3-1 Order of Operations / 4 days
3-2 Combining Terms / 3 days
3-3 Distributive Property / 3 days
3.1 – 3.3 Quiz / 1 day
3-4 One-Step Equations/Inequalities / 4 days
3-5 Two-Step Equations/Inequalities / 3 days
3.1 – 3.5 Quiz / 1 day
3-6 Solving Formulas / 4 days
3-7 Multi-Step Equations / 5 days
3-8 Writing Equations / 3 days
Test Review / 1 day
Test / 1 day
Cumulative Review / 1 day
Unit Project / 1 day
Total days in Unit 3 - Pre-Algebra Concepts = 35 days
Review Question
What set(s) of numbers does ‘-4’ belong? Integers, Rationals
Discussion
This unit is called Pre-Algebra. It is getting us ready for the next unit: Algebra. We are going to learn four important skills in this unit that we will need for Algebra: order of operations, combining terms, distributive property, and solving equations. I am going to be reminding you of these skills all unit.
Think about getting ready in the morning. Notice there is a particular order in which you get ready. You must shower before you put your clothes on. You must put your socks on before your shoes.
How do you know the order in which to do these things?
There is a particular order in which we must do math problems. I want you to know this order as well as you know the order of putting your clothes on.
The order in math is as follows: parentheses, exponents, multiplication and division, then addition and subtraction. The following saying will help you remember.
Please Excuse My Dear Aunt Sally
Notice how the words are grouped. Multiplication and division are the same and addition and subtraction are the same. To break these ties go left to right.
SWBAT simplify a numeric expression using the order of operations
Definition
Numeric Expression – problem that only involves numbers that doesn’t have an equal sign
Example 1: 6 – 2 + 1
4 + 1
5
Example 2: 42 – 3 ∙ 2 + 1
16 – 3 ∙ 2 + 1
16 – 6 + 1
10 + 1
11
Example 3: 2(3 + 2)2 – 7 ∙ 2
2(5)2 – 7 ∙ 2
2(25) – 7 ∙ 2
50 – 14
36
Example 4: 52 – 3(12 – 32)
52 – 3(12 – 9)
25 – 3(3)
25 – 9
16
You Try!
1. 1 + 14 ÷ 2 ∙ 4 29 2. 23 – (1 + 3)2 + 2 9
3. (16 + 8)/(15 – 13)2 6 4. 2 + 18 ÷ 32 ∙ 3 8
5. 18 – 4 ∙ 3 + 2 8 6. 10(8(15 – 7) – 4 ∙ 3) 520
What did we learn today?
Notice that the answers to the homework problems will start to appear in your book during this unit. This was done intentionally. This was done so that you will check your answers and try to make corrections before class. Also, you will know which problems are giving you difficulty. This will allow you to ask pertinent questions about your assignment.
1. Give a real life example where the order in which something is done matters. Discuss how the result of the example would be different if the you changed the order you did things.
2. Why is there a “tie” between addition and subtraction?
List the operations you would perform in the order you would have to perform them.
3. 8 · 9 – 3 + 5 4. 7 – 4 ÷ 2 · 3 + 1
Evaluate each numeric expression using the order of operations.
5. 22 – 5 + 2 19 6. 24 – 2 ∙ 32 6
7. 12 ÷ 3 + 21 25 8. 12 – 3 + 21 ÷ 3 16
9. 9 + 18 ÷ 3 15 10. 8 + 5(6) – 22 34
11. 32 – 2 · 2 + 3 8 12. 12 – 24 ÷ 12 + 5 15
13. 17 + 2 – 12 · 4 ÷ 16 16 14. 40 ÷ 5 – 3 · 2 2
15. 14 + 8 ÷ 2 + 4 · 2 26 16. 6 · 3 ÷ 9 · 3 – 2 4
17. (16 + 11) – 12 ÷ 3 23 18. 13 – (45 + 21) ÷ 11 7
19. 6 · 5 – 25 ÷ 5 – 23 17 20. 10 + (32 ÷ 23) ÷ 2 12
21. If you have more than one set of parentheses, how do you know what operation to do first?
Review Question
-5/12
Discussion
Today we are going to continue our discussion using order of operations. We are going to “spiral back” to Unit 2 by including integers, decimals, and fractions.
The order in math is as follows: parentheses, exponents, multiplication and division, then addition and subtraction. The following saying will help you remember.
Please Excuse My Dear Aunt Sally
Notice how the words are grouped. Multiplication and division are the same and addition and subtraction are the same. To break these ties go left to right.
SWBAT simplify a numeric expression using the order of operations including integers, decimals, and fractions
Example 1: 4.2 + 12.6 ÷ 6 – 1.85
4.2 + 2.1 – 1.85
6.3 – 1.85
4.45
Example 2:
Example 3: -10 ÷ 2 + 5 · 3
-5 + 5 · 3
-5 + 15
10
You Try!
1. -4 + 12 ÷ 22 -1 2. 19.8 – 2(1.2 + 2.4) 12.6
3. 2(3 – 5) – 23 -12 4. -60 ÷ 6 + 4 ∙ 3 2
5. 7/16 6. 6 – (10 – 4 ∙ 2) 4
What did we learn today?
Evaluate each numeric expression using the order of operations.
1. 8 + 9 – 3 + 5 19 2. 7.2 · 5.1 + 2.4 39.12
3. 8 – 3 · 23 -16 4. (-9 + 4)(18 – 7) -55
5. (-10 + 5) – (5 + 12) -22 6. 9.84 ÷ 2.4 – 2.2 1.9
7. -32 · 4 ÷ 2 -64 8. 18 – (9 + 3) + 22 10
9. 62 + 5 · 2 + 3 49 10. 13/12
11. 10 + 8 – 8 · 4 ÷ 2 2 12. 11/6
13. 4 + 8 ÷ 2 + 4 · 5 28 14. 6 · 3 ÷ 9 · 2 + 1 5
15. (-15 + 21) ÷ 3 2 16. -2(-5 – 9) ÷ 4 7
17. 5 · 6 + 25 ÷ 5 – 23 27 18. (-40 ÷ 4) ÷ 5 – 10 -12
19. 10 20. 59
Review Question
What is a numeric expression?
A problem that involves numbers without an equal sign
Discussion
What do you think makes Algebra different from all of the other math topics that you have learned so far?
Variables
SWBAT simplify an algebraic expression using the order of operations
Definitions
Variable – letter used to represent an unknown
Use a variable that makes sense
* use m for money
* use w for weight
Algebraic Expression – variables, operations, and numbers but no equal sign
*all of the order of operation problems that we have been solving were examples of numeric expressions
Use for examples one and two: x = 4, y = 7, z = 2
Example 1: 6x – 2z
6(4) – 2(2)
24 – 4
20
* notice when two numbers are written next to each other it represents multiplication
Example 2:
7 + 8 – 5
15 – 5
10
* notice when two variables are written next to each other it represents multiplication
You Try!
x = 1, y = 2, z = 3
1. 7x – 2z 1
2. (z + 3y) – 3 6
3. 2
4. (3x – y) + y2 5
5. 2.14z + 10.52
6. 4/15
What did we learn today?
Evaluate each algebraic expression if x = 7, y = 3, and z = 9.
1. 2x – y 11 2. 6(x +y) – 10 50
3. 9 4. x2 – 3y + z 49
5. 2y – (x – y)2 -10 6. 2.4(x – y) – y 6.6
7. 4z – (2y + x) 23 8. x(y3 + 2z – 4) 287
9. 20/9 10. 34
11. 23/12 12. x – y + z – 2x -1
13. Explain the difference between the following two algebraic expressions (3y)2 and 3y2. Use numerical values for y to illustrate your explanation.
Review Question
What is an algebraic expression?
A problem that involves variables without an equal sign
What does it mean when two variables are next to each other?
Multiplication
Discussion
In your foreign language class, you translate sentences from English into a foreign language. In class today, we will be translating sentences from English into Algebraic expressions. You need to think of Algebra as a foreign language
.
Pass out the “Translating English into Algebra” worksheet. Fill in each column with words in English that mean addition, subtraction, multiplication, and division. Then share all of your words until you have a complete list of appropriate words.
Translating English into Algebra
Addition / Subtraction / Multiplication / DivisionSWBAT translate a sentence from English into an algebraic expression
Our goal today is to translate one sentence into a simple algebraic expression. Eventually we will translate an entire paragraph into a complicated algebraic equation.
Example 1: A number divided by six. → n/6
* notice we chose ‘n’ for our variable because we are talking about a Number
* notice we are translating one sentence into a simple expression
Example 2: Twice an integer. → 2i
* What is an integer?
* notice we chose ‘i’ for our variable because we are talking about an Integer
Example 3: Eight more than a number. → n + 8 or 8 + n
* notice we chose ‘n’ for our variable because we are talking about a Number
* notice you can write the expression either way because addition is commutative, that is, it can be written either way and still give the same result
Example 4: Eight less than a number. → n – 8
* notice we chose ‘n’ for our variable because we are talking about a Number
* notice 8 – n is incorrect because subtraction is not commutative, that is, the order in which you write the problem matters
* use the example of 8 less than ten is two
You Try!
1. Three feet shorter than the ceiling. c – 3
2. The quotient of x and 3. x/3
3. John’s salary plus a $200 bonus. s + 200
4. Three minutes faster than Jimmy’s time. t – 3
5. Twice the amount of money plus four dollars.2m + 4
What did we learn today?
Evaluate each algebraic expression if x = 3, y = 4, and z = 5.
1. 6x – 3y 6 2. -6(x + y) -42
3. 19/12 4. 2x2 + 3z 33
5. 3x – (2y + z) -4 6. 2.36(x – z) -4.72
7. 4z – (2y + x) 9 8. x(y3 + z + 4) 219
9. 2/15 10. 3
11. 2 12. 12
Translate each phrase into an algebraic expression.
13. Six minutes less than Bob’s time.
14. Four points more than the Cougars scored.
15. Joan’s temperature increased by two degrees.
16. The cost decreased by ten dollars.
17. Seven times a number.
18. Twice a number decreased by four.
19. Twice the sum of two and y.
20. The quotient of x and 2.
Review Question
1. -4 + 5 = 1
2. -4 + (-5) = -9
3. -4 – 5 = -9
Discussion
In the previous unit, we combined monomials (terms) using multiplication and division.
1. (3x5)(2x4) = 6x9 2.
Notice how we could combine any two things together. Today, we will be combining terms using addition and subtraction. This is going to be different. You can only combine certain terms together. Let’s try to figure it out together.
Use actual pieces of chalk and eraser for the visuals.
What is one eraser plus two erasers? Three erasers
What is 1e + 2e? 3e
It is the same thing. You have to know Algebra just as well as English.
What is four pieces of chalk minus two pieces of chalk? Two pieces of chalk
What is 4c – 2c? 2c
It is the same thing. You have to know Algebra just as well as English.
SWBAT combine terms using addition and subtraction
Example 1: 3x + 5x = 8x
Example 2: 8y – 2y = 6y
Example 3: -8m + 5m = -3m
Example 4: 5b – 2b + b
3b + b
4b
* The order of operations still applies.
You Try!
1. 3x + 6x 9x 2. 7y – 2y 9y
3. -4y – 3y -7y 4. -5a + 11a 6a
5. 2x + 4x – 3x 3x 6. 2b – 5b + b -2b
7. 12x – (4x + 3x) 5x 8. (-5y + 10y – 3y) + 12y 14y