TI-AIE Visuaising, comparing and contrasting: number systems
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Contents What this unit is about 5 What you can earn in this unit 5 1 Comparing and contrasting activities 5 4 of 9 Wednesday 27 May 2015
What this unit is about What this unit is about Mathematics has severa number systems: natura numbers, whoe numbers, integers, rationa numbers, irrationa numbers and rea numbers. Students often think that the distinctions between these number systems are somewhat obscure and trivia. This unit wi expore the simiarities and differences of these number systems, and how mathematica operations such as addition, mutipication, exponentiation, etc., function within these. The Nationa Curricuum Framework for Teacher Education (NCFTE, 2009) requires mathematics essons to be interesting, student-focused and student-participatory, and to buid the students understanding of mathematics. That is not easy to accompish. This unit aims to support teachers to achieve this by working on compare and contrast tasks and using visuaisation. The unit wi expain each of these approaches and provide exampes of how they can be appied in the cassroom. The unit wi aso discuss how to make sma changes to existing cassroom practice using a textbook. What you can earn in this unit Some ideas for how to expore and anayse different ways of visuaising numbers and number operations, and reaise their imitations. To recognise the structure and effects of compare and contrast activities, and how visuaisation can hep your students make sense of mathematics. This unit inks to the teaching requirements of the Nationa Curricuum Framework (NCF, 2005) and NCFTE (2009) outined in Resource 1. 1 Comparing and contrasting activities Comparing and contrasting are good activities to make peope aware of mathematica properties and its appications. The action of comparing and contrasting forces students to think about the properties of the mathematica objects and to notice what is the same and what is different. Whie doing so, students make connections they might not normay consider. They are prompted into mathematica thinking processes such as generaising, conjecturing about what stays the same and what can change, and verifying these conjectures. This is an exampe of the nationa curricuum requirement of etting students use abstractions to perceive reationships, see structures, reason things out and argue the truth or fasity of statements. Compare and contrast activities, such as in Activity 1, can hep students to: consoidate their earning remind them of the different purposes of the various number systems become aware of the subte simiarities and differences of the systems. 5 of 9 Wednesday 27 May 2015
1 Comparing and contrasting activities Activity 1 requires your students to think about which statements are aways, sometimes or never true. The aim is to become aware of differences between the number systems and to deveop a mathematica argument for their reasoning. By doing so, they become more precise in their use of mathematica anguage and notice the more subte simiarities and differences. In the ater questions there is a first step towards visuaisation by referring to a number ine. Before attempting to use the activities in this unit with your students, it woud be a good idea to compete a (or at east part) of the activities yoursef. It woud be even better if you coud try them out with a coeague, as that wi hep you when you refect on the experience. Trying the activities yoursef wi mean that you get insights into a earner s experiences that can in turn infuence your teaching and your experiences as a teacher. When you are ready, use the activities with your students. After the esson, think about the way that the activity went and the earning that happened. This wi hep you to deveop a more earner-focused teaching environment. Activity 1: Aways, sometimes or never true? Preparation The statements can be offered as a ist, or can be written on cards and shared out randomy. A fuer ist of questions, and an exampe of a worksheet that can be cut up as cards, can be found in Resources 2 and 3. An overview of different number systems and their properties can be found in Resource 4. Here are some suggestions for ways of working on this activity: You can ask students to work on this on their own, share their ideas with the cass or share their ideas with their partners/cassmates. The statements can be offered for a to be done but this coud become tedious, so ask students to compete ony part of them; for exampe, to do odd/even/prime numbered ones. You coud aso ask students to make their own choice. For exampe, if they seect three that they woud ike to tacke and two that they woud not want to do, you coud ask them to do a five of these with the support of their partner(s). Letting students make their own choice often empowers stimuating active participation and engagement in the cassroom. The activity Ask your students which of the foowing statements are aways true, sometimes true or never true, and why? 1 The sum of two natura numbers is a natura number. 2 The sum of two integers is NOT an integer. 3 The difference of two irrationa numbers is an irrationa number. 4 The product of two irrationa numbers is an irrationa number. 5 The quotient of two whoe numbers is a whoe number. 6 The quotient of two rea numbers is a rea number. 7 There are an infinite number of pairs of whoe numbers whose sum is 0. 8 There exists a pair of whoe numbers whose product is 1. 9 The product of two rea numbers is not a non-repeating, non-terminating decima. 6 of 9 Wednesday 27 May 2015
1 Comparing and contrasting activities 10 The exact ocation of a natura number cannot be determined on a number ine. 11 The difference of two integers is to the eft of each of the two integers on a number ine. 12 There are finite natura numbers between any two rea numbers. 13 The number a2 is a natura number if a is a natura number. 14 The number ab is greater than both a and b. Video: Tak for earning You may aso want to have a ook at the key resource Tak for earning. Case Study 1: Mrs Aparajeeta refects on using Activity 1 This is the account of a teacher who tried Activity 1 with his secondary students. I was a bit scared about trying out this activity for severa reasons: the unusua structure of the activity requiring the students to deveop their own reasoning students taking to each other when this is not usuay aowed in my cassroom my fear that they might not remember anything about the different number systems at a! Before we started on this activity, I asked six students for a number between one and 14. These six numbers became the questions they had to do. I was very uncomfortabe about aowing the students to tak to each other as I thought it woud become very noisy with 79 students in the cassroom; I woud not be abe to contro what was discussed and it coud become chaotic. What if the principa happened to wak past at such chaotic moment! On the other hand I do beieve that taking heps earning, because when you tak, you have to organise your thinking otherwise the istener wi not understand you. So I pushed mysef to be brave and started the esson: I asked the students to work the answers out for themseves in sience and to write down their reasoning. Then I asked them to discuss their thoughts and reasoning with their cassmates, in pairs. But they had to do it quiety, so the others coud not overhear what they were saying and stea their ideas! I tod them that after some time I woud seect at random someone to expain what the thinking of the pair had been. Other students coud then comment in a constructive way, just ike we sometimes do in whoe-cass discussions. To address my fear that they woud not be abe to remember anything, I wrote the page numbers of their textbook where they coud find information about the number systems on the backboard and they were free to refer to these. 7 of 9 Wednesday 27 May 2015
1 Comparing and contrasting activities To my surprise it worked reay we! The cassroom did not end in chaos. There was noise from the taking in pairs, but it was not oud: they seemed to ike the sense of competition and secrecy. I istened into the conversations by waking around and by standing quiety against the wa, observing. What I iked very much was that the conversations were about mathematics, and that disagreements ed to pointed discussions about the mathematica properties, ficking through the textbook to check their reasoning and using a number ine to iustrate what they were saying. The whoe cass discussion was ess ivey than I had anticipated because there was itte disagreement by then. However, I was impressed with the quaity of the mathematica anguage used. The cass discussion thus became a consoidation activity. Their responses showed me they seemed to reay understand the differences in properties, and the consequences of these differences when doing operations in the different number systems. What I found very hard in this esson was my changed roe as teacher: I was no onger standing at the front expaining and teing them what to do. Actuay, I taked very itte, and it was difficut not to interfere in the discussion and show them how to do it. However, it meant that this was now their earning and their thinking, which was a powerfu experience. Woud I change anything next time? Yes, I think I woud try to et them choose their own questions, athough I am not sure how to ead the whoe-cass discussion then as they might have chosen a different ones. Perhaps I coud ask them to prepare sma presentations on their findings. I think I wi aso prepare the questions in card format and use these when we have five or ten minutes eft over at the end of a esson. I think that coud offer the repetition needed to keep a this knowedge in the mind, without it meaning doing more exercises from the book. Refecting on your teaching When you do such an activity with your cass, refect afterwards about what went we and what went ess we. Consider the questions that ed to the students being interested and being abe to progress and where you needed to carify. Such refection aways heps with finding a script that heps you engage the students to find mathematics interesting and enjoyabe. If they do not understand and cannot do something, they are ess ikey to become invoved. Use this refective exercise every time you undertake the activities, noting, as Mrs Aparajeeta did, some quite sma things that made a difference. 8 of 9 Wednesday 27 May 2015
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