7 d ecimAls - Maths English LB Grade6 Book lowres

7.1 Fifths and tenths and hundredths

In this unit you will measure lengths with Greysticks. Because the Greystick is longer than the Yellowstick, we can divide it into many more smaller parts than we could divide a Yellowstick. This makes it possible to measure lengths more accurately.

We shall focus on fifths, tenths and hundredths and will learn a different notation for fractions.

Answer the questions below. The strips and Greysticks are given on the next page.

1. What can we call the small parts in Greysticks A, B and C?

2. How long is the green strip? Write your answer in more than one way.

3. What do we call the small parts in Greystick F?

4. What do we call the small parts in Greystick G?

5. How long is the yellow strip?

6. How long is the red strip? Give two or more possible answers.

7. Write these fractions as tenths:

(a) two fifths (b) three fifths (c) eight twentieths (d) five fiftieths 8. Write these fractions as hundredths:

(a) two fifths (b) three fifths

9. Add the following and give your answers in hundredths:

(a) 6 tenths + 7 tenths

(b) 23 hundredths + 5 hundredths (c) 35 hundredths + 73 hundredths (d) 14 tenths + 3 hundredths

(e) 123 tenths + 42 hundredths

A B C D E F G H I J K L

7.2 A different notation for fractions

You can write the number 2₁₀³ as 2,3 and the number 1¹₂ as 1,5.

1. If 2₁₀³ is written as 2,3, why do you think 1¹₂ is written as 1,5?

Discuss this with one or two of your classmates.

2₁₀³ and 2,3 are two different notations for the same number.

2,3 is the decimal notation.

10 has no whole number part and so it is written as 0,3.

A comma separates the whole number part from the fraction. The first position after the comma indicates the number of tenths in the number. The second position is for the hundredths.

2. Write the length of each of these strips in fraction notation and in decimal notation. Measure in Yellowsticks. This is one Yellowstick:

One Yellowstick

(a)

(b)

3. How can you turn a tenths ruler into a fiftieths ruler?

4. How can you turn a tenths ruler into a hundredths ruler?

The number 1₂¹ can be written as 1,5 because 1,5 is 1 and 5 tenths.

5. (a) On the right is a green strip between two

Greysticks. Write the length of the green strip in fraction notation. Give two answers.

(b) Write the length of the green strip in decimal notation.

6. Write the following fractions in decimal notation:

(a) ₁₀⁷ (b) ₁₀₀⁷²

(e) ₁₀₀³ (f) ₁₀²⁷

7. Write the following in fraction notation:

(a) 2,57 (b) 0,3

(e) 5,30 (f) 1,22

8. (a) What fraction of this rectangle is purple?

(b) What fraction of this rectangle is white?

Give your answers in fraction notation as well as in decimal notation.

9. What fraction of this rectangle is

(a) green (b) purple (c) white?

Give your answers in fraction notation as well as in decimal notation.

10. What fraction of this rectangle is

(a) green (b) purple (c) white?

Give your answers in fraction notation as well as in decimal notation.

11. What fraction of this rectangle is

(a) green (b) purple (c) white?

Give your answers in fraction notation as well as in decimal notation.

7.3 Place value parts and number names

We write 300 + 50 + 6 + ₁₀⁷ + ₁₀₀² as 356,72.

This notation, 300 + 50 + 6 + ₁₀⁷ + ₁₀₀² , is called the expanded notation or place value expansion of 356,72.

1. Write down in words how you would read the number 356,72 aloud.

2. Simon says 356,72 is three hundred and fifty-six comma seventy- two.

(a) Is Simon correct?

(b) Explain your answer.

We can also call 300 + 50 + 6 + ₁₀⁷ + ₁₀₀² the place value parts of 356,72.

We read 356,72 as three hundred and fifty-six comma seven two.

The number name of 356,72 is three hundred and fifty-six and seven tenths and two hundredths.

The digit 3 in 354,76 tells us that there are 3 hundreds in the number.

The digit 3 in 534,76 tells us that there are 3 tens in the number.

The digit 3 in 543,76 tells us that there are 3 units in the number.

The digit 3 in 547,36 tells us that there are 3 tenths in the number.

The digit 3 in 547,63 tells us that there are 3 hundredths in the number.

This table shows how the above numbers are made up of place value parts. The table also shows the different numbers that are indicated by the digit 3 in different positions.

Hundreds Tens Units Tenths Hundredths

354,76 3 5 4 7 6

534,76 5 3 4 7 6

543,76 5 4 3 7 6

547,36 5 4 7 3 6

547,63 5 4 7 6 3

3. Write the number name and the place value parts of each of the following numbers:

(a) 362,74 (b) 1 208,50 (c) 70,36

(d) 154,12 (e) 592,04 (f) 735,83

7.4 Counting in tenths in both notations

1. Write the next ten numbers in each sequence:

(a) ₁₀¹; ₁₀²; ₁₀³; … (b) 0,1; 0,2; 0,3; … 2. Calculate.

(a) What is ₁₀⁹ + ₁₀¹? (b) What is 10 − 0,1?

3. Write the next ten numbers in each sequence:

(a) 99,5; 99,6; 99,7; … (b) 11,4; 11,3; 11,2; … (c) 9,8; 9,6; 9,4; … (d) 11,4; 11,3; 11,2; … (e) 5,7; 5,5; 5,3; … (f) 3,9; 3,6; 3,3; …

4. Follow the arrows and count in tenths in this flow diagram. Find the numbers for (a), (b), (c) etc. and write them in a list.

(a) (b)

(f) (e) (d) (c)

(g) (h) (i) (j)

3 10

1 10

+ 10 ¹

+ 10

1 + 10

+ 10 ¹

+ 10

+ 10 ¹

+ 10

5. Follow the arrows and count in 0,1s. Find the numbers for (a), (b), (c) etc. and write them in a list.

6. Draw an open number line like the one below. Measure carefully and write the 0, 1 and 2 at the correct places below the line.

Now, without making any measurements, place the following numbers carefully on your number line. Estimate where they should be:

1,2; 0,3; 0,9; 1,5; 0,75 7. Complete the sequences:

(a) 0,2; 0,4; 0,6; ; ; ; ; ; ; ; . (b) 0,3; 0,6; 0,9; ; ; ; ; ; ; ; . (c) 0,4; 0,8; 1,2 ; ; ; ; ; ; ; ; . (d) 0,5; 1; 1,5 ; ; ; ; ; ; ; ; . (e) 0,6; 1,2; 1,8; ; ; ; ; ; ; ; .

(a) + 0,1 0,4 + 0,1 (b) + 0,1 (c)

+ 0,1

(g) + 0,1 (f) + 0,1 (e) + 0,1 (d)

(h) + 0,1 (i) + 0,1 (j) + 0,1 (k)

+ 0,1

0 1 2

8. Complete the sequences below. Copy the number line if you need to draw arrows on it to help you find the numbers.

(a) 0,2; 0,4; 0,6; ; ; ; ; ; ; ; . (b) How many 0,2s are in 1?

(Adding on in 0,4s) (e) How many 0,4s are in 2?

(f) 0,5; ; ; ; ; ; ; ; ; ; . (Adding on in 0,5s)

(g) 0,6; ; ; ; ; ; ; ; ; ; . (Adding on in 0,6s)

7.5 Counting in hundredths in both notations

1. Write the next ten numbers in each sequence:

(a) ₁₀₀¹ ; ₁₀₀² ; ₁₀₀³ ; … (b) 0,01; 0,02; 0,03; … (c) ₁₀₀⁵ ; ₁₀₀¹⁰; …

2. (a) How many groups of 5 hundredths are there in 1?

(b) What is 1 − ₁₀₀¹ ? (c) What is 1 − 0,01?

0 1 2 3

3. Write the next ten numbers in each sequence:

(a) 101,05; 101,04; 101,03; … (b) 11,04; 11,03; 11,02; … (c) 9,05; 9,07; 9,09; … (d) 10,07; 10,06; 10,05; … (e) 7,13; 7,16; 7,19; … (f) 6; 5,96; 5,92; …

4. Use the given number lines, if you need to, to help you to complete the sequences.

(a) Count in 0,25s from 0,25 to 2,5.

0,25; 0,50; ...

0 1 2

(b) Count in 0,05s from 0,05 to 1,1.

0,05; 0,1; 0,15; ...

0 0,5 1

0,15; 0,30; 0,45; …

0 0,5 1

5. Count in hundredths in this flow diagram. Find the correct numbers for (a), (b), (c) etc. and write them in a list.

7.6 From fractions to decimals to fractions

1. We have to write a fraction as tenths or hundredths in order to be able to write it as a decimal fraction to two decimal places.

(a) Which other fractions, besides tenths and hundredths, are easy to write as decimals?

(b) Explain how you will go about writing each of these fractions as decimals.

2. Write the following numbers in decimal notation.

(a) 2₁₀¹ (b) 5₁₀⁷

(e) 124¹₂ (f) 17¹₄

(g) 23₁₀₀¹³ (h) 4₁₀₀⁷

(b) (c)

(g) (f) (e) (d)

(h) (i) (j) (k)

94 + 100

100 (a) + ¹

100 + ¹

100

+ ¹

100 + ¹

100

+ ¹ 100

3. Write the following numbers in expanded fraction notation to show the place value parts of each number.

(a) 3,2 (b) 4,27

(e) 50,30 (f) 3,25

(g) 56,20 (h) 20,50

(i) 11,75 (j) 0,8

4. First complete each sequence in decimals, and then rewrite the sequence in fraction notation.

(a) 10; 9,8; 9,6; ; ; ; ; ; ; 10; 9₁₀⁸ ; 9₁₀⁶ ; ; ; ; ; ; ; (b) 0,15; 0,3; 0,45; ; ; ; ; ; ₁₀₀¹⁵ ; ₁₀³ ; ; ; ; ; ; ;

7.7 Comparing decimals

1. Below are the results of two events at an athletics championship.

For each event, arrange the names in order, starting with the winner.

(a) Boys under 19: 100 m sprint (time in seconds)

Temba Tshembe 11,9 s Con September 11,59 s Gavin Solomon 11,63 s NoahTshabalala 11,23 s Ivan Williams 11,4 s Manfred Ngcobo 11,57 s (b) Girls under 19: Long jump (distance in metres)

Kato Zuma 4,23 m Jane Sithole 4,51 m Lindi Xolani 4,5 m Pumla Makae 4,7 m Nthabi Faku 4,07 m Denise Galant 4,72 m

2. In each case, say which decimal you think is bigger and why.

(a) 0,6 or 0,06 (b) 4,6 or 4,60 (c) 0,3 or 0,43 (d) 0,3 or 0,23 (e) 7,42 or 7,24 (f) 5,6 or 5,57 (g) 0,4 or 0,40 (h) 3,45 or 3,5

3. Sometimes we can take away a zero in a number and it does not change the value of the number. But sometimes the value of the number does change if the zero is removed.

In each case, say whether or not we can take the zero away without changing the value of the number. Give a reason for your answer.

(a) 3,08 (b) 72,40 (c) 20,56

(d) 2,05 (e) 23,60 (f) 0,43

4. In each case, give a number that is between the two given numbers.

(a) 4,5 and 4,7 (b) 3,9 and 3,11 (c) 7,8 and 7,9 (d) 14 and 14,1 (e) 0 and 0,1

5. How many numbers are between 7,5 and 7,6?

7.8 Reading scales

Read the value indicated by each of the arrows on the number lines. For some of them you have to estimate as accurately as possible.

1. 0

(a) 1 2

(b) (c) (d) (e)

2. 0

(a) 1 2

(b) (c) (d) (e)

3. 0

(a) 1 2

(b) (c) (d) (e)

4. 0

(a) 1 2

(b) (c) (d)

5. 0

(a) 1 2

(b) (c) (d)

6. 0

(a) 1 2

(b) (c) (d)

7. 0

(a) 1 2

(b) (c) (d)

8. 2

(a) 3 4

(b) (c) (d)

9. 6

(a) 7

(b) (c) (d)

10. 9

(a) 10 11

(b) (c) (d)

11. Read the value indicated by each of the arrows on the scales. For some of them you have to estimate as accurately as possible.

Scale A Scale B

Scale C Scale D

7.9 Addition of decimals

Our number system is a decimal system. Ten is the basis of our number system. We count 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

The next number, 11, is 10 + 1. We extend the set of whole numbers to form the rational numbers, which include numbers less or smaller than 1.

Addition and subtraction of fractions written in decimal notation works in the same way as addition and subtraction of whole numbers.

1. In each case, add the fractions and then rewrite all of the fractions in decimal notation.

(a) ₁₀³ + ₁₀⁴ (b) ₁₀⁴ + ₁₀₀⁷ (c) ₁₀₀³⁶ + ₁₀₀⁵³ (d) ₁₀₀⁶ + ₁₀₀⁸

(a) (b)

(c) (d)

(e)

11a.

(a)

(b) (c) (d)

(e)

11b.

(a) (b)

(c) (d)

(e)

11c.

(a) (b)

(c)

(d) (e)

11d.

(a) (b)

(c) (d)

11e.

(a) (b)

(c) (d)

(e)

11a.

(a)

(b) (c) (d)

(e)

11b.

(a) (b)

(c) (d)

(e)

11c.

(a) (b)

(c)

(d) (e)

11d.

(a)

(b)

(c) (d)

11e.

(a) (b)

(c) (d)

(e)

11a.

(a)

(b) (c) (d)

(e)

11b.

(a) (b)

(c) (d)

(e)

11c.

(a) (b)

(c)

(d) (e)

11d.

(a) (b)

(c) (d)

11e.

(a) (b)

(c) (d)

(e)

11a.

(a)

(b) (c) (d)

(e)

11b.

(a) (b)

(c) (d)

(e)

11c.

(a) (b)

(c)

(d) (e)

11d.

(a) (b)

(c) (d)

11e.

2. First write all of the numbers in expanded notation. Then add the numbers and write the answer in decimal notation.

(a) 14,35 + 23,41 (b) 12,14 + 324,7 (c) 56,05 + 32,67 (d) 41,30 + 18,77

(e) 276,54 + 13,86 + 103,29

(f) 532,66 + 81,92 + 202,43 + 47,64

3. In this question you have to find the numbers that must be added to get to the target. Write your answers for (a), (b), (c) etc. in a list.

Now use your calculator to check your answers.

3,4 + (a) 3,9 + (b) 4,1 + (c) 4,4

+ (d)

7,1 + (g) 6 + (f) 5,7 + (e) 5

7,6 + (i) 8,1 + (j) 9,2 + (k) 10,3

+ (h)

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