concept of fractions

7/31/2019 Concept of Fractions

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Concept of Fractions, Naming Fractions

Materials

Teacher needs: 3-4 candy bars, same size but different kinds, knife

Student worksheets

Students need: scissors, crayons, glue

Note: Depending on how students do on the diagnostic test, the teacher may opt to combine some ofthese early lessons. It is much better to go step by step, however, even with the best students.

Anticipatory Set

Learning: Today you will learn what a fraction is and how to name fractions. You will also learn themeaning of "numerator" and "denominator." Write those terms on the board, with numerator on top.

Purpose: Who remembers some of the reasons people use fractions? Did anyone find any more?

Transfer: You probably already use fractions more than you realize. Name some ways youpersonally have used fractions.

Motivation: A suggestion: after students have completed the lesson successfully, offer to break thecandy into fractional parts and share it.

Check for Understanding: Some sample questions are below. It is important to check forunderstanding frequently throughout every lesson. I will indicate good places to check forunderstanding with a pound sign: #.

What will you learn today?

Why are fractions important?

What will you get if you do a good job today?

Teach

A fraction is an equal part of a whole.# (The asterisk means to ask: What is a fraction? In a classroom situation, all students mayanswer at once. This kind of back and forth communication keeps students engaged and ontask.)

For example, if I have one whole candy bar and I want to share it with a friend, what can Ido? (Divide it in half.)

Right. I'll do that. (Cut.)

Now can I share it with a friend?(Yes.)

The part I give to my friend is written like this: Write 1/2 on the board, with a horizontal ratherthan diagonal line.

The top number in a fraction is called the numerator. Point to the "1" on the board.# What is the top number called?

The bottom number in a fraction is called the denominator. Point to the "2" on the board.# What is the bottom number called?
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Write some fractions on the board and point to numerators and denominators at random,having children name them. Then erase the words and have the children continue to namethem.

The denominator stands for the number of pieces I cut the whole into. How many pieces arethere? (2) What does the denominator stand for?Another way to look at it is the denominator stands for how many people I have to share the

whole candy bar with. In this case, I'm sharing it with two people, so 2 is the denominator. The numerator stands for the number of pieces I gave my friend. How many pieces did I give

my friend? (1) What does the numerator stand for?

Now, what if I have two friends and myself and only one candy bar? (Divide it in threepieces.)How much of the candy bar do I get to eat? (1/3, review "numerator" and "denominator")How much do I give away? (2/3)

What if I have three other friends? Continue with examples until kids have the idea.

Non-example: The parts you divide a whole into must be equal. Say I have one friend to share thiscandy bar with. Divide the candy into unequal pieces. This wouldn't be fair, would it? Pieces are nothalves unless the whole candy bar is divided into two equal pieces?What is important to remember when dividing a whole into fractions? (They must be equal.)

Guided Practice

Now, you'll divide circles into fractional parts. # What will you do now? Pass out the papers.Let's read the directions together. List materials and steps. Get out the materials.

Standards: Remember to cut neatly and only use a spot of glue. Be very careful not to get glue onthe desk.

During guided practice, circulate among the students, giving assistance and reinforcement as needed.

Closure

Let's go over what we learned today.What is a fraction? (An equal part of a whole.)What part of the candy bar am I holding up? (1/2, 2/3, etc.)In the fraction 3/4, which is the numerator? Which is the denominator?

Identifying Fractions

Materials:

Several colored paper circles for the teacher to cut in half. Student worksheets.

Review:

Yesterday, we learned what a fraction is and how to name a fraction. Let's see ifyou remember. Show a whole circle, then fold it and cut it in half.
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How much of the circle is this? (1/2: write it on the board) What is the word for the bottom number of a fraction? (Denominator) What does the denominator stand for? (The number of pieces a whole is cut

into.) What is the word for the top number of a fraction? (Numerator)

What does the numerator stand for? (The number of pieces you're talking about.) So I cut this circle into two equal pieces and I have one. That's 1/2. What is a fraction? (An equal part of a whole.)

Anticipatory Set:

Learning: Today we are going to identify and write the names of many fractions. # Whatare we going to do?

Purpose: You need to learn to work with fractions other than halves, thirds, and fourths,so you can work problems with all kinds of fractions. # Why do you need to learn to workwith fractions other than halves, thirds and fourths?

Transfer: You will use the same things you learned yesterday, only with otherfractions. # What will you learn?

Teach

Let's say I take this circle and cut it like this. Cut into fourths. How much isthis? Hole up 1/4. How much is this? Hole up 3/4.

Now say I cut one like this. Cut into eighths.How many equal pieces have I cut it into? (Eight)What number in a fraction tells me how many equal pieces I've cut it into, thenumerator or the denominator? (Denominator) Write 8 on the board for thedenominator.How many pieces do I have? Hold up 3/8.Does the numerator or the denominator tell me how many pieces I'm talkingabout? (Numerator) Write 3 for the numerator. The final fraction is 3/8.

It works the same for all fractions.

To find the denominator, look at how many equal pieces the whole had beendivided into.To find the numerator, find how many pieces you're talking about.

Guided Practice

Now you will name and write fractions. # What will you do?I will give you a worksheet like this. Show worksheet.It has problems like this. Write a circle on the board divided into sixths, with 5 of

the 6 pieces shaded.


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Listen carefully to the directions: "Write the correct fraction for the shaded parts."What would be the answer for the problem on the board? (5/6) Do otherexamples as necessary.

Circulate among the students during guided practice to make sure theyunderstand. Congratulate those who are doing a good job.

Closure

Today you got better and faster at identifying and naming fractions. # What did you dotoday?What does the denominator stand for? (Number of equal pieces of the whole.)What does the numerator stand for? (Number of pieces you are talking about.


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Adding Fractions

There are 3 Simple Steps to add fractions:

Step 1: Make sure the bottom numbers (thedenominators)are the same

Step 2: Add the top numbers (thenumerators). Put the

answer over thedenominator.

Step 3: Simplify the fraction (if needed).

Example 1:

1+

1

4 4

Step 1. The bottom numbers (the denominators) are already the same. Go

straight to step 2.

Step 2. Add the top numbers and put the answer over the

same denominator:

1+

1=

1 + 1=

2

4 4 4 4

Step 3. Simplify the fraction:

2=

1

4 2

In picture form it looks like this:

1/4 +

1/4 =

2/4 =

1/2


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(If you are unsure of the last step seeEquivalent Fractions.)

Example 2:

1+

1

3 6

Step 1: The bottom numbers are different. See how the slices are different

sizes?

1/3 +

1/6 = ?

We need to make them the same before we can continue, because

we can't add them like that.

The number "6" is twice as big as "3", so to make the bottom numbers the

same we can multiply the top and bottom of the first fraction by 2, like this:

2

1=

2

3 6

2
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Important: you multiply both top and bottom by the same

amount, to keep the value of the fraction the same

Now the fractions have the same bottom number ("6"), and our question

looks like this:

2/6 +

1/6

The bottom numbers are now the same, so we can go to step 2.

Step 2: Add the top numbers and put them over the same denominator:

2+

1=

2 + 1=

3

6 6 6 6


2/6 +

1/6 =

3/6

Step 3: Simplify the fraction:

3=

1

6 2


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In picture form the whole answer looks like this:

2/6 +

1/6 =

3/6 =

1/2

With Pen and Paper

And here is how to do it with a pen and paper (press the play button):

Play with it!

Try theAdding Fractions Animation.

Example 3:

1+

1

3 5

Again, the bottom numbers are different (the slices are different sizes)!
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1/3 +

1/5 = ?

But let us try dividing them into smaller sizes that will each be the same:

5/15 +

3/15

The first fraction: by multiplying the top and bottom by 5 we ended up

with 5/15 :

5

1=

5

3 15

5

The second fraction: by multiplying the top and bottom by 3 we ended up

with 3/15 :

3

1=

3

5 15

3


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The bottom numbers are now the same, so we can go ahead and add the top

numbers:

5/15 +

3/15 =

8/15

Making the Denominators the Same

In the previous example how did we know to cut them into 1/15ths to make

the denominators the same? Read how to do this using either one of these

methods:

Common Denominator Method, or the

Least Common Denominator Method

They both work, use whichever you prefer!

Adding Mixed Fractions

I have a special (more advanced) page onAdding Mixed Fractions.

Subtracting Fractions

You might like to readAdding Fractionsfirst.

There are 3 simple steps to subtract fractions

Step 1. Make sure the bottom numbers (the denominators)

are the same
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Step 2. Subtract the top numbers (the numerators). Put the

answer over the same denominator.

Step 3. Simplify the fraction.

Example 1:

3

1

4 4

Step 1. The bottom numbers are already the same. Go straight to step 2.

Step 2. Subtract the top numbers and put the answer over the samedenominator:

3

1=

31=

2

4 4 4 4


2=

1

4 2

(If you are unsure of the last step seeEquivalent Fractions.)

Example 2:

1

1

2 6

Step 1. The bottom numbers are different. See how the slices are different

sizes? We need to make them the same before we can continue, because

we can't subtract them like this:

1/2 -

1/6 = ?


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To make the bottom numbers the same, multiply the top and bottom of the

first fraction (1/2) by 3 like this:

3

1=

3

2 6

3

And now our question looks like this:

3/6 - 1/6

The bottom numbers (the denominators) are the same, so we can go to step

2.

Step 2. Subtract the top numbers and put the answer over the same

denominator:

3

1=

31=

2

6 6 6 6


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3/6 -

1/6 =

2/6


2=

1

6 3

With Pen and Paper


Subtracting Mixed Fractions

I have a special page onAdding and Subtracting Mixed Fractions.

FRACTIONS

Subtraction

Back toFractions Units

Subtracting fractions is done differently than the usual numbers.

Normally, while adding or subtracting fractions you will find twotypes of problems:

Type 1: where the

fractions being added or

subtracted have the same

denominator eg.

Type 2: where the

fractions being added or

subtracted have different

denominators eg.

As you know, fractions represent parts of the whole. So, when theseparts are from the whole broken into same number of parts it is easy
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to subtract them.

For Type 1 problems, we just need to subtract the top parts

(numerator) of the fractions and leave the denominator as such. So,

to solve the problem in the above example the solution will be:

For Type 2 problems where the denominator is different, we can not

subtract these fractions by simply subtracting the numerators. In

order to solve these problems first we will need to make into

fractions with the same denominator. See how this is done

inexamplesbelow.

Examples

Example 1

changing both fractions to a common

Comments:

To solve this problem, first we will need to look

at the denominators 5 and 2 and see what

multiples of these numbers are common.

Multiples of 5 are:
http://www.staff.vu.edu.au/mcaonline/units/fractions/fractsub.html#Example%20Question:http://www.staff.vu.edu.au/mcaonline/units/fractions/fractsub.html#Example%20Question:http://www.staff.vu.edu.au/mcaonline/units/fractions/fractsub.html#Example%20Question:http://www.staff.vu.edu.au/mcaonline/units/fractions/fractsub.html#Example%20Question:


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denominator (see comments on the

right)

5 10 15 20 25

Multiples of 2 are:

2 4 6 8 10 12 14

so, in multiples of 5 and 2 the least commonmultiple is 10.

now, we will change 4/5 and 1/2 into their

equivalent fractions with denominator 10:

Example 2

Changing them to simple fractions we

find:

Now, changing both fractions to a

common denominator (see comments on

the right)we can solve it as:

This fraction can be further simplified as:

Comments:

It is always good to convert these mixed fractions

into simple fractions before working out

subtraction. We can do this using the techniqueshown on the right. Multiply the denominator

with the whole number and add the product to thenumerator. Re-write the fraction with this sum as

numerator.

Next, we need to look at the denominators and

see what multiples of them are common:

Multiples of 3:

3 6 9 12 15 18 21

Multiples of 5:

5 10 15 20

So, we find that 15 is the least common multiple.
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Have a Go

Problem 1 Problem 2
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Practice Questions

Question 1 Question 2


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Solution 1

First we will simply to make simple fractions

Now, we need to find the least common multiple of 3 and 4:

3= 3 6 9 12 15

4= 4 8 12 16

Now we need to convert both fractions to a common denomintor of 12

back to Have a Go

Solution 2

Here is a mixed problem. First we will change everything to a simple fraction then

solve the sum one step at a time.
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Now, find the least common multiple for 6, 3 and 4

6= 6, 12, 18

3= 3, 6, 9, 12, 15, 18

4= 4, 8, 12, 16

So, converting the fractions to a common denominator of 12 we get:

back to Have a Go

Multiplying Fractions

Multiply the tops, multiply the bottoms.

There are 3 simple steps to multiplyfractions

1. Multiply the top numbers

(the numerators).

2. Multiply the bottom numbers
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(the denominators).

3. Simplify the fraction if needed.

Example 1

1

2

2 5

Step 1. Multiply the top numbers:

1

2=

1 2=

2

2 5

Step 2. Multiply the bottom numbers:

1

2=

1 2=

2

2 5 2 5 10


2=

1

10 5

(If you are unsure of the last step seeEquivalent Fractions)

With Pen and Paper



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Example 2

1

9

3 16

Step 1. Multiply the top numbers:

1

9=

1 9=

9

3 16

Step 2. Multiply the bottom numbers:

1

9=

1 9=

9

3 16 3 16 48


9=

3

48 16

Multiplying Fractions

To Multiply Fractions:

Multiply the numerators of the fractions Multiply the denominators of the fractions Place the product of the numerators over the product of the

denominators Simplify the Fraction

Example: Multiply 2/9 and 3/12

Multiply the numerators (2*3=6) Multiply the denominators (9*12=108) Place the product of the numerators over the product of the

denominators (6/108) Simplify the Fraction (6/108 = 1/18)

The Easy Way. It is often simplest to "cancel" before doingthe multiplication. Canceling is dividing one factor of thenumerator and one factor of the denominator by the samenumber.

For example: 2/9 * 3/12 = (2*3)/(9*12) = (1*3)/(9*6) =

(1*1)/(3*6) = 1/18


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Dividing Fractions

Has A Weird Rule

Dividing fractions can be a little tricky. It's the only operation thatrequires using the reciprocal. Using the reciprocal simply meansyouflip it over, or invert it.

For example, the reciprocal of2/3 is 3/2.

After we give you the rule, we will attempt to explain WHY you have touse the reciprocal in the first place. But for now...

Here's the Rule for division...

To divide fractions, convert the division process to a multiplicationprocess by using the following steps.

1. Change the "" sign to "x" and invert the fraction to the right ofthe sign.

2. Multiply the numerators.3. Multiply the denominators.4. Re-write your answer in its simplified or reduced form, if needed

Once you complete Step #1 for dividing fractions, the problem actuallychanges from division to multiplication.

1/2 1/3 = 1/2 x 3/1

1/2 x 3/1 = 3/2

Simplified Answer is 1 1/2

Now that's all there is to it. The main thing you have to remember whenyou divide is to invert the fraction to the right of the division sign, andchange the sign to multiplication.

The "divisor" (like 1/3 in our example) has some otherconsideration that you should keep in mind...

Special Notes!


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Remember to only invert the divisor.

The divisor's numerator or denominator can not be "zero".

We must convert the operation to multiplication BEFORE

performing an cancellations.

I promised to try to explain why the rule requires inverting the divisor.

Here goes..

Why Dividing FractionsRequires Inverting The Divisor

Let's use our simple example to actually validate this strange Rule fordivision.

If you really think about it, we are dividing a fraction by a fraction,which forms what is called a "complex fraction". It actually looks likethis...

When working with complex fractions, what we want to do first is get ridof the denominator(1/3), so we can work this problem easier.

You may recall that any number multiplied by its reciprocal is equal to 1.

And since, 1/3 x 3/1 = 1, we can use the reciprocal property of 1/3(3/1) to make the value of the denominator equal to 1.

But, you might also remember that whatever we do to the denominator,we must also do to the numerator, so as not to change the overall"value".

So let's multiply both the numerator and denominator by 3/1...


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Which gives us...

Here's what happened...

By multiplying the numerator and denominator by 3/1, we were then ableto use the reciprocal property to eliminate the denominator. Actually,without our helpful Rule, we would have to use all of the steps above.

So, the Rule for dividing fractions really saves us a lot of steps!

Now that's the simplest explanation I could come up withfor WHYand HOW we end up with a Rule that says we must invert thedivisor!

Best regards

Dividing Fractions

Turn the second fraction upside down, then just multiply.

There are 3 Simple Steps to Divide Fractions:

Step 1. Turn the second fraction(the one you want to divide by)upside-down

(this is now areciprocal).

Step 2.Multiplythe first fraction by that

reciprocal

Step 3.Simplifythe fraction (if needed)

Example 1
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1

1

2 6

Step 1. Turn the second fraction upside-down (it becomes a reciprocal):

1becomes

6

6 1

Step 2. Multiply the first fraction by that reciprocal:

1

6=

1 6=

6

2 1 2 1 2


6= 3

2

With Pen and Paper


Does it make sense?

Does1

1

really equal 3 ?2 6

You can change a question like "What is 20 divided by 5?" into "How many 5s

fit into 20?"

In the same way our fraction question can become:


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1

1How many

1in

1?

2 6 6 2

Now look at the pizzas below ... how many "1/6th slices" fit into a "1/2

slice"?

How many in ? Answer: 3

So now you can see that1

1

= 3 really does makes sense!2 6

Example 2

1

1

8 4

Step 1. Turn the second fraction upside-down (the reciprocal):

1becomes

4

4 1

Step 2. Multiply the first fraction by that reciprocal:

1

4=

1 4=

4

8 1 8 1 8


4=

1

8 2

And that is all you have to do.


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But maybe you want to know why we do it this way ...

Why Turn the Fraction Upside Down?

Well ... what Does a Fraction Do?

A fraction says to:

multiply by the top number

divide by the bottom number

xample:3

/4 means to cut into 4 pieces, and then take 3 of those.o you:

ivide by 4

ultiply by 3

Example: 3/4 of 20 is:

20 divided by 4, then times 3 = (20/4) 3 = 5 3 = 15

Or you could multiply before dividing:

20 times 3, then divide by 4 = (20 3) / 4 = 60/4 = 15Either way gets the same result

Dividing

But when you DIVIDE by a fraction, you are asked to do the opposite of

multiply ...

So you:

divide by the top number

multiply by the bottom number

Example: dividing by 5/2 is the same as multiplyingby 2/5


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Because:

Dividing by 5, then Multiplying by 2

is the same as

Multiplying by 2, then Dividing by 5

So instead of dividing by a fraction, it is easier to turn that fraction upside

down, then do a multiply.

Tenths as Decimals

Decimals are a method of writing fractional numbers without writing a fractionhaving a numerator and denominator.

The fraction 7/10 could be written as the decimal 0.7. The period or decimalpoint indicates that this is a decimal.

The decimal 0.7 could be pronounced as SEVEN TENTHS or as ZERO POINTSEVEN.

If a decimal is less than 1, place a zero before the decimal point. Write 0.7 not.7

Tenths as Decimals

Decimals are a method of writing fractional numbers without writing a fractionhaving a numerator and denominator.

The fraction 7/10 could be written as the decimal 0.7The period or decimal point indicates that this is a decimal.

The decimal 0.7 could be pronounced as SEVEN TENTHS or as ZERO POINTSEVEN.

There are other decimals such as hundredths or thousandths. They all arebased on the number ten just like our number system.

A decimal may be greater than one. The decimal 3.7 would be pronouncedas THREE AND SEVEN TENTHS.

Decimals

A Decimal Number (based on the number10

) contains a Decimal Point.


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Place Value

To understand decimal numbers you must first know aboutPlace Value.

When we write numbers, the position (or "place") of each number is

important.

In the number 327:

the "7" is in the Units position, meaning just 7 (or 7 "1"s),

the "2" is in the Tens position meaning 2 tens (or twenty),

and the "3" is in the Hundreds position, meaning 3 hundreds.

"Three Hundred Twenty Seven"

As we move left, each position is 10 times bigger!

From Units, to Tens, to Hundreds

... and ...

As we move right, each position is 10 times smaller.

From Hundreds, to Tens, to Units

But what if we continue past Units?

What is 10 times smaller than

Units?

1

/10 ths (Tenths) are!
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But we must first write a decimal point,

so we know exactly where the Units position is:

"three hundred twenty seven and four tenths"

but we usually just say "three hundred twenty seven point four"

And that is a Decimal Number!

Have a play with decimal numbers yourself:

View Larger

Decimal Point

The decimal point is the most important part of a Decimal Number. It is

exactly to the right of the Units position. Without it, we would be lost ... andnot know what each position meant.
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Now we can continue with smaller and smaller values, from tenths,

to hundredths, and so on, like in this example:

Large and Small

So, our Decimal System lets us write numbers as large or as small as we

want, using the decimal point. Numbers can be placed to the left or right of a

decimal point, to indicate values greater than one or less than one.

17.591The number to the left of the decimal point is

awhole number(17 for example)

As we move further left, every number place gets 10 times

bigger.

The first digit on the rightmeans tenths (1/10).

As we move further right, every number place

gets 10 times smaller (one tenth as big).

Play with it ...
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See decimals on the Zoomable Number Line

Definition of Decimal

The word "Decimal" really means "based on 10" (From Latin decima: a

tenth part).

We sometimes say "decimal" when we mean anything to do with our

numbering system, but a "Decimal Number" usually means there is a

Decimal Point.

Ways to think about Decimal Numbers ...

... as a Whole Number Plus Tenths, Hundredths, etc

You could think of a decimal number as a whole number plus tenths,

hundredths, etc:

Example 1: What is 2.3 ?

On the left side is "2", that is the whole number part.

The 3 is in the "tenths" position, meaning "3 tenths", or 3/10

So, 2.3 is "2 and 3 tenths"

Example 2: What is 13.76 ?

On the left side is "13", that is the whole number part.

There are two digits on the right side, the 7 is in the "tenths" position,

and the 6 is the "hundredths" position

So, 13.76 is "13 and 7 tenths and 6 hundredths"

... as a Decimal Fraction

Or, you could think of a decimal number as a Decimal Fraction.

A Decimal Fraction is a fraction where the denominator (the bottom number)

is a number such as 10, 100, 1000, etc (in other words apower of ten)
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So "2.3" would look like this:23

10

And "13.76" would look like this:1376

100

... as a Whole Number and Decimal Fraction

Or, you could think of a decimal number as a Whole Number plus a Decimal

Fraction.

So "2.3" would look like this: 2and3

10

And "13.76" would look like this: 13 and76

100

Those are all good ways to think of decimal numbers.

concept of fractions

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