Tips
Students usually miss these questions because they make simple mistakes. Here are some tips to help avoid simple mistakes: * Look at all the possible answers before answering.

Double check your answer.

Take your time.

Examples:

Question:

What is the tens digit for the following number? 347,516

Answer:

1

Question:

What is the millions digit for the following number? 2,347,516

Answer:

2

Question:

What is the ten thousands digit for the following number? 347,516

Answer:

4

Question:

What is three hundred thousand, twenty-two written as a number?

Answer:

300,022

Place Value - Decimals

0.781

ones

decimal

tenths

hundredths

thousandths

0

.

7

8

1

Tips
Students usually miss these questions because they make careless mistakes. Here are some tips to help avoid careless mistakes: * Look at all the possible answers before answering.

Double check your answer.

Take your time.

The first place to the right of the decimal is the tenths place.
The second place to the right of the decimal is the hundredths place.
The third place to the right of the decimal is the thousandths place.

Examples:

Question:

What digit is in the tenths place for the following number? 16.79

Answer:

7

Question:

What digit is in the hundredths place for the following number? 4.81

Answer:

1

Question:

What digit is in the thousandths place for the following number? 97.813

Answer:

3

Converting Fractions, Decimals, & Percents

Converting from a decimal to a percent: MULTIPLY the decimal by 100.

Converting from a percent to a decimal: DIVIDE the percent by 100.

Examples:

Convert 0.25 to a percent: (0.25 × 100) = 25%

Convert 0.75 to a percent: (0.75 × 100) = 75%

Convert 25% to a decimal: (25 ÷ 100) = 0.25

Convert 75% to a decimal: (75 ÷ 100) = 0.75

Fractions, decimals, and percents can all be used to represent part of a whole. It is important to know how to convert common fractions and common percents. See the table of common fractions and percents below.

Fraction=Decimal= Percent

1/10 = 0.1 = 10%

1/5 = 0.2 = 20%

1/2 = 0.5 = 50%

3/4 = 0.75 = 75%

4/5 = 0.8 = 80%

9/10 = 0.9 = 90%

Comparing Whole Numbers

You need to know the following symbols:

1. < means "less than"
2. > means "greater than"
3. = means "equal to"

Examples:

To show that 1 is "less than" 2 write: 1 < 2

To show that 4 is "greater than" 2 write: 4 > 2

To show that 2 is "equal to" 2 write: 2 = 2

Tips when comparing or ordering numbers:

1. Convert fractions to decimals because they are easier to compare.
2. Double check your answer in order to avoid any mistakes.
3. Remember with symbols < & >, the symbol points to the smaller value.

Comparing Fractions

You need to know the following symbols: < means less than

means greater than

and = means equal to

For example the following statements are true: 1 < 2
4 > 2
2 = 2

To figure out which of two given fractions is larger do the following:

1. Draw two rectangles of the same size.
2. Fill in the fractional part of the rectangles for each given fraction.
3. The fraction corresponding to the rectangle which has more filled in is the larger fraction.

Example:

Question: Which fraction is the largest 2/3, 2/4, 5/7, or 11/15?

Answer:

Fill in the fractional part of the rectangle for each fraction. You must use rectangles of the same size.

2/3

=

2/4

=

5/7

=

11/15

=

As you can see, 11/15 is the largest followed by 5/7, then 2/3, and finally 2/4.

Number Lines

On a number line, all numbers to the right of zero are positive, and all the numbers to the left of zero are negative.

Each number represents the coordinate of the point that it labels, while the point is the graph of the number. A number line with the points 0, 2, and 5 graphed on it is shown below.

Distance Between Two Points on a Number Line

To find the distance between two points on a number line, subtract the left (or bottom) number from the right (or top) number.

Example: An ant walked from point A to point B. How far did the ant travel?

The vertical distance traveled by the ant is 3 cm - 1 cm = 2 cm.
The horizontal distance traveled by the ant is 5 cm - 1 cm = 4 cm.
The total distance traveled by the ant is 4 cm + 2 cm = 6 cm

Number Lines with Fractions

On a number line, all numbers to the right of zero are positive, and all the numbers to the left of zero are negative.

Sometimes you will have to identify a point on a number line.

Fractions: A number line with the points 0, 11/2, 33/4 and 5 graphed on it is shown below.

Odd vs. Even

Even numbers are numbers that CAN be divided by 2.

Odd numbers are numbers that CANNOT be divided by 2.

To figure out if a number is odd or even, look at the digit in the ones place. If the digit in the ones place is 0, 2, 4, 6, or 8, then the number is even. If the digit in the ones place is 1, 3, 5, 7, or 9, then the number is odd.

Here are some examples of even numbers:

22

100

58

44

Here are some examples of odd numbers:

21

103

55

49

Here is an example of an even number of shapes.

Notice that the shapes are in pairs. An even number of items can always be divided into two equal groups.

Here is an example of an odd number of shapes.

Notice that the shapes are in pairs except the last shape. An odd number of items cannot be divided into two equal groups.

Factors, Multiples, & Squares

Factors

A factor is a number that can divide into another number exactly without a remainder.

24 ÷ 1 = 24
24 ÷ 2 = 12
24 ÷ 3 = 8
24 ÷ 4 = 6

24 ÷ 24 = 1
24 ÷ 12 = 2
24 ÷ 8 = 3
24 ÷ 6 = 4

Therefore, 1, 2, 3, 4, 6, 8, 12, and 24 are factors of 24.

Multiples

A multiple of one number has the number as a factor.

The multiples for 6 are the following: {6, 12, 18, 24, 30, 36, 42, 48, ...}
All the multiples of 6 have 6 as a factor.

Squares

Any number raised to the power of 2 can be modeled using a square! That's why we call raising a number to the second power "squaring the number".

The area of the square above is 25, and the length of a side is 5; therefore, 5 squared is equal to 25. Example:

The above model shows a seven by seven array of blocks. Based on the model above, what is 7 squared?

Solution: Since 7 squared can be modeled by 7 rows each containing 7 columns, we want to find the sum of 7 sevens or
7 + 7 + 7 + 7 + 7 + 7 + 7 = 49.
Alternatively, count the number of colored squares to find 7 squared. Since there are 49 squares filled in, 7 squared = 49.

Example:

The model above has sixteen blocks arranged in four rows and four columns. Based on the model above, what is the square root of 16?

Solution: The blocks form a square with sides of length 4. The area of the square is equal to the number of blocks in the square. However, the area of a square is also equal to one side squared (or side × side).
Thus, we can see that
42 = 4 × 4 = 16.

If 42 = 16, then the square root of 16 is 4.

Multiplication as Repeated Addition

There are several ways to solve multiplication problems. One way is to consider multiplication as "repeated addition".

Example 1: 5 × 4 = ?
Multiplying 5 by 4 is the same as adding together 5 groups of 4.
5 × 4 = 4 + 4 + 4 + 4 + 4 = 20

Add & Subtract Decimals

Adding Decimals

Place the numbers so that the decimal points are aligned vertically.

Add each column, starting on the right and working left. If the sum of a column is greater than ten, then carry the one to the next column on the left.

Examples:

A. 265.4 + 18.5 =

B. 843.92 + 271.426 =

C. 342.76 + 157.2137 + 46.27 =

Solution

----

Subtracting Decimals

Place the numbers so that the decimal points are aligned vertically.

Subtract each column, starting on the right and working left. If the number being subtracted is larger than the number it is being subtracted from, then add ten to the number and subtract one from the number in the next left column. This is called borrowing.

Examples:

A. 265.4 - 18.5 =

B. 853.92 - 221.416 =

Solution

Add & Subtract Decimals

Use these tips to solve number sentence problems:

1. Simplify the sentence down as much as possible before trying to solve.

2. Try to guess the correct answer before looking at the potential answers.

3. Plug each answer into the sentence to test it.

Example 1:

Which number for @ makes the sentence true?

4.1 + @ = 12.9
Remember, subtraction is the opposite of addition. So, to solve this problem, subtract 4.1 from 12.9. When using decimals, make sure the decimals are lined up.

Finally, substitute 8.8 into the original number sentence to check your answer. 4.1 + 8.8 = 12.9

Ratios, Proportions, & Percents

A ratio represents a comparison between two values.

A ratio of two numbers can be expressed in three ways. A ratio of "one to two" can be written as:

1 to 2

1:2

1/2

A proportion is made up of two ratios with an "=" (equal) sign between them. For example:

1

|| ||

|| 2 || || = || || 4 ||

|| ||

|| 8 || || ||

Example: Billy is building a model of a 10-foot-long truck. The model is 1/10 the size of the truck. How long is Billy's model?

Solution: First make a proportion for the situation with one unknown:

x

|| ||

|| 10 || || = || || 1 ||

|| ||

|| 10 || ||

x is equal to 1; therefore, Billy's model is 1 foot long. ----

A percent is a ratio or fraction whose second term is 100. Percent means parts per hundred. In mathematics, we use the symbol % for percent. For example, the ratio 40:100 can be written 40%.

Example: The price of a hamburger is $1.00. If the sales tax is 8%, what is the total cost of one hamburger?

Answer: Since one dollar is made up of 100 cents, 8% of $1 is 8¢, or $0.08. Now add the tax onto the original cost.

## Place Value

millionshundred thousandsten thousandsthousandshundredstensonesTipsStudents usually miss these questions because they make simple mistakes. Here are some tips to help avoid simple mistakes: * Look at all the possible answers before answering.

Examples:Question:Answer:Question:Answer:Question:Answer:Question:Answer:## Place Value - Decimals

onesdecimaltenthshundredthsthousandthsTipsStudents usually miss these questions because they make careless mistakes. Here are some tips to help avoid careless mistakes: * Look at all the possible answers before answering.

The first place to the right of the decimal is the

tenthsplace.The second place to the right of the decimal is the

hundredthsplace.The third place to the right of the decimal is the

thousandthsplace.Examples:Question:Answer:Question:Answer:Question:Answer:## Converting Fractions, Decimals, & Percents

Convertingfrom adecimalto apercent: MULTIPLY the decimalby 100.Convertingfrom apercentto adecimal: DIVIDE the percentby 100.Examples:0.25to apercent:(0.25 × 100) = 25%0.75to apercent:(0.75 × 100) = 75%25%to adecimal:(25 ÷ 100) = 0.2575%to adecimal:(75 ÷ 100) = 0.75Fraction=Decimal=Percent## Comparing Whole Numbers

1.

<means "less than"2.

>means "greater than"3.

=means "equal to"Examples:1 is "less than" 2write:1 < 24 is "greater than" 2write:4 > 22 is "equal to" 2write:2 = 21. Convert fractions to decimals because they are easier to compare.

2. Double check your answer in order to avoid any mistakes.

3. Remember with symbols

<&>, the symbol points to the smaller value.## Comparing Fractions

< means

less than- means

and = meansgreater thanequal toFor example the following statements are true:

1 < 2

4 > 2

2 = 2

1. Draw two rectangles of the same size.

2. Fill in the fractional part of the rectangles for each given fraction.

3. The fraction corresponding to the rectangle which has more filled in is the larger fraction.

Example:Question:Which fraction is the largest 2/3, 2/4, 5/7, or 11/15?Answer:Fill in the fractional part of the rectangle for each fraction. You must use rectangles of the same size.

As you can see, 11/15 is the largest followed by 5/7, then 2/3, and finally 2/4.

## Number Lines

Each number represents the coordinate of the point that it labels, while the point is the graph of the number. A number line with the points 0, 2, and 5 graphed on it is shown below.

Distance Between Two Points on a Number LineTo find the distance between two points on a number line, subtract the left (or bottom) number from the right (or top) number.

Example:An ant walked from point A to point B. How far did the ant travel?The vertical distance traveled by the ant is

3 cm - 1 cm =

2 cm.The horizontal distance traveled by the ant is

5 cm - 1 cm =

4 cm.The total distance traveled by the ant is

4 cm + 2 cm =

6 cm## Number Lines with Fractions

positive, and all the numbers to the left of zero arenegative.Sometimes you will have to identify a point on a number line.

Fractions:A number line with the points 0, 11/2, 33/4 and 5 graphed on it is shown below.## Odd vs. Even

Evennumbers are numbers that CAN be divided by 2.Oddnumbers are numbers that CANNOT be divided by 2.To figure out if a number is odd or even, look at the digit in the

onesplace. If the digit in the ones place is0, 2, 4, 6, or 8, then the number iseven. If the digit in the ones place is1, 3, 5, 7, or 9, then the number isodd.Here are some examples of

evennumbers:- 2
- 100
- 5
- 4

Here are some examples of284oddnumbers:- 2
- 10
- 5
- 4

Here is an example of an1359evennumber of shapes.Notice that the shapes are in pairs. An even number of items can always be divided into two equal groups.

Here is an example of an

oddnumber of shapes.Notice that the shapes are in pairs except the last shape. An odd number of items cannot be divided into two equal groups.

## Factors, Multiples, & Squares

Factorsfactoris a number that can divide into another number exactly without a remainder.2424 ÷ 2 =

1224 ÷ 3 =

824 ÷ 4 =

6124 ÷ 12 =

224 ÷ 8 =

324 ÷ 6 =

4Multiplesmultipleof one number has the number as a factor.The multiples for 6 are the following:

{6, 12, 18, 24, 30, 36, 42, 48, ...}

All the multiples of 6 have 6 as a factor.

## Squares

The area of the square above is 25, and the length of a side is 5; therefore, 5 squared is equal to 25.

Example:The above model shows a seven by seven array of blocks. Based on the model above, what is 7 squared?

Solution:Since 7 squared can be modeled by 7 rows each containing 7 columns, we want to find the sum of 7 sevens or7 + 7 + 7 + 7 + 7 + 7 + 7 =

49.Alternatively, count the number of colored squares to find 7 squared. Since there are 49 squares filled in, 7 squared =

49.Example:The model above has sixteen blocks arranged in four rows and four columns. Based on the model above, what is the square root of 16?

Solution:The blocks form a square with sides of length 4. The area of the square is equal to the number of blocks in the square. However, the area of a square is also equal to one side squared (orside × side).Thus, we can see that

42 = 4 × 4 = 16.

If 42 = 16, then the square root of 16 is

4.## Multiplication as Repeated Addition

Example 1:5 × 4 = ?Multiplying 5 by 4 is the same as adding together 5 groups of 4.

5 × 4 =

4 + 4 + 4 + 4 + 4 = 20## Add & Subtract Decimals

Adding Decimalsdecimal points are aligned vertically.Examples:A.265.4 + 18.5 =B.843.92 + 271.426 =C.342.76 + 157.2137 + 46.27 =SolutionSubtracting Decimalsdecimal points are aligned vertically.borrowing.Examples:A.265.4 - 18.5 =B.853.92 - 221.416 =Solution## Add & Subtract Decimals

- Which number for

4.1 +Example 1:@makes the sentence true?@= 12.9Remember, subtraction is the opposite of addition. So, to solve this problem, subtract 4.1 from 12.9. When using decimals, make sure the decimals are lined up.

Finally, substitute

8.8into the original number sentence to check your answer.4.1 +

8.8= 12.9## Ratios, Proportions, & Percents

ratiorepresents a comparison between two values.proportionis made up of two ratios with an "=" (equal) sign between them. For example:Example:Billy is building a model of a 10-foot-long truck. The model is 1/10 the size of the truck. How long is Billy's model?Solution:First make a proportion for the situation with one unknown:xxis equal to 1; therefore, Billy's model is 1 foot long. ----percentis a ratio or fraction whose second term is 100. Percent means parts per hundred. In mathematics, we use the symbol % for percent. For example, the ratio 40:100 can be written 40%.

$1.00 + $0.08 =Example:The price of a hamburger is $1.00. If the sales tax is 8%, what is the total cost of one hamburger?Answer:Since one dollar is made up of 100 cents, 8% of $1 is8¢, or $0.08. Now add the tax onto the original cost.$1.08

4.1 +@= 12.9@= 12.9 - 4.112.9@= 8.8