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Teach fractions and decimals for understanding

Operations with Decimals are in Teach for Understanding: Whole numbers Basic fraction concepts

1 Folding paper and naming fractions p3 2 Circle fractions1 p4 3 Tangrams p5 4 Circle fractions2 p6 5 Square fractions p7 6 Triangle fractions p8 7 Strip fractions p9 8 Number lines up to 1 p10 9 Fractions of a dozen p11 10 Pattern block fractions p12 11 Rod fractions p13

Improper fractions & mixed numbers 12 Improper fractions (pizzas) p14 13 Improper fractions (halves) p15 14 Improper fractions (quarters) p16 15 Improper fractions (thirds) p17 16 Improper fractions (pattern blocks) p18 17 Improper fractions (rods) p19 18 Mixed numbers (number line) p20 19 Mixed numbers (dice game) p21

Fractions of whole numbers 20 Fractions of numbers (counters) p22 21 Guessing and checking fractions p23 22 Clock face fractions p24 23 Fractions of an hour p25 24 Fractions of dampers p26 25 Fractions of rods p27

Comparing fractions 26 Comparing fractions (strips) p28 27 Comparing fractions (counters) p29 28 Comparing fractions (circles) p30 29 Rectangle shapes p31

Equal fractions 30 Equal fractions (circles) p32 31 Equal fractions (squares) p33 32 Equal fractions (triangles) p34 33 Equal fractions (strips) p35 34 Equal fractions (counters) p36 35 Equal fractions (clocks) p37 36 Equal fractions (graphs) p38

Fraction operations 37 Adding and subtracting (counters) p39 38 Adding and subtracting (strips) p40 39 Adding and subtracting (squares) p41 40 Adding and subtracting (triangles) p42 41 Multiplying fractions (counters) p43 42 Multiplying fractions (strips) p44 43 Multiplying fractions (squares) p45

Fractions and division 44 Fractions and dividing p46 45 Sharing pizzas evenly p47 46 Sharing money evenly p48 47 Steps p49 48 Remainders as fractions (counters) p50 49 Dividing by fractions p51

Decimals – place value 50 Decimals include tenths p52 51 Decimals include hundredths p53 52 Decimals include thousandths p54

Fractions & percentages 53 Percentage squares p55 54 Percentages and fractions p56 55 Bounce fractions and percentages p57

Fractions, decimals and percentages 56 Fractions and decimals (wall) p58 57 Fractions and decimals (ruler) p59 58 Fractions of a dollar p60 59 Guitar fractions and decimals p61 50 Decimals and percentages p62 61 Changing fractions to decimals p63 62 Changing decimals into fractions p64

Ratio 63 Whole number comparisons p65 64 Mixed number comparisons p66 65 Fraction comparisons & reciprocals p67

Proportion & percentage problems 66 Equal ratios p68 67 Recognising proportional situations p69 68 Solving proportion problems p70 69 Solving percentage problems p71 70 Percentage increases and decreases p73

Teach Fractions and decimals for understanding page

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Resources for learning The curriculum described in this section does not use textbooks. Instead it calls on the wealth of high quality learning resources that are available, mainly through MAV.

Lesson plans Maths300 This is available from MAV. There is an annual subscription, for a user name and password for on-line access to over 170 lesson plans, many with high quality associated software. RIME This collection of lesson plans is also available from MAV. There are three books in the series, also available on CD, with extra spreadsheets. Choose RIME (Measurement, Space, Chance & data). RIME 5&6: A set of RIME-style lessons written specifically for upper primary. From MAV.

Teaching advice Continuum Assessment for Common Misunderstandings Scaffolding Numeracy in the Middle Years

Problem solving Maths With Attitude For each content dimension and for VELS levels 3, 4, 5 and 6, this is a repackaging of the best Maths300 lessons and the best 20 problem solving tasks, with a useful guide. From MAV. Mathematics Task Centre This collection of problem solving tasks are available from Doug Williams www.blackdouglas.com.au/taskcentre, or as part of the Maths With Attitude kits, from MAV. Action Numeracy – Middle Primary The following stimulus books include material on fractions: Digging deep, Expeditions, The future of forests Action Numeracy – Upper Primary The following stimulus books include material on fractions, decimals or percentages: Bikes, Exploring space, The facts of living, Technology to the rescue, Water and food, Who wants to be a millionaire?

Worksheets Active Learning (Number & Algebra) This is a set of graded worksheets in a book from MAV. They are also available on a CD containing the contents of all three books in the series, plus extra worksheets describing how to use the hundreds of spreadsheets also on the CD. Active Learning 2 (Number & Algebra): More of the same in a book or a CD. Tuning in with task cards – lower, middle, upper primary Each book is a set of 150 workcards to guide students into hands-on activities. From the Curriculum Corporation.

Computers Interactive Learning One CD from MAV, containing hundreds of spreadsheets requiring no knowledge of Excel, and covering all levels and dimensions.Very useful for homework! Learning Objects (FUSE) Free high quality software from a large Federal Government project. Available to government schools from www.education.vic.edu.au/fuse (via a password) or from your CEO or AISV office.

Resources from MAV may be purchased with a credit card or school order number on-line,

using the MAV’s web site: www.mav.vic.edu.au/shop.

http://www.mav.vic.edu.au/cgi-bin/csv/search/new-member-csvsearch.pl?search=maths300+subscription&method=exact

http://www.mav.vic.edu.au/shop/members-price.html

http://www.mav.vic.edu.au/cgi-bin/csv/search/new-member-csvsearch.pl?search=RIME+for+Years+5+%26+6&method=exact

http://www.mav.vic.edu.au/cgi-bin/csv/search/new-member-csvsearch.pl?search=RIME%3A+CD+complete&method=exact

http://www.education.vic.gov.au/studentlearning/teachingresources/maths/mathscontinuum/number/default.htm

http://www.mav.vic.edu.au/cgi-bin/csv/search/new-member-csvsearch.pl?search=Maths+With+Attitude&method=exact

http://www.blackdouglas.com.au/taskcentre/iceberg.htm

http://www.mav.vic.edu.au/cgi-bin/csv/search/new-member-csvsearch.pl?search=Active+Learning&method=exact

http://www.mav.vic.edu.au/cgi-bin/csv/search/new-member-csvsearch.pl?search=Active+Learning+2&method=exact

http://www.mav.vic.edu.au/cgi-bin/csv/search/new-member-csvsearch.pl?search=Tuning+in&method=exact

http://www.mav.vic.edu.au/cgi-bin/csv/search/new-member-csvsearch.pl?search=Inter-active+learning&method=exact

https://fuse.education.vic.gov.au/

https://fuse.education.vic.gov.au/

http://www.blackdouglas.com.au/taskcentre/iceberg.htm

http://www.education.vic.gov.au/studentlearning/teachingresources/maths/common/

http://www.education.vic.gov.au/studentlearning/teachingresources/maths/snmy/default.htm

http://www.mav.vic.edu.au/cgi-bin/csv/search/new-member-csvsearch.pl?search=action+numeracy&method=exact


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1 Folding paper and naming fractions This introductory experience should be both enjoyable and successful. It will be successful if the students feel they have learned something about fractions. It is strongly recommended that you join the students in these introductory activities. As you go talk about the basic ideas of the fractions. • Discuss what is the ‘whole’ (the strip, square or rectangle, of length of string). • Count the equal pieces into which it is split (the denominator, but you probably won’t use that word).

Make sure they know how we say the word for that fraction. This is not so obvious before sixth, seventh, etc. From then on we just add ‘th’ to the word for the number of parts (tenth). But a half is not a ‘twoth; a third is not a ‘threeth’ and a quarter is not a ‘fourth’ though that is sometimes used. Remember that some children will need help with the special words we use in mathematics.

Suggested activities 1 Folding strips Cut strips of paper from blank A4 sheets, the long way (‘landscape’). Children fold these into halves, quarters, eighths, and label the parts when the strip is unfolded. 2 Folding squares Use more blank paper; some teachers prefer to use brightly coloured red or yellow kindergarten squares. (Indigenous colours are preferred where a choice exists.) Fold a rectangle diagonally as shown and cut off the extra. This makes a square.

• By folding in two perpendicular directions these can be folded into rectangles or squares.

• If they are folded at 45° the result is triangles. When it is opened out, students can colour in 34 etc.

3 Halving string How far is half way across the room? Students use a length of string that equals the width of the room. Fold it in half once to get half and check. Repeat with quarters or eighths of the initial length. In this way they mark the position of each fraction along the wall. (Some equal fractions will appear.)



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2 Circle fractions 1 It is strongly recommended that you join the students in these introductory activities. As you go talk about the basic ideas of the fractions. • Discuss what is the ‘whole’ (the strip, square or rectangle, of length of string). • Count the equal pieces into which it is split (the ‘denominator’, but you probably won’t use that word).

• Make sure they know how we say the word for that fraction. This is not so obvious before sixth, seventh, etc. From then on we just add ‘th’ to the word for the number of parts (tenth). But a half is not a‘twoth; a third is not a ‘threeth’ and a quarter is not a ‘fourth’ though that is sometimes used. Remember that some children will need help with the special words we use in mathematics.

Suggested activities Children need more than one variation on this theme in order to abstract from them all the essential idea of the fraction. In activity 1 we folded strips and squares. In activity 2 we subdivide circles by folding. It is recommended that you actually use circles, at this stage. Make flat circular shapes two for each child, and provide scissors to cut them. Please actually do the activity with the children. It makes all the difference. 1 Cutting halves, quarters and eighths Fold and cut your circles into halves first. Discuss the meaning. Then fold and cut each half into two parts and show that there are now four equal parts, called quarters. Discuss one quarter, two quarters and three quarters. Show that four quarters is the whole circle. Finally fold and cut each quarter to make eighths, and discuss this in the same way.

2 Cutting thirds, sixths and twelfths Follow the same procedure with thirds. (It is not so easy! Think clock: 12, 4 and 8 o’clock.) Discuss one, two and three thirds. Note that it doesn’t matter which you take. Divide each third to make sixths. Discuss similarly. Divide each sixth to make twelfths. You might note the similarity to a clock face.



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3 Tangrams The tangram is a famous spatial puzzle. It is claimed that it comes from China. The large square (10 cm by 10 cm) is cut into seven separate pieces, with these relative sizes.

Suggested activities • Students should first cut up the square puzzle into seven pieces.

• Find which ones cover the others. For example, the two small triangles cover the middle triangle, the square and also the parallelogram.

• Determine the fraction of the total for each part of the puzzle.

• Make a kangaroo, by putting together the pieces to form the same shape as the black silhouette.

Here is the solution.

1 1

1

1

1

1

1

4

4

8

8

8

16

16middle triangle

triangle

triangle

large

large

parallel

square

-ogra m



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4 Circle fractions 2 You will need circle pieces, all using the same radius – 2 halves, 3 thirds, 4 quarters, 6 sixths and 8 eighths. We are still developing the basic concept of a fraction. You should not be surprised if some children still do not understand yet. In the first three activities we were mainly concerned only with the denominator – the number of pieces into which the ‘whole’ was split. We divided strips, squares and circles into halves, quarters etc.

Suggested activities In the next few activities we are concerned equally with both numerator (that tells us how many of the given size there are) and the denominator (that tells us the size of each piece). • Have the children sort the shapes into similar piles. • Then have them put them together to form complete circles.

• They should name the fractions used to make up each circle: halves, thirds, quarters, sixths and eighths.

• Some equal fractions It is quite possible that children will notice that some different fraction names seem to show the same amount of the circles. For example 24 and 36 and 48 are all equal to 12 .



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5 Square fractions You will need square pieces – 2 halves, 3 thirds, 4 quarters (squares), 4 quarters (triangles), 6 sixths (rectangles), all using the same size of square, such as 10 cm square. This is clearly similar to the previous activity but uses the basic shape of a square. Here we are concerned with both numerator (that tells us how many of the given size there are) and the denominator (that tells us the size of each piece).

Suggested activities • Have the children sort the squares into similar piles.

• Then have them put them together to form complete squares of the same sized pieces.

• They should name the fractions used to make up each square: halves, thirds, quarters, sixths and eighths.

• Then have them put them together to form complete squares of differently sized pieces. There are many ways to do this. They should name the fractions used to make up each square: for example 1 half and 2 quarters.

• Using this approach they should soon be able to recognise equal fractions (that is, fraction names that have the same value).

• Children should make up outlines of fractions made from plastic pieces, and challenge others to name the fraction shown



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6 Triangle fractions You will need plastic triangle pieces – 2 halves, 3 thirds, 4 quarters (equilateral triangles), 6 sixths, 8 eighths. It is clear from the designs which ones belong together. They are cut from this design.

This is clearly similar to the previous activities but uses the basic shape of an equilateral triangle. Here we are concerned with both numerator (that tells us how many of the given size there are) and the denominator (that tells us the size of each piece).

Suggested activities • Have the children sort the triangles into similar piles.

• Then have them put them together to form complete triangles of the same sized pieces.

• They should name the fractions used to make up each triangles: halves, thirds, quarters, sixths and eighths. • Then have them put them together to form complete triangles of differently sized pieces. There are many

ways to do this. They should name the fractions used to make up each triangles: for example 1 half and 2 quarters.

• Using this approach they should soon be able to recognise equal fractions (that is, fraction names that have the same value).

• Children should make up outlines of fractions made from plastic pieces, and challenge others to name the fraction shown



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7 Strip fractions A set of strips (fraction wall – see next page) shows the fractions of halves, thirds, quarters, fifths, sixths, eighths, ninths, tenths and twelfths. These are used at this stage to reinforce the basic idea and to allow simple comparison of size by means of comparing length.

Suggested activities • Students should use the strips to find and compare pairs of fractions. e.g. 34 and 56 .

• Many students have the misconception that fractions with larger denominators must be bigger, when exactly the opposite is the case. Explore this using comparisons such as these: 23 and 24 and also 25 and 26 .

• There are very many chances for students to find equal fractions. This is not the main aim of this activity, but it is a major goal later, so progress towards that at this stage is welcome.

1 whole strip

1

2--

1

2--

1

4--

1

4--

1

4--

1

4--

1

8--

1

8--

1

8--

1

8--

1

8--

1

8--

1

8--

1

8--

1

3--

1

3--

1

3--

1

6--

1

6--

1

6--

1

6--

1

6--

1

6--

1

12-----

1

12-----

1

12-----

1

12-----

1

12-----

1

12-----

1

12-----

1

12-----

1

12-----

1

12-----

1

12-----

1

12-----

1

9--

1

9--

1

9--

1

9--

1

9--

1

9--

1

9--

1

9--

1

9--

1

5--

1

5--

1

5--

1

5--

1

5--

1

10-----

1

10-----

1

10-----

1

10-----

1

10-----

1

10-----

1

10-----

1

10-----

1

10-----

1

10-----



10

8 Number lines up to 1 For many students putting fractions on a number line is the first step to recognising fractions as numbers. Clearly the number line is closely related to the fraction wall above, however collapsing the strips onto a line is not conceptually an easy thing.

Suggested activities • Start with simple fractions: halves, quarters, eighths, and name them all. • Work with ‘separate families’, such as [thirds, sixths, twelfths], [fifths, tenths, twentieths]

• On the same line, combine the simpler fractions from different ‘families’, e.g. halves, thirds, sixths, fifths. This naturally leads to observations about equal fractions (e.g. 3 sixths is one half) or comparing (e.g. 2 fifths is less than one half).



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9 Fractions of a dozen You will need a lot of counters. It might be also useful to have some loops of string. We are still working on the basic idea: denominator tells us how many equal parts to split the whole thing into, and the numerator tells us how many of those equal parts to look at. This time we take a natural unit of number — the dozen — which happens to have many simple fractions. For this reason 12 was the basis for imperial measurement: for money (12 pennies made 1 shilling) and measurement (12 inches made 1 foot).

Suggested activities • Talk about dozens. Eggs will certainly come to mind, but we will use something more Australian, and less

breakable — counters.

• Give each child a lot of counters and a loop of string. Ask them to put one dozen (12) counters into the loop. Now get them to split the counters inside the loop into two parts. How many counters make one half? If necessary, demonstrate this.

• Continue in this way with many other fractions, getting the child to say the fractions. Use examples such as two thirds, three quarters, five sixths, etc. Include equal fractions, such as four sixths and two thirds.

• Ask the children to try to make one fifth of the dozen. Discuss with them why it is not possible. (Because 12 cannot be split evenly into a whole number of 5s; there is always 2 over.)

• Deal with comparisons, which should be based on the number of counters coloured. So, for example, 26 is

less than 24 because 26 has 4 counters coloured, but 24 has 6 counters coloured.

• Ask for those fractions that are equal, based only on the fact that they have the same number of counters coloured. For example, 36 and 24 both have 6 counters coloured.

• If time, extend to 20 counters in the ring. Find halves, quarters, fifths this time, and tenths. It will not be possible to make thirds or sixths.

• Use the 20 counter rings to compare these fractions. Please help students to actually do it.

25 or 12 1

5 or 14 510 or 25 3

4 or 35

510 or 34 2

5 or 24 35 or 7

10 45 or 34

• Write at least one other fraction equal to this one using the 20 counters only.

12 1

4 25 6

10



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10 Pattern block fractions This activity and the next (using cuisenaire rods) demonstrate the flexible use of materials. Mathematically the major idea is that the value of a fraction depends on what it is that represents the 1.

So one half of is quite different from one half of . You will need a set of Pattern Blocks. For the purposes of learning fractions it is best to remove the red squares and the tan (narrow) rhombuses (diamonds). You may need to borrow them from one of the junior levels of the school where they are frequently used for making geometric patterns. Their potential for representing fractions is less well known. It is best if each students has a small set, but that several students at the same table can share and compare.

Suggested activities • Let them make patterns first. This is good, because: (a) you won’t be able to stop them anyway! and (b) it

gets them familiar with the very relationships that will be needed to explore the fraction potential of the material.

• Ask: What shapes are half of other shapes? Find as many as you can and show them. For each one, show why it is half.

• What shapes are one-third of other shapes? Show two-thirds. • What shapes are one-sixth of other shapes? Show two-sixths, three-sixths, four-sixths, five sixths.

• We can make shapes to show 1 that involve more than one piece.

For example: Find all the fractions you can, based on this.

Find all the fractions you can, based on each of these.



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11 Rod fractions For this activity your students must use cuisenaire rods. As with the previous activity, this shows the importance of knowing what represents the 1 when you are finding a fraction.

Suggested activities • Let the children ‘play’ with the rods for a while. They will discover relationships as they do it, and can be

guaranteed to build a staircase!

• Ask: What rods are half of other rods? Find as many as you can and show them. For each one, show why it is half. (There are five: ‘1’ and ‘2’, ‘2’ and ‘4’, ‘3’ and ‘6’, ‘4’ and ‘8’, ‘5’ and ‘10’.)

• What rods are one-third of other rods? (There are three: ‘1’ and ‘3’, ‘2’ and ‘6’, ‘3’ and ‘9’.)

• Show two-thirds. (There are three: ‘2’ and ‘3’, ‘4’ and ‘6’, ‘6’ and ‘9’.)

• What rods are one-quarter of other rods? (There are two: ‘1’ and ‘4’, ‘2’ and ‘8’.) • Show two-quarters (‘12’ and ‘4’, ‘4’ and ‘8’). Show three-quarters (‘3’ and ‘4’, ‘6’ and ‘8’).

• Students search for pairs of rods which have other fraction relationships.

There are many unit fractions, such as 110 , 19 , 18 , etc, using the white rod and each of the others for the

larger. They will find all the thirds, quarters, fifths, sixths, sevenths, eighths, ninths and tenths.

Some of these appear several times, which suggests equal fractions. (24 = 36 etc.)



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12 Improper fractions (pizzas) Improper fractions, and their equivalent mixed numbers, are important ideas that are not well understood. Relating improper fractions and mixed numbers is not an easy idea for many children. There is a great emphasis placed on a fraction being less than the ‘whole thing’. So the idea of fractions greater than the ‘whole thing’ is rather a surprise, at least at first. This deals with that idea by defining the ‘whole thing’ as a pizza. Children are familiar with having many pizzas, and fractions of pizzas are familiar. ‘Mixed numbers’ combine whole numbers and a fraction.

Suggested activities • Use paper or card circles for pizzas. Initially at least this is better than drawing. Deal with halves.

12 , 1, 11

2 , 2, 212 , 3, 31

2 etc.

• Convert any mixed number (with halves) into a number of halves by cutting the circles into halves. So 312

will have 7 halves, written as 72 .

• Repeat with quarters, e.g. 214 is 9 quarters. Repeat with other denominators (thirds, sixths, etc.)

• So that they all understand the idea, do not try to teach any kind of rule at this stage. If a student figures it out for themselves get them to tell you in their own words.



15

13 Improper fractions (halves) This deals with that idea by defining the ‘whole thing’ as the number of counters that will fit into an outline. In this activity we use pairs, so the ‘unit’ is 2, so each counter is one-half. This is all that is needed to see the improper fraction. For example, if there are 7 counters, each worth 12 , there are 7 halves, which we can write as 72 .

Students can put the counters into the outlines, and there will possibly be some left over. In the case of 7 counters, there will be three outlines filled and one tile over. Students will easily see that this represents 3 whole units, but calling the one left over is not always so obvious.

1

23

7

2=

This is where the idea of one thing as a fraction of another comes in. The one counter left over is 12 of the pair of

counters that we called the ‘whole thing’ in the outline. So 72 is the same as 312 .

You will need a collection of large counters. Instead of the paper outlines you could cut an egg carton into six pieces so that you have natural sets of two to define the unit.

Suggested activities • Use counters and outlines. Demonstrate that the outline (on paper or with the egg carton) will hold two

counters and so defines the 1.

• Go through the process of adding one counter at a time, saying the improper fraction, putting them into the outlines, and naming the mixed number (whole number part and fraction part).

12 , 1, 11

2 , 2, 212 , 3, 31

2 etc.




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14 Improper fractions (quarters) This time we deal with quarters. For example if the outline holds 4, each counter will be 12 . Then students can put the counters into the outlines and there will be some which will not fill an outline. These will be the fraction part of the mixed number (e.g. 12 ). Students can also say how many quarters they used, in this case 15.

3

43

15

4=

You will need a collection of large counters. (An alternative is plastic one-inch squares.) Instead of the paper outlines you could cut an egg carton into three pieces so that you have natural sets of four to define the unit.

Suggested activities • Use counters and outlines. Demonstrate that the outline (on paper or with the egg carton) will hold four



• Some students will realise that two-quarters is the same as one-half. This is to be encouraged. Have a student who realises this explain it to the others.

14 , 24 or 12 , 34 , 1, 11

4 , 124 or 11

2 , 134 , 2, 21

4 , 224 or 21

2 , 234 , 3 etc.




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15 Improper fractions (thirds) Use counters and outlines. (Outlines could be parts of an egg carton cut into four equal parts, so there are three spots to each part.) The outline will hold three counters and so defines the 1.

Suggested activities • Use counters and outlines. Demonstrate that the outline (on paper or with the egg carton) will hold three



13 , 23 , 1, 11

3 , 123 , 2, 21

3 , 223 , 3 etc.




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16 Improper fractions with pattern blocks This activity continues the idea of naming improper fractions as mixed numbers.

Suggested activities • Use the rhombus as the 1. One at a time, add each small triangle, getting the students to name the

improper fraction and the mixed number.

12 , 1, 11

2 , 2, 212 , 3, 31

2 etc.

• Use the hexagon as the 1. One at a time, add each trapezium (half hexagon), getting the students to name the improper fraction and the mixed number.

12 , 1, 11

2 , 2, 212 , 3, 31

2 etc.

• Use the hexagon as the 1. One at a time, add each rhombus (diamond), getting the students to name the improper fraction and the mixed number.

13 , 23 , 1, 11

3 , 123 , 2, 21

3 , 223 , 3 etc.

• Use the hexagon as the 1. One at a time, add small triangle, getting the students to name the improper fraction and the mixed number.

16 , 26 or 13 , 36 or 12 , 46 or 23 , 56 , 1, 11

6 , 1226 or 11

3 , 136 or 11

2 , 146 or 12

3 , 156 , 2, etc.



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17 Improper fractions with rods This activity continues to develop the ideas of improper fractions and mixed numbers. You need enough rods for each child at the table to be able to work alone, comparing results with others.

Suggested activities • Choose the small red rod as the unit. In this case, some answers will include halves.

Add many white (1) cuisenaire rods, one at a time. Say the name as both improper fractions and mixed numbers The answers will be 12 , 1, 11

2 , 2, 212 , 3, 31

2 etc.

• Choose the small light green rod as the unit. In this case, some answers will include thirds. Add many white (1) cuisenaire rods, one at a time. Say the name as both improper fractions and mixed numbers. The answers will be 13 , 23 , 1, 11

3 , 123 , 2, 21

3 , 223 , 3, 31

3 etc.

Choose the small pink rod as the unit. In this case, some answers will include quarters (and maybe halves). Add many white (1) cuisenaire rods, one at a time. Say the name as both improper fractions and mixed

numbers. The answers will be 14 , 24 or 12 , 34 , 1, 11

4 , 124 or 11

2 , 134 , 2, 21

4 , 224 or 21

2 , 234 , 3 etc.

• For an extra challenge, students should choose any two different rods. They make the smaller one the unit (1). Then they express the larger one as a mixed number. (Some students may need to replace the larger rod by small white ones to see the link to the previous work.), For example, if you choose black (8) and dark green (6), the black is 12

6 (and also 113 ).



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18 Mixed numbers on number lines For a proper understanding of fractions, and mixed numbers in particular, it is most important that students spend time working with number lines. The aim of this activity is to get familiar and comfortable with the positions of mixed numbers on number lines, and introducing ‘skip counting’. For this activity it would be good to have many spare copies of the number lines. Suggested problems for you to use are given below.

Suggested activities • Hand out copies of number lines: one marked in halves, one in thirds, one in quarters etc. Discuss the

pattern with students.

1 The first line has whole numbers every two marks, to show halves. Talk about finding 112 etc.

2 The second line has whole numbers every three marks, to show thirds. Talk about finding 113 etc.

The pattern continues. Make sure students can tell you about it.

3 On the first line, put a dot at the first mark after 0 and write its name. (It is 12 .)

4 Continue to write the names of the positions every two marks along. They are 112 , 21

2 , 312 etc.

5 On the second line, put a dot at the first mark after 0 and write its name. (It is 13 .)

Continue to write the names of the positions every two marks along.

They are 1, 123 , 21

3 , 3, 323 , 41

3 , 5, etc

6 Continue in this way for the other lines. 7 Repeat for every third mark on each of the number lines.

• After some fluency begins, students could be asked to speak the names of the mixed numbers obtained by ‘skip-counting’ in this way.

• Mark a large-scale number line outdoors. The marks can be one step-length apart. Children should choose the fraction they are using (e.g. quarters) and write the whole numbers into position on the line with chalk. They then walk up and down the line saying the position they are in at each step.



21

19 Mixed numbers – dice game This game is designed to give children some greater appreciation of the positions of fractions and mixed numbers on the number line. You will need one die. Each two or three children need one copy of a sheet with parallel number lines from 0 to 3. They are marked in halves, thirds, quarters, fifths and sixths. Each child needs a marker which is small enough to show a position on the number line.

0 1 2 3

0 1 2 3

0 1 2 3

0 1 2 3

0 1 2 3

Suggested activities • In turn each child rolls the die twice. The first roll is the numerator, and the second is the denominator. (If

the second roll is a 1, the child should roll again.) The child then finds that position on the appropriate number line on the sheet. If the numerator is larger than the denominator, the child will have to convert the fraction to a mixed number, for example 32 becomes 11

2 .

• When all the children have rolled the die and placed their markers, the child with the largest number wins. This is easily found by seeing which is further from 0 on the number lines. (There are some equal fractions, such as 12 and 36 . If two children have equal fractions and they are both winners, then those children only roll the die twice again, and choose the winner from that ‘play-off’.)

• Extension: Each child rolls the die three times, and adds the first two rolls to make the numerator. This will produce larger mixed numbers. Some of these will fall on the unlabeled part of the line, and children will have to work out the position for themselves, with the others checking! Others will be so large that they do not fit onto the page. For example rolls of 6, 5 and 2 gives 11

2 and therefore the mixed number is 512 . This

will mean that children will have to find their own methods of comparing and proving to the others that they have won!



22

20 Fractions of whole numbers The most common use of fractions is finding fractions of whole numbers. It is at this point that you will start to find out who really does understand the ideas and who still needs at lot of help. Each student needs a set of counters.

Suggested activities • Start with 6 counters. Ask the child to split them into thirds. (There may be some children who will react by

splitting the 6 counters into 3s. Be patient!)

• Discuss how to show one-third (2 counters), and two-thirds (4 counters). Discuss the meaning of three-

thirds (all 6 counters).

• The activity asks students to use 12 counters. It is quite important for most children that they actually use objects, e.g. counters. Students colour in the circles to show the selected counters. There are two ways to conceptualise this question:

- For example, for 23 of 12 counters share into three equal parts, and then choose (colour) two of those parts;

- For example, for 23 of 12 cookies organize them into groups of 3, and choose (colour) 2 out of every 3 circles.

• They find all the halves, thirds, quarters, sixths and twelfths. This also reveals that there are many equal fractions. Hopefully they will discover this for themselves.

• Use array models. For example for 6 counters, use two rows of 3 to represent one unit. Then finding 12

should be easy and finding 13 or 23 should be easy too.

• For some children you might be able to ask if they can see a short cut. All children should find their way to this eventually, but do not rush them; it is best if it is understood, or even discovered by the child.

• When they split 12 counters into 6 equal parts (for sixths) they are dividing 12 by 6. Then when they select 5 of those parts, they are multiplying by 5.

• Discuss that you can ‘multiply first then divide’ or ‘divide first then multiply’.

(For example, 56 of 12 can be 12 x 5 ÷ 6 = 60 ÷ 6 = 10 or 12 ÷ 6 x 5 = 2 x 5 = 10.)



23

21 Guessing and checking fractions Teach for understanding This set of activities aim to develop further the ideas of finding fractions of whole numbers. This is done by dividing a length into a number of equal parts and then choosing some of those parts. There are two ways of doing this. One is estimation. This means that the child looks at the line, mentally divides it into equal parts, and then chooses a number of those parts. This then gives the position of the fraction of the whole line. For example, to find 23 of a length, the child imagines it split into three equal parts, and then puts a mark at the end of two of those thirds. Estimation also motivates the calculation part for the next step. This is to measure the length of the line (in centimetres) and divide this number into so many equal parts, and then multiply to get the number of centimetres in the required fraction. For example, the child measures a line 60 cm long. Then to find 23 of 60 cm, the child divides 60 by 3 (20 cm for each third) and multiplies by 2 (40 cm for 23 .)

Then a ruler or tape is used to find the correct position of the 23 and compare it with the estimation.

You will need a strip of blank tape or length of string (or heavy cord). A good length is 120 cm, since this can be divided by 2, 3, 4, 5, 6, 8 or 10. It is useful to have a few clothes pegs to act as markers. It would also be useful to have a measuring tape (longer than a 30 cm ruler) such as a 150 cm sewing tape.

Suggested activities • Two students hold the tape tight, and a third student places a clothes peg at the point they think is 23 from

one end. They then measure the tape, calculate the fraction of its length and compare the position of the peg with the ‘correct’ answer. (The process is discussed above.)

• Repeat the activity with a wide variety of fractions, such as one-half, all the thirds, all the quarters, all the fifths, all the sixths, all the eighths and all the tenths. (You can go even further if you also let the children use calculators to divide.)

• You can extend the activity easily by asking child to estimate, say, two thirds of the distance across the room. After measuring the distance, the child will need to use a calculator to divide by 3 and multiply by 2. This provides excellent conceptual development (through the estimation) and links it to the computation process (through the use of the calculator).



24 22 Clock face fractions

This is another chance to practise finding fractions of whole numbers. In this case the whole number is always 12, and the application is a very real and useful one – fractions of one clock face of 12 hours. It would be useful if you had a real clock face or one designed for teaching time.

Suggested activities • Use the page of clock faces (above). Students find the number of hours that match certain fractions of the

clock face. Start with halves and quarters; these will be familiar because of common usage.

• Thirds are more difficult. Divide the clock face into three equal parts, each 4 hours. So 23 is 8 hours. Try to get the children to see that they can divide 12 by 3 then multiply by 2.

• Continue with sixths, eighths and twelfths, by getting the fraction of 12.



25

23 Fractions of an hour This is another chance to practise finding fractions of whole numbers. In this case the whole number is always 12, and the application is a very real and useful one – fractions of one clock face of 12 hours. It would be useful if you had a real clock face or one designed for teaching time.

Suggested activities • Use the page of clock faces (below). Students find the number of minutes that match certain fractions of one

hour. Start with halves and quarters; these will be easy because of common usage.

• Thirds are more difficult. Divide the 60 minutes into three equal parts, each 20 minutes. So 23 is 40 minutes. Try to get the children to see that they can divide 60 by 3 then multiply by 2.

• Continue with sixths, fifths and tenths, by getting the fraction of 60.



26

24 Fractions of circles Finding fractions of whole numbers is relatively easy when the denominator divides exactly into the whole number, such as 23 of 6. It is much more difficult when the denominator does not divide exactly into the whole

number, such as 23 of 5. This activity works with this idea using small numbers of circles. (Some students may be familiar with the ways that pizzas are cut. This will help.) Each student should have access to at least three sets of circles – divided into thirds, quarters, sixths, eighths and twelfths (see below). Organise these into complete sets, so they may be used for the problems.

Suggested activities • Have students find fractions of two plastic circles where the answer is a fraction or mixed number.

Start with 34 of 2. Because of the quarters we choose two circles divided into four parts.

There are 8 quarters, so 34 of 2 circles will be of 8 quarters = 6 quarters. This is 112 circles.

• There is another way to find this: get three quarters of each circle and add the results.

Either way you should get 34 of 2 is 6 quarters or 112 .

• Continue with 23 of 2. Because of the thirds, we choose two circles divided into three parts.

There are six thirds, so 23 of 6 thirds is 4 thirds. (Alternative: get 23 of each circle and add.)

Either way, this is 113 .

• Continue with 58 of 2. Because of the eighths, we choose two circles divided into eight parts.

There are 16 eighths, so 58 of 16 eighths is 108 which is 12

8 or 114 .



27

25 Fractions of rods A similar activity can be done with number lines. A number of units is selected and we get a fraction of it. It is necessary to choose a line divided into suitable parts.

Here is an example, using a 15 cm line. 13 of 5 is 123 .

0 1 2 3 4 52

31 1

33

1 third 1 third 1 third Suggested activities • Use Cuisenaire rods as units to make this clearer. They are useful because they have width and therefore

unit lengths of 1 cm. Demonstrate

• For 13 of 5 we can choose light green rods of length 3 units. We choose this length because it may be split

into 3 equal parts. We need five of them. Here they are for 13 of 5, placed end to end along a ruler – to a

length of 15 cm. You can see that one third of the total length is 123 .

0 1 2 3 4 523

1 13

3

1 third 1 third 1 third

• Instead of a ruler you may replace the light green rods with white rods (cubes of length 1) to show how the splits occur. Again, one third of the total length is 12

3 .

0 1 2 3 4 52

31 1

33

1 third 1 third 1 third

The general idea is to choose a rod for the unit whose length may be divided by the denominator. Students divide the total length of a number of rods (laid out end to end) by the denominator and multiply by the numerator.

• Discuss another example. It uses the yellow rods (5), because we are making fifths. 45 of 3 splits into 15 parts. So it is 45 of 15 fifths, which is 12 fifths or 22

5 .

1 fifth 1 f ifth 1 fifth 1 fifth 1 fifth



28

26 Comparing fractions (strips) An important part of understanding fractions is to be able to compare their sizes. This also improves understanding of the roles of denominator (the number of equal parts of the whole) and the numerator (how many of those equal parts are used in the fraction). Students can use fraction strips to compare fractions. This activity has three different comparisons. It is important that students understand, particularly the second type.

• Type 1: with the same denominator. Clearly the larger numerator shows the bigger fraction. • Type 2: with the same numerator. Not so clearly, the smaller denominator shows the bigger fraction. This is a surprise to many students, as they focus on the larger number (in the denominator) and think at first that this makes the fraction larger. In fact, a larger denominator means that the whole is split into more equal parts, so each part is smaller.

For example 23 is bigger than 24 . The larger the denominator, the smaller the fraction, if the numerators are the same. • Type 3: when one fraction has both numerator and denominator increased by the same number to form a second fraction, the larger numbers show the bigger fraction.

For example, 23 is bigger than 12 , and 34 is bigger than 23 . The increased fraction is always bigger.

The fraction strips are an ideal way for the students to learn this. They all have the same length but are clearly split into equal parts.

Suggested activities • Find and compare the sixths. 5 sixths is longer than 4 sixths, is longer than 3 sixths etc.

• Now compare one sixth and one quarter. If you write the fractions (16 and 14 ) you will find some students who

think 16 is bigger, thinking ‘6 is bigger than 4’. Let them use the strips to compare.

1

6

1

4

• Discuss why 14 is actually larger than 16 .

• Try one with a numerator of 2: compare 23 and 25 . Discuss why 23 is larger than 25 . Try to get a general understanding expressed by the children. Try other examples of two fractions with the same numerator to help them generalise the rule.

• The last step is to compare fractions in which both numerator and denominator of one are increased by 1 to form the other fraction. The fraction with the increased numbers will be larger, providing that the original fraction is less than 1.

In fact this applies to increasing numerator and denominator by any (same) number.

For example, if 23 is changed to 45 then the increased fraction (45 ) will be greater, providing that the original

fraction is less than 1. Note: If the original fraction is greater than 1, the increased fraction is the lesser. A more general rule is that the increased fraction will always be closer to 1.



29

27 Comparing fractions (counters) In this activity students compare two fractions by finding fractions of the same larger whole number and comparing the results. For example, by using 12, 34 of 12 (9) is bigger than 23 of 12 (8).

This leads directly to using the same denominator as a method for comparing, adding and subtracting:

So 34 = 912 and 23 = 8

12 . It will also have the advantage of reviewing ‘fractions of whole numbers’ which was done earlier. Any children who were absent, or who didn’t quite understand earlier should be given extra help at this stage.

Suggested activities • Each child has a small pile of counters.

Put 4 counters aside and call them 1. (Make sure it is understood that the set of four is the unit, 1.) So each counter shows 14 .

Now ask one child to make 12 and another child to make 34 . Ask: Which is bigger, 12 or 34 ?

• Now make 6 counters represent 1. So each counter shows 16 .

Now ask one child to make 12 and another child to make 23 .

Ask: Which is bigger, 12 or 23 ? Which is bigger, 56 or 23 ?

• Which is bigger, 12 or 25 ? Why is this not possible with 6 counters? (Because we cannot make fifths.) How many counters should represent 1? Why? (10, because we can make both half and fifths.) Which is bigger, 12 or 12 ? Which is bigger, 12 or 12 ?

• How many counters should represent 1if we want to work out which is bigger, 12 or 12 ?

(We use 12, so we can make quarters and thirds.)

Which is bigger, 34 or 23 ? Which is bigger, 34 or 23 ?

• How many counters should represent 1 if we want to work out which is bigger, 34 or 56 ?

(We use 12, so we can make quarters and sixths. Note that 24 will also work, but 12 is easier.)




30

28 Comparing fractions (circles) This model of fractions may be used to make comparisons. You will need circle pieces, one set for each student. A complete set has a circle made from each of these fractions: 14 , 16 , 18 , 1

10 , 112 .

Suggested activities • Each child makes a circle using quarters. Now ask one child to make 12 and another child to make 34 . Ask:

Which is bigger, 12 or 34 ?

• Each child makes a circle using sixths.

Now ask one child to make 12 and another child to make 23 . Ask: Which is bigger, 12 or 23 ?

Another question: Which is bigger, 56 or 23 ?

• Which is bigger, 12 or 25 ? Why is this not possible with a circle in sixths? (Because we cannot make fifths.) Which circle should we use? Why? (10 because we can make both half and fifths.) Which is bigger, 12 or 25 ? Which is bigger, 7

10 or 45 ?

• Which circle should we use if we want to work out which is bigger, 34 or 23 ?

(We use the circle in twelfths, so we can make quarters and thirds.)


• Which circle should we use if we want to work out which is bigger, 34 or 56 ? (We use the circle in twelfths, so we can make quarters and sixths. Note that 24 will also work, but 12 is easier.) Which is bigger, 34 or 56 ? Which is bigger, 14 or 16 ?



31

29 Rectangle shapes This activity is about the shapes of rectangles. Although they are all rectangles (even the squares!) some are tall and thin, others are short and fat, and so on. Rectangles that are ‘closer to squares’ have heights more nearly equal to their widths.

Rectangles that are the same shape will have equal fractions formed with heightwidth .

The fraction heightwidth is closer to 1 the closer the rectangle is to a square.

Here is an example. For these rectangles, height is less than width.

The rectangle on the right is closer to a square, so fraction 35 is bigger than the fraction 24 .

2 cm

4 cm

3 cm

5 cmFraction 2

4

Fraction 3

5

Each rectangle has one diagonal drawn as well. Rectangles that have the same shape will have diagonals with the same slope.

Larger fractions have steeper slopes for their diagonals. For example the diagonal for 35 is steeper than the

diagonal for 12 .

Suggested activities • Explore rectangles wherever you can find them.

• The activity with rectangles should now be extended to real rectangles around the room: doors, cupboards, blackboard, paper, posters, books, tables, floor, walls, etc.



32

30 Equal fractions (circles) This is the first of seven activities on equal fractions. Confidence with this topic is very useful, both for further work with fractions, such as adding and subtracting, and with real life situations, where equal fractions may be used to solve ratio problems. Equal fractions have been met before on many occasions. These activities formalise the idea, through the comparison of several models: circles, squares, triangles, strips, whole numbers, clocks and rectangles. Each student will need circle pieces, to make complete circles in halves, thirds, quarters, fifths, sixths, eighths, tenths and twelfths.

Suggested activities • Ask someone to show you 16 of a complete circle. Write the fraction. Now find two equal pieces that can

take its place. This shows that 16 = 212 . Write this out. Ask the students to look for patterns.

• Ask someone to show you 56 of a complete circle. Write the fraction. Now find equal pieces that can take its

place. This shows that 56 = 1012 . Write this out. Tell the students to look for patterns.

• Ask someone to show you 15 of a complete circle. Write the fraction. Now find two equal pieces that can take

its place. This shows that 15 = 210 . Write this out. Remind the students to look for patterns.

• After working through the problems, the pattern should be fairly obvious.

Have kids tell you how to create a fraction equal to another one. • Discuss the idea of reducing fractions to the lowest numbers: ‘simplest terms’. For example, make the

fraction 68 . Now find a way to make the same amount of circle with fewer equal pieces. You can use three

quarter pieces, so 68 = 34 .

• Use equal fractions to compare pairs of fractions.

For example, 56 is bigger than 34 , because 1012 is bigger than 9

12 . Check with the circles.



33

31 Equal fractions (squares) Each student will need square and rectangle pieces, to make complete squares in halves, thirds, quarters, fifths, sixths, eighths, tenths and twelfths.

Suggested activities • Ask someone to show you 16 of a complete square. Write the fraction.

Now find two equal pieces that can take its place. This shows that 16 = 212 . Write this out.

Tell the students to look for patterns.

• Ask someone to show you 46 of a complete square. Write the fraction.

Now find fewer equal pieces that can take its place. This shows that 46 = 23 . Write this out.

Tell the students to look for patterns. • After working through some problems, a pattern should be fairly obvious.

Have kids tell you how to create a fraction equal to another one. Discuss the idea of reducing fractions to the lowest numbers: ‘simplest terms’.

For example, make the fraction 68 .

Now find a way to make the same amount of square with fewer equal pieces. You can use three quarter pieces, so 68 = 34 .



12 . Check with the square pieces.



34

32 Equal fractions with equilateral triangles Each student may need the triangle pieces, to make complete triangles in halves, thirds (two ways), quarters, sixths and eighths. The diagram uses sixths, eighths and twenty fourths.

Suggested activities • Ask someone to show you 14 of a complete triangle. Write the fraction.

Now find two equal pieces that can take its place. This shows that 14 = 28 . Write this out. Tell the students to look for patterns.

• Ask someone to show you 13 of a complete triangle. Write the fraction.

Now find equal pieces that can take its place. This shows that 26 = 13 . Write this out. Tell the students to look for patterns.

• After working through some problems, the pattern should be fairly obvious. Have kids tell you how to create a fraction equal to another one.

• Discuss the idea of reducing fractions to the lowest numbers: ‘simplest terms’. For example, make the fraction 68 . Now find a way to make the same amount of square with fewer equal

pieces. You can use three quarter pieces, so 68 = 34 .



12 . Check with the triangle pieces.



35

33 Equal fractions with strips Use the fraction strips. These have the same length split into these numbers of equal parts: 2, 3, 4, 5, 6, 8, 9, 10, 12. They therefore display very many equal fractions. The focus in this activity is more on the ‘families of equal fractions’. The members of the same family are all equal to each other, so we can recognise equality if we can reduce them to the simplest fraction in the family. For example, 69 and 10

15 both belong to the family 23 .

Suggested activities • Find all the fractions the same length as 12 , and so on.

• Compare the fractions 28 and 312 . Putting the eighths and twelfths side by side will make it obvious that they

are the same length.

1

8---

2

8---

3

8---

4

8---

5

8---

6

8---

7

8---

1

12------

2

12------

3

12------

4

12------

5

12------

6

12------

7

12------

8

12------

9

12------

10

12------

11

12------

• Why are they the same? Let the children discuss it until they realise that both are equal to 14 .

They are ‘in the same family’.

• Why are 48 and 612 equal? (Which family are they in?)

Why are 68 and 912 equal?

• Use the sixths and ninths to ask: Why are 46 and 69 equal?



36

34 Equal fractions with counters Equal fractions have been met before on many occasions. These activities formalise the idea, through the comparison of several models: circles, squares, triangles, strips, whole numbers, clocks and rectangles. The emphasis in this activity is firmly on reducing a fraction to ‘lowest terms’ and possibly using this to then find another fraction from the same family that is equal to the first (and to the second). Each student should have access to a large number of counters (or equivalent counters). This enables students to find equal fractions using fractions of whole numbers.

Suggested activities • Make a pile of ten counters. Find 4

10 of it (4). What simpler fraction is equal to this one?

• (The counters can be grouped into 2s, showing that 4 counters is 25 of 10.)

• Make a set of 18 counters. Make a small pile from 69 of them (12).

What simpler fraction is equal to this one?

(The counters can be grouped into 6s, showing that 12 counters is 23 of 18.)

• Is the 46 of 18 the same amount? (Yes.) What simpler fraction is equal to this one? (Also 23 , obtained by grouping in 3s.)



37

35 Equal fractions (clocks) This is the fifth of seven activities on equal fractions. Confidence with this topic is very useful, both for further work with fractions, such as adding and subtracting, and with real life situations, where equal fractions may be used to solve ratio problems. As well as providing further reinforcement for the basic idea of equal fractions, this activity offers practice in fractions of 60, and in many fractions of an hour.

Suggested activities • Use the clocks and find equal fractions by finding minutes for many fractions. Use the spreadsheet.

• Here is one example. 40 minutes is 23 of 60 and also 46 of 60. It is also equal to 812 of 60.



38

36 Equal fractions (graphs) This is the last of the activities on equal fractions. Students who get this far are expected to be ready for an extended challenge. The spreadsheet introduces the idea of using graph coordinates to represent a fraction.

The numerator is the distance up (‘rise’) The denominator is the distance right (‘run’)

Then the slope of the line is the size of the fraction: numeratordenominator , sometimes called rise

run .

Finding equal fractions then becomes a matter of looking for points that lie along a line from the bottom left corner.



39

37 Adding and subtracting (pizzas) This activity provides a gentle introduction to adding and subtracting, by way of stories about children eating pizzas. The adding comes from finding ‘how much pizza is eaten’. The subtracting comes from finding ‘how much more one person’s share is than another’s’ and ‘how much is left from the whole pizzas’. At this stage it is strongly recommended that children do all problems using concrete materials, or by colouring diagrams. Do not press for short cuts, or mention common denominators, or make other moves towards a final algorithm. Great damage can be done by pushing too early for these pen and paper methods when a bit of common sense and thinking will help children understand what is going on. Many fraction problems can be grasped and solved mentally if a good mental image is used. Concentrate on building this image of what it all means. In this activity we use circles. You will need several sets of the plastic circles.

Suggested activities • Make up a story about two of them eating pizzas. One eats 14 , one eats 12 .

Make the fractions with pizza pieces in quarters. There are three questions: Who ate the most? How much more did this person eat? How much did they eat in total?

They only bought one pizza. Note that after taking away the 14 , the 12 means half of the whole pizza, not 12 of what was left! How much was left at the end?

• Two other children were given three pizzas to share. One child ate 34 , one ate 112 .

Make the fractions with pizza pieces in quarters. Who ate the most? How much more did this person eat? How much did they eat in total? How much was left from the three pizzas?

• Show two other fractions. Ask the children to make up stories, and answer the same three questions. (These problems will need circles in sixths. Equal fractions (halves or thirds to sixths) may be required. These should occur easily at this stage after the extensive work on this concept.)

23 and 56 from two pizzas. 11

3 and 56 from three pizzas. 23 and 116 from two pizzas.

• Let the children try the activity when they are ready to succeed.



40

38 Adding and subtracting (strips) Students will need more than one way of thinking about it if they are to create their own mental abstraction of the process. For this work, students should have their own set of fraction strips. There are important ideas here. Please make sure they understand what is going on through the activity.

Suggested activities • First, use common denominators. For example, using tenths, try these additions and subtractions.

310 + 1

10 710 + 1

10 310 – 1

10 710 – 1

10

For adding, count along the strip, for example 3 tenths, and then 1 more. The answer is 4 tenths. Now we look for another fraction of the same length; 4 tenths is the same as 3 fifths. (Place them side-by-side to check.) For subtracting, count along the strip for the first number then count backwards.

• Try 12 + 14 . Discuss the fact that the first fraction is about a strip divided into 2 parts, but the second fraction is about a strip divided into 4 parts. What can we do?

We change 12 into 24 . These are equal fractions. (Put the strips side-by-side to show this.)

Then 12 + 14 becomes 24 + 14 , which is 3

4 .

• Try 59 – 13 . Discuss the fact that the first fraction is about a strip divided into 9 parts, but the second fraction is about a strip divided into 3 parts. What can we do?

We change 13 into 39 . These are equal fractions. (Put the strips side-by-side to show this.)

Then 59 – 13 becomes 59 – 39 , which is 29 .

• Provide more examples if needed.



41

39 Adding and subtracting (squares) Students will need more than one way of thinking about it if they are to create their own mental abstraction of the process. For this work, students should have their own set of plastic squares. There are important ideas here. Make sure they understand what is going on before you let them try the activity. At this stage we go directly to the problems with mixed denominators. You should use several examples to make sure children have the ideas.

Suggested activities • Try to add 12 and 13 by putting them next to one another.

The problem is not in the adding, but in finding the proper name for the fractions they produce. The picture shows that you can swap 12 for 36 and 13 for 26 . This makes naming the total easy.

• Try to subtract 512 from 34 by putting the 5

12 on top of 34 and trying to name the rest.

The problem is not in the subtracting, but in finding the proper name for the fraction left. The picture shows that you can swap 34 for 9

12 . This means we are taking 5 twelfths from 9 twelfths, and makes the fraction left 4 twelfths.

This time we can see that 412 is the same size as 13 .

• At this stage we are focusing entirely on the understanding. The ideas about common denominators and

equal fractions are what is essential. Please do not attempt to teach a routine method for solving this kind of problem to any child until you are absolutely sure that the children will clearly understand the reasons behind each step.



42

40 Adding and subtracting (triangles) This is the fourth activity on adding and subtracting. Students will need more than one way of thinking about it if they are to create their own mental abstraction of the process. For this work, students should have their own set of plastic triangles. There are important ideas here. Make sure they understand what is going on before you let them try the activity.

Suggested activities • The first two questions use common denominators. Try a few of these with triangles that you make up at the

time. For example, using twelfths, try these additions and subtractions.

512 + 1

12 712 + 1

12 512 – 1

12 712 – 1

12

Show how each of these answers can be replaced by a simpler fraction: 12 , 23 , 13 and 12 .

• Try 12 + 14 . What can we do?

We change 12 into 24 . These are equal fractions. (Two quarter-triangles fit on top of the half-triangle.) Then 12 + 14 becomes 24 + 14 , which is 34 .

• Try 12 – 13 . What can we do?

We change 12 and 13 into 36 and 26 . These are equal fractions. (Show these.)

Then 12 – 13 becomes 36 – 26 , which is 16 .

• Provide more examples if needed.



43

41 Multiplying fractions (pizzas) Although the activity is called ‘Multiplying’ we are concentrating on the meaning of ‘x’, which is ‘of’.

• The meaning of 12 of 23 is simple.

We make the 23 first, then find 12 of it. In this case the fraction was easy to find, without using equal

fractions. It is 13 .

• The meaning of 23 of 12 is also simple.

We make the 12 first, then find 23 of it. This time we need to convert to sixths (equal fractions) to name the

answer. It is also 13 .

• These two examples also show another idea: 12 of 23 is the same as 23 of 12 .

The questions give the same answer in either order. The students you are teaching should have their own plastic circles to help them work out the answers. Remember to focus entirely on the main idea, which is in three parts.

• Make the second fraction. • Find the first fraction of it. • Name the answer.

You should start with some very simple examples that focus clearly on the idea of ‘of’.

Suggested activities • Show 12 . Find a piece that is half of it. (It is 14 .) This shows that 12 of 12 is 14 . This is how it is written.

• Show 14 . Find a piece that is half of it. (It is 18 .) This shows that 12 of 14 is 18 .



• Show 12 . Find three-quarters of it. Change the 12 for 48 . Then 34 of 12 is 38 .

• Show 12 . Find two-thirds of it. Change the 12 for 36 . Then 23 of 12 is 26 or 13 .

• Students try more problems of this type, using their plastic circles. Make sure they can be done using the circles.

The idea that the answer is the same when the fractions are reversed might be left as a discovery!



44

42 Multiplying fractions (strips) This is the second activity on the meaning of ‘x’, which is ‘of’.

• The meaning of 12 of 23 is simple. We make the 23 first, then find 12 of it.

whole strip

In this case the fraction was found without using equal fractions. It is 13 .

• The meaning of 23 of 12 is also simple.

whol e strip

We make the 12 first, then find 23 of it. This time we need to convert to sixths (equal fractions) to name the

answer. It is also 13 .

Note that 12 of 23 is the same as 23 of 12 . The fractions give the same answer in either order.

The students you are teaching should have their own plastic strips to help them work out the answers. Remember to focus entirely on the main idea, which is in three parts.

• Make the second fraction. • Find the first fraction of it. • Name the answer. You should start with some very simple examples that focus clearly on the idea of ‘of’.

Suggested activities • Show 12 . Use the quarters-strip to find half of it. So 12 of 12 is 14 . This is how it is written.

• Show 14 . Use the eighths-strip to find half of it. (It is 18 .) This shows that 12 of 14 is 18 .

• Show 23 . Use the thirds-strip to find half of it. (It is 13 .) This shows that 12 of 23 is 13 .

• Show 12 . Find a piece that is three-quarters of it. Change the 12 for 48 . Then 34 of 12 is 38 .

• Show 12 . Find a piece that is two-thirds of it. Change the 12 for 36 . Then 23 of 12 is 26 .

(This can also be simplified to 13 , but this is not essential at this stage.)

• The idea that the answer is the same when the fractions are reversed might be left as a discovery!



45

43 Multiplying fractions (squares) This is the third activity on the meaning of ‘x’, which is ‘of’.

• The meaning of 12 of 23 is simple. We make the 23 first, then find 12 of it.

In this case the fraction was found without using equal fractions. It is 13 .

• The meaning of 23 of 12 is also simple. We make the 12 first, then find 23 of it.

This time we need to convert to sixths (equal fractions) to name the answer. It is also 13 .

• Note that 12 of 23 is the same as 23 of 12 . The fractions give the same answer in either order.

The students you are teaching should have their own plastic squares to help them work out the answers. Remember to focus entirely on the main idea, which is in three parts.

• Make the second fraction. • Find the first fraction of it. • Name the answer. You should start with some very simple examples that focus clearly on the idea of ‘of’.

Suggested activities • Show 12 on a square. Show half of it. What fraction of the square is this?

(It is 12 of 12 is 14 . This is how it is written.)

• Show 14 on a square. Show half of it. What fraction of the square is this?

(It is 18 .) This shows that 12 of 14 is 18 .

• Show 23 on a square. Show half of it. What fraction of the square is this?

(It is 13 .) This shows that 12 of 23 is 13 .

• Show 12 . Find three-quarters of it. Change the 12 for 48 . Then 34 of 12 is 38 .

• Show 12 . Find two-thirds of it. Change the 12 for 36 . Then 23 of 36 is 26 .

(This can also be simplified to 13 , but this is not essential at this stage.)

• The idea that the answer is the same when the fractions are reversed might be left as a discovery!



46

44 Fractions and dividing This is the first activity aimed at developing an understanding of fractions as the result of dividing. This uses the linear model. Normally we relate fractions to parts of one whole thing. However, fractions also arise from dividing a number of things (the numerator) by another number (the denominator). Below are examples. You need many strips of paper and some sticky tape. Cut A4 sheets into strips 2 cm wide. This gives 10 strips per sheet. Go through this carefully with the students before they tackle the sheet.

Suggested activities • Lay out three strips in a row. Tape them together.

Fold to divide into 2 parts. 3 ÷ 2 = 112 . The crease is 11

2 strip lengths from the end.

crease

strip 1 strip 2 strip 3

• Now fold the same three strips to divide into four parts.

So 3 ÷ 4 = 34 ; the creases are 34 of a strip length apart.

crease


crease crease • Now fold the same three strips to divide into six parts.

So 3 ÷ 6 = 36 = 12 ; the creases are 12 of a strip length apart.

crease


crease crease creasecrease • Now fold the same three strips to divide into five parts. (This may require a bit of trial and error.)

So 3 ÷ 5 = 35 ; the creases are 35 of a strip length apart.

crease


crease crease crease For this last one is not so easy to guess the fraction. It can be checked by folding strip 1 into five equal parts.

• Use 2 metres of string. Fold it into quarters. Each part is 24 = 12 of a metre (50 cm.) So 2 ÷ 4 = 12 .

• Use 2 metres of string. Fold it into sixths. Each part is 26 = 13 of a metre (3313 cm.) So 2 ÷ 6 = 13 .

• Use 2 metres of string. Fold it into fifths. Each part is 25 of a metre (40 cm.) So 2 ÷ 5 = 25 .



47

45 Sharing circles evenly This aims to develop an understanding of fractions as the result of dividing using the circle model. Normally we relate fractions to parts of one whole thing. However, fractions also arise from dividing a number of things (the numerator) by another number (the denominator). Below are examples. Each student needs one set of circles: in thirds, quarters, sixths, eighths and twelfths. Work together in sets, since you will need several similar circles.

Suggested activities • Lay out three circles in quarters. “Here are three circles. They are to be shared by two people. How much

does each get?” (Each gets 6 quarters, which is 112 .)

A

A

A

A

A

A

B

B B

B B

B

A

A

A

B

B

B

C

C D

C D

D

• The three circles are to be shared by four people. How much does each get?” (Each gets 34 .)

• The three circles are to be shared by six people. How much does each get?” (Each gets 36 = 12 .)

A

B

A

B

C

D

C

D F

E E

F

A A

A

B

BB

C

C C

D D

D

E

EE

F

F

F GG

G

HH

H

• The three circles are to be shared by eight people. How much does each get?”

(Now we need circles cut in eighths; see above right. Each gets 23 .)

• Note that we needed 24 part circles in order to share three circles among 8 people. How can you tell how many parts you will need? They may need to work together to have enough circles.



48

46 Sharing money evenly We all need to be able to handle money. It is a good thing to use to practise with fractions. Sharing whole numbers of dollars evenly is a good way to get fractions, and the answers will be expressed as decimals also, reinforcing decimal and fraction equivalences. If at all possible this should be done with real coins, at least for the demonstration and discussion with a small group.

Suggested activities • “$4 is shared between 5 people.” Here it is split into 20 cent coins. There are 20 coins.

Each of the five people gets 4 coins. So each gets 80 cents = $ = $0.80.

• “$3 is shared between 4 people.” Here it is split into 20 cents, then into 20, 10 and 5 cent coins.

• Each of the 4 people gets 75 cents = $ = $0.75.

Make sure they can write the answer as both a fraction and as a decimal (that is, as dollars).



49

47 Steps A great emphasis placed on the idea that a fraction is found by dividing a 1 by the denominator then multiplying by the numerator. Therefore it surprises many students that 2 ÷ 3 is the same as . This uses the steps model. Make sure students count the steps, not the footprints.

Suggested activities Draw these diagrams, preferably on large paper, and discuss them with the children. • When you walk along a concrete footpath you will take a number of steps to cover a number of concrete

slabs. Three examples are drawn below. •

1

2

3

4

1

2

3

6

5

4

8

5

6

7

A B

C

A shows 1 slab covered in 2 steps, therefore 2 in 4 steps and 3 in 6 steps.

Thus 3 ÷ 6 = 2 ÷ 4 = 1 ÷ 2 = 12 , Each step is 12 a slab.

B shows 2 slabs covered in 3 steps, hence 4 in 6 steps. Each step is 23 of a slab. Thus 2 ÷ 3 = 23 .

C shows 3 slabs covered in 4 steps, hence 6 in 8 steps. Each step is 34 of a slab. Thus 3 ÷ 4 = 34 .



50

48 Remainders as fractions (counters) When we divide any remainder can be treated in at least three ways, depending on the problem. For example, this problem needs the remainder to be rounded upwards. A: “Cars will take 4 children each. There are 11 children; how many cars will we need?” (3 cars) This problem needs a whole number remainder.

B: “A ribbon is 11 cm long. We want to cut 4 cm pieces from it. How much is left at the end?” (3 cm of 34 of a ribbon.) This problem needs a fraction, either as a common fraction or a decimal.

C: “We have $11 to pay for fabric at $4 for each metre. How much can we buy?” (2.75 m, or 2 34 m)

Here is the last problem in pictures. The circles show $1 coins. When we divide and there is a remainder it is the fraction part of a mixed number.

‘How man y 4s in 11?’

There are tw o groups of 4 and 3 over.

The 3 is of a group of 4.

So the answer is 2

4

43

11 ÷ 4 means

3

Note: There are two different meanings for division. “How many 4s in 11?” is a different question from “Share $11 in four equal parts.”

It is good for children to see both ways to think of a division.

Suggested activities • Use a set of counters. Make a group of 7. “How many groups of 2 can you make from the 7?”

The group size is 2. The remainder is 1; it is half of the group size, so the answer is 3.

• Here is the same problem expressed as a ‘sharing’ problem. “Share $7 equally between 2 people.” The answer is clearly $3.50, or $3.

• Use the problem above, in the teacher’s explanation. $11 ÷ $4 = 2 people, preferably with real money. There is $3 left over.

• Here is the same problem expressed as a ‘sharing’ problem. “Share $11 equally between 4 people.” The answer is clearly $2.75, or $23

4 .



51

49 Dividing by fractions The rule ‘invert and multiply’ is usually based on very little understanding. There is another way to tackle dividing by fractions that makes much more sense. It uses intuitive ideas of ratio. Since we are dividing one quantity by another, we cannot use the ‘sharing’ idea of division. Instead we can ask ‘how many times does the second fraction fit into the first?’. There is a graded set of problems we can use. a Problems that can be easily solved by drawing or visualising with models. At this stage, the answer is always

a whole number. For example 1 ÷ 14 = 4. How many 14 ‘s in 1? or 2 ÷ 23 = 3. How many 23 ‘s in 2?

b Problems where the answer is a mixed number. Again drawing (or models) is the way to develop

understanding. In a ÷ b, the second number (b) has to be thought of as 1. Because the answer is not a whole number the issue is how to deal with the ‘remainder’. It has to be expressed as a fraction of the second number (b). For example 34 ÷ 12 = 11

2 or 2 ÷ 34 = 223

How many 12 ‘s fit into 34 ? How many 34 ‘s fit into 2?

The one quarter over is half of the 12 . The two parts of the ‘remainder’ makes 23 of the 34 .

If both numbers are expressed with the same denominator it becomes clear that the answer is the ratio of

the two numerators.

For example 34 ÷ 12 or 2 ÷ 34 .

becomes 34 ÷ 24 becomes

84 ÷ 34

which is 32 or 112 which is 83 or 2

23

c Problems where the answer is a fractional number. Again drawing (or models) is the way to develop understanding. Because the second number is larger than the first, the answer is a fraction.

For example 12 ÷ 34 = 23 or 23 ÷ 34 = 89

“What fraction of 34 is 12 ?” “What fraction of 34 is 23 ?”

If both numbers are expressed with the same denominator the answer is the ratio of the two numerators, as

before.

For example 12 ÷ 34 or 23 ÷ 34

becomes 24 ÷ 34 becomes

812 ÷ 9

12

which is 23 which is 89

Suggested activities • Let each group of students use a set of 12 counters. Pose the problems above and discuss the problems

with the students. Lead them to see that the division is comparing the size of two groups (using ratio), and this can be done by comparing their numerators.

• Let each group of students use a set of fraction strips and pose and discuss above.

• Let them use whatever other shape they wish to try the same problems, and come to the same conclusions.



52

50 Decimals include tenths The problems that many students have with decimals arise from lack of understanding, not lack of skills. The basic idea is place value, but now the places are extended to the right and the values divided by ten each time. It is convenient that we have familiar examples of decimals: money and metric units of length. Using these applications, and renaming of the base 10 materials, we can make sure students grasp the main ideas.

Suggested activities • Work with ten-cent coins, and write as dollar amounts; ten of them make $1.00.

• Note that added zeros do not change the value: $0.10 is the same as $0.1.

• Count by tenths: (0.1, 0.2, 0.3 ...), by 0.2, by 0.3 etc. • Name the places on a number line. For example:

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5

• Find the decimals above on a metric ruler or tape. • Measure a length in millimetres, and write it in centimetres. (Example: 35 mm = 3.5 cm)



53

51 Decimals include hundredths The most common practical application of decimals with two places is money. However many children will also know metric length, using centimetres. It is wise to make frequent links to these practical applications.

Suggested activities • Work out the numbers halfway between the tenths. Use money (halfway from 60 c to 70 c), a number line

and a tape. For example halfway from 60 c to 70 c is 65 c.

• Mark the ‘5-cent’ positions on the number line. Name positions as above.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5

0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 1.05 1.15 1.25 1.35 1.45 1.550.05 • Find the centimetre positions on a metric ruler or tape. Name them.

• Consider order. Which is the larger number: 0.4 or 0.15? • Compare how much money they represent and their positions on a number line or tape.

• Recognise that the number of tenths is more important than the number of hundredths, just as with whole numbers tens are more important than ones. The left-most place has more value.

• Use the number chart. Each row is one tenth of the chart, so 0.23 means two rows and 3 more numbers. • Measure a length in centimetres, and write it in metres. (Example: 35 cm = 0.35 m, 153 cm = 1.53 m)



54

52 Decimals include thousandths The most common application of decimals to three places is metric length (e.g.1.234 m, or 1.234 km), and many children will have experienced other metric measures in this way: mass (1.234 kg) or capacity (1.234 L). It is wise to make frequent links to these practical applications.

Suggested activities • We have the model of millimetres for this. Students refer to a ruler to help complete gaps in number lines.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15

0.015 0.025 0.035 0.045 0.055 0.065 0.075 0.085 0.095 0.105 0.115 0.125 0.135 0.1450.005

• Use Base 10 blocks (if you have them) and name the biggest block as the 1.

Then a flat shows 0.1, a long shows 0.01 and a mini shows 0.001. Make models of different decimal numbers, such as 0.105, 0.025 and 0.3 and decide which has the most wood (and shows the largest number).

• Measure a length in millimetres, and write it in metres. (435 mm = 0.435 m, 1504 mm = 1.504 m) • Measure a capacity (eg. water) in millilitres, and write it in litres.

(435 mL = 0.435 L, 1504 mL = 1.504 L)

• Use flashcards for metric conversion and for scale readings.



55

53 Percentage squares This is the first of a number of activities designed to introduce the concept of percentage – “out of 100”.

Suggested activities • A square made from small squares (10 by 10) has some of the small squares coloured. Students estimate

the percentage of the large square that is coloured. They are then count to check their guesses.

• The activity is extended with activities inviting some creativity – colouring in a pattern but colouring the right number of small squares.



56

54 Percentages and fractions This activity involves reading the percentage and fraction scales. However there are a number of good estimation activities which should be done first. You will need a metre ruler, which is blank on one side, with centimetres marked on the other side.

Suggested activities • Put a chalk mark or otherwise indicate a spot on the blank side of the ruler. (A rubber band around the ruler

is useful as it also shows the position on the centimetre scale.)

Ask the students to guess the percentage from the zero end. Then turn the ruler over and show them the answer.

40 cm = 40% • Students mark where they think a given percentage of the total width of the blackboard will be.

For example, guess where 20% from the left end of the board will be. They each put a mark on the board with their initials. They measure the width of the blackboard (e.g. 452 cm). They then turn the percentage into a decimal. (For example, 20% = 0.2) They calculate the distance by multiplying the decimal by the width of the board on a calculator. (For example, 0.2 x 452 cm = 90.4 cm.) If this is too difficult, do it for them. They measure the correct distance from the left of the board and compare their answers with it.

Measure width

Measure correct distance



57

55 Bounce fractions and percentages This tries to answer this question: Do balls always bounce up to the same percentage of their previous height? Use a real experiment. You need a ball, preferably a ‘super ball’. You also need a metre ruler or sewing tape fixed to a wall.

Suggested activities • Students drop the ball from exactly 1 metre. The bottom of the ball should be level with the 1.00 m. They get

their eyes level with the height to which the ball rises. In this way they estimate the height to which the bottom rises on the first bounce.

• They work out the percentage (bounce height ÷ drop height x 100). • Repeat 1 and 2, dropping the ball from 90 cm. Find the percentage.

• Repeat from several other heights. The percentages should be similar.



58

56 Fractions and decimals (wall) This activity introduces decimals as one particular way to write a fraction. (Decimals are also introduced as part of the Whole numbers activities, which stresses their place value properties.) Students will come to see that decimals are equal to other fractions, so that they are just another way to write them. Decimals are very convenient for some things, and very awkward for others. The attached strip ‘wall’ includes a ‘ruler’ marked in decimals and percentages. Students can use a ruler vertically to measure the length of a particular fraction as a decimal; and as a percentage. They will also discover that the percentages are just the same as the first two places of the decimals.

0 . 1 0 . 2 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 90 . 3 1 . 0 00 . 0 0 . 1 5 0 . 2 5 0 . 3 5 0 . 4 5 0 . 5 5 0 . 6 5 0 . 7 5 0 . 8 5 0 . 9 50 . 0 5

9

10------

8

10------

7

10------

6

10------

5

10------

4

10------

3

10------

2

10------

1

10------

1

20------

3

20------

2

20------

4

20------

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20------

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9

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20------

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1

2---

1

5---

2

5---

3

5---

4

5---

1

4---

2

4---

3

4---

1

12------

2

12------

3

12------

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5

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6

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7

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8

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9

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10

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11

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1

6---

2

6---

3

6---

4

6---

5

6---

2

3---

1

3---

1

8---

2

8---

3

8---

4

8---

5

8---

6

8---

7

8---

For example, the line above shows that 14 is the same as 5

20 , 312 etc. and the same as 0.25 and also 25%.

The same result can be obtained using a calculator. You just divide the numerator by the denominator, for example 1 ÷ 4 = 0.25, and 5 ÷ 20 = 0.25 also. This becomes a very quick and useful way to decide whether two fractions are equal, or which one is the greater.

Suggested activities • Show students how to find some fractions that are equal, using a ruler or other straight edge. • Explore the tenths, as multiples of 0.1 and as common fractions. This should stress the equivalence.

• After finding equal fractions on the ‘wall’, students may use their calculators to check the answers.



59

57 Fractions and decimals (ruler)

Use a metre ruler or a sewing tape, usually 150 cm long. The strips on the right of this page can be used. This is a rich experience Cut out each of the four strips, and stick them together. with the short marks at the end lined up. Each strip is 25 cm, so the combination will make one metre. The marks show:

• quarters (at the ends of the separate strips), • fifths (20 cm intervals), • tenths (10 cm intervals), • twentieths (5 cm intervals), • 25 ths (at 4 cm intervals) • 50 ths (at 2 cm intervals).

Suggested activities • Measure some distances as fractions of

a metre, and give the answers as decimals by reading the number of centimetres from the tape: for example, 40 cm = 0.40 m.

• With the students count up and label the marks as shown above.

• Choose a number of the fractions and convert them to decimals by measuring their distance from 0 with the measuring tape or ruler.

• For example, 45 measures 80 cm, so 45 = 0.8.

• Discuss the fact that we can put in, or leave out the zero after the 8. (0.8 is the same as 0.80.)

• Let the students work through some questions using a tape and the strip as a reference. Remember the main aim is to develop understanding of decimals. However a review of the basic ideas of fractions will certainly take place as well.



60

58 Fractions of a dollar This is the third activity that uses decimals to represent fractions of something. The first one used fractions of the ‘wall’. The second used real metric lengths, using a metre as the unit. For this one it is important to use real money. It shows we are dealing with the real world. You need at least two 50 c coins, ten 10 c coins, five 20 c coins, and a few 5 c coins.

Suggested activities • Have children find fractions of a dollar, giving the answers as decimals:

for example, 40 cents = $0.40. • Start with 10 cents, and discuss that it is written as $0.10, but could be written as $0.1.

How many lots of $0.10 do you need to make $1.00? (10). This means that $0.10 is 110 of a dollar.

• Extend this to $0.20 (both 210 and 15 ). Extend it to $0.30, and to $0.40, and to $0.50.

• Discuss what amount you need to have four times to make $1.00, that is, a quarter of a dollar. Some children have trouble with this because it is not a single coin. (Unless you’re American!) Extend this to 34 of a dollar.



61

59 Guitar fractions and decimals This is another real-world example of decimals linked to fractions. It will benefit greatly if you demonstrate with a real guitar. Playing is not needed. The frets on a guitar are very close to exact fractions of the total free string length.

0.1 0.2 0.4 0.5 0.6 0.7 0.8 0.90.3

B ridgeAA#BCC #DD#EFF#GG#AA#BCC #DD#E

Musical Note

St ring

The frets are shown by the vertical lines above. On a real guitar, the frets allow the player to shorten the length of string that vibrates. There are many interesting relationships involving fractions. They were first discovered by Pythagoras (yes, the right-angled triangle chap!) over two millennia ago. Pythagoras did not use decimals.

The first major discovery is that 12 of the open string length always creates a note exactly an octave higher. So on the finger-board above the A in the middle is exactly half of the total length, from the bridge to the A on the right end. For the same reason the distance to B in the middle is half the distance to B near the right end, and so on. This explains why the frets get closer together as you move to the left. The answers from a real guitar make the other fractions clear enough. Note A G# G F# F E D# D C# C B A# A Decimal 0.53 0.56 0.6 0.63 0.67 0.71 0.75 0.8 0.83 0.89 0.94 1.0 0.5 Fraction 1

2 59 3

5 23 3

4 45 5

6 89 1

Suggested activities • Students measure in millimetres. They first measure the open string length. All other lengths will be fractions

of that. • Students measure the lengths from the bridge to each fret. They divide the length to each fret by the open

string length.

• They experiment to find the simple fraction closest to the value obtained.

Note for teachers only The more accurately we measure the less the Pythagoras’ fractions hold up. There are two reasons. 1 The string is stretched when it is played. So the fret positions must take account of this. 2 The scale used on a modern guitar uses tuning that Pythagoras never heard. It is called ‘equal temperament’.

It means, for example, that E is at a fraction of 0.6674 and not at 0.6666.The C# is not at 0.8000, but at 0.7837, and so on. We are used to having most notes slightly ‘out of tune’, so we can play in all keys. Pythagoras worked with a ‘purer’ form of tuning, used in musical instruments (such as organs) until the Renaissance or even later.

Note: The values quoted in the note above use an exponential formula, above our students.



62

60 Decimals and percentages There are several levels of complexity in the relationship between these two forms of fractions. Of course they all use place value. • The basic idea is that percentages (“out of 100”) are just hundredths. So 0.23 is 23%, 1.23 is 123%, and

0.234 is 23.4%. Note that whole numbers are always hundreds of percents (e.g. 2 = 200%).

• The conversions also go the other way: 35% = 0.35, 135% = 1.35, 3000% = 30.

• It gets more complex when fractions are also included in the percentage. These are not uncommon, e.g. 121

2 % is one eighth, and is often expressed as a percentage. This tell us that the decimal has 1212

hundredths, so it must be 0.125, the 0.005 being 5 thousandths or half of 1 hundredth, since that is 10 thousandths.

• Examples with repeating decimals will be covered in 10.6.

Suggested activities • Use a place value chart

This should clearly show the hundredths, and also the percentages.

NUMBER tens ones • tenths hundredths thousands

1 2 3 4 5

PERCENTS 1000s 100s tens ones • tenths

• Discuss the examples above.

• Discuss how to use the % key on a simple calculator. Most of them multiply the decimal by 100.



63

61 Changing fractions to terminating or repeating decimals This is a complex idea with various levels of complexity. Some decimal numbers have their exact value after a small number of decimal places. (0.56, 0.25, 0.875 etc.) Some decimal numbers can never be written out in full to have their exact value because there is always a little bit more. These decimals are of two kinds: those that repeat a pattern of digits over and over (repeating or recurring decimals), and those that never repat but keep on varying the sequence. The easiest way to distinguish between the first two types is to look at their fraction form, in ‘simplest form’.

o If the denominator is a power of 5 or 2 (or both) the fraction will eventually terminate. o If the denominator includes as a factor a prime number (other than 2 or 5) then it will repeat.

The third type of numbers is irrational numbers, because they cannot be expressed as a ratio (i.e. fraction). Helping students to understand all this requires time and a set of guided explorations. The best tool is a computer, so the spreadsheets below are recommended. A calculator frequently does not show enough digits for a pattern to become clear.

Suggested activities • Fractions are changed into decimals by dividing the denominator into the numerator. For convenience it is

best to start by comparing the simple 1 and 2 digit denominators. 1 ÷ 2 is easy, but the fundamental issue appears with 1 ÷ 3. There is always 1 remainder, so the same division (10 ÷ 3) is always going to be repeated. The decimal can never be written out in full. Demonstrate with a short division problem. 0.33333333333… 3 ) 1.00000000000…

• Compare 2 ÷ 3, 4 ÷ 3, 5 ÷ 3 to see the pattern.

• Dividing by 4 and 5 are easy and terminate quickly, but 1 ÷ 6 show up a different pattern again. This time the first non-zero digit is 1 and then 6 is repeated. Compare 0.1666… to half of one third 0.333….

• Explore the pattern for sevenths. There is a cycle of 6 digits that appear in the same order, but with a different starting number.

• Eighths are easy. Explore ninths. Note that nine ninths is 0.999…, and this must be 1.

• Once the idea is clear, there are many fascinating explorations with repeating decimals. For example, how can you predict how long the repeating cycle will be? The pattern shown in the ‘Ten clocks’ spreadsheet reveals hidden symmetry in these numbers.

• An additional challenge is converting percentages such as 3313 % to a decimal. The 33% shows that the first

two digits are 0.33, and the 13 % shows that the remaining digits continue the pattern in 3s. Similarly 816 % will

become 0.0816666…



64

62 Changing decimals into fractions There are many occasions when it is convenient to be able to recognise a decimal (repeating or terminating) and convert it into a fraction. Because these situations generally come as part of something else, it becomes valuable for a keen student to memorise as many of these as possible.

Suggested activities • Memorise the common terminating decimals with denominators: 2, 4, 8, 16, 5, 10, 20, 25, 40, 50, 80.

• Memorise the simple repeating patterns with these denominators: 3, 6, 9, 11, 12, 15, 18

• Understand and learn to perform the process to convert a repeating decimal to a fraction. Example: 0.714285 714285 … is a repeating decimal with a cycle of 6 repeating digits. Call it F. Then 1000 000 times F is 714285.714285 714285 … and the difference between these (subtracting) is just 714285. This is equal to 999 999 times F. If you divide 714285 by 142857 and 999 999 by 142857 you will see that this is the same as five sevenths!



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63 Whole number comparisons A ratio is a measure of how many times bigger (or smaller) one thing is than another. It may be a single number (e.g. 2), but is sometimes expressed as a two-number expression using a colon (e.g. 2:1). In either case, the numbers may be whole numbers, fractions or decimals. Often the single number is a percentage. So 50% means one-half, or 0.5 or 1:2. Research and teaching experience indicate that students find whole number comparisons much easier than others. This occurs when the answer to the question: “B is how many times A?” is a whole number.

Suggested activities • Coin ratios

Compare the value of common coins – from 5 c to $2. (Avoid 20c to 50c).

• Body ratios (rounded) Students use tapes to compare wrist circumference with neck or head, finger span with arm length etc. and look for (rounded) whole number ratios.

• Informal measures Height of a wall (using the number of rows of bricks), mass of wall, words in a book (using words per line, lines per page, pages per book)

• Simple rates These are compared to numbers of seconds: Pulse rates, Speeds,

• Comparing units How many millimetres in a centimetre? etc. How many inches in a foot? feet in a yard?



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64 Mixed number comparisons This activity introduces decimals as one particular way to write a fraction. (Decimals are also introduced as part of the Whole numbers activities, which stresses their place value properties.)


Compare the values of common coins – from 5 c to $2. (Include 20c to 50c).

• Rates Students measure: each others speeds for walking and running, bike riding or car speeds. Explore usage rates for water, electricity, phone calls Compare surface area to volume for different 3D shapes (using unit cubes) and look at biological applications. Simple interest is a rate (say 10%, which compares the interest paid with the principal, per year –another rate)

• Olympic and other sporting records Students calculate the sped of runners, swimmers, etc.

• Rectangle aspect ratios The most common use of this is on TV screens; some pictures don’t fit your screen. The aspect ratio of A4 paper is a comparison of height to width. It is 1.414.

• Comparing units Compare metric to British, using rounded values. How many: • centimetres to the inch? • feet to the metre? • kilometres to the mile



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65 Fraction comparisons and reciprocals This activity introduces decimals as one particular way to write a fraction. (Decimals are also introduced as part of the Whole numbers activities, which stresses their place value properties.)


Compare the value of common coins – from 5 c to $2 by asking “What fraction is the smaller (e.g. 5 c) of the larger (e.g. $1)?”

• Reciprocals Notice that most people find what fraction a shorter length is than a longer one by working out how many times the shorter fits into the longer, and then getting the reciprocal. For example, what fraction is the shorter than the longer line?

• Simple rates Find your average step length (in metres). Find the thickness of a sheet of paper by measuring many, and dividing. Look up currency conversions, and convert both ways.



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66 Proportions show equal ratios Ratios compare two numbers, such as 4 compared to 2. Such a ratio is often written as a singe number (2) showing how many times bigger one is than the other.

But there are other pairs of numbers where one is 2 times the other: 6 to 3, 15 to 7.5, 34 to 38 and so on. (Notice

that any kinds of numbers can be part of a ratio.) A proportion is an equation that says that two ratios are equal. It is often written so that it looks like equivalent

fractions. For example, 42 = 105 , or 15

7.5 =

34

38

. Clearly the last case explains the need for the colon notation, so 15 :

7.5 = 34 : 38 .

The concept of equal ratios is a very useful one, with very many real-life applications, such as scales on maps and plans, pricing, etc. Even percentages can be seen as an example of a proportion; for example the ratio 3 to 4

is equal to the ratio 75 to 100, so 34 = 75%.

Suggested activities • Percentages as proportions

Convert ratios less than 1 (such as 34 ) into percentages under 100. Show that ratios over 1 (e.g 54 )

become percentages of 100 (e.g. 125%) and discuss what this means.

• Aspect ratio of rectangles The aspect ratio is height ÷ width. For a rectangle in ‘portrait’ orientation this is over 1, but in ‘landscape’ orientation it is under 1. ‘Similar’ rectangles will present the same-shape rectangle at different scales and provide examples of proportions.

• Step length Assuming constant step length, you can use the length of 10 steps to find the distance for other number of steps, such as 20, 30, 40 and 25 steps etc .

• Speeds Assuming constant speed, you can use the time to walk 10 m to find the time for other number of steps, such as 20 m, 30 m, 40 m and 25 m etc. This allows you to find distances by measuring times.

A-series of paper An interesting extension of the aspect ratio is to consider the aspect ratio for A4 paper. It will be close to 1.414, and it is possible to show that it must be √2.



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67 Recognising proportional situations It is sometimes assumed that all a teacher needs to do to teach proportion is to provide a simple algorithm, or procedure, so that all problems may be solved. Research suggests that the matter is far more complex than this. The first step is for the student to be able to identify when a proportion situation exists – and when it does not. This requires some understanding of the ‘multiplicative’ character of the relationship between the variables.

Suggested activities • Simple multiplication tables

Provide a simple table based on multiplication only, such as this one.

n 1 2 5 9 2.5 7.5

p 4

There are very many patterns to be found in such a table. • Conversion tables

Provide a more complex table based on multiplication only, such as this one (rounded values).

inches 2 4 6 3 9 12

centimetres 5

There are very many patterns to be found in such a table. Note that this assumes 1 inch = 2.5 cm.

• Using linear graphs – through the origin A graph of either of the tables above, or any similar ones, will go through the origin. Any two pairs of numbers on the graph will form a proportion. For example, for y = 4x, we have (8,2) and (20,5). The gradient (4) is the ratio of any y-value to any x-value.

• Methods of solving proportion problems The method used by a student to solve a proportion problem can indicate the level of development of ‘multiplicative thinking’ of the student. Some students will use entirely additive methods, some will use methods that mix adding with dividing and multiplying, and others will use purely ‘multiplicative thinking’ (dividing and multiplying). The important step is to use a problem that is more complex than whole number comparisons (e.g. 6 to 4 = ? to 10), and to ask the student to explain their reasoning.

• The ‘double – double’ property One way to recognize a proportion situation is to check that whenever one variable doubles, the other also doubles. It should apply to any change using multiplication (x3) or division (÷2).



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68 Solving proportion problems The ability to solve direct proportion problems is extremely useful, both in real life (prices etc) and in sciences such as chemistry and physics.

Suggested activities • Get the relationships clear

Solving a proportion problem means finding a missing number. Three numbers will be known, and for one variable you will have two values. It is useful to carefully set out the four numbers in a grid, with the known pairing first. Here are some examples: a 6 items for $10 b 6 items cost $10 9 items for $? (two values of item numbers) ? items cost $25 (Two values of prices)

• Look for simple ratios to use, where you can Many proportion problems can be solved mentally, using the simple ratios involved. For example, - 6 items cost $10; ? items cost $30? (Just multiply by 30/10 = 3). - 6 items cost $10; 30 items cost $? (Just multiply by 30/6 = 5). How some ratios might not be so simple: - 6 items cost $10; ? items cost $25? (Multiply by 25/10 = 2.5. Some people might just double and then add on half the number you can get for $10.)

• For more complex problems, use the unitary method A common method is to use an intermediate step to find out the rate: how much for one. Which rate you choose will depend on what you have to find. In these examples the intermediate step is ‘how much money for 1 item?’ - 6 items cost $24 7 items cost $? (Each item costs $4, so 7 items will cost $4 x 7 = $28). - 4 items cost $15 7 items cost $? (Each item costs $15/4; the rate is $3.75 per item. So you pay $3.75 x 7). In these examples the intermediate step is ‘how many items for $1?’ - 8 items cost $5 ? items cost $30? (For each $1 you get 8/5 = 1.6 items. So the answer is 1.6 x 30 items = 48.) - 6 items cost $4 ? items cost $9? (For each $1 you get 6/4 = 1.5 items. So the answer is 1.5 x 9 items = 13.5. You can buy 13.)

• Estimate proportions It is important to be able to estimate approximate answers, as real-life applications - such as supermarket shopping - require quick estimates.

• Some common applications - At the same time of day the length of a shadow is proportional to the height of an object. - For a given material the mass is proportional to the volume; the ratio ‘mass/volume’ is called density. - Many problems involving similar figures, including trigonometry (using similar right-angled triangles).



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69 Solving percentage problems There are basically three different types of percentage problems.

• Finding an amount that is a percentage of a quantity. For example, find the amount that is 25% of $60.

• Find what percentage an amount is of another quantity For example, find the percentage that $15 is of $60.

• Given the amount and the percentage, find the original quantity (100%). For example, given that $15 is 25% of a quantity, find that quantity.

These problems are very similar to proportion questions, but instead of the fourth number being able to change, it is always 100. The suggestions below build on understanding, not rules like ‘multiply by 100 over 1’.

Suggested activities • Use diagrams to get the relationships clear

Solving a percentage problem means finding a missing number. Two numbers will be known, and 100 is one of the four numbers you need. Diagrams will probably help. Draw the 100% and the part of it. Put in the numbers you know. For example, what is 25% of $60. The $60 is the 100%, and we need to find s smaller part of it, labeled 25%. MONEY $?? $60

PERCENTAGES 25% 100% Using simple proportion, you divide the 100 by 4 to get 25, so you divide the $60 by 4 to get $15.

Some people find a dual number line useful – money (or other) on one side and percentages on the other. • Build understanding by using simple numbers first

Here are the other two types of percentage problems formulated in this way.

For example, what percentage is $15 of $60. The $60 is the 100%, and we need to find s smaller part of it, labeled 25%. MONEY $15 $60 PERCENTAGES ??% 100% Using simple proportion, you divide the $60 by 4 to get $15, so you divide the 100% by 4 to get 25%. For example, given that $15 is 25% of a quantity, find that quantity. The unknown is the 100%, and we know that 25% is $15 . MONEY $15 $?? PERCENTAGES 25% 100% Using simple proportion, you multiply the 25% by 4 to get 100%, so you multiply the $15 by 4 to get $60.

• For more tricky numbers, simplify the numbers to be sure of the procedure first. This is particularly needed for the third kind of problem, finding the value for 100%. For example, given that $30 is 15% of a quantity, find that quantity. MONEY $30 $?? PERCENTAGES 15% 100% The problem is that it is not intuitively easy to see how many times the 15% is the 100%. If the 15% were 10%, then you divide the 10 into the 100 to get the answer.

So this is what you do with the 15%. So 100% ÷ 15% = 100 ÷ 15 = 10015 . So this is the multiplier.

Using simple proportion, you multiply the 15% by 10015 to get 100%, so you multiply the $30 by 100

15

to get $200. • Include problems for which the ‘part’ percentage is over 100%.



72 Sometimes students think that percentages must always be under or equal to 100, because “You can’t get more than 100% on a test.” But of course a percentage is just another name for a fraction, so the fraction 1.5 means the same as 150%. Diagrams will probably help. Draw the 100% and the part of it. Put in the numbers you know. For example, what percentage is $60 of $25. Establish first that it is more, so the answer is more than 100%. The $25 is the 100%, and we need to find the percentage corresponding to $60.

• MONEY $25 $60 PERCENTAGES 100% ??% Using simple proportion, you multiply the 25 by 4 to get 100, so you multiply the $60 by 4 to get 240%.



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70 Percentage increases and decreases Percentage increases include price increases, profits, and increases of other kinds, including interest. There are two types of problems: finding the percentage that corresponds to the increase, or finding the increase that corresponds to the percentage. There are two ways to solve such problems.

• Subtract to find the actual increase, and find the increase as a percentage of the smaller (original) amount. For example, a price increases from $25 to $35. What is the percentage increase? Subtract to find the increase itself: $35 - $25 = $10. Find the percentage that $10 is of the smaller original amount: $10 is 40% of $25.

• Find the final amount as a percentage of the smaller (original) amount, then subtract 100%. For example, a price increases from $25 to $35. What is the percentage increase? $35 is 140% of $25. So the increase ($10) must be 40%.

Percentage decreases include price decreases, discounts, and decreases of other kinds including depreciation. There are two types of problems: finding the percentage that corresponds to the decrease, or finding the decrease that corresponds to the percentage. There are two ways to solve such problems.

• Subtract to find the actual decrease, and find the decrease as a percentage of the larger (original) amount. For example, a price decreases from $25 to $20. What is the percentage decrease? Subtract to find the decrease itself: $25 - $25 = $10. Find the percentage that $10 is of the larger original amount: $10 is 20% of $25.

• Find the final amount as a percentage of the larger (original) amount, then subtract from 100%. For example, a price decreases from $25 to $20. What is the percentage decrease? $20 is 80% of $25. So the decrease ($10) must be 20%.

Suggested activities Use diagrams to get the relationships clear Again it is really just a matter of working out what is what, and a diagram always helps. The original amount is always 100%.

INCREASE MONEY $25 $10 increase $35

PERCENTAGES 100% ?% increase ?% Use the $25 and 100% and one other pair of numbers.

DECREASE MONEY $20 $10 decrease $25

PERCENTAGES ?% ??% decrease 100%

Use the $25 and 100% and one other pair of numbers.

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