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Why are the powers of 10 used for writing numbers in scientific notation? 

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Why are the powers of 10 used for writing numbers in scientific notation? 

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When you are writing it out in standard form, it is easy to move your decimal places right or left depending on the exponent. If you had any other number, it would be trickier to change numbers from and to scientific notation.

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What difference do parentheses make when using exponents? Give examples in your explanation.

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What difference do parentheses make when using exponents? Give examples in your explanation.

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When there is an exponent next to parentheses and a coefficient and variable inside, the exponent applies to both coefficient variable. If there was no parentheses, the exponent would only apply to the variable.

Ex. (3a)^2= 3a x 3a
3a^2= 3 x a x a

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Definition - When answers are too long to be written all the way out, they are written in scientific notation.

Example Problems - 9.13x10^3= 9,130
9.13x10^-3= 0.00913

The image shows another example of how to write something in standard form when in scientific notation.

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Definition - Exponent that is zero. Always equals one

Example Problems -
1,000,000^0 = 1
.23749283749274^0 = 1
x^0 = 1

This image shows how anything to the power of zero is one

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Definition - How many times you multiple the base
Example Problem - 5^3 = 5x5x5=125

This image shows how the base to the power of six is equal to 2x2x2x2x2x2

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Definition - The number you multiply with

A base usually has an exponent, which is when you multiply the base by itself that many times (Describe how the word relates to other vocabulary/concepts)

The picture shows the base apart from an exponent

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2(lw)+2(lh)+2(wh)
2(3x9)+2(9x5)+2(5x3)=228

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b^2+2(bH)
(H=Slant Height)

6^2x2(6x5)=2160

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2(3.14xr^2)+3.14dH
2(3.14x5^2)+3.14x10x10=471

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bhxH(S1+S2+S3)
4x3+11(3+5+4)=144

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bhxH
10x10x12=1200

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4/3x3.14xr^3

4/3x3.14x6^3=904.32

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3.14xr^2xH/3

3.14x7^2x18/3=923.16

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3.14xR^2xH

3.14x5^2x10=12324.5

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0.5xbxhxH

0.5x4x3x11=236.5

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3.14x2R
3.14x40=125.6

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3.14xR^2

3.14x20^2=1256

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0.5xh(b1+b2)
0.5x6(4+9)=39

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bh
6x5=30

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bh
10x13=130
20x26=520

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1/2(BH)

1/2(3x5)= 7.5

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What happens to the area when the dimensions of a two-dimensional shape are changed?  Include an example of two different shapes that support your explanation.

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What happens to the area when the dimensions of a two-dimensional shape are changed?  Include an example of two different shapes that support your explanation.

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When you multiply the radius of a circle by two, the area will quadruple.
When you multiply one side of a square by two , the area also quadruples.
So when you multiply the sides by two, the area will quadruple.

And if you multiply the scale factor you use on the dimensions by itself, you get the scale factor of the area.

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Explain the difference in finding the area of inscribed shapes and composite shapes.  Include an example problem and answer for both types to support your explanation.

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Explain the difference in finding the area of inscribed shapes and composite shapes.  Include an example problem and answer for both types to support your explanation.

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When finding the area of composite figures, you have to find the area of the two shapes that make the figure and add them together.

Ex. The first figure below is a composite figure. The area of the rectangle is 35ft and the area of the triangle is 17.5ft. The total area would be 52.5ft.

To find the area of an inscribed figure, you find the area of the shaded region by subtracting the area of the inside shape from the area of the bigger shape.

Ex. In the second figure below, the area of the circle is about 50.24. The area of the square is 64. The area of the shaded region would be 13.76

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<ABD is adjacent to angle <DBC

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This represents this theorem because 3+5>7

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This represents the triangle sum theorem because all the angles in the triangle equal 180

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This is a drawing of supplementary angles because 130+50=180

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The red line is a transversal because it intersects both black lines

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This is two complementary angles because 35+55=90

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This link explains what an adjacent angle is and how to find them

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Two angles are Adjacent when they have a common side and a common vertex (corner point) and don't overlap.

Angle ABC is adjacent to angle CBD

Because:

they have a common side (line CB)

they have a common vertex (point B)

What Is and Isn't an Adjacent Angle

These ARE Adjacent Angles

they share a vertex and a side

NOT Adjacent Angles

they only share a vertex, not a side

NOT Adjacent Angles

they only share a side,
not a vertex

Don't Overlap!

ALSO the angles must not overlap.

NOT Adjacent Angles

angles a and b overlap

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Two angles are Adjacent when they have a common side and a common vertex (corner point) and don't overlap. Angle ABC is adjacent to angle CBD Because: they have a common side (line CB) they have a common vertex (point B) These ARE Adjacent Angles they share a vertex and a side NOT Adjacent Angles they only share a vertex, not a side NOT Adjacent Angles they only share a side, not a vertex ALSO the angles must not overlap.

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Two angles that have a common vertex and a common side

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This link explains what complementary angles are and how they are different from supplementary angles

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Two angles are Complementary when they
add up to 90 degrees (a Right Angle )

These two angles (40° and 50°) are Complementary Angles, because they add up to 90°:

Notice that together they make a right angle

But the angles don't have to be together.

These two are complementary because 27° + 63° = 90°

Play With It ...

(Drag the points)

images/geom-comp-supp.js?mode=comp

Right Angled Triangle

In a right angled triangle, the two non-right angles are complementary, because in a triangle the three angles add to 180°, and 90° has already been taken by the right angle.

When two angles add to 90°, we say they "Complement" each other.

Complementary comes from Latin completum meaning "completed" ...
because the right angle is thought of as being a complete angle.

Spelling: be careful, it is not "Complimentary Angle" (with an "i") ...
that would be an angle you get for free!

Complementary vs Supplementary

A related idea is Supplementary Angles - those add up to 180°

How to remember which is which? Well, alphabetically they are:

Complementary add to 90°

Supplementary add to 180°

You can also think:

"C" of Complementary is for "Corner" (a Right Angle), and

"S" of Supplementary is for "Straight" (180° is a straight line)

Or you can think:

when you are right you get a compliment (sounds like complement)

"supplement" (like a vitamin supplement) is something extra, so is bigger

5231, 5233, 819, 2109, 820, 2110, 5240, 9063, 3305, 3306

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Two angles are Complementary when they add up to 90 degrees (a Right Angle )

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Complementary angles are two or more angles that equal 90 degrees

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This link explains what supplementary angles are and how they are different from complementary angles

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These two angles (140° and 40°) are Supplementary Angles, because they add up to 180°:

Notice that together they make a straight angle.

But the angles don't have to be together.

These two are supplementary because
60° + 120° = 180°

Play With It ...

(Drag the points)

images/geom-comp-supp.js?mode=supp

When the two angles add to 180°, we say they "Supplement" each other. comes from Latin supplere, to complete or "supply" what is needed

Complementary vs Supplementary

A related idea is Complementary Angles, they add up to 90°

How to remember which is which? Well, alphabetically they are:

Complementary add to 90°

Supplementary add to 180°

You can also think:

"C" of Complementary is for "Corner" (a Right Angle), and

"S" of Supplementary is for "Straight" (180° is a straight line)

Or you can think:

when you are right you get a compliment (sounds like complement)

"supplement" (like a vitamin supplement) is something extra, so is bigger

817, 1804, 5232, 5234, 818, 3303, 1805, 3304, 5235, 5236

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Two Angles are Supplementary when they add up to 180 degrees.

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Supplementary angles are two or more angles that equal 180 degrees

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This link explains what this theorem is and what it means

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Triangle Inequality Theorem: The rule explained with pictures and examples

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The Triangle Inequality Theorem-explained with pictures, examples, an interactive applet and several practice problems, explained step by atep

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The Triangle Inequality Theorem states that any two sides of the triangle must be greater than the third side

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This link explains what the triangle sum theorem is and gives examples of it.

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This concept teaches students the Triangle Sum Theorem, and how to use it.

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The triangle sum theorem states that all three angles in a triangle must equal to 180 degrees.

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This link gives the definition of a transversal

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A line that intersects two or more other lines

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- Linear Equation parts - Missing that the x and y are the coordinates.
- Similar shapes also have congruent corresponding angles and are the same shape but different size.

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Explain what has to be true for 2 figures to be considered similar.  Show an example picture that represents your explanation.

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Explain what has to be true for 2 figures to be considered similar.  Show an example picture that represents your explanation.

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The figures have to be proportional to be similar

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What are the different parts of a linear equation?  Include the variable used for each part and where you find it on the equation.

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What are the different parts of a linear equation?  Include the variable used for each part and where you find it on the equation.

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The linear equation is y=mx+b. The M is your slope, and the b is your y intercept

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What are the 4 types of slope?  Include the name, example value, and drawing for each

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What are the 4 types of slope?  Include the name, example value, and drawing for each

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The image shows two similar figures. The scale factor would be 3

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A scale factor is a number that can be multiplied or divided to get a proportional side

This link explains what scale factors are

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This image shows how a problem would be solved involving indirect measurement

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Def. Indirect Measurement is using proportions to measure an unknown thing

This link tells you what indirect measurement is

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Indirect Measurement - Concept - Geometry Video by Brightstorm

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Time-saving video that explains how to use similar triangles to measure the height of objects - called indirect measurement. Examples show two methods of indirect measurement. Concept explanation.

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This is an example of a proportion

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Def. Proportions show that two ratios are equal

The link explains what proportions are

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Proportion says that two ratios (or fractions) are equal.

Example:

We see that 1-out-of-3 is equal to 2-out-of-6

The ratios are the same, so they are in proportion.

Example: Rope

A rope's length and weight are in proportion.

When 20m of rope weighs 1kg, then:

So:

Sizes

When shapes are "in proportion" their relative sizes are the same.

Here we see that the ratios of head length to body length are the same in both drawings.

So they are proportional.

Making the head too long or short would look bad!

Example: International paper sizes (like A3, A4, A5, etc) all have the same proportions:

So any artwork or document can be resized to fit on any sheet. Very neat.

Working With Proportions

NOW, how do we use this?

Example: you want to draw the dog's head ... how long should it be?

Let us write the proportion with the help of the 10/20 ratio from above:

Now we solve it using a special method:

Multiply across the known corners,
then divide by the third number

And we get this:

So you should draw the head 21 long.

Using Proportions to Solve Percents

A percent is actually a ratio! Saying "25%" is actually saying "25 per 100":

We can use proportions to solve questions involving percents.

The trick is to put what we know into this form:

Example: what is 25% of 160 ?

The percent is 25, the whole is 160, and we want to find the "part":

Multiply across the known corners, then divide by the third number:
Answer: 25% of 160 is 40.
Note: we could have also solved this by doing the divide first, like this:

Either method works fine.

We can also find a Percent:

Example: what is $12 as a percent of $80 ?

Fill in what we know:

Multiply across the known corners, then divide by the third number. This time the known corners are top left and bottom right:

Answer: $12 is 15% of $80

Or find the Whole:

Example: The sale price of a phone was $150, which was only 80% of normal price. What was the normal price?

Fill in what we know:

Multiply across the known corners, then divide by the third number:

Answer: the phone's normal price was $187.50

Using Proportions to Solve Triangles

We can use proportions to solve similar triangles.

Example: How tall is the Tree?

Sam tried using a ladder, tape measure, ropes and various other things, but still couldn't work out how tall the tree was.

But then Sam has a clever idea ... similar triangles!

Sam measures a stick and its shadow (in meters), and also the shadow of the tree, and this is what he gets:

Now Sam makes a sketch of the triangles, and writes down the "Height to Length" ratio for both triangles:

Multiply across the known corners, then divide by the third number:
Answer: the tree is 5.4 m tall.
And he didn't even need a ladder!

The "Height" could have been at the bottom, so long as it was on the bottom for BOTH ratios, like this:

Let us try the ratio of "Shadow Length to Height":

Multiply across the known corners, then divide by the third number:
It is the same calculation as before.
A "Concrete" Example

Ratios can have more than two numbers!

For example concrete is made by mixing cement, sand, stones and water.

A typical mix of cement, sand and stones is written as a ratio, such as 1:2:6.

We can multiply all values by the same amount and still have the same ratio.

10:20:60 is the same as 1:2:6

So when we use 10 buckets of cement, we should use 20 of sand and 60 of stones.

Example: you have just put 12 buckets of stones into a mixer, how much cement and how much sand should you add to make a 1:2:6 mix?

Let us lay it out in a table to make it clearer:

You have 12 buckets of stones but the ratio says 6.

That is OK, you simply have twice as many stones as the number in the ratio ... so you need twice as much of everything to keep the ratio.

Here is the solution:

And the ratio 2:4:12 is the same as 1:2:6 (because they show the same relative sizes)

So the answer is: add 2 buckets of Cement and 4 buckets of Sand. (You will also need water and a lot of stirring....)

Why are they the same ratio? Well, the 1:2:6 ratio says to have:
In our mix we have: So it should be just right!
That is the good thing about ratios. You can make the amounts bigger or smaller and so long as the relative sizes are the same then the ratio is the same.

5674, 5676, 5678, 5680, 5682, 5684, 5686, 5688, 5690, 5692

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Proportion says that two ratios (or fractions) are equal. We see that 1-out-of-3 is equal to 2-out-of-6 The ratios are the same, so they are in proportion. A rope's length and weight are in proportion. When 20m of rope weighs 1kg, then: So: When shapes are "in proportion" their relative sizes are the same.

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This image shows that the unit rate for this problem would be 5

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Def. A unit rate is a rate that has one as the denominator.

Link. Tells what unit rate is and what rates are

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An interactive math lesson about unit rates.

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A solution set is a set of numbers that would make your inequality correct

An explanation of how you would solve a problem with solution sets-- You would have to solve your inequality and anything that fits your answer would be your solution set Ex. If you got x<4 your solution could be 3,2,1,0, -1, or anything lower than that

A synonym to solution set would be answers because it is what would fit the inequality

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Def. An inequality is a mathematical sentence that says that two things are not equal

Image. This image shows the four most common symbols shown in an inequality

Link: this link explains the definition of an inequality and how they are different from equations

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Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents.

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When do you flip the inequality sign on an algebraic inequality?  Give an example problem and solution that shows this.

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When do you flip the inequality sign on an algebraic inequality?  Give an example problem and solution that shows this.

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You flip the sign in an inequality when the inequality is a multi-step problem and you multiply or divide by a negative number in your last step to get just your variable.

Ex. I flipped the sign in this inequality because I had to divide by negative five

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For each symbol, list 3 descriptive words that are used to describe each symbol, give an example statement for each symbol, and a drawing/number line that represents the statement.

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For each symbol, list 3 descriptive words that are used to describe each symbol, give an example statement for each symbol, and a drawing/number line that represents the statement.

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This image shows what an additive inverse would be

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How do you estimate the square root of a number?  Give an example problem and solution that shows the steps.

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How do you estimate the square root of a number?  Give an example problem and solution that shows the steps.

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To estimate the square root of a number, you must find the perfect squares that your number is in between. Then, you find their square roots and you know your number is between those numbers

Ex.  √27 is in between √25 and √36. √25 is 5 and √36 is 6 so you know √27 will be in between 5 and 6 (it is really  5.19615242271)

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Explain the steps to finding the fraction of a repeating decimal and give an example problem and solution that shows the steps

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Explain the steps to finding the fraction of a repeating decimal and give an example problem and solution that shows the steps

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(The bold is my example problem)

First, you have to add an extra decimal both before and after your repetition.
Ex. 1.272727...  is now
1.27.2727....
Next, you set up an equation based off of this. (if x is your decimal) and subtract each side

100x = 127.27....
.-     x =      1.27....
-----------------------------
99x = 126

Then you solve like you would a two step equation

99x = 126
(divide by 99 on each side)

x = 126/99

And then you simplify

126/99 = 1 and 3/11

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This image shows some examples of irrational numbers

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This link explains both what irrational numbers and rational numbers are and what the difference between those are

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An Irrational Number is a real number that cannot be written as a simple fraction:

Irrational means not Rational (no ratio)

Let's look at what makes a number rational or irrational ...

Rational Numbers

A Number can be written as a Ratio of two integers (ie a simple fraction).

Example: 1.5 is rational, because it can be written as the ratio 3/2

Example: 7 is rational, because it can be written as the ratio 7/1

Example 0.333... (3 repeating) is also rational, because it can be written as the ratio 1/3

Irrational Numbers

But some numbers cannot be written as a ratio of two integers ...

...they are called Irrational Numbers.
π = 3.1415926535897932384626433832795... (and more)
We cannot write down a simple fraction that equals Pi.

The popular approximation of = 3.1428571428571... is close but not accurate.

Another clue is that the decimal goes on forever without repeating.

Cannot Be Written as a Fraction
It is irrational because it cannot be written as a ratio (or fraction),
not because it is crazy!
So we can tell if it is Rational or Irrational by trying to write the number as a simple fraction.

Example: 9.5 can be written as a simple fraction like this:

So it is a rational number (and so is not irrational)

Here are some more examples:

Square Root of 2

Let's look at the square root of 2 more closely.

When we draw a square of size "1",
what is the distance across the diagonal?

The answer is the square root of 2, which is 1.4142135623730950...(etc)

But it is not a number like 3, or five-thirds, or anything like that ...

Famous Irrational Numbers

Pi is a famous irrational number. People have calculated Pi to over a quadrillion decimal places and still there is no pattern. The first few digits look like this:

3.1415926535897932384626433832795 (and more ...)

The number e ( Euler's Number) is another famous irrational number. People have also calculated e to lots of decimal places without any pattern showing. The first few digits look like this:

2.7182818284590452353602874713527 (and more ...)

The Golden Ratio is an irrational number. The first few digits look like this:

1.61803398874989484820... (and more ...)

Many square roots, cube roots, etc are also irrational numbers. Examples:

√3

1.7320508075688772935274463415059 (etc)

√99

9.9498743710661995473447982100121 (etc)

But √4 = 2 is rational, and √9 = 3 is rational ...

... so not all roots are irrational.

Note on Multiplying Irrational Numbers

Have a look at this:

So be careful ... multiplying irrational numbers might result in a rational number!

Fun Facts ....

Apparently Hippasus (one of Pythagoras' students) discovered irrational numbers when trying to write the square root of 2 as a fraction (using geometry, it is thought). Instead he proved the square root of 2 could not be written as a fraction, so it is irrational.

But followers of Pythagoras could not accept the existence of irrational numbers, and it is said that Hippasus was drowned at sea as a punishment from the gods!

434,435,1064,2022,3987,1065,3988,2023,2990,2991

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An Irrational Number is a real number that cannot be written as a simple fraction: Irrational means not Rational (no ratio) Let's look at what makes a number rational or irrational ... A Number can be written as a Ratio of two integers (ie a simple fraction).

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An irrational number is a number that cannot be written as a simple fraction

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This link describes what an additive inverse is.

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Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents.

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A number's additive inverse is another number that when added together makes zero. Usually the opposite of your number. Ex. The additive inverse of 8 is -8 because 8 + -8 = 0

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This image shows the different ways you can show a repeating decimal

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This link explains what a Repeating decimal is (or reoccurring decimal)

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A decimal number with a digit (or group of digits) that repeats forever.

Often show by "..."

Examples:

1/3 = 0.333... (the 3 repeats forever)
1/7 = 0.142857142857... ( the "142857" repeats forever)
77/600 = 0.128333... (the 3 repeats forever)

The part that repeats can also be shown by placing dots over the first and last digits of the repeating pattern, or by a line over the pattern.

Also called a "Repeating Decimal".

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A decimal number with a digit (or group of digits) that repeats forever. Often show by "..." Examples: 1/3 = 0.333... (the 3 repeats forever) 1/7 = 0.142857142857... ( the "142857" repeats forever) 77/600 = 0.128333...

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A repeating decimal is a decimal where one or more digits after the decimal point continues on forever

Ex. 0.9876876876876......

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This image shows a few examples of square roots and numbers squared

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This link explains what square roots are and also how to square a number

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First learn about Squares, then Square Roots are easy.

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A number's square root is another number that when multiplied by itself once is your original number

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The cube root of eight would be two because 2 times 2 times 2 is eight

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A number's cube root is another number that when multiplied by itself three times creates your original number

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This link explains what cube roots are and also how to cube a number

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To understand cube roots, first we must understand cubes ...

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- Fraction to decimal conversion. Not all denominators can be converted to 100.  The correct rule is to divide the numerator by the denominator.

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How would you explain to someone how to convert a mixed number to an improper fraction?  Give an example.

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How would you explain to someone how to convert a mixed number to an improper fraction?  Give an example.

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Multiply your integer by your denominator, and then add your numerator. That answer will be the numerator of your improper fraction. The denominator of your improper fraction will be the same as your mixed number. {Ex. 4 and 3/4 would be equal to 4x4+3/4 which is equal to 19/4}

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How would you explain to someone how to complete a multiplication problem that includes decimals?  Give an example.

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How would you explain to someone how to complete a multiplication problem that includes decimals?  Give an example.

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Count the places right to the decimal in both numbers, and write that number down or remember it. Then, multiply normally, ignoring the decimals. Finally, put the decimal however many places to the left you counted before. {Ex. 0.5 times 0.4 is two decimal places. 4 times 5 is 20, so after you add the decimals, your answer is 0.20}

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How would you explain to someone how to convert a decimal to a fraction?  Give an example.

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How would you explain to someone how to convert a decimal to a fraction?  Give an example.

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You say the decimal aloud to convert it to a fraction and then you simplify. {Ex. 0.42 is equal to 42/100 which in turn is equal to 21/50}

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How would you explain to someone how to convert a fraction to a decimal?  Give an example.

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How would you explain to someone how to convert a fraction to a decimal?  Give an example.

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To convert a fraction to a decimal, you need to change the denominator to a hundredth and then say it aloud to see what it would be as a decimal.
{Ex. 1/5 is equal to 20/100, which is equal to 0.20 (twenty hundredths)}

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This link tells you what mixed numbers are and how they compare to improper fractions

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1 3 4

(one and three-quarters)

A Mixed Fraction is a whole number and a proper fraction combined.

Such as 1

Examples

See how each example is made up of a whole number and a proper fraction together? That is why it is called a "mixed" fraction (or mixed number).

Names

We can give names to every part of a mixed fraction:

Three Types of Fractions

There are three types of fraction:

Mixed Fractions or Improper Fractions

We can use either an improper fraction or a mixed fraction to show the same amount.

Converting Improper Fractions to Mixed Fractions

To convert an improper fraction to a mixed fraction, follow these steps:

Divide the numerator by the denominator.

Write down the whole number answer

Then write down any remainder above the denominator.

Example: Convert to a mixed fraction.

Divide:

Write down the 2 and then write down the remainder (3) above the denominator (4).

Answer:

That example can be written like this:

Example: Convert to a mixed fraction.

Answer:

Converting Mixed Fractions to Improper Fractions

To convert a mixed fraction to an improper fraction, follow these steps:

Multiply the whole number part by the fraction's denominator.

Add that to the numerator

Then write the result on top of the denominator.

Example: Convert 3 to an improper fraction.

Multiply the whole number part by the denominator:

Add that to the numerator:

Then write that result above the denominator:

We can do the numerator in one go:

Example: Convert 2 to an improper fraction.

Are Improper Fractions Bad ?

NO, they aren't bad!

For mathematics they are actually better than mixed fractions. Because mixed fractions can be confusing when we write them in a formula: should the two parts be added or multiplied?

Mixed Fraction:

What is:

it may be:

1 + 2 + 1 4

Or it may be:

1 + 2 × 1 4

Improper Fraction:

What is:

It is:

4 9 13 4 4 = 4 +

But, for everyday use, people understand mixed fractions better.

4955, 4956, 916, 917, 1374, 1375, 3538, 3536, 3537, 3539

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1 3 4 (one and three-quarters) A Mixed Fraction is a whole number and a proper fraction combined. Such as 1 See how each example is made up of a whole number and a proper fraction together? That is why it is called a "mixed" fraction (or mixed number).

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The image shows how 7/4, an improper fraction, is equal to 1 and 3/4, a mixed number

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Mixed numbers are whole numbers that are accompanied by fractions. (Ex. 5 and 3/5)

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The link explains what they are, how they are represented and what other types of fractions there are

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7 4

(seven-fourths or seven-quarters)

An Improper Fraction has a top number larger than (or equal to) the bottom number.

More Examples

See how the top number is bigger than (or equal to) the bottom number?
That makes it an Improper Fraction, (but there is nothing wrong about Improper Fractions).

Three Types of Fractions

There are three types of fraction:

Fractions

A Fraction (such as ) has two numbers:

The top number (the Numerator) is the number of parts we have.
The bottom number (the Denominator) is the number of parts the whole is divided into.

So we can define the three types of fractions like this:

Improper Fraction

So an improper fraction is a fraction where the top number (numerator) is greater than or equal to the bottom number (denominator): it is top-heavy.

4 4

Can be Equal

What about when the numerator is equal to the denominator? For example ?

Well it is the same as a whole, but it is written as a fraction, so most people agree it is a type of improper fraction.

Improper Fractions or Mixed Fractions

We can use either an improper fraction or a mixed fraction to show the same amount.

Converting Improper Fractions to Mixed Fractions

To convert an improper fraction to a mixed fraction, follow these steps:

Divide the numerator by the denominator.

Write down the whole number answer

Then write down any remainder above the denominator.

Example: Convert to a mixed fraction.

Divide:

Write down the 2 and then write down the remainder (3) above the denominator (4).

Answer:

That example can be written like this:

Example: Convert to a mixed fraction.

Answer:

Converting Mixed Fractions to Improper Fractions

To convert a mixed fraction to an improper fraction, follow these steps:

Multiply the whole number part by the fraction's denominator.

Add that to the numerator

Then write the result on top of the denominator.

Example: Convert 3 to an improper fraction.

Multiply the whole number part by the denominator:

Add that to the numerator:

Then write that result above the denominator:

We can do the numerator in one go:

Example: Convert 2 to an improper fraction.

Are Improper Fractions Bad ?

NO, they aren't bad!

For mathematics they are actually better than mixed fractions. Because mixed fractions can be confusing when we write them in a formula: should the two parts be added or multiplied?

Mixed Fraction:

What is:

Is it:

1 + 2 + 1 4

= 3 1 4 ?

Or is it:

1 + 2 × 1 4

= 1 1 2 ?

Improper Fraction:

What is:

It is:

4 9 13 4 4 = 4 +

But, for everyday use, people understand mixed fractions better.

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7 4 (seven-fourths or seven-quarters) An Improper Fraction has a top number larger than (or equal to) the bottom number. See how the top number is bigger than (or equal to) the bottom number? That makes it an Improper Fraction, (but there is nothing wrong about Improper Fractions).

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This image shows how improper fractions are equal to or greater than one whole.

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Improper fractions are when the fraction's numerator is greater than or equal to the fraction's denominator. (Ex. 8/5)

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The link explains what equivalent fractions are and why they are the same or represent the same amount

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Equivalent Fractions have the same value, even though they may look different.
These fractions are really the same:

Why are they the same? Because when you multiply or divide both the top and bottom by the same number, the fraction keeps it's value.
"Change the bottom using multiply or divide,
And the same to the top must be applied"
The rule to remember is:

Here is why those fractions are really the same:

× 2× 2

1

=

2

=

4

2

4

8

× 2× 2

And visually it looks like this:

See Fractions on the Number Line ...

... it shows many equivalent fractions.

Also see the Chart of Fractions with many examples of equivalent fractions.

Dividing

Here are some more equivalent fractions, this time by dividing:

÷ 3

÷ 6

18

=

6

=

1

36

12

2

÷ 3

÷ 6

Choose the number you divide by carefully, so that the results (both top and bottom) stay whole numbers.

If we keep dividing until we can't go any further, then we have simplified the fraction (made it as simple as possible).

Summary:

You can make equivalent fractions by multiplying or dividing both top and bottom by the same amount.

You only multiply or divide, never add or subtract, to get an equivalent fraction.

Only divide when the top and bottom stay as whole numbers.

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Equivalent Fractions have the same value, even though they may look different. These fractions are really the same: Why are they the same? Because when you multiply or divide both the top and bottom by the same number, the fraction keeps it's value.

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The image shows how all fractions are the same amount, even though they are represented with different numbers

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Fractions that are equivalent (equal) or are the same amount

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This link explains what denominators and common denominators are, why they are important and how we use them in math

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Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents.