Recognise and represent decimal fractions

MATHEMATICS

Grade 4

NUMBERS, FEACTIONS, DECIMALS AND NUMBER PATTERNS

Module 9

RECOGNISE AND REPRESENT FRACTIONS Activity 1: To recognise and represent decimal fractions [LO 1.3] RECOGNIZING DECIMALS 1. What are decimals? 1.2 Think back to Place Value: Thousands; Hundreds; Tens and Units. Complete the following: 1 000  10 = 100 100  10 = 10  10 = 110 = ?

1.3 Now check your answers on the calculator. The calculator says that 1 10 = 0,1. What does 0,1 then mean? Discuss with a friend. 1.4 Draw lines on the bar below to show tenths. One has been divided by 10. We say that 0,1 is one tenth. It is the only way in which calculators can write one tenth, because of the way that they have been programmed. Now label each section on the bar below 0,1.

What is the meaning of 0,1? We have found that 1  10 = 0,1. Think back to Fractions: we said: 1  2 = 12 size 12{ { {1} over {2} } } {} So 1  10 = 110 size 12{ { {1} over {"10"} } } {} 1  10 = 110 size 12{ { {1} over {"10"} } } {} = 0,1 0,1 is just another way of writing 110 size 12{ { {1} over {"10"} } } {} Study the diagram: Thousands1 000 Hundreds100 Tens10 Units1 Tenths

Thousands1 000 Hundreds100 Tens10 Units1 Tenths 110 size 12{ { {1} over {"10"} } } {} 7 1 9 3 6 5 0 6 9 1

How do we show the end of the whole number when there are no headings? We use a DECIMAL COMMA. What are the numbers that have been written in the columns? 7 193, 6 = 7 × 1 000 + 1 × 100 + 9 × 10 + 3 × 1 + six tenths 5 069,1 = 5 × 1 000 + 0 + 6 × 10 + 9 × 1 + 110 size 12{ { {1} over {"10"} } } {} Our calculators cannot write common fractions as we can; they are only machines that have been programmed to use place value, so they can only write decimal fractions. Remember: We use a DECIMAL COMMA to show the END OF THE WHOLE NUMBER and the BEGINNING OF THE DECIMAL FRACTION 2. Now write the following decimal numbers in their expanded form under the correct heading in the columns below: 2.1 (a) 1 456,3 (b) 4 601,9 (c) 8,5 (d) 31, 7 (e) 456,2 X 1 000 × 100 × 10 × 1 × 0,1(tenths) (a) (b) (c) (d) (e)

The FIRST digit after the decimal comma is always TENTHS. 2.2 Now write them again in their expanded form: (a) 1 456,3 = 1 × 1 000 + 4 × 100 + 5 × 10 + 6 × 1 + 3 × 0,1 Activity 2: To compare fractions [LO 1.5] 1. Carefully consider the value of each digit and use the correct sign from:<;  ; = to > compare the following: 1.1 1,5 _____1,7 1.4 45,9 ____62,3 1.2 6,3 ____ 6,1 1.5 13,2 ____8,6 1.3 24,7____ 42,3 1.6 57,5 ____58,2 2. Encircle the largest number: 43,7; 41,9; 43,1; 49,1; 41,5 3. Write down the number that is: Answer Answer 3.1 one more than 9,9 3.1 3.5 0,1 less than 7,1 3.5 3.2 0,1 more than 5,3 3.2 3.6 0,1 more than 99,0 3.6 3.3 0,1 less than 6 3.3 3.7 0,1 more than 5,8 3.7 3.4 0,1 less than 8,3 3.4 3.8 0,1 less than 10 3.8

Activity 3: To convert from fractions to decimal fractions and vice versa [LO 1.5] Group discussion. 1. Read the following conversation between John and Sarah.

Was Sarah’s answer correct? It did not seem to help John completely. Where did the ,5 come from? Discuss. Try to explain why half = 0,5 on a calculator. Which of you were wide awake? All of you? Did you all know? Wonderful! Yes, it’s because the calculator counts in tenths and five-tenths = one-half. The poor calculator has to use equivalent fractions to make tenths from things like halves and quarters and any fractions that are not tenths (hundredths and thousandths, but they come later). With a calculator: 2. Make decimal fractions from the following: 2.1 34 size 12{ { {3} over {4} } } {} = 3  4 = 0, ____ 2.2 25 size 12{ { {2} over {5} } } {} = 25 = 2.3 35 size 12{ { {3} over {5} } } {} = 2.4 45 size 12{ { {4} over {5} } } {} = 2.5 55 size 12{ { {5} over {5} } } {} = 2.6 14 size 12{ { {1} over {4} } } {} =

We can convert any ordinary fraction to decimals in that way. 3. Make one-third into a decimal: 13 size 12{ { {1} over {3} } } {} = 1  3 =_____ Can you think of a reason why the answer is the way it is? Without a calculator: 4. Write down equivalent fractions for each of the following and then write them as decimal fractions: Fraction Fraction as tenths Decimal fraction half one third Can’t

Fraction Fraction as tenths Decimal fraction two-thirds Can’t one-quarter three-quarters one-fifth two-fifths three-fifths four-fifths one-sixth Can’t one-eighth

(Some of the above have more than one decimal place but it is good to know about them.) 5. What about the thirds and sixths and others that cannot be made into tenths? Use division. one-third = 1  3 = two-thirds = 2  3 = Use your own method for the division or use a calculator. 13 size 12{ { {1} over {3} } } {} = 1  3 Or one way: ? x 3 = 1 0 x 3 = 0,0 0,3 x3 = 0,9 0,03 x 3 = 0,09 0,99 (which is nearly 1) so: (0 x 3) + (0,3 x 3) + (0,03 x 3) 0 + 0,3 + 0,03 = 0,333 (and the calculator will go on dividing: 0,333) We say: 0,3 recurring or 0,3֯(The dot means recurring.) TEST YOUR PROGRESS 1. Solve without a calculator: 1.1 17 × 26 1.2 153  9 2. Share 11 sausage rolls equally amongst 10 boys. How much sausage roll will each boy receive? 3. Share 12 sausage rolls equally amongst 10 boys. How much sausage roll will each boy receive? 4. Mike drinks 112 size 12{ size 11{1 { {1} over {2} } }} {} mugs of milk for breakfast. His sister, Sharon, drinks 34 size 12{ { {3} over {4} } } {} of a mug of milk. How much milk have they drunk altogether? 5. Write the following in expanded notation: 5.1 64,8 = 5.2 341,2 =

6. Write as decimals: Three and four-fifths = ……………………… One and three-tenths = …………………. Five and one-quarter = ………………… 412 size 12{ size 11{4 { {1} over {2} } }} {} = …………………. 7. From<; > ; = write down the correct sign to make the following true: 2,4 ____ 4,2 1,7 _____2,1 8. Write down the number that is: Answer one tenth more than 45,9 one tenth less than 10

Assessement 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.

Memorandum ACTIVITY 1: recognising and representing decimal fractions 1.1 Missing numbers: 10; 1; one-tenth 1.2 Calculator answers: 10; 1; 0,1 0,1 means one-tenth 2.1 x 1 000 x 100 x 10 x 1 x 0,1 (a) 1 4 5 6 3 (b) 4 6 0 1 9 (c) 8 5 (d) 3 1 7 (e) 4 5 6 2

2.2 (b) 4 x 1 000 + 6 x 100 + 0 x 10 + 1 x 1 + 9 x 0,1 (c) 0 x 1 000 + 0 x 100 + 0 x 10 + 8 x 1 + 5 x 0,1 or just: 8 x 1 + 5 x 0,1 (d) 0 x 1 000 + 0 x 100 + 3 x 10 + 1 x 1 + 7 x 0,1 or just: 3 x 10 + 1 x 1 + 7 x 0,1 (e) 0 x 1 000 + 4 x 100 + 5 x 10 + 6 x 1 + 2 x 0,1 or just: 4 x 100 + 5 x 10 + 6 x 1 + 2 x 0,1 ACTIVITY 2: comparing decimal fractions 1.1 < 1.2  1.3 < 1.4 < 1.5  1.6 < 2. Encircled number: 49,1 3.1 10,9 3.2 5,4 3.3 5,9 3.4 8,2 3.5 7 3.6 99,1 3.7 5,9 3.8 9,9 ACTIVITY 3: converting from fractions to decimal fractions and vice versa 1. Discussion 2. With a calculator 0,75 2.2 0,4 2.3 0,6 2.4 0,8 2.5 0,8 2.6 0,25 3. 0,33333 4. Fraction Fraction as tenths Decimal fraction half Five tenths 0,5 One-third Can’t 0,3333 Two-thirds Can’t 0,6666 One-quarter Can’t; 0,25 Three-quarters Can’t; 0,75 One-fifth Two-tenths 0,2 Two-fifths Four-tenths 0,4 Three-fifths Six-tenths 0,6 Four-fifths Eight-tenths 0,8 One-sixth Can’t 0,1666 One-eighth Can’t; 0,125

0,333 0,666

TEST YOUR PROGRESS 1.1 442 1.2 17 2. one and one-tenth or 1,1 sausage rolls 3. one and two-tenths or 1 and a fifth sausage rolls (or 1,2) 4. two and a quarter mugs 6 x 10 + 4 x 1 + 8 x 0,1 3 x 100 + 4 x 10 + 1 x 1 + 2 x 0,1 3,8 1,3 5,25 4,5 < < 46 9,9