1. numerical coefficient
Language of algebra
The first step in learning to "speak algebra" is learning the definitions of the most commonly used words.
Algebraic Expressions An algebraic expression is one or more algebraic terms in a phrase. It can include variables, constants, and operating symbols, such as plus and minus signs. It's only a phrase, not the whole sentence, so it doesn't include an equal sign.
Algebraic expression: 3x – 9 + 2x - 7x + 5
In an algebraic expression, terms are the elements separated by the plus or minus signs. This example has five terms, 3x, 9, 2x, 7x and 5. Terms may consist of variables and coefficients, or constants.
3n – 6n + 7
constant term
variable
numerical coefficient
Grade 8

2. numerical coefficient
Language of algebra
Variables In algebraic expressions, letters represent variables. These letters are actually numbers in disguise. In this expression, the variable is n. We call these letters "variables" because the numbers they represent can vary—that means, we can substitute one or more numbers for the letters in the expression.
Coefficients Coefficients are the number part of the terms with variables. In 3n - 6n +7, the coefficient of the first term is 3. The coefficient of the second term is 6, and the coefficient of the third term is 7.
If a term consists of only a variable, its coefficient is 1, for example, x by itself means ‘1x’.
3n – 6n + 7
constant term
variable
numerical coefficient
Grade 8

3. numerical coefficient
Language of algebra
Constants
Constants are the terms in the algebraic expression that contain only numbers. That is, they're the terms without variables. We call them constants because their value never changes, since there are no variables in the term that can change its value. In the expression 3n - 6n + 7 the constant term is “7”.
3n – 6n + 7
constant term
variable
numerical coefficient
Grade 8

4. Language of algebra
Real Numbers In algebra, we work with the set of real numbers, which we can model using a number line.
Real numbers describe real-world quantities such as amounts, distances, age, temperature, and so on. A real number can be an integer, a fraction, or a decimal.
Grade 8

5. Translating Words into an Algebra Expression
Here is a statement…
the sum of three times a number and eight
How would you translate it to algebra?
In this expression, we don't need a multiplication sign or brackets. Phrases like "a number" or "the number" tell us our expression has an unknown quantity, called a variable. In algebra, we will use letters to represent numbers.
The words "the sum of" tell us we need a plus sign because we're going to add three times a number to eight. The words "three times" tell us the first term is a number multiplied by three.
Grade 8

6. Algebraic Translation
Operation
Algebraic Expression
Read
Addition
n + 5
n plus five
the sum of n and 5
n increased by 5
5 more than n
Subtraction
n – 5
n - 5
the difference of n and 5
n diminished by 5
5 less than n
Multiplication 5n
five n
the product of 5 and n
5 multiplied by n
Division
n ÷ 5 n 5
n over 5
the quotient of n and 5
n divided by 5
one fifth of n
Exponentiation n2
n squared
the square of n
n with exponent 2
n to the second power
This table should be copied so students can see some of the ways an algebraic expression can be interpreted
Grade 8

7. Read these and translate to ALGEBRA
1) The sum of a and 10
2) The quotient of 4 and y
3) The difference of 9 and b
4) The sum of double a and 8
5) The square of the difference of a and 2
Grade 8

8. Simplifying Algebraic Expressions Addition and Subtraction
A sum or difference can be simplified by adding or subtracting like terms, such as the constant terms together and the terms with the same variable together. Example  3n n + 6
Bring like terms together
3n + 2n
5n + 7
6x + 4 – 4x – 2
6x – 4x + 4 – 2
2x + 2
Grade 8

9. Simplifying Algebraic Expressions Addition and Subtraction
Adding a whole number means adding each of its parts; (3n + 2) + (2n + 3)
3n n + 3
Remove brackets, rewrite expression
Bring like terms together
3n + 2n
5n + 5
Subtracting a whole number means subtracting each of its parts; (7n + 6) – (4n + 3)
Remove brackets, rewrite expression
7n + 6 – 4n – 3
Bring like terms together
7n – 4n + 6 – 3
3n + 3
Grade 8

10. Find the sum or difference for each
1. (2a + 6) + (3a – 4)
Remove brackets & rewrite expression
Bring like terms together
2. (6n – 5) – (4n + 4)
3. (12s – 7) – (9s – 5)
Grade 8

11. Multiplication & Division
Remember: When you add or subtract algebraic expressions you only combine like terms!
Multiplication & Division
You can MULTIPLY or DIVIDE unlike terms!
5a • 2 =
6a ÷ 2 =
2y • -3=
-40y/-4 =
-4x • -6=
15x ÷ 3 =
Grade 8

12. Area & Perimeter 2a
Perimeter = 4 4
Area =
length • width 2a
Grade 8

13. Area & Perimeter
Perimeter = 5x 6
8 – 3x
b • h 2
Area =
Grade 8

14. Simplifying Algebraic Expressions Multiplication and Division
Multiplying a whole number means multiplying each of its parts 4(3n + 2)
4  3n + 4  2
12n + 8
Dividing a whole number means dividing each of its parts; (8n + 6) 2
8n ÷ ÷ 2
4n + 3
Grade 8

15. Find the product/quotient for each:
 (18s – 12)
 -3(5x – 4)
 3(6n + 3) ÷ 9
Grade 8

16. Substitution ‘Finding a numerical value of an expression’
If we know the value of a variable, we can substitute or “plug it in” and the algebraic expression becomes a numerical value.
What is the value of 2a2 + 4a - 6 if a = 3 or a = 5
Given a = 3, then 2a2 + 4a – 3 becomes:
Given a = 5, then 2a2 + 4a – 6 becomes:
←numerical value
←numerical value

17. Substitution ‘Finding a numerical value of an expression’
To calculate the numerical value of an algebraic expression, the required steps are:
Write the algebraic expression;
Replace the variable with the selected value;
Calculate according to the order of operations;
Write your calculations vertically, with only one operation per line.

18. Here are examples of equations:
An equation is a mathematical sentence containing an equal sign. It tells us that two expressions mean the same thing, or represent the same number. An equation can contain variables and constants. Using equations, we can express math facts in short, easy-to-remember forms and solve problems quickly.
Here are examples of equations:
3z + 2 = 14       x – 9 = 20       p + 2p = 3
The most important skill to develop in algebra is the ability to translate a word problem into the correct equation, so that you can solve the problem easily.

19. Equations Let's try a few examples:
1) A number n times 3 is equal to 120.
2) Ten less than four times a number equals seventy.
3) Lori worked fifteen hours, she earned $120.00, write an equation to represent her earnings.

20. Equations 4) Nine more than five times a number is twenty four.
5) Four times a number decreased by seven equals seventeen.
6) Three times a number plus five equals five times a number less nine.

21. There are 4 kinds of Level I equations that we will be solving.
When solving a Level I equation, we use the opposite operation that is displayed to determine the value of our variable (letter).
For example, addition is the opposite operation of subtraction, and multiplication is the opposite operation of division.
There are 4 kinds of Level I equations that we will be solving.
Level I (addition) x + 5 = 9
Try these...
x + 9 = 23
11 + x = -15
n + 7 = 18
– 5 – 5
x = 4

22. Level I (subtraction) x – 9 = 6
Level I Equations
Level I (subtraction) x – 9 = 6
x = 15
Try these...
x – 4 = -8
x – 11 = 18
x – 9 = -5

23. Level I (multiplication) 6x = -42
Level I Equations
Level I (multiplication) 6x = -42
To solve a Level I multiplication equation we use division to help us find the value of x.
6x =
x = -7
Try these...
4x = 48
5.5x = 55
-8x = -56

24. Level I Equations x = 7 12 x = -6 -9 x = -4 16
Level I (division) x =
In division we use a special type of opposite operation called cross-multiplication.
x =
x  1= 4  -9
x = -36
Try these...
x = 7 12
x = -6 -9
x = -4 16

25. Level I Equations 1) x + 7 = 12 2) x – 6 = 11 3) x = 9 6 4) 7x = 35

26. Level II Equations
A Level II equation will require you to do two opposite or inverse operations to solve for the variable (letter). Always do the opposite of any addition or subtraction first, then proceed to do the inverse operation of any multiplication or division.
E.g.: 5x – 9 = 8
E.g.: x + 7 =
– – 7
5x = 17
x =
5x =
x =
x = 3.4
x • 1 = 5 • 2.3
x = 11.5

27. Level II Equations
1) 3x + 2 = 14
3) x + 3 =
2) 4x + 8 = 4

28. Level II Equations
4) 5x - 3 = 18
6) x =
5) -13x + 3 = 6

29. Level III Equations
A Level III equation is one with more than one group of variables. To solve you must first gather like terms on each side of the equal sign.
Example #1:
Example #2:
Bring like terms
together
3x + 2 = x – 5
4x – 6 = 2x + 10
– – 2
3x = x – 7
4x = 2x + 16
– x – x
– 2x – 2x
2x = -7
2x = 16
2x =
2x =
x = -3.5
x = 8

30. Level III Equations #1) 4x + 3 = 9x + 7 #3) 7x – 3 – 2x = 8

31. Level III Equations #4) 3x + 8x = 44 #5) -2x - 8x = -100

32. Level IV Equations
A Type IV equation has one or more sets of brackets. Use the number on the outside of the brackets as a multiplier and multiply it by everything inside the brackets. Then proceed to solve as if it was a Type III or Type II equation.
Example #2:
Example #1:
4(2x – 5) = 2(3x + 6)
3(x + 7) = -39
4 · 2x – 4 · 5 = 2 ·3x + 2 ·6
3 · x + 3 · 7 = -39
8x – 20 = 6x + 12
3x + 21 = -39
– 21 – 21
8x = 6x + 32
3x = -60
– 6x – 6x
2x = 32
3x =
2x =
x = -20
x = 16

33. Level IV Equations  3(x – 8) = 6  7(x – 2) = 5(3x – 8)

34. Level IV Equations
4) 4(x + 2) = 3( x – 3)
5) 0.4(3x + 8) = 0.2(x + 5)

35. Level V Equations Example #1: Example #2: 3x + 5 = 7 4 2 x = -3 6 2
A Level V equation is made up of two ‘fractions’, one on either side of the equal sign. To solve these equations we first have to ‘cross-multiply’, this means that we multiply the means and extremes. Then we place one result on each the right side of the equation and the other on the left side. We then proceed to solve as if it was a Level III or Level II equation.
Example #1:
Example #2:
3x + 5 =
x =
2(3x + 5) = 7 • 4
x • 2 = -3 • 6
6x = 28
2x = -18
– 10 – 10
2x =
6x = 18
6x =
x = -9
x = 3

36. Level V Equations
1) x =
2) 3x =
3) 5x + 2 = x