Chapter 3
Lesson 1
Factors and Prime Numbers
A prime number is a number with exactly two factors, 1 and itself.
The prime numbers are 2,3,5,7,11,13,17.
A composite number is a whole number that has more than two factors. The number 1 is neither prime nor composite.
Example: 36
36= 1*36, 2*18, 3*12, 4*9, 6*6
36 is composite not prime because it has more than two
factors.
Lesson 2
Exponents
The number in a power that tells how many times a factor is repeated in a product.
Example: 3exp2*7exp2
=3*3*7*7
=441
Example: 3exp2*7exp2
=3*3*7*7
=441
Lesson 3
Prime Factorization
A prime factorization expresses a composite number as a product of prime factors.
Example:97=1*97
97 is prime because it only has two factors, 1 and itself.
Example:97=1*97
97 is prime because it only has two factors, 1 and itself.
Lesson 4
Divisibility Rule
One number is divisible by another if the quotient is a whole number and the remainder is 0.
Example:135 5
No.135 is not divisible by 5 because the last digit does not end with 5 or 0.
Example:135 5
No.135 is not divisible by 5 because the last digit does not end with 5 or 0.
Lesson 5
Greatest Common Factor
The greatest whole number that is a common factor of two or more numbers. It is also called the greatest common divisor.
Example:6, 15
6= 1*6, 2*3
15=1*15, 3*5
GCF=3
Example:6, 15
6= 1*6, 2*3
15=1*15, 3*5
GCF=3
Lesson 6
Least Common Factor
The least number that is a multiple of two or more numbers.
Example:9, 18
18=1*18, 2*9, 3*6
9=1*9, 3*3
18=2, 6, 18
Common=3, 9
9=3
LCM=2*3*3*6*9*18
=20,916
The least number that is a multiple of two or more numbers.
Example:9, 18
18=1*18, 2*9, 3*6
9=1*9, 3*3
18=2, 6, 18
Common=3, 9
9=3
LCM=2*3*3*6*9*18
=20,916
Lesson 8
Write Equivalent Fractions
Equivalent fractions represent the same number.
Example:4/6
Multiplication
4= 2*4=8
6= 2*6=12
4/6=8/12
Division
4= 4/2=2
6= 6/2= 3
4/6=2/3
For the multiplication way 4/6 is the same as 8/12. For the division way 4/6 is the same as 2/3.
Equivalent fractions represent the same number.
Example:4/6
Multiplication
4= 2*4=8
6= 2*6=12
4/6=8/12
Division
4= 4/2=2
6= 6/2= 3
4/6=2/3
For the multiplication way 4/6 is the same as 8/12. For the division way 4/6 is the same as 2/3.
Lesson 9
Simplest Form
A fraction whose numerator and denominator have the number 1 as the only common factor.
Example:12/24
12= 2*2*3=12
24= 2*2*2*3=24
2
2*2*3= 2
12
The simplest form of 12/24 is 2/12.
A fraction whose numerator and denominator have the number 1 as the only common factor.
Example:12/24
12= 2*2*3=12
24= 2*2*2*3=24
2
2*2*3= 2
12
The simplest form of 12/24 is 2/12.
Lesson 10
Compare and Order Fractions
Comparing and ordering fractions is a way to find the common denominator and make equivalent fraction.
Example:6/7 = 6/7
6/7 is equal to to the other 6/7 because they have the same numerator and denominator.
Comparing and ordering fractions is a way to find the common denominator and make equivalent fraction.
Example:6/7 = 6/7
6/7 is equal to to the other 6/7 because they have the same numerator and denominator.
Lesson 12
Fractions, Mixed Numbers, and Decimals
To compare decimals, fractions, and mixed numbers,express the numbers in the same form.
Example:5/10 0.6 0.55
divide 5 by 10, that equals 0.5.
Now order 0.5, 0.6 and 0.55 from least to greatest.
1. 0.5
2. 0.55
3. 0.6
To compare decimals, fractions, and mixed numbers,express the numbers in the same form.
Example:5/10 0.6 0.55
divide 5 by 10, that equals 0.5.
Now order 0.5, 0.6 and 0.55 from least to greatest.
1. 0.5
2. 0.55
3. 0.6
Lesson 13
Terminating and Repeating Decimals
A decimal quotient that terminates or stop because the repeating block of digits consists only of zeroes.
Example:
Terminating Decimals
18/20
Divide 20 by 18, it equals to 0.9 and there is no remainder so this is a terminating decimal.
A decimal quotient that terminates or stop because the repeating block of digits consists only of zeroes.
Example:
Terminating Decimals
18/20
Divide 20 by 18, it equals to 0.9 and there is no remainder so this is a terminating decimal.
Lesson 14
Add and Subtract Fractions with Like Denominators
We can add and subtract fractions only when they have a common denominator.
Example:2/9 + 5/9
2 + 5
9 9
=7/9
The answer is 7/9 because the denominator is the same but the numerator is not the same so we need to add or subtract the numerator, and 2+5 equals to 7. So, we just take the denominator which is 9, and take 7 which is the numerator.
We can add and subtract fractions only when they have a common denominator.
Example:2/9 + 5/9
2 + 5
9 9
=7/9
The answer is 7/9 because the denominator is the same but the numerator is not the same so we need to add or subtract the numerator, and 2+5 equals to 7. So, we just take the denominator which is 9, and take 7 which is the numerator.
Chapter 4
Lesson 1
Add Fractions With Like Denominators
- Build each fraction (if needed) so that both denominators are equal.
- Add the numerators of the fractions.
- The new denominator will be the denominator of the built-up fractions.
- Reduce or simplify your answer, if needed.
Example: 2/9 + 5/9
2/9 + 5/9= 7/9
2/9 + 5/9= 7/9
Subtract Fractions With Like Denominators
just add or subtract the numerators , and write the result over the same denominator.
Example: 6/7 - 3/7
6/7 - 3/7= 3/7
just add or subtract the numerators , and write the result over the same denominator.
Example: 6/7 - 3/7
6/7 - 3/7= 3/7
Lesson 2
Add Fractions and Mixed Numbers
- Step 1: Keep the denominator the same. Step 2: Add or subtract the numerators. ...
- Step 1: Find the Lowest Common Multiple (LCM) between the denominators. ...
- Step 1: Convert all mixed numbers into improper fractions. ...
- In this second method, we will break the mixed number into wholes and parts.
Example:3/5 + 4/15
Denominator:
5=1 x 5
15=3 x 5
3 x 5=15
Numerator:
3+4=7
3/5 + 4/15= 7/15
Denominator:
5=1 x 5
15=3 x 5
3 x 5=15
Numerator:
3+4=7
3/5 + 4/15= 7/15
Lesson 3
Subtract Fractions and Mixed Numbers
- Make sure the bottom numbers (the denominators) are the same.
- Subtract the top numbers (the numerators). Put the answer over the same denominator.
- Simplify the fraction (if needed).
Example:5/6 - 1/2
Denominator:
6=2 x 3
2=1 x 2
1 x 3= 3
Numerator:
5 - 6= 1
5/6 - 1/2= 1/3
Denominator:
6=2 x 3
2=1 x 2
1 x 3= 3
Numerator:
5 - 6= 1
5/6 - 1/2= 1/3
Lesson 7
Reciprocals
The reciprocal of a fraction inverted.
Example: 1/6=6/1
The opposite of 1/6 is 6/1.
The opposite of 1/6 is 6/1.
Lesson 8
Divide by a Fraction
When dividing fractions, most people learn the rule "invert and multiply", where one would change the division sign to a multiplication sign and invert the second fraction in the problem.
Example: 2/3 divided by 3/4
3 x 3
4 x 2
3 x 3
2 x 2 x 2
3 x 3=9
2/3 divide by 3/4= 9 or 1 whole.
3 x 3
4 x 2
3 x 3
2 x 2 x 2
3 x 3=9
2/3 divide by 3/4= 9 or 1 whole.
Lesson 9
Divide Fractions and Whole Numbers
First step: Write the whole number and the mixed number as improper fractions.
- Second step: Write the reciprocal of the divisor, 2/5, and multiply.
- Third step: Simplify, if possible.
- Fourth step: Perform the simple multiplication of the numerators and the denominators.
Example: 5 divided by 1/7= 5/1 divided by 1/7
5/1 x 7/1= 35/1 or 35
5/1 x 7/1= 35/1 or 35
Lesson 10
Divide Mixed Numbers
- Change each mixed number to an improper fraction.
- Multiply by the reciprocal of the divisor, simplifying if possible.
- Put answer in lowest terms.
- Check to be sure the answer makes sense.
Example:1/2 divided by 3= 1/2 divided by 3/1
1/2 x 1/3
Numerator:
1 x 1=1
Denominator:
2 x 3=6
1/2 x 1/3=1/6
1/2 divided by 3=1/6
1/2 x 1/3
Numerator:
1 x 1=1
Denominator:
2 x 3=6
1/2 x 1/3=1/6
1/2 divided by 3=1/6
Lesson 11
Metric System of Measurements
A system of measurement in which the basic units are the meter, the second, and the kilogram. In this system, the ratios between units of measurement are multiples of ten.
Example:
Chapter 5
Lesson 1
Integers
A whole number; a number that is not a fraction
Example: -9 - +10
-9 - +10= -1
-9 - +10= -1
Lesson 2
Compare and Order Integers
If there are two numbers we can compare them. One number is either greater than, less than or equal to the other number.
Example no.1 : -6, 5, -1, -4
Order from least to greatest:
-6, -4, -1, 5
Example no.2 : 11_-12
11 > -12
Order from least to greatest:
-6, -4, -1, 5
Example no.2 : 11_-12
11 > -12
Lesson 4
Add Integers
To add integers with different signs, keep the sign of the number with the largest absolute value and subtract the smallest absolute value from the largest. Subtract an integer by adding its opposite.
Example: -8 + 7
-8 - 7= -1
Ans. -8 + 7= -1
-8 - 7= -1
Ans. -8 + 7= -1
Lesson 5
Subtract Integers
Subtract an integer by adding its opposite.
Example: -10 - 7
-10 + 7= -17
Ans. -10 - 7= -17
-10 + 7= -17
Ans. -10 - 7= -17
Lesson 6
Multiply Integers
To multiply integers, always multiply the absolute values and use these rules to determine the sign of the answer.
Example: -4 x -7
-4 x -7= 28
Ans. -4 x -7= 28
-4 x -7= 28
Ans. -4 x -7= 28
Lesson 7
Divide Integers
To or divide integers, always divide the absolute values and use these rules to determine the sign of the answer.
Example: -36 / -6
-36 / -6= -6
Ans. -36 / -6= -6
-36 / -6= -6
Ans. -36 / -6= -6
Lesson 9
Rational Numbers
A rational number is a number that can be expressed in the form a/b, where a and b are integers and b is not zero.
Example no.1 :
Example no.2 :
Chapter 6
Expressions and Equations
Definition of expressions and equations:
Lesson 1
Use Addition Properties to Evaluate Expressions
A knowledge of the addition properties can help you evaluate mathematical expressions.
Properties for Addition
1) Commutative Property- The order in which numbers are added does not affect their sum.
3 + 8= 8 + 3
The Commutative Property allows you to add two or more numbers in any order.
3 + 8= 8 + 3
The Commutative Property allows you to add two or more numbers in any order.
2) Associative Property- The way in which numbers are grouped does not affect their sum.
(2 + 6) + 4 = 2 + (6 + 4)
The Associative Property allows you to group three or more numbers in any way to add them.
(2 + 6) + 4 = 2 + (6 + 4)
The Associative Property allows you to group three or more numbers in any way to add them.
3) Identity Property- The sum of any number and zero is that number.
5 + 0= 5
5 + 0= 5
Addition Properties
Lesson 2
Use Multiplication Properties to Evaluate Expressions
Properties for Multiplication
Lesson 3
Order of Operations
1) Always do operations in parentheses first.
2) Rewrite any exponents.
3) Multiply and divide from left to right.
4) Add and subtract from left to right.
2) Rewrite any exponents.
3) Multiply and divide from left to right.
4) Add and subtract from left to right.
Example:
Lesson 4
Use the Distributive Property to Evaluate Expressions
The property which states that when two addends are multiplied by a factor the product is the same as if each addend was multiplied by the factor and those products were added.
Distributive Property
The product of a number of a and a sum ( b + c) is equal to the sum of a x b and a x c.
With a Sum
3 x (4 + 5)
3 x (4 + 5)= 3 x 9= 27
With a Difference
3 x (5 - 4)
3 x (5 - 4)= 3 x 1= 3
With a Sum
3 x (4 + 5)
3 x (4 + 5)= 3 x 9= 27
With a Difference
3 x (5 - 4)
3 x (5 - 4)= 3 x 1= 3
Meaning and Example for Distributive Property:
Lesson 5
Evaluate Expressions with Fractions
To evaluate an expression means to find its value.
Example:
Lesson 6
Write Addition and Subtraction Expressions
Equations with Addition and Subtraction
Write Multiplication and Division Expressions
Equations with Multiplication and Division
Lesson 14
Equations with Fractions
an equation containing the unknown in the denominator of one or more terms
Chapter 7
Lesson 1
Ratios
A ratio shows the relative sizes of two or more values.
Lesson 2
Rates
A rate is a ratio that compares two quantities with different units of measure.
Lesson 3
Equivalent Fractions
Fractions that show different numbers with the same value.
Lesson 4
Solve Proportions
Solving proportions is simply a matter of stating the ratios as fractions, setting the two fractions equal to each other, cross-multiplying
Lesson 7
Scale Drawings
The ratio of the size in a drawing or model to the actual size of an object.
Lesson 9
Decimal and Percentages
Percents and Decimals. The word percent means “hundredths” or “divided by .” The symbol % is used to represent percent.
Lesson 10
Fractions and percents
Decimals are a type of fractional number. The decimal 0.5 represents the fraction 5/10. The decimal 0.25 represents the fraction 25/100. Decimal fractions always have a denominator based on a power of 10.
Lesson 11
Fractions, Decimals and Percents
Converting decimals to percentages. Once a number is written as a decimal, it is easy to convert it to a percentage . Remember that 'per cent' means 'per hundred', so converting from a decimal to a percentage can be done by multiplying by 100.
Chapter 8
Lesson 1
Find a percent of a Number
If you want to know what percent A is of B, you simple divide A by B, then take that number and move the decimal place two spaces to the right.
Lesson 2
Find a Percent
When finding a percent, you use multiplication when the percent and the specific number is given. You use division when the two numbers are given.
Lesson 3
Find a number when a Percent is known
Lesson 5
Mental Math: Estimate With Percent
Percentage of a number using mental math. Since 10% is 1/10, and it's so easy to find 1/10 of any number, to find 20% of a number, first find 10% of it, and double that.
Lesson 7
Simple Interest
Simple interest is a quick and easy method of calculating the interest charge on a loan.
Lesson 8
Sales, Tax and Discount
discount. An amount of discount is a percent off the original price. amount of discount = discount rate • original price. sale price = original price −discount. The sale price should always be less than the original price.
Chapter 9
Lesson 7
Triangles and Angle Sums
Lesson 8
Quadrilateral and Angle Sums
Lesson 12
Polygons
A polygon is any 2-dimensional shape formed with straight lines.
Chapter 10
Lesson 2
Area of Parallelogram
A flat shape with 4 straight sides where opposite sides are parallel.
Lesson 3
Area of Triangle
The area of each triangle is one-half the area of the rectangle.
Lesson 5
Circumference
the size of something as given by the distance around it. the length of the closed curve of a circle.
Lesson 7
Surface Area
the surface area of a cube is the area of all 6 faces added together.
Lesson 10
Volumes of Rectangular Prism
A solid (3-dimensional) object which has six faces that are rectangles.
Lesson 11
Volumes of Other Solids