing the problem by using equivalent fractions. “What fraction did Bob and Sue each have, and what fraction did they eat altogether?” Can allow students to use Cuisenaire rods to help them reach an answer.

Teacher writes anecdotal notes on their Teacher Focus Group Activity. Identifying whether students were successful in using equivalent fractions within an addition equation. What strategies did they use? Did they use concrete materials or draw a picture?

Observe whether students were able to successfully use a fraction wall by identifying appropriate fractions.

Session 4

Understanding the link between equivalent fractions and real life contexts. Using equivalent

Whole class investigation. “How are equivalent fractions used in real life situations?” “Why is it important to understand and use

Students design a park using a grid to help Terry the town planner design a site plan for the park. Students assign regions on a grid for different

Students pair up and discuss their different park site plans by discussing what fraction each section of their park is. Students have to find other students that have used the same or

Students having difficulties can use concrete materials eg. Square tiles, cubic cm etc. To plan their park, before transferring it onto a grid. A size limit of the grid can be given

Observe if students are successful in comparing their park site plans by using equivalent fractions to see if parts of their park are bigger/smaller

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fractions to compare sizes.

equivalent fractions?” Teacher models how to use graph paper/grid by partitioning it and labelling the fraction. Introduce task, Design a Park. “What things would we need to include in the plan?”

parts of the park eg. Sandpit, pond etc. Students express equivalent fractions for each section of their park. Eg. My pond is 1/8 of the whole park or 2/16. (Adapted from: Department of Education and Early Childhood Development Website. (2010)

equivalent fraction for the same section of a park. Pairs can present to class discussing similarities and differences between their different site plans using equivalent fractions as evidence for similarities/differences.

if students require it, eg. The park is a 10x10 grid. For extension students can increase the size of their park or choose their own size. Or modifications can be made eg. “What happened if the pond flooded and doubled in size, what fraction of the whole park would it be now?

than their peers. Assess if students could identify at least 3 similarities or differences between their park plan and their partners.

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References:

Afamasaga-Fuata’i, K., & Jigyel, K. (2007). Students’ Conceptions of Models of Fractions and Equivalence. Australian Mathematics Teacher, 63(4), 17-25.

The Australian Curriculum (2010). Retrieved September 16, 2011 from: http://www.australiancurriculum.edu.au/Mathematics/Curriculum/F-10

Blecher, T., Coates G. D., Franco J., & Mayfield-Ingram, K. (1997). Assessment and Equity. In J. Trentacosta & M.J. Kennedy (Eds.), Multicultural and Gender Equity in the Mathematics Classroom (pp. 195-200) Reston, VA: National Council of Teachers of Mathematics

Department of Education and Early Childhood Development. (2010) Early Fraction Ideas with Models. Retrieved September 20, 2011 from: http://www.education.vic.gov.au/studentlearning/teachingresources/maths/mathscontinuum/number/N25001P_2.htm

Downton, A., Knight, R., Clarke, D., & Lewis, G. (2006). Mathematics assessment for learning: Rich tasks & work samples. Melbourne: Mathematics Teaching and Learning Centre, Australian Catholic University. (As used in tutorial 5)

EDMA310/360 Learning & Teaching Mathematics 2 (2001) Readings, Activities & Resources. (Colour in Fractions p. 171 & Fraction Wall p. 172)

Math Goodies Fraction Activities. (2009) Retrieved September 20, 2011 from: http://www.mathgoodies.com/lessons/fractions/equivalent.html

Naiser, E., Wright, W., Capraro, R.

(2004) Teaching Fractions: Strategies Used for Teaching Fractions to Middle Grades Students: Journal of Research in Childhood Education 18. 3, Spring, 193-198.

National Library of Virtual Manipulatives. Retrieved September 16, 2011 from: http://nlvm.usu.edu/en/nav/frames_asid_105_g_2_t_1.html

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Thomas, J. (1997). Teaching Mathematics in the Multicultural Classroom: Lessons from Australia. In J. Trentacosta & M. J. Kennedy (Eds.) Multicultural and Gender Equity in the Mathematics Classroom (pp. 34-45). Reston, VA: National Council of Teachers of Mathematics.

Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2010). Elementary & middle school mathematics: Teaching developmentally (7th ed.). Boston, MA: Allyn & Bacon.

Victorian Essential Learning Standards (2007) Progressions Points, Retrieved September 2, 2011 from: http://vels.vcaa.vic.edu.au/support/progression/maths.html

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Appendix 2.

Adapted from: Downton, A., Knight, R., Clarke, D., & Lewis, G. (2006). Mathematics assessment for learning: Rich tasks & work samples. Melbourne: Mathematics Teaching and Learning Centre, Australian Catholic University. (As used in tutorial 5)

Everything about my fraction: Write down a list of all the fractions you know. From this list choose your favourite fraction. Answer the following questions about “your” fraction: Write down everything you know about your fraction. Write down the decimal equivalent of your fraction. Represent your fraction on a number line. Represent the decimal form of your fraction on a number line. Find at least three other fractions equivalent to your fraction. Represent your fraction as part of a metre.

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Appendix 5.

Virtual Manipulative making connections between fractions, decimals and percentages.Retrieved from:http://illuminations.nctm.org/ActivityDetail.aspx?ID=11

Extension Computer Activity for students who find the first two ICT activities easy and require some further extension work.

Explore different representations for fractions including improper fractions, mixed numbers, decimals, and percentages. Additionally, there are length, area, region, and set models. Adjust numerators and denominators to see how they alter the representations and models. Use the table to keep track of interesting fractions.

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Appendix 3.Virtual Manipulative Activities used in Session 2.

Fractions Equivalent.http://nlvm.usu.edu/en/nav/frames_asid_105_g_2_t_1.html

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Appendix 4.Virtual Manipulative Activities used in Session 2.

Fractions-parts of whole http://nlvm.usu.edu/en/nav/frames_asid_105_g_2_t_1.html

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Appendix 1.Pizza partioned into 8 pieces. Retrieved from: http://www.mathgoodies.com/lessons/fractions/equivalent.html

Pizza partioned in fourths, showing one fourth.Retrieved from: http://www.mathgoodies.com/lessons/fractions/

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