m30505 Comparing fractions 1.1 2009/07/25 08:20:38.296 GMT-5 2009/07/25 09:27:35.067 GMT-5 gert bezuidenhout gert bezuidenhout gertb@mweb.co.za gert bezuidenhout gert bezuidenhout gertb@mweb.co.za gert bezuidenhout gert bezuidenhout gertb@mweb.co.za Mathematics and Statistics en

Activity 1: To compare fractions [LO 1.3] 1. Each of the following three bars represents one whole. The top bar shows thirds. The middle bar shows twelfths. The last bar shows sixths. You may use them to replace size 12{ * } {} with the correct sign from: < and  to make the statements true: 1.1 23 size 12{ { {2} over {3} } } {} size 12{ * } {}56 size 12{ { {5} over {6} } } {} 1.2 56 size 12{ { {5} over {6} } } {} size 12{ * } {}1112 size 12{ { {"11"} over {"12"} } } {} 1.3 912 size 12{ { {9} over {"12"} } } {} size 12{ * } {}23 size 12{ { {2} over {3} } } {} 1.4 312 size 12{ { {3} over {"12"} } } {} size 12{ * } {}26 size 12{ { {2} over {6} } } {} 2. Here again, each of the bars represents one whole. The top bar shows . The lower bar shows . You may use these bars to complete the following: two-fifths < tenths six-tenths < fifths four-fifths  tenths two-tenths < fifths Which is greater: four-tenths or four-fifths? . Which is greater: three-tenths or two-fifths? . Which is less: three-fifths or five-tenths? . 3. Here again, each of the bars represents one whole. Look carefully at the bars above and then complete the following: half  eighths. two-eighths < quarters three-quarters  five . Activity 2: To count forwards and backwards in fractions [LO 1.3] 1. Group Discussion Read the following and discuss who was correct: The educator said, “Count in halves from 0 to 10.”

Who was correct? Actually both ways of counting were correct. Let’s look at the way Peter did it. 12 size 12{ { { size 11{1}} over { size 11{2}} } } {}; 22 size 12{ { {2} over { size 11{2}} } } {}; 32 size 12{ { {3} over { size 11{2}} } } {}; 42 size 12{ { {4} over { size 11{2}} } } {}; 52 size 12{ { {5} over { size 11{2}} } } {} What do you notice? Yes, after the first two, the top part of the fraction is bigger than the bottom part. What does this mean? Discuss. Yes, it means that there is at least one whole hidden in there. 22 size 12{ { {2} over { size 11{2}} } } {} = one whole; 32 size 12{ { {3} over { size 11{2}} } } {} = 112 size 12{ size 11{1 { {1} over { size 11{2}} } }} {} What do four halves mean? What do five halves mean? When the top part of the fraction is larger than the bottom part, we call it an IMPROPER FRACTION. 52 size 12{ { {5} over { size 11{2}} } } {} is an IMPROPER FRACTION; the top bit is larger than the bottom bit. Sometimes it is necessary to make improper fractions in calculations. However, most educators like the final answer to a calculation to be a mixed number. 2. ORAL WORK: Now do these counting exercises. You may use either improper fractions or mixed numbers. Ask a friend to check your answers. 2.1 (a) Count in halves from 0 to 10. (b) Count backwards in halves from 100 to 90 2.2 (a) Count in thirds from 6 to 10. (b) Count backwards in thirds from 30 to 25. 2.3 (a) Count in quarters from 12 to 16. (b) Count backwards in quarters from 100 to 96. 2.4 (a) Count in fifths from 50 to 55. (b) Count backwards in fifths from 10 to 6. 2.5 (a) Count in sixths from 24 to 26. (b) Count backwards in sixths from 36 to 30. 2.6 (a) Count in sevenths from 0 to 4. (b) Count backwards in sevenths from 21 to 17. 2.7 (a) Count in eighths from 0 to 3. (b) Count backwards in eighths from 10 to 8. 2.8 (a) Count in tenths from 3 to 8. (b) Count backwards in tenths from 100 to 97. Activity 3: To recognise equivalent fractions [LO 1.5, 2.1] Two boys study a measuring beaker half full:

Who is correct? Yes, they both are. There is only one beaker and one quantity of Coke, but it can be called 12 size 12{ { { size 10{1}} over { size 10{2}} } } {} and 5001 000 size 12{ { { size 10{"500"}} over { size 10{"1 000"}} } } {} ; thus, different names for the same quantity. We say 12 size 12{ { { size 10{1}} over { size 10{2}} } } {} and 5001 000 size 12{ { { size 10{"500"}} over { size 10{"1 000"}} } } {} are EQUIVALENT FRACTIONS. The thousandths are smaller pieces, but there are 500 of them; enough to make a half. 1. Now see if you can work out the equivalent fractions here (Use the bars in the diagram if necessary): Half a sausage roll is equivalent to …………. quarters of an identical sausage roll. Equivalence occurs when the whole may have been cut into a different number of parts, but there are enough of them to make the same quantity as there is in the other fraction. We write it in words or with digits, thus 12 size 12{ { { size 10{1}} over { size 10{2}} } } {} = 24 size 12{ { { size 10{2}} over { size 10{4}} } } {} and we use the = sign. 1.2 Half a sausage roll is equivalent to sixths of an identical sausage roll. Half a sausage roll is equivalent to eighths of an identical sausage roll. Now make up one of your own: Half a sausage roll is equivalent to of an identical sausage roll. 2. Halves. The top bar shows halves. Shade in one half. Now see if you can find fractions in the other bars equivalent to a half. Write them all down below. Try to spot some pattern in your answers. Discuss this with a friend. 3. Thirds. The top bar shows thirds. Shade in one third. Now see if you can find fractions in the other bars equivalent to one third. Write them all down below. Try to spot some pattern in your answers. Discuss this with a friend. 3.3 Find all the fractions that are equivalent to two-thirds. Write them down below: 4. Fifths. The top bar shows fifths. Shade in one fifth. Now see if you can find fractions in the other bars equivalent to one fifth. Write them all down Try to spot some pattern in your answers. Discuss this with a friend. 4.3 Now see if you can find fractions in the other bars equivalent to two fifths. Write them all down. Try to spot some pattern in your answers. Discuss this with a friend. 4.4 Now see if you can find fractions in the other bars equivalent to three fifths. Write them all down. Try to spot some pattern in your answers. Discuss this with a friend. 4.5 Now see if you can find fractions in the other bars equivalent to four fifths. Write them all down. Try to spot some pattern in your answers. Discuss this with a friend. 5. Patterns. 5.1 Spot the pattern and fill in the missing parts: FRACTION EQUIVALENT FRACTIONS 1 2 size 12{ { { size 11{1}} over { size 11{2}} } } {} 4 size 12{ { {} over {4} } } {} 8 size 12{ { {} over {8} } } {} 8 size 12{ { {8} over {} } } {} 16 size 12{ { {"16"} over {} } } {} Try to write down the patterns that you saw. Spot the pattern and fill in the missing parts: FRACTION EQUIVALENT FRACTIONS 1 3 size 12{ { { size 11{1}} over {3} } } {} 6 size 12{ { {} over {6} } } {} 9 size 12{ { {} over {9} } } {} 6 size 12{ { {6} over {} } } {} 12 size 12{ { {} over {"12"} } } {} 5.4 Try to spot patterns for making equivalent fractions for other fractions. Discuss them in class. Activity 4: To use equivalent fractions [LO 1.5, 1.7] 1. Joan spent three-quarters of her holiday at home and her brother, Willie, spent five-eighths of the same holiday at home. Which of them spent more time at home that holiday? 2. David’s rabbits ate 35 size 12{ { {3} over {5} } } {} of a bunch of carrots. Roy’s rabbits ate 710 size 12{ { {7} over {"10"} } } {} of an identical bunch of carrots. Which boy had more carrots left over? 3. Len’s mother made three identical tins of shortbread. She cut the first one into three pieces; the second one into six pieces and the third one into twelve pieces. Len ate one piece from the first tin. His brother, Bruce, ate three pieces from the second tin and their father ate four pieces from the third tin. 3.1 Who ate the most shortbread? 3.2 Which of them ate the same quantity of shortbread? 4. Amos looked after the vines at the end of his grandfather’s vegetable garden. When the grapes were ripe, Amos picked 15 kilograms of delicious Hanepoot grapes. His grandfather said he could put them in packets that held 112 size 12{ size 11{1 { {1} over {2} } }} {} kg of grapes each, and sell them for R6 each, or he could put the grapes in boxes which held 5 kg of grapes, and sell each box for R20. Amos wanted to make as much money as possible. 4.1 How many packets would Amos need if he chose packets? 4.2 How many boxes would he need if he chose boxes? 4.3 Would he make more money by using the packets or the boxes, and if so, how much more would he make? Explain your answer.

Learning outcomes(LOs) LO 1 Numbers, Operations and RelationshipsThe learner will be able to recognise, describe and represent numbers and their relationships, and to count, estimate, calculate and check with competence and confidence in solving problems. Assessment standards(ASs) We know this when the learner: 1.1 counts forwards and backwards in a variety of intervals; 1.3 recognises and represents the following numbers in order to describe and compare them: common fractions with different denominators, common fractions in diagrammatic form, decimal fractions and multiples of single-digit numbers; 1.3.2 common fractions with different denominators, including halves, thirds, quarters, fifths, sixths, sevenths and eighths; 1.3.3 common fractions in diagrammatic form; 1.3.4 decimal fractions of the form 0,5; 1,5 and 2,5; etc., in the context of measurement; 1.3.6 multiples of single-digit numbers to at least 100; 1.5 recognises and uses equivalent forms of the numbers including common fractions and decimal fractions; 1.5.1 common fractions with denominators that are multiples of each other; 1.5.2 decimal fractions of the form 0,5; 1,5 and 2,5, etc., in the context of measurement; 1.7 solves problems that involve comparing two quantities of different kinds (rate); 1.7.1 comparing two or more quantities of the same kind (ratio); 1.8 estimates and calculates by selecting and using operations appropriate to solving problems that involve addition of common fractions, multiplication of at least whole 2-digit by 2-digit numbers, division of at least whole 3-digit by 1-digit numbers and equal sharing with remainders; 1.8.3 addition of common fractions in context; 1.8.6 equal sharing with remainders; 1.9 performs mental calculations involving: 1.9.2 multiplication of whole numbers to at least 10 x 10; 1.12 recognises, describes and uses:, and 1.12.1 the reciprocal relationship between multiplication and division (e.g. if 5 x 3 = 15 then 15 ÷ 3 = 5 and 15 ÷ 5 = 3; 1.12.2 the equivalence of division and fractions (e.g. 1 ÷ 8 = ⅛); 1.12.3 the commutative, associative and distributive properties with whole numbers. Learning outcomes(LOs) LO 2 Patterns, Functions and AlgebraThe learner will be able to recognise, describe and represent patterns and relationships, as well as to solve problems using algebraic language and skills. Assessment standards(ASs) We know this when the learner: 2.1 investigates and extends numeric and geometric patterns looking for a relationship or rules; 2.1.1 represented in physical or diagrammatic form; 2.1.2 not limited to sequences involving constant difference or ratio; 2.1.3 found in natural and cultural contexts; 2.1.4 of the learner’s own creation; 2.2 describes observed relationships or rules in own words; 2.3 determines output values for given input values using verbal descriptions and flow diagrams; 2.3.1 verbal descriptions; 2.3.2 flow diagrams.

ACTIVITY 1: comparing fractions 1.1 < 1.2 < 1.3 . 1.4 < 2.1 five (or six, seven, eight, nine) tenths 2.2 four 2.3 seven (or six, five, four, three, two, one) tenths two (or three, four, five) fifths four-fifths two-fifths 2.7 five-tenths three (or two, one) two (or three, four) eighths ACTIVITY 2: counting in fractions 1.1 Group discussion ACTIVITY 3: equivalent fractions 1.1 shading; 2 quarters 1.2 3 sixths 1.3 four eighths 1.4 own shading half = two-quarters = four-eighths shading one-third = two-sixths = four-twelfths = eight twenty-fourths two-thirds = four-sixths = eight-twelfths = sixteen twenty-fourths 4. Fifths 4.1 shading 4.2 ; ; ; 4.3 ; ; ; 4.4 ; ; ; 4.5 ; ; ; Discuss patterns 5. Patterns 5.1 Missing parts: 2; 4; 16; 32 Pattern Missing parts: 2; 3; 18; 4 Class discussion: patterns for making equivalent fractions. ACTIVITY 4: using equivalent fractions 1. Joan; = 2. David; = , David had more carrots left over. 3.1 Len: one third; Bruce three-sixths (i.e. half); Dad: four-twelfths so Bruce ate the most. Len and his father. 4.1 1 x 10 = 15; 10 packets 4.2 5 x 3 = 15; 3 boxes 4.3 10 x R6 = R60; 3 x R20 = R60He’d get the same amount of money whether he used boxes or packets. 3. 1 - = - = 4. + = + = = 1 0 1 2