This lesson helps primary school students to better understand decimal numbers.

The most common test is the question “Which number is bigger? 0.650 or 0.65”. Most kids
will give the wrong answer “0.650 is greater than 0.65”. Surprisingly most will get this
question right “Are these two numbers the same? 0345 and 345”.

Why the zeros before the whole part and the zeros after the decimal part of a decimal

number do not matter.

~~000000000000~~345.65~~000000000000000~~

Decimal numbers are written according to some rules. The decimal rules are also consistent

with normal whole numbers. A decimal number can be thought of as two numbers plus
together. The first number is the whole part, and the other one is the decimal part. Therefore
3.45 is 3 plus with .45

The leading zeros

Let’s look at a normal whole number: 345

Hundreds	Tens	Units(ones)
3	4	5

We can break the number up to see how the number 345 is constructed.The construction of a number 345 actually means

3 of 100s + 4 of 10s + 5 of ones.

Now imagine extending this number 345 to show some hidden numbers. These numbers have been taken away because they have no real value at all

Thousands	Hundreds	Tens	Units(ones)
0	3	4	5

Similarly the construction of the number 0345 is

0 of 1000s + 3 of 100s + 4 of 10s + 5 of ones

We can see that 0 of 1000s means zero. So we do not count the number of 0s leading a number.

The trailing zeros after the decimal part of a decimal number

Let’s look at this number 0.650

Decimal point	Tenths/10th	Hundredth/100th	Thousandth/1000th
.	6	5	0

The construction of this decimal part of a decimal number means

6/10 + 5/100 + 0/1000

We can see that 0 out of 1000 is nothing. So we can ignore this 0. What it means is that 0.65 is the same as 0.650

Similarly 0.6500 is the same as 0.65 because it means

6/10 + 5/100 + 0/1000 + 0/10000

Something to Do

Questions	Answers
Is 0.34 = 0.340 ?	True
Is 345 = 3450 ?	False
Is 0345 = 345 ?	True
Which one is greater?
.09 or 0.10	0.10
0.0999999 or 0.10000000	0.10000000
3.01 or 2.99	3.01
0345 or 346	346
Is 16 x 10 = 016 ?	False
Is 160.0 = 160 ?	True

Then link this knowledge with decimal fractions. If the students understand fractions well

enough to know that multiplying both numerator and denominator with the same number will
not alter the value of the fraction, this proof will help them to understand the above rules in a
different light.

If we know that 1/2 is the same as 1x2 / 2x2 = 2/4, then 1/10 is the same as 1x10 / 10x10 =

10/100

0.1 is 1/10

0.10 is 10/100

Logically, by deduction, if 10/100 is the same as 1/10, 0.10 must be the same as 0.1

The proof can be repeated with numbers 0.1 and 0.100

Work Sheet 1

1) 0.6 + 0.2 =

2) 0.9 - 0.1 =

3) Write three tenths as a decimal.

4) Write 0.4 in fractions.

5) 1.0 - 0.9 =

6) What is two tenths add 0.6?

7) 0.1 + 0.7 =

8) 0.8 + 0.1 =

9) 0.3 - 0.1 =

10) 0.5 + 0.1 - 0.2 =

11) 0.7 - 0.3 =

12) What is three tenths add 0.6?

13) Write 0.3 as a fraction.

14) 0.4 + 0.1 + 0.1 =

15) 0.9 - 0.3 =

16) What is 0.1 add five tenths?

17) Write ten tenths as a decimal.

18) 0.3 + 0.3 - 0.2 =

19) What is nine tenths subtract 0.4?

20) 0.6 + 0.1 =

21) 0.5 + 0.5 =

22) What is two tenths add 0.5, subtract 0.3?

23) Write 1.0 as a fraction

24) What is 0.5 add one tenth?

25) 0.6 + 0.2 + 0.1 - 0.4 =

Work Sheet 2

1) 0.3 + 0.5 =

2) 0.7 - 0.3 =

3) Write two tenths as a decimal.

4) Write 0.9 as a fraction.

5) 1.0 - 0.3 =

6) What is three tenths add 0.5?

7) 0.1 + 0.9 =

8) 0.2 + 0.4 =

9) 0.7 - 0.4 =

10) 0.6 + 0.4 - 0.2 =

11) 0.7 - 0.5 =

12) What is five tenths add 0.2?

13) Write 0.2 as a fraction.

14) 0.3 + 0.2 + 0.2 =

15) 0.4 - 0.1 =

16) What is 0.7 add one tenth
17) Write six tenths as a decimal.

18) 0.4 + 0.1 - 0.2 =

19) What is five tenths subtract 0.2?

20) 0.2 + 0.6
21) 0.4 + 0.3 + 0.2 =

22) What is three tenths add 0.7, subtract 0.2?

23) Write 0.9 as a fraction.

24) What is 0.2 add five tenths
25) 0.8 + 0.1 - 0.3 - 0.4 =

Larger Decimals

There are two activities in this section:

A) How to introduce decimals larger than 1.0 but smaller than 2.0

1) Draw a rectangle on the board and split it into ten sections:

2) Colour in one section at a time and ask a child to identify how much of the rectangle is

coloured (giving their answer in fractions and decimals).

3) When you have coloured in all ten sections, draw another rectangle and split it into ten

sections again. Then colour in one of the sections of this new rectangle.

4) Explain that you now have one unit and one tenth coloured. Ask how they think we might

be able to write down that number (in fractions or decimals), i.e. 1 and 1/10 or 1.1. Repeat

this activity, colouring in a few more sections of the second rectangle. Discuss how these

numbers can be written.

5) Give each student a copy of the following Decimals Strips :

Decimals Strips

1/10

Ask them to cut out one whole strip and colour it in. Then they should cut out another strip

and only colour a few of the sections in. After sticking the strips in their books, they can then

write information about these strips, describing the number that they have shown with their

colouring, e.g.

One unit and 9 tenths are coloured.
This can also be written as 1 9/10 or 1.9 (which we say as "One Point Nine")

Matching larger fractions and decimals

This activity can be carried out as a whole class with the table (shown below) drawn on the

board (and with individual children coming to the front to complete the activity) or

individually with children completing the following worksheet :

Matching Fractions and Decimals

Fill in the blank spaces to complete the table.

Fraction (words)	Fraction (figures)	Decimal (figures)	Decimal (words)
One Unit and One Tenth	1 1/10	1.1	One Point One
One Unit and Two Tenths	1 2/10	1.2	One Point Two
		1.3
	1 4/10
			One Point Five
One Unit and Six Tenths
		1.7
	1 8/10
		1.9

The aim of the activity is to develop children's understanding of the relationship between

these larger fractions and decimals.

Fraction (words)	Fraction (figures)	Decimal (figures)	Decimal (words)
One Unit and One Tenth	1 1/10	1.1	One Point One
One Unit and Two Tenths	1 2/10	1.2	One Point Two
		1.3
	1 4/10
			One Point Five
One Unit and Six Tenths
		1.7
	1 8/10
		1.9
			Two Point Zero

Your students should complete the table by filling in the blank spaces. Numbers in each row

have the same value (i.e. One Unit and One Tenth = 1 1/10 = 1.1 = One Point One). The third
row should therefore look like this:

One Unit and Three Tenths

1 3/10

1.3

One Point Three

Addition and Subtraction of Larger Decimal

This activity involves a worksheet containing addition and subtraction questions involving
numbers from 0.0 to 2.0. The worksheet can be used in the following ways:

1. Each child can work through the worksheet individually at his own pa

2. You could have a decimals race, with individual children (or small groups of children)

trying to work out the answers to the sheet in the quickest time (although accuracy is
most important).

Decimals Sheet

1) 0.8 + 0.2 =

2) 0.9 - 0.4 =

3) 0.5 + 0.8 =

4) 0.6 + 0.9 =

5) 1.7 - 0.9 =

6) 1.5 + 0.3 =

7) 0.5 + 0.7 =

8) 1.8 + 0.1 =

9) 1.3 - 0.1 =

10) 1.5 + 0.1 - 0.2 =

11) 1.7 - 0.6 =

12) 2.0 - 0.7 =

13) 1.9 - 0.9 =

14) 0.4 + 0.6 + 0.4 =

15) 1.9 - 1.2 =

16) 0.7 + 0.4 =

17) 0.8 + 0.7 =

18) 0.9 + 0.2 + 0.3 =

19) 0.4 + 1.3 =

20) 0.7 + 0.1 =

21) 0.5 + 0.5 =

22) 1.5 + 0.5 =

23) 1.6 - 0.4 =

24) 1.5 - 0.6 =

25) 1.6 + 0.2 - 0.4 =

26) 2.0 - 1.9 =

27) 1.5 - 0.7 =

28) 1.7 - 0.2 + 0.3 =

29) 0.3 + 1.6 =

30) 1.2 + 0.4 =

31) 1.5 + 0.2 =

32) 1.6 - 0.8 =

33) 1.9 - 1.4 =

34) 1.2 - 0.5 =

35) 1.7 + 0.3 =

36)     0.3 + 1.5 =

37)     0.7 + 1.2 =

38)    0.9 + 1.0 =

39)     2.0 - 0.5 + 0.2 =
40) Write 1 unit and 2 tenths as a decimal.

41) Write 1.7 as a fraction.
42) Write 2 units as a decimal
.
43) Write 0.7 as a fraction.

44) Write 1 unit and 5 tenths as a decimal.
45) Write 1.9 as a fraction

Answers:

1) 1.0	10) 1.4	19) 1.7	28) 1.8	37) 1.9
2) 0.5	11) 1.1	20) 0.8	29) 1.9	38) 1.9
3) 1.3	12) 1.3	21) 1.0	30) 1.6	39) 1.7
4) 1.5	13) 1.0	22) 2.0	31) 1.7	40) 1.2
5) 0.8	14) 1.4	23) 1.2	32) 0.8	41) 1 7/10
6) 1.8	15) 1.7	24) 0.9	33) 0.5	42) 2.0
7) 1.2	16) 1.1	25) 1.4	34) 0.7	43) 7/10
8) 1.9	17) 1.5	26) 0.1	35) 2.0	44) 1.5
9) 1.2	18) 1.4	27) 0.8	36) 1.8	45) 1 9/10

Even Bigger Decimals!

Thus far, we have covered decimals which are smaller than 2.0. The following activities
should help when teaching your students about decimals larger than 2.0 :

Number Line

Draw a number line on the board (from 0.0 to 10.0, with markings for each tenth). Point at a
marking on the number line and ask the students what that number is. They can answer in
decimals or fractions (depending on your choice) and you could ask them to aim to reply as
quickly as possible.

What's the Number

Draw a number line on the board (from 0.0 to 10.0, with markings for each tenth). Explain
that you are starting at a certain point (e.g. 5.2). Tell the children that you move five tenths
forward and 9 tenths backwards. Ask them where you are now. Repeat, giving more
complicated instructions each time.

Match the Numbers

This activity requires students to draw lines from a fraction to the decimal of the same value.
Some of the decimals and fractions may be written in words rather than figures.

Match the Numbers

1.2				Four Tenths					2.1
			19.6
							5.7		5 6/10
	Twelve and One Tenth
				13.5				Nineteen and Six Tenths
	99 9/10
		2 1/10
						1 2/10		13 5 /10

	5 7/10			Five Point Six			12.1

									5.9

	Five and Nine Tenths				0.4			Ninety-Nine Point Nine

Matching Fractions and Decimals

This activity is similar to previous activities in this section, requiring your students to
complete the spaces in a table. Numbers are not in numerical order.

Match the Numbers
Complete the table:

Fraction (words)	Fraction (figures)	Decimal (figures)	Decimal (words)
One Unit and Two Tenths	1 2/10	1.2	One Point Two
Two Units and Four Tenths	2 4/10		Two Point Four
	3 5/10	3.5
Five Units and Six Tenths
		5.9
			Six Point Seven
	7 1/10
Eight Units
		8.6
	12 5/10

Answers are as follows:

Fraction (words)	Fraction (figures)	Decimal (figures)	Decimal (words)
One Unit and Two Tenths	1 2/10	1.2	One Point Two
Two Units and Four Tenths	2 4/10	2.4	Two Point Four
Three Units and Five Tenths	3 5/10	3.5	Three Point Five
Five Units and Six Tenths	5 6/10	5.6	Five Point Six
Five Units and Nine Tenths	5 9/10	5.9	Five Point Nine
Six Units and Seven Tenths	6 7/10	6.7	Six Point Seven
Seven Units and One Tenth	7 1/10	7.1	Seven Point One
Eight Units	8	8	Eight
Eight Units and Six Tenths	8 6/10	8.6	Eight Point Six
One Ten, Two Units and Five Tenths	12 5/10	12.5	Twelve Point Five

Decimals Revision Quiz

This quiz can be carried out as a worksheet exercise or as a test. You can call out the questions and let you students write answers in their books / on paper.

Decimals Quiz

Addition	Subtraction	Rounding to the nearest unit
0.1 + 0.7 = __ 0.5 + 0.9 = __ 1.9 + 5/10 = __ 3.6 + 9/10 = __ 8.9 + 6.4 = __ 17.6 + 9 4/10 = __ 31.6 + 19.3 = __ 12.7 + 72.1 = __ 19.6 + 18.4 = __ 57 1/10 + 29 9/10 = __	0.5 - 0.2 = __ 0.9 - 0.5 = __ 1.2 - 0.6 = __ 4.7 - 3.1 = __ 5.4 - 3.2 = __ 12.7 - 9.8 = __ 31.6 - 10.9 = __ 54.0 - 17.5 = __ 92.1 - 15 6/10 = __ 25 7/10 - 12 8/10 = __	6.3 9.4 9.9 0.6 12.3 31.7 57.6 91.2 84.5 97.0
Answer these questions... I have one unit and seven tenths. Write this number in decimals. What is nine units and five tenths in decimals? Why do we write the decimal point in the number 6.4? How many tens, units and tenths are there in 75.6? Why do we write the zero in 0.2?

Answers are as follows...

Addition	Subtraction	Rounding to the nearest unit
0.1 + 0.7 = 0.8 0.5 + 0.9 = 1.4 1.9 + 5/10 = 2.4 3.6 + 9/10 = 4.5 8.9 + 6.4 = 15.3 17.6 + 9 4/10 = 27 31.6 + 19.3 = 50.9 12.7 + 72.1 = 84.8 19.6 + 18.4 = 38 57 1/10 + 29 9/10 = 87	0.5 - 0.2 = 0.3 0.9 - 0.5 = 0.4 1.2 - 0.6 = 0.6 4.7 - 3.1 = 1.6 5.4 - 3.2 = 2.2 12.7 - 9.8 = 2.9 31.6 - 10.9 = 20.7 54.0 - 17.5 = 36.5 92.1 - 15 6/10 = 76.5 25 7/10 - 12 8/10 = 12.9	6.3 (6) 9.4 (9) 9.9 (10) 0.6 (1) 12.3 (12) 31.7 (32) 57.6 (58) 91.2 (91) 84.5 (84 or 85) 97.0 (97)
Answer these questions... I have one unit and seven tenths. Write this number in decimals. (1.7) What is nine units and five tenths in decimals? (9.5) Why do we write the decimal point in the number 6.4? (to separate the units and the tenths) How many tens, units and tenths are there in 75.6? (7 tens, 5 units and 6 tenths) Why do we write the zero in 0.2? (to remind us that the number is less than one)

Introducing Hundredths

1) Draw a square on the board. Tell the children that you want them to think of that square as
    one unit.

2) Split the square into ten rectangular sections. Ask them what fraction of the unit each of these
     sections is worth (i.e. tenths).

3) Now split each of the ten sections into ten squares (there should now be 100 small squares  inside
    the large square). Explain that we can split tenths up even more. The small sections are called
    hundredths.

4) Explain the following concepts:

· There are ten hundredths in one tenths and one hundred hundredths in one unit.

· Hundredths are written to the right of the tenths column when writing numbers in figures, i.e.

Units	Decimal Point	Tenths	Hundredths
3	.	6	3
9	.	5	2

· Hundredths are said in the following way:

7.53 = Seven Point Five Three (NOT seven point fifty three)

9.39 = Nine Point Three Nine (NOT nine point thirty nine)

· 4.70 is the same as 4.7. We don't have to write in the zero, but it is good practice to.

Which is Bigger?

This is a simple exercise which uses the worksheet . It requires students to look at the pair of
numbers in each row and decide which of them is bigger. They can indicate their answer by circling the larger number.

Which is Bigger?

Circle the number in each row which is of greater value.

1.12

4.32

7.61

9.12

5.30

6.08

4.42

8.19

12.79

64.7

12.14

8.42

7.32

10.11

5.32

6.10

4.4

8.20

13.79

63.75

Answers are as follows: (larger numbers are underlined)

1.12
4.32
7.61
9.12
5.30
6.08
4.42
8.19
12.79
64.7

12.14
8.42
7.32
10.11
5.32
6.10
4.4
8.20
13.79
63.75

Sequencing Activity

Another simple exercise, which asks children to put a set of numbers in order (from
smallest to largest).

Put these numbers into order, with the smallest on the left and the largest on the right.

7.52

6.19

2.53

1.48

0.14

10.54

4.5

Write the answer in the boxes below:

The correct answer is:

0.14

1.48

2.53

4.5

6.19

7.52

10.54

Matching Exercise

Matching Decimals and Fractions

Complete the table by filling in the spaces. Numbers on each row have the same value.

Fraction (words)	Fraction (figures)	Decimal (figures)	Decimal (words)
6 units, 8 tenths and 9 hundredths	6 8 9/100	6.89	Six Point Eight Nine
		2.71
			Three Point Four Two
	7 9 1/100
1 unit, 3 tenths and 4 hundredths
		19.42
			Eight Point Six Four
	9 1 2/100
		3.02
3 units, 4 tenths and 0 hundredths

This exercise is similar to previous exercises involving tenths. Your students should fill in the
blank spaces on the table, showing how numbers of the same value can be written in different
ways. The completed table should look like this:

Fraction (words)	Fraction (figures)	Decimal (figures)	Decimal (words)
6 units, 8 tenths and 9 hundredths	6 89/100	6.89	Six Point Eight Nine
2 units, 7 tenths and 1 hundredth	2 71/100	2.71	Two Point Seven One
3 units, 4 tenths and 2 hundredths	3 42/100	3.42	Three Point Four Two
7 units, 9 tenths and 1 hundredth	7 91/100	7.91	Seven Point Nine One
1 unit, 3 tenths and 4 hundredths	1 34/100	1.34	One Point Three Four
1 ten, 9 units, 4 tenths and 2 hundredths	19 42/100	19.42	Nineteen Point Four Two
8 units, 6 tenths and 4 hundredth	8 64/100	8.64	Eight Point Six Four
9 units, 1 tenth and 2 hundredths	9 12/100	9.12	Nine Point One Two
3 units, 0 tenths and 2 hundredths	3 2/100	3.02	Three Point Zero Two
3 units, 4 tenths and 0 hundredths	3 4/10	3.4	Three Point Four

If the children have completed the previous matching exercises, it is important to remind
them that the rows in this table are not in order. The other tables went in numerical order (e.g.
1/10, 2/10, 3/10 etc). The numbers on this table are jumbled.

Decimals Place Value

Numeral	M	HTh	TTh	Th	H	T	U	.	0.1 1/10	0.01 1/100	0.001 1/1000
								.
								.
								.
								.
								.
								.
								.
								.
								.
								.
								.
								.

On the left of the sheet, different numbers should be written (involving figures ranging from
thousandths to millions). Your students should then rewrite this number on the other side of
the sheet, by splitting the number into the following parts:

Millions
Hundreds of thousands
Tens of thousands
Thousands
Hundreds
Tens
Units
Tenths
Hundredths
Thousandths

Decimals Problems

This self-explanatory worksheet contains a variety of questions involving decimals.

Decimals

1) Put these decimals in order with the smallest first.

a) 5.25, 15.3, 5.87, 5.78, 5.2.

b) 1.5, 1.375, 1.4, 1.3, 1.35, 1.425.

c) 7.765, 7.675, 6.765, 7.756, 6.776.

2) Add decimals by rounding eg 7.4 + 9.8 could be added as 7.2 + 10.Work out the following
    showing rounding in your answers.

a) 6.9 + 7.6 =

b) 10.7 + 14.3 =

c) 29.3 + 15.8 =

3) Convert the following metric units.

a) 3.5 kg into g

b) 11.25 l into ml.

c) 750g into kg

d) 300ml into l

e) 3 cm into m.

4) The train leaving platform 1 at 14.25pm will arrive at 1607. How long will the journey
    take?

5) The price of an article in a shop has been reduced by 5%. If the article’s original price was
   $25.00, what is the new cost of the article ?

6) If 3 articles costing the same amount each comes to $ 870, what is the cost of 2 articles?

7) A rectangle measures 3m long by 1.5m wide. What is its area in m².

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Saturday, April 28, 2012

Better Understanding of Decimal Numbers