Arithmetic: Decimals, Fractions and Percentages

Transcription

1 A B UNIT Lesson Plan Fractions and Place value and conversion of decimals to fractions T: Can you remember the place value table? Can you tell me the headings we use? Ps:... hundreds, tens, units, tenths, hundredths, thousandths,... T: Right. Now, I'll say some decimals, and you have to write them on the table. Then someone can volunteer to put them in order, smallest first. T says, volunteer Ps write into table: 9 thousandths tenths hundredths Hundreds Tens Units tenths hundredths thousandths P: < 0. 0 < 0. Converting fractions to decimals T: What other way can you write tenths? (As a fraction) T: So, can you write it in this form? P (writes on BB): 9 T: And the other two numbers? (, ) 0 T: How do we write these numbers as decimals? (0., 0.009, 0.0) T: Can you think of yet another way to say tenths? (e.g. 00 thousandths, since T: And what about (T writes on BB): 0. Ps dictate, T writes on BB: 0 tenths hundredths = + = + = = hundredths T: Let's do the same with Ps dictate, T writes on BB: 00 = ) tenths 6 thousandths = + = = 06 0 Conversion of Decimals and Fractions T makes Ps recall their knowledge of decimals, and find the connection with fractions. T sketches a table on BB; Ps say the numbers aloud and volunteer Ps write them on the table. Another volunteer P writes these numbers on BB in order of size. Ps write in Ex.Bs. Agreement. T and Ps revisit the concept of improper fractions and mixed numbers. T: If the decimal is larger than, will that give us any problems? Ps dictate, T writes on BB: units tenths = 0 = + = = tenths mins

2 UNIT Fractions and Lesson Plan Practice converting decimals to fractions T: Did you realise how easy it was to convert decimals to fractions? Now you can try some for yourselves, like this: Someone comes to the front and writes a decimal on BB. They call another person to read it out and write it as a fraction. Then this person will write a decimal on the BB, call another person, and so on. Are you ready? Let's start with decimals smaller than. Who'd like to write the first one? e.g. P writes on BB: 0. and chooses P. P (says): hundredths (writes):, etc. Converting fractions to decimals Can you do this in reverse? We'll see. OS., Q Q Q8 Q Q Q Q6 Q T: But what about = 09. = = 6. = 00. = = 0. = 0. 0 =. 0 mins? We'll look at this one later. 0 8 mins Decimals to fractions OS. T: Look at question. Why do you think the second equals sign and fraction line are there? (Because the fractions can be written in several forms) T: For example? ( 8 = 80 6 = = = 0, etc.) T: Why is it that a fraction can be written in several forms? (Because its value stays the same if we multiply or divide both its numerator and denominator by the same non-zero number) Conversion of Decimals and Fractions P who wrote the number on BB decides if answer is correct, T and other Ps agree or not. Correct solution is written in Ex.Bs. After about decimals have been converted, T asks Ps to now choose decimals larger than. When all Ps have had a turn, T ends the activity. Praises Ps. Brief whole class activity; Ps should find this straightforward after the previous work. Problems appear on OHP. For the first few fractions, volunteer Ps come to front, explain and write solutions on OS; other Ps write in Ex.Bs. The remaining problems are worked on individually in Ex.Bs, monitored, helped. Checking at OHP; T points to P, P answers. Agreement, feedback, T writes solution on OS self-correction. T encourages more and more Ps to give examples, writes them. on BB, then makes Ps recall the rule using correct mathematical language.

3 T: Which form do you prefer to work with? UNIT Lesson Plan Fractions and T: Why aren't we given another fraction line in question? ( 00. =, and this is its simplest form) T: Right. Now, write the other decimals as fractions and find their simplest forms. Ps at OHP: Q = = Q 06. = = Q. = = Q = = 0 Decimals to fractions - practice PB., Q (b), (g) PB., Q (a), (e), (j), (l) 0 00 mins ((b) 9, (g) 0 ) ((a), (e) 0, (j) 00, (l) 0 ) PB., Q (a), (c) ((a), (c) 9) mins Conversion of Decimals and Fractions They agree that is the simplest form of 8, so T writes it after the equals sign on OS. Note that it could be written in other ways, e.g. 0, Volunteer Ps come to OHP, explain and write solutions on OS. Other Ps and T agree or not. T praises. Individual work, monitored, helped. Checking at BB: volunteer Ps come to BB and write solution, other Ps agree or not that this is the simplest form. Self-correction. Set homework PB., Q (c), (i), (k) PB., Q (b), (d), (f) PB., Q (a), (b), (f) PB., Q (b), (d), (i) PB., Q6 (b), (e)

4 UNIT Fractions and Lesson Plan Conversion of Decimals and Fractions Checking homework A PB., Q (c) (i) 0 (k) 8 PB., Q (b) 0.0 (d) 0. (f) 0.0 Verbal checking: T reads out question, points to a P, P answers, T agrees/disagrees correction. Praising/self-correction. PB., Q (a). (b). (f).8 B PB., Q (b) 0.08 = 8 (i) 0.8 = 8 0 PB., Q6 (b).0 = 0 = (d) = = 6 0 (e).008 = mins = 00 Warm-up practice T: Some parts of the homework were not straightforward. Let's do some easier ones in our heads. T (asks and writes on BB): 0. = ( 0. = ( = ( 0 0. = (. = ( 0. = ( 0 = ) = ) 0 = ) 0 = ) = ) = ) 8 Checking at BB: T points to P, P dictates, T writes answer on BB, other Ps agree or not (e.g. at 0.8) suggest a simpler form. Self-correction. Mental work to warm up and prepare for the topic to come. T writes decimals on BB. Ps volunteer, T waits for slower Ps' to reach answer and then points to P. P answers in two steps; other Ps agree/disagree correct answer. T writes solution on BB; praises. A mins Looking at denominators T: On the BB you can see some decimals converted into fractions, written in their simplest forms. What do you notice about the denominators of the simplest form fractions? What kind of numbers can or cannot be denominators of fractions written in their simplest forms? Why? Ps dictate, T writes on BB (Ps in Ex.Bs): = 0 = 00 = 0 = 0 = 0 0 = 00 = 0 8 = 0 Work remains on BB from homework and, and now Ps examine the denominators in the answers. T questions Ps and arrives at reasons why numbers such as sevenths and thirds cannot be denominators after simplification of fractions derived from converting decimals. Finally Ps give the pairs of multipliers that give, or 0 as a product.

5 A B UNIT Fractions and Lesson Plan T: So, can we write as a decimal? (Yes, we can convert it into tenths) T: How do we do that? ( 6 = = 06. ) T: And what about? (We can convert it into hundredths) T: Will we have any problem with improper fractions, for example,? (No, 0 = =. ) Further practice T: Good. Now with the next fractions, determine the equivalent fractions with denominators as either, or 0, and then write them as decimals. OS., extended with Q9 0 = = Q 0 = = mins Conversion of Decimals and Fractions Problems appear on OHP. Ps volunteer; especially for easier questions, T encourages slower Ps to come to OHP and do a conversion, with help if necessary. If Ps do not understand, T must explain again, with patience! Individual work with denominators PB., Q (c), (d), (g), (j), (k) ((c) 0.8 (d) 0. (g) 0.00 (j) 0. (k) 0.0) PB., Q (a), (b), (e), (f) ((a). (b).0 (e). (f).00) mins Revision - adding and subtracting decimals T: Can you remember how to add and subtract decimals? Let's do some easy calculations mentally = (0.8) = (.9). +. = (.). + = (.) = (.). +.8 = (8.) = (0.) = (0.9) = (0.) 0. = (0.8). 0.6 = (0.).. = (.8) 6 mins 6 Adding fractions and decimals T: What do we do if we are given this problem (writes on BB): 0. + =? Ps: We have to add them! T: That's right - but how? What have we learnt earlier in this lesson? Individual work, monitored, helped. Checking at BB with detailed discussion about getting, or 0 as denominators. Volunteer Ps come out, write and explain solutions. Agreement, feedback, selfcorrection. Mental work with most Ps contributing. T asks, chooses volunteer Ps, but waits for slower ones especially for the easier questions. Agreement. P : We have to convert into a decimal.

6 UNIT Fractions and Lesson Plan 6 T: So, what do we do? P : = and = 0. T: Good. Who would like to write it on BB? P (writes on BB): 0. + = 0. + = = 0. T: Let's look at some other ones (writes on BB): +. (= +. = =. ) 0 T (writes on BB): ( = = = 0. 0) 00 0 T: Now try some on your own. (a) (b) +.. (c). + Solutions 6 (a) (b) Set homework () PB., Q (f), (h), (i), (l) () PB., Q (c), (d) () (a) + 0. (b) = +. = = 9.. =. =.. = 0. 6 (c). + =. + = =. mins Conversion of Decimals and Fractions Agreement. Probably only stronger Ps will volunteer to solve these problems, but T can encourage slower ones. Individual work. T writes problems on BB, then monitors and helps Ps as they work. Checking at BB with discussion, self-correction and much praising. Agreement.

7 Checking homework () PB., Q (f) (h) 0. (i) 0.0 (l) 0. () PB., Q (c).6 (d). A B C UNIT Lesson Plan Fractions and () (a) (b) + 0. = = =.. =. mins Division and multiplication of decimals T: You've shown you can do addition and subtraction with decimals. But what about division and multiplication? First, let's do some mental calculations: 6. (=.) (=.) 08. (=0.). (= 0.). (=.) 69. (=.). (=.8) 08. (0.) Written practice T: Now write these calculations and their solutions in your Ex.Bs: (a). = (0.) (b) 6. = (.) (c) 6. 8= (0.) (d) 6 (.) (e) (0.) Revision of Unit : Fractions T: Have you seen the decimals in the last two answers anywhere else during this lesson? T: Why were these the answers to these questions? Conversion of Decimals and Fractions T has asked one of Ps to write solutions on BB as soon as P arrives. Agreement, feedback, selfcorrection. Mental work. T asks, points to P, helps slower Ps, P answers. Agreement. Practice with multiplication included here for practice. T writes problems on BB and then calls Ps (encouraging slower ones) to give answers and explanations. Agreement. Solutions to homework remain on BB, so Ps look for. and 0. among the numbers.. and 0. ) ( 6 T: And how did we get these answers? ( 6 =. = 0. ) T: Why are these the answers? What is meant by? (The unit is divided into equal parts, and we have of the ) T: Is there another way of describing this? T makes Ps remember what they learnt in Unit. ( also means that we divided units into equal parts, and are referring to three of them) T: So, what is of? ( ) T: What is of 6? ( 6 ) T: What is 6? ( 6 ) T: What is? ( 0 mins )

8 UNIT Fractions and Lesson Plan Whole class practice OS.. (a)... (b) = (6.8) T: Wouldn't it have been easier to convert this fraction into a decimal in the usual way? So why didn't we do that? Never mind, we can use what we have learnt in this unit to check the answer. Volunteer P at BB: 68 = = 68. T: This method is much quicker. Which method will you choose to convert the next fraction? Ps: We'll divide. T: OK, it's up to you. P at OHP:. (a)... (b) 8 = (.6) T: Who'd like to show the other method? Ps:? T (encouraging): Can you get, or 0 by multiplying 8 by a whole number? Look at the last lesson's work in your Ex.Bs. ( 8 = 0 ) T: So? ( = ) 8 0 T: Do the multiplication in your Ex.Bs! T (after a short pause): Have you got the same answer? (Yes, = 6, so 8 =. 6) T: Which was the quicker method? (The calculation of 8) T: That's how it goes, sometimes one way is better, sometimes the other. 6 mins Practice writing fractions as decimals T: Write the following fractions as decimals by using division, then check your solutions by determining the equivalent fractions with the denominators as either, or 0. 6 PB., Q6 (a), (b), (c) ((a) = = = 06. (b) 8 = = = (c) = = = 6. ) Conversion of Decimals and Fractions Task appears on OHP. T makes Ps dictate division ; T writes on OS, Ps in Ex.Bs. Praising and then discussion. T calls a P to work through the division at OHP. Individual work, monitored, helped. Checking at BB with selfchecking. Agreement, feedback, selfcorrection. mins

9 UNIT Lesson Plan Fractions and Using equivalent fractions T: Now we'll use the reverse process for the following fractions. Write them as decimals by determining the equivalent fractions with the denominators as either, or 0, then check using division. (a) (b) 0 Ps: These fractions cannot be written as tenths, hundredths or thousandths. T: So they don't have decimal forms? Ps: No! T: So, why don't you try it the other way; write them as decimals by using long division. P (at BB): = P (at BB): 0 = P (at BB): = (c) mins 6 Individual practice T: Write the following fractions as decimals, using division to determine each number to the third non-zero digit. Conversion of Decimals and Fractions T writes the fractions on BB, and lets Ps find out that they cannot do as asked. T calls volunteer Ps to BB to do the divisions; other Ps write them in Ex.Bs. They all discuss what they have discovered about these fractions, that they have decimal forms, but without an end. Ps realise that, after some decimal places, the digits are repeated. When calculating, P must be encouraged to continue until the repetitions are obvious. Finally T tells Ps that they will deal with this problem later, and, until then, it will be enough to determine the first or nonzero digits after the decimal point, depending on the requirements of the task. Individual work, monitored, helped. (a) 6 (b) Solutions (a) 6 = (b) = mins Set homework () PB., Q6 (d), (e), (f), using both methods. () Write the following fractions as decimals using division to determine each number to the third non-zero digit: (a) 0 (b) 0 Checking at BB: T writes solutions on BB. Feedback, self-correction.

10 Checking homework () PB., Q6 (d) 6 =. 6 UNIT Lesson Plan Fractions and = =. PB., Q6 (e) 6 = = = 6. PB., Q6 (f) 0 8 = = =. 0 Detailed checking of Q6 questions at BB. Agreement, feedback, selfcorrection. () (a) (b) 0 = 0 = = = 0 0 = mins Preparing for percentages T: Now that we can convert fractions into decimals and decimals into fractions, we'll make things a little more difficult. Imagine that there is a country where the units are our hundredths. When we need to deal with them, we have to convert all our numbers into hundredths. Let's see how it might work. T: 0.8 Ps: = 8 Only solutions needed for (). Feedback, self-correction. 0.0 = 0.99 = 99 =. = = 0 = = 8 = = = = 60. = = etc. mins

11 UNIT Fractions and Lesson Plan Introducing percentages T: Do you think there is such a country? Ps: No! T: But there is! And not only one country. In today's society, most countries think in hundredths when speaking about changes. What other name do we give to these hundredths? Some Ps:. T: Yes, we call them percentages. The word comes from two Latin words, per meaning 'through' and centum meaning 'hundred'. What is the symbol we use for 'percent'? Volunteer P at BB: (%) T: Good. What fraction do we mean if we say: T: 9% Ps: = 9 % = 0 0% = = T writes on BB, Ps answer, T agrees, praises, writes Ps answers on BB, Ps in Ex.Bs. T asks for the fractions in their simplest form and helps Ps understand what is meant by the.% and.%. 0 0% = =..% = = 0..% = = = 0 0 T: Can you say where you have seen the word 'percentage' or the % symbol used? (Change in wages, price reductions, etc.) 0 mins Practical work with percentages T: Let's look at some examples of percentages. OS.6 () 8 = 8% 90 () = 90 = 90% 88 () = 88 = 88% () () (6) = = 6% 0 = = 0% = = % T illustrates the relevance of percentages. Work in pairs or whole class activity. Problems appear on OHP, and each pair of Ps has a copy. T allows time for discussion between each pair on the first shape, then T chooses a volunteer P to show answer, explain and write the fraction and percentage under the shape on OS. Other pairs agree/disagree praising, self-correction. Continue in this way for other shapes.

12 () UNIT Fractions and Lesson Plan (8) (9) 0 = = 0% = = % = = % 0 mins Finding percentages PB., Q and (a), (d), (f) Q. (a) % (d) 0% (f) % Q. (a) % (d) 0% (f) % 6 Converting percentages to fractions OS. 9. 9% = 0. 0 % = = ( 6 mins or for shading). % = = 0 ( for shading) mins Individual work with fractions PB., Q (a), (b), (d) (a) % (d) (b) Set homework PB., Q and Q (b), (c), (e) PB., Q (a), (b) PB., Q (c), (e) PB., Q6 (b), (c), (f) 0% mins % Individual work, monitored, helped. T divides class into two parts, according to seating. Half the Ps have to write down the percentage of shapes (a), (c) and (f) that is shaded; the other half have to write down what percentage is not shaded. For checking, T calls out one P from each group to write solution on BB at the same time. After agreement, T makes Ps compare the percentages ( %) Self-correction. Problem appears on OHP. Volunteer Ps come to BB to convert percentages into fractions, giving the simplest forms, and the easiest form for shading. Individual work. T gives Ps minutes to copy and shade a suitable number of the squares; T helps slower Ps. Checking: T sketches solution on BB feedback, self-correction, praising.

13 UNIT Fractions and Lesson Plan Checking homework PB., Q (b) 6% (c) 8% (e) 80% PB., Q (b) 6% (c) % (e) 0% PB., Q (a) for example: (b) for example: Solutions to questions and appear on OHP T has prepared. Solutions to other questions are checked in words, with detailed discussion following. % shaded 0% shaded PB., Q (c) for example: (e) for example: 90% shaded PB., Q6 (b) 0% (c) % (f) 60% % shaded mins Mental work with fractions and percentages T: Can you remember how we found fractions of quantities? For example, of? Ps answer, T writes on BB: of = = = Mental work to prepare for the lesson's topic. T: What about of 0? Ps answer, T writes on BB: of 0 = 0 T: 0 ( ) = = 8 of 9 Ps: 9 of = 6 = 6 After writing down the answers to the first two questions in detail, T asks questions, points to volunteer Ps to answer. 6 of 6 = 6 of T: T writes again on BB (Ps answer): = 6 0 of =

14 UNIT Fractions and Lesson Plan e.g. = of = = 6 or = 0 = 0. 0 = 6 of = = 0. =. 8 T: What kind of fractions have we been working with here? Ps: Hundredths. T: How else can we describe them? Ps:. T: Right; let's carry on. mins Calculating percentages OS.. % of 00 kg = of 00 kg 00 e.g. = of 00 kg = kg = kg 0 0 or = 00 kg = kg. 0% of 0 m = 0 of 0 m 0 e.g. = of 0 m = m = m or = 0 0 m = ( 0. 0) m = m. % of 900 = of e.g. = of 900 = = or = 900 = ( 9 ) =. 0% of 80 = 0 of 80 e.g. = = 80 of 80 = ( 8 ) = or = 80 0 = ( ) = mins Finding the quickest method PB., Q9 (b), (c), (f), (h) Solutions: (b) 0% of 0 kg = 0 00 of 00 kg = 0 kg = kg (c) 60% of 0 p = 60 0 of 0 p = of 0 p = = 0 p T leads Ps to find different ways (with or without simplifying) to the solution. T asks, Ps answer, T writes on BB, Ps in Ex.Bs. Problems appear on OHP. Having prepared for this topic in, volunteer Ps come to OHP to write and explain solutions. For the first question, T suggests that Ps think carefully which method is the quicker and whether it helps to simplify the fraction. When P has given all the method () solutions, T calls another P to give solutions by method (), at BB. Agreement. Finally, after discussion T and Ps decide that as a general rule, the % of a quantity can be calculated by dividing the quantity by and then multiplying the quotient by x. Individual work, monitored, helped. Checking: T and Ps agree the shortest way; these solutions are written on BB by Ps.

15 UNIT Fractions and Lesson Plan (f) % of 0 m = 0 of 0 m = m = 0 m (h) 6% of = 6 = ( 0. 6 ) = 0. 0 mins Real-life examples with percentages. T: This activity deals with exchange rates for various currencies. T: Why won't many of the conversions listed be relevant today? Ps: The Euro is now the currency of most countries in the EC. T: That's right; it was introduced in 00, but we can still use the exchange rates given here to practice our calculations. A Simple currency exchange., Q (a), (d) Q (a), (b) B P (Q(a)): 60. $ = 6$ P (Q(d)): L. 0 = 680 L P (Q(a)): 9. 0 = ( ). 6 (to d.p.) P (Q(b)):. = ( ). (to d.p.) Charging commission., Q P (Q(b)):. DM 00 = 6 DM 6. = 6.. = =. 8 (DM) P (Q(a)):. 60 $ 00 = 0 $ 0 =. = =. 6 ( $ ) ( ) = P (Q(c)): 9. 0 F Fr F Fr 9. 0 = 8 F Fr Mental calculation: 9. = = 80 and = 6. so 9.. = and 80 = 8 Agreement.. appears on OHP and each P has a copy. T works through the example in three steps. T and Ps discuss the 'The Abroad' table - what is meant by 'exchange rate' - how to convert pounds to foreign currencies, etc. Volunteer Ps are called to do conversion at BB; other Ps in Ex.Bs (recalling and practising multiplication and division, to decimal places, of decimals). After discussing commission charges, T calls a stronger P for Q (b) to calculate at BB the amount obtained. Then T makes a slower P repeat the procedure with the 'easier' numbers in Q (a), at BB. Encouraging, helping, agreement, praising. All Ps write in Ex.Bs. (Discussion as to whether it is quicker to calculate 98% instead of % and a subtraction.) Q (c) can be given as individual work. Checking in detail at BB, then T asks if anyone has a way of doing this in their head. If noone volunteers, T shows Ps the method.

16 C UNIT Fractions and Lesson Plan Different rates for buying and selling (analysing the 'MEP Bank' table) mins Set homework (). - for all Ps (each P has a copy). ()., Q and Q - for stronger and enthusiastic Ps. There will not be enough time to go into this in depth; interested Ps can investigate further in their homework (.). Two Ps can show and explain solutions at the beginning of the next lesson.

17 UNIT Fractions and Lesson Plan 6 Decimals, Fractions and Checking homework () A Looking at the extra homework exercises:., Q P (writes and explains at BB):. DM 0 = 0 DM B 0 =. = = 6. ( DM)., Q P: (a) ( to d.p. ) to d.p. ( ) = (b) You have lost =.. Checking homework (). 6 mins mins Mental work - familiarisation with percentages. (B) T asks who would like to work out the exchange problem if the bank has different rates for buying and selling foreign currency and also charges commission (maybe with different percentages for buying and for selling). Then Q and Q are explained and solved by successful Ps at BB. T agrees, praises; other Ps write in Ex.Bs.. sheet appears on OHP and T asks which route was chosen. Answering. Agreement, step by step. Self-correction. Mental work/game. This activity is a good start to this lesson. Ps must learn to automatically interchange decimals, fractions and percentages in their heads, e.g. when they see they should 6 mins Conversions: decimals, fractions, percentages OS.8 P (writes and explains at BB): P : % = 0. = = 0 = = % = 0. 9 P : 09. = 9% = = automatically think %. This activity helps to develop this ability. Praise from T whenever possible. After the 'warm-up game', T and Ps need to discuss in detail how to do these conversions. Table appears on OHP. T chooses a row and a P comes to front of class to say the alternative forms. After agreement and praising, T makes these Ps fill in the gaps in their row on OS. Slower Ps should be encouraged to repeat the procedures with the other numbers. T and Ps can help

18 UNIT Fractions and Lesson Plan 6 Practice with conversions PB., Q 6 Self-marked test M mins mins = 60% ( marks) = ( marks) ( mark). % ( mark). 8 = ( marks) 6. 6 = 0 ( marks). % 8% = 8% ( marks) 8. = = % 0 ( marks) Set homework M. PB., Q of 0 = of 0 = 6 = ( marks) 6 0 of 0 kg = of 0 kg = 6 kg = 6 kg ( marks) 0 0 Marks out of possible 0: 8-0 Excellent - Good 8- Fair less than 8 Poor mins 0.0 % 0. % 0. 0% 0. % 0. % 0.8 8% Decimals, Fractions and them, or, if they have real problems, T must explain the conversions again. Meanwhile, other Ps can proceed with PB., Q, which can be checked at the end of the lesson. Individual work. T monitors and helps Ps' work. Checking: T puts complete table on OHP (prepared in advance) or sketches it on BB. Self-correction, each P should note their mistakes and learn from them. Individual work. Ps have the opportunity to test themselves. Each P has a copy of the test and 9- minutes to complete it. After this time, T puts answers, with marks), on OHP (prepared in advance). Ps can correct and mark their tests, noting their weak areas. Praising and encouraging.