What is Algebra? Return to Table of Contents Slide 5 / 191 The - - PDF document
Slide 1 / 191 Slide 2 / 191 5th Grade Algebraic Concepts 2015-10-16 www.njctl.org Slide 3 / 191 click on the topic to go Table of Contents to that section What is Algebra? Order of Operations Grouping Symbols Writing Simple
Slide 1 / 191
Algebraic Concepts
2015-10-16 www.njctl.org
Slide 2 / 191
click on the topic to go to that section
Table of Contents
· What is Algebra? · Grouping Symbols · Writing Simple Expressions & Interpreting Numerical Expressions · Graphing Patterns & Relationships in the Coordinate Plane · Order of Operations · Function Tables · Writing & Interpreting Expressions Application Problems · Glossary
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click on the topic to go to that section
Table of Contents
· What is Algebra? · Grouping Symbols · Writing Simple Expressions & Interpreting Numerical Expressions · Graphing Patterns & Relationships in the Coordinate Plane · Order of Operations · Function Tables · Writing & Interpreting Expressions Application Problems · Glossary
[This object is a pull tab]
Teacher Notes
Vocabulary Words are underlined with a dashed line in the presentation. This is linked to the page at the end
word defined on it.
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The word "algebra" is taken from a book title al-jebr w'al-muqabalah (circa 825) by the Persian mathematician known as al-Khowarismi. This is considered to be the first book written about Algebra. al-jebr (from the book title) means "reunion of broken parts". Kind of like puzzle pieces. Al'Khwarizmi was a Persian mathematician who wrote on Hindu-Arabic numerals (the numeral system we use today). He was also
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Algebra is a way of solving Math problems.
It is basically like looking at the problem like a puzzle. Putting together the pieces you have, in order to figure out what is missing. What is the missing piece to this number puzzle?
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$40
______ (hours)
Sarah earns $10 for every hour she works.
What is the missing value?
Algebra help us tie together many mathematical ideas.
(Like hours worked and money earned.)
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Things change. To describe things that change or vary, mathematicians invented Algebra.
Algebra makes it easier to say exactly how two changing things (like dollars earned and hours worked) are related.
Sarah earns $10 for every hour she works.
$10 x = $40
(hours)
If we change the amount Sarah earns, the number of hours she worked will change too.
$10 x = $60
(hours)
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Important Vocabulary: An expression is like a phrase and names a number. An equation is a number sentence that describe a relationship between two expressions. H x 6 is an example of an algebraic expression. An algebraic expression uses operation symbols (+,-,x,÷) to combine variables and numbers. A letter that stands for a number is called a variable. Some common variables are: l = length, w = width, h = height and x or y.
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Slide 10 / 191 Imagine this: You put your shoes on, and then your socks. Wait... What? What's wrong with this picture? It's out of order.
click
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In life, there is an order in which we do things. Like putting our socks on first, before our shoes. Math is no different. When we perform operations (+,-,x, ) there is an order. Slide 12 / 191
In an expression with more than one operation, use the rules called Order of Operations.
Note: From left to right means first come, first served. Like reading a book. Start on the left, and work your way to the right. First, completing all x and . Then all + and -.
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Name the operation that should be done first. 6 x 3 + 4 multiplication 3 + 4 x 6 multiplication 5 - 3 + 6 subtraction 9 - 6 3 division
click click click
click
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1 Do you add or multiply first? 6 + 3 x 2 + 7
A
add
B
multiply
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1 Do you add or multiply first? 6 + 3 x 2 + 7
A
add
B
multiply
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Answer
B
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2 Do you divide or add first? 12 + 3 ÷ 12 ÷ 4
A
add
B
divide
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2 Do you divide or add first? 12 + 3 ÷ 12 ÷ 4
A
add
B
divide
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Answer
B
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3 10 - 6 x 6 + 4 x 10
Which operation do you do... A B C D E F G H I
+
+
+
1st? 2nd? 3rd?
Choose an operation for each step.
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3 10 - 6 x 6 + 4 x 10
Which operation do you do... A B C D E F G H I
+
+
+
1st? 2nd? 3rd?
Choose an operation for each step.
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Answer
C,E,G
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Evaluate the expression using the Order of Operations. 4 + 3 x 7 Step 1 Multiply 3 x 7. Step 2 Rewrite the expression. 4 + 21 Step 3 Add 4 + 21. So, 4 + 3 x 7 = 25.
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Evaluate the expression. 25 - 4 x 5 + 4 Step 1 Multiply 4 x 5. Step 2 Rewrite the expression. 25 - 20 + 4 Step 3 Subtract 25 - 20 Step 4 Rewrite the expression. 5 + 4 Step 5 Add 5 + 4. So, 25 - 4 x 5 + 4 = 9.
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4 Evaluate 6 + 3 x 2 + 7 =
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4 Evaluate 6 + 3 x 2 + 7 =
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Answer
19
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5 Evaluate 12 + 12 ÷ 4 ÷ 3 =
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5 Evaluate 12 + 12 ÷ 4 ÷ 3 = [This object is a pull tab]
Answer
13
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6 Evaluate 100 - 6 x 6 + 4 x 10 =
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6 Evaluate 100 - 6 x 6 + 4 x 10 =
[This object is a pull tab]
Answer
104
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7 Evaluate 50 ÷ 10 + 15
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7 Evaluate 50 ÷ 10 + 15
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Answer
20
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Sarah has 8 dollars. Jonathan has 5 dollars more than Sarah. He spends half of his money. Write an expression that represents this scenario.
Sara's $ plus $5 divided in half
Think, pair, share: Is this correct? Explain your answer on your paper.
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Sara's $ plus $5 divided in half If we follow the order of operations, we would divide the "$5 more" before we added it to Sarah's money.
So how can we represent this scenario? We can use parenthesis. Slide 26 / 191
Parentheses ( ) are used to group calculations to be sure that they are done in a certain order. When you use parentheses ( ), you are saying, "DO THIS FIRST."
Sara's $ plus $5 divided in half The parenthesis tell us to add $5 to Sarah's money before we divide it in half.
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Lets look at the results of each expression.
Step 1: Step 2:
8 + 2.5
10.5
John has $10.50.
13 2
6.5
John has $6.50.
Which makes more sense?
Sarah has 8 dollars. Jonathan has 5 dollars more than Sarah. He spends half of his money.
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Evaluate each expression using the order of operations. 10 - 2 x 4
Remember, to do what is inside the parenthesis first.
(10 - 2) x 4 What do you notice? 10 - 8 2 8 x 4 32 Step 1 click click Step 2 Slide 29 / 191
Let's solve (17 - 4) x 3 The parentheses tell you to subtract 17 - 4 first. (17 - 4) x 3 Then multiply by 3. 13 x 3 The answer is 39. 39 OR Let's solve 17 - (4 x 3) The parentheses tell you to multiply 4 x 3 first. 17 - (4 x 3) Then subtract. 17 - 12 The answer is 5. 5
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Evaluate the expression. (10 + 6 x 6) - 4 x 10 Step 1 Start with computations inside the parentheses. Using the Order of Operations, multiply first and then add. (10 + 6 x 6) (10 + 36) 46 Step 2 Rewrite the expression with parentheses evaluated. 46 - 4 x 10 Step 3 Multiply 4 x 10. Step 4 Rewrite the expression. 46 - 40 Step 5 Subtract. So, (10 + 6 x 6) - 4 x 10 = 6.
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8 What is the value of this expression? 5 + 3 x (7 - 1)
Remember to do inside the parentheses( ) first. A 23
B
25 C 48 D 64
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8 What is the value of this expression? 5 + 3 x (7 - 1)
Remember to do inside the parentheses( ) first. A 23
B
25 C 48 D 64
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Answer
A
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9 What is the value of this expression? (8 + 4) ÷ 3 x 6 A 6 B 9 C 24
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9 What is the value of this expression? (8 + 4) ÷ 3 x 6 A 6 B 9 C 24
[This object is a pull tab]
Answer
C
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10 Use the Order of Operations.
Write each step and evaluate the expression. 5 x (12 - 5) + 7
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11 Evaluate (14 - 5) + ( 10 ÷ 2)
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12 Which expression equals 72?
A 36 ÷ 4 - 3 x 2 B (36 ÷ 4 - 3) x 2 C 36 ÷ (4 - 3 x 2) D 36 ÷ (4 - 3) x 2
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13 Enter your answer.
From PARCC sample test
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14 Evaluate (8 x 9) - (6 x 7)
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15 Without actually calculating, compare the two
expressions using <, >, or =. On your paper, explain how you know . 10 x (4 x 3) 10 x (5 x 6) A B C
< > =
Problem derived from
( (
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16 Select each expression that is equal to 3/5 x 6. On your
paper, explain why the others are not equal using words, pictures, or numbers. A B C D
3 x (6 5) 3 (5 x 6) (3 x 6) 5 3 x 6
5
Problem derived from
( (
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17 Danny wrote an equation using two operations and one
set of parentheses, with an answer of 24. Select one letter from each column to complete Danny's equation below. A15 B8 C+ D- E3 F4 Gx H I2 J5
= 24
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18 Use what you know about the order of operations to
select the correct parenthesis locations to make this equation true. A B C D E F
( ( ( ( ( (
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19 Which of the following does not equal 134.86 + 52.14?
A (134+0.14)+(52+0.86) B (134+0.86)+(52+0.14) C (134+52)+(0.86+0.14) D(134.86-0.14)+(52.14-0.86)
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Besides parentheses ( ), brackets [ ] and braces { } are other kinds of grouping symbols used in expressions. To evaluate an expression with different grouping symbols, perform the operation in the innermost set of grouping symbols first. Then evaluate the expression from the inside
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Evaluate the expression 2 x [(9 x 4) - (17 - 6)] Step 1 Do operations in the parentheses ( ) first. multiply, subtract and rewrite 2 x [36 - 11] Step 2 Next do operations in the brackets [ ]. subtract and rewrite 2 x 25 Step 3 Multiply 2 x 25 = 50. So, 2 x [(9 x 4) - (17 - 6)] = 50.
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Evaluate the expression 3 x [(9 + 4) - (2 x 6)] Step 1 Do the operations in the parentheses ( ) first. add, multiply and rewrite 3 x [13 - 12] Step 2 Next do operation in the brackets [ ]. subtract and rewrite 3 x 1 Step 3 Then multiply 3 x 1 = 3. So, 3 x [(9 + 4) - (2 x 6)] = 3.
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Let's evaluate an expression together. Remember the Order of Operations, and solve parentheses ( ) first, then brackets [ ]. 5 x [(11 -3) - (13 - 9)] 5 x [8 - 4] 5 x 4 20
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Your turn...Evaluate the expression. Write each step. 8 x [(7 + 4) x 2] Step 1 8 x [11 x 2] Step 2 8 x [22] Step 3 176
click click click
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20 Evaluate an expression, with multiple grouping
symbols, from the inside out.
True False
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21 In the following expression, what operation would you
do first? 4 x [(15 - 6) x (7 - 3)]
A
multiiply
B
add
C subtract
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22 Evaluate the expression. Rewrite each step.
40 - [(8 x 7) - (5 x 6)]
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23 Evaluate the expression.
60 ÷ [(20 - 6) + (14 - 8)]
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Follow the same rules to solve expressions with braces { }. Perform the operation in the innermost set of grouping symbols first. Then, evaluate the expression from inside out. Evaluate the expression 2 x {5 + [(10 - 2)] + (4 - 1)]} Step 1 Do operations in parentheses ( ) first. subtract and rewrite 2 x {5 + [8 + 3]} Step 2 Next do operations in brackets [ ]. add and rewrite 2 x {5 + 11} Step 3 Then solve operations in braces { }. add and rewrite 2 x 16 Step 4 Multiply 2 x 16 = 32. So, 2 x {5 + [(10 - 2)] + (4 - 1)]} = 32.
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Let's evaluate an expression together. Remember the Order of Operations and to solve parentheses ( ), brackets [ ] and braces{ } from the inside out. 7 + {32 + [(7 x 2) - (2 x 5)]} 7 + {32 + [14 - 10]} 7 + {32 + 4} 7 + 36 43
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24 Evaluate the expression.
3 x {30 - [(9 x 2) - (3 x 4)]}
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25 Evaluate the expression.
10 + {36 ÷ [(14 -5) - (10 - 7)]}
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26 Which expressions equals 8?
A {5+[6-(3 x 2)] -1} B {[5 + (6 - 3) x 2] - 1} C {5+ 6 - [3 x (2 - 1)]}
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Word problems use expressions that you can write with
A variable is a letter that represents an unknown number. Any letter can be used for a variable. Writing algebraic expressions for words helps to solve word problems. These are a few common words that are used for operations. add (+) subtract (-) multiply (x or ) divide (÷) sum difference product quotient increased by minus times divided by plus less doubled per more than decreased by tripled
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"more than" means add "more than x" means add 17 to 5
17 more than 5
It is written as 5 + 17. four times the sum of 17 and 5
"four times" means multiply by 4 "sum of 17 and 5" means add 17 and 5
It is written as 4(17 + 5).
The words mean multiply 4 by (17 + 5) Examples:
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Practice writing a simple algebraic expression for these words. Addition Subtraction 9 increased by 12 336 less 24 9 + 12 336 - 24 322 more than 4 129 decreased by 106 322 + 4 129 - 106 8 plus 92 subtract 13 from 155 8 + 92 155 - 13 Multiplication Division 8 times 6 16 divided by 4 8 6 16 ÷ 4 9 multiplied by 5 the quotient of 24 and 14 9 x 5 24 ÷ 14
click click click click click
click
click
click
click click
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27 Which phrase is the correct algebraic expression?
the sum of 3 and 9
A
3 + 9
B
9 + 3
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28 Which phrase is the correct algebraic expression?
4 more than 5 A B 5 + 4 4 + 5
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29 Which phrase is the correct algebraic expression?
12 decreased by 7
A
12 - 7
B
7 - 12
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30 Three students were working on this problem.
Each of them had a different answer. Which answer is correct?
A B C
From PARCC sample test
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31 Which phrase is the correct algebraic expression?
13 less than 15
A
13 - 15
B
15 - 13
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32 Which phrase(s) is the correct algebraic expression?
product of 2 and 4
A
4 + 2 B 2 4
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33 Which phrase is the correct algebraic expression?
9 divided by 3
A
3 ÷ 9 B 9 ÷ 3
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34 Which phrase is the correct algebraic expression?
three times the sum of 8 and 7
A
8 x (3 + 7)
B
3 (8 + 7)
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35 Which is the correct algebraic expression?
12 divided by the sum of 5 and 2
A
12 (5 + 2)
B
(5 + 2) 12
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Let's practice writing phrases for these algebraic expressions. Remember, key words or phrases help decide which operation(s) to use when making your translations. Operation Key Words/Phrases Add (+) sum, more than, increased by Subtract (-) difference, less than, decreased by Multiply (x) product, times, twice, doubled, of Divide ( ) quotient, half, per
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Add (+) sum, more than, increased by Examples: 5 + 4 5 and 4 more 270 + 19 19 added to 270 100 + 160 160 increased by 100
click click
click
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Subtract (-) difference, less than, decreased by Examples: 400 - 199 400 decreased by 199 65 - 12 12 less than 65 50 - 31.5 difference of 50 and 31.5
click
click click
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Multiply (x) product, times, twice, doubled, of Examples: 9 x 4 9 times 4 45 2 product of 45 and 2 2 12 twice 12
click click click
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Divide ( ) quotient, half, per Examples: 12 6 12 divided by six 8 2 half of 8 15 5 5 per 15
click click click
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36 The phrase,16 less than 20, is the same as 20 - 16.
True False
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37 The phrase, 23 subtract from 233, is the same as 23 - 233.
True False
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38 Is the product of a 5 and 12, the same as 5 12?
Yes
No
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39 Which phrase is correct for the expression 21 7?
A
21 decreased by seven
B
the quotient of 21 and seven C the quotient of seven and 21
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40 Which phrase(s) are correct for the expression 3 2 + 9?
A three times 2 plus nine B three times 9 plus 2 C triple 2 added to nine
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41 The sum of 6 and 5 is 11,
is the same as 6 + 5 = 11.
True False
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Return to Table
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Often, in "real world" problems, a scenario is given in words, and you must translate it into a number sentence. Let's study four examples. Example 1 Patty bought just enough nuts to put five on each brownie she
4 nuts on each brownie, how many brownies did she make? Thus, the correct number sentence would be: 20 = 4 brownies So she made 5 brownies.
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Example 2 Pedro bought five of each kind of cookie that a bakery made. If we don't know the number of kinds of cookies the bakery had, we can use a blank box. What number sentence could represent how many cookies Pedro bought? number of cookies = x 5
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Example 3 Shandra sold five fewer boxes of Girl Scout cookies than Lisa. Let's use a blank box for the number of boxes Lisa sold, since we don't know how many she sold. What number sentence could represent how many boxes of cookies Shandra sold? Shandra = - 5
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Example 4 Nick brought 5 new packs of baseball cards today. We don't know how many packs he had yesterday, so we will use a blank
now? Today = + 5
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42 For a recycling project, 4 students each collected the same
amount of plastic bottles. They collected 32 in all. Which equation, when solved, will tell how many bottles each student collected? A 32 x 4 = B 4 - 32 = C 4 x = 32
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43 David has 46 sweaters in his closet. He has some sweaters
in his dresser as well. David has 64 sweaters in all. Which equation, when solved, will show how many sweaters are in David's dresser? A 46 + = 64 B 64 + 46 = C 64 + = 46
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44 A teacher opened a box of raisins and divided them evenly
among 16 students. Each student got 6 raisins. Which equation, when solved, will tell how many raisins were in each box? A - 16 = 6 B 6 = 16 C 16 = 6
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45 Dana took some almonds from a bowl. She ate ten of them
and had 18 almonds left. Which equation, when solved, will tell how many almonds Dana took from the bowl? A - 10 = 18 B 10 = 18 C + 10 = 18
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46 True/False: The following two scenarios can be
represented by the same expression. True False Enter your answer. Then, on your paper, explain how you know that the answer is true or false.
Lia works 7 hours a day for days. Gil has 7 baseballs in bags.
Problem derived from
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47
The chart shows expressions that students wrote in math class. Select the expressions that are equal. A B C D
Jake 38+6 Liz 56-24 Ryan 8x6 Kim 4x8
Amy wrote the expression 17+15. Her equation is also equivalent to these expressions. On your paper, use what you know about expressions to explain why this is true.
Part I Part II
Problem derived from
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In John's class, the boys are outnumbered 3 to 1. So, for every boy there are 3 girls. If there is 1 boy, there are 3 girls. If there are 2 boys, there are 6 girls. If there are 5 boys, there are ______ girls. (1,3) (2,6) (5,___)
(1,3) (2,6) (5,___) { } ; ;
This set of numbers is called a relation, and it describes the relationship between boys and girls in John's class.
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A relation is a set of pairs of numbers, called ordered pairs, that represent a relationship between two things. The members of a set can be: · pairs of things (like socks) · people (like boys and girls) · people and things (like students and the types of books they read) · numbers (like 5 and 10).
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They are many ways to show that two different items are related. Let's practice using tables, equations and graphs to describe a function or relation. boys girls 1 3 2 6 3 9 · a list of ordered pairs
{(1,3); (2,6); (3,9)}
Number of boys Number of girls
· a graph · a table · a word description For every boy, there are 3 girls · an algebraic rule or equation girls = boys x 3
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A function shows the relationship between an Input amount and an Output amount. Input Output Rule
My Function Machine Slide 97 / 191
The value of the output relies on
Input Output Rule The rule is the relationship between the input and the output. It says what happens to the input inside the machine. The value of the output always depends on the value of the input.
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Let's Practice figuring out the rule.
Step 1. Assign a value to the input.
Step 2. Hit Enter to see the output. The input and output values will show
this table. Step 3. Once you have enough input/output values to figure out the rule, select + or * and the addend or factor. Step 4. Check Your Rule
Click here for online practice.
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A function table shows you the relationship between pairs of
applies to all the pairs of numbers on the table. You can think of the rule as a black box or machine. Usually, the Input is labeled (x) and the Output is labeled (y).
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The function table can be set up vertically or horizontally to show the relation. x = first number (input) y = second number (output). If the rule is add 5, here are the tables: 5 1 6 2 7 3 8
1 2 3 5 6 7 8
input(x) output(y)
input(x)
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The variables x and y are usually used for an unknown value, but other letters can be used. Examples: m = miles c = cost h = hours l = length
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Let's apply a rule to the function table.
Pull Pull
Teacher Note
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Let's apply a different rule to the function table.
Pull Pull
Teacher Note
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48 Rule: Add 4
The missing value is 14.
True False 6 10 7 11 8 12 10 12 16
input(x) output(y)
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49 Rule: Multiply 3
The missing value is 12. True False
2 6 3 9 6 18 8 24
input(x) output(y)
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50 Rule: Add 9
What is the missing value?
1 2 3 4 9 10 11 12 13
input(x)
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51 Rule: Divide by 2
What is the missing value? 48
24 16 8 8 4 2 1
input(x) output(y)
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52 Rule: Subtract 8
What is the missing value that the arrow is pointing to?
8 10 19 22 40 2 11 14 32
input(x)
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53 Rule: Subtract 8
What is the missing value that the arrow is pointing to?
8 10 19 22 40 2 11 14 32
input(x)
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Let's find the rule for the function table. Add, subtract, multiply
Let's study the pattern between the input and output values.
Pull Pull
Teacher Note
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Look at the difference between the numbers. The rule is y = x 8 or divided by 8.
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Let's practice finding the rule or function on the tables. Remember to look at the given Input-Output pairs.
2 9 3 10 5 12 6 13
Input (x) Output(y) Each Output is greater than the Input. Try a rule with addition or multiplication. 2 9 3 10 5 12 6 13 The rule is Add 7, or y = x + 7.
15 3 20 4 25 5 40 8
Input(x) Output(y) Each Output is less than the Input. Try a rule with subtraction or division. 15 3 20 4 25 5 40 8 The rule is Divide by 5, or y = x 5.
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54 Is the rule or function
y = x 9? Yes
No 9 1 18 2 27 3 36 4
Input(x) Output(y)
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55 Is the rule
y = x - 6?
Yes No
6 8 10 14 2 4 8
input(x)
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56 Is the rule or function
y = 2x - 1?
True False 10
19 8 15 14 27 6 11
Input(x) Output(y)
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57 What is the rule or function?
A Subtract 2 B Add 3 C Add 2
5 8 7 10 6 9 8 11
Input(x) Output(y)
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58 What is the rule or function?
A y = x + 2 B y= 2x C y = x 2 2 20 10 14 7 10 5
Input(x) Output(y)
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59 What is the rule or function?
A y = 2x + 2 B y = 3x + 2 C y = 2x - 2 1 5 2 8 3 11 4 14
Input(x) Output(y)
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A function table can be used to solve Miguel and Mary's problem with their pens. Solution: "Miguel has 7 fewer pens than Mary." means: Number pens Miguel has is 7 fewer than the number pens Mary has y = - 7 x So you get , y = x - 7
Pull Pull
Teacher Note
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A function table can be used to solve Sharon's problem with number of miles she will run given any number of hours. Solution: For the number of miles, multiply by the rate(miles per hour). number of miles = miles per hour x number of hours y = 5 x x So you get, y = 5x
Pull Pull
Teacher Note
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Lori is traveling by taxi in New York City. Using a function table, she can calculate the cost of her trip. Use the letter m, for the miles traveled and c, for the cost of the taxi ride. miles traveled, m
1 2
3 4 5 6 7
cost of taxi ride, c
$6 $7
$8 $9 $1 $11 $12
How would you describe the function in words? _______ each mile is five more dollars What would be the equation to calculate the cost? _______ c = m + 5 Using the equation, how much would if cost to travel 20 miles? _______ $25 = 20 + 5
Pull Pull
Teacher Note
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60 Given the rule used in the function table, will 15
people fit in 3 vans? Yes
No
number of people, p
5 10 15 20
number of vans, v
1 2 3 4
Pull Pull
Teacher Note
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61 Use the function table and equation.
How many vans are needed for 35 people?
A 5 vans B 35 vans C 7 vans
number of people, p
5 10 15 20
number of vans, v
1 2 3 4
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62 If each package contained 2 cookies, what is the tenth
number of cookies in the package?
5 10 7 14 8 16 10 20
(p) number
packages (c) number
cookies
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63 Use the function table and equation.
How many hours will it take for the car to travel 495 miles?
A 7 hours B 9 hours C 11 hours D 15 hours
time (hr) 1
2 3 4
distance (miles) 55
110 165 220
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64 Use the function table and equation.
How much money will you earn in 4 weeks?
Hours Worked (h) Money Earned (m) 1 $6.25 2 $12.50 3 $18.75 4 $25.00 5 $31.25
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65 Given the rule add 3 and
starting at 0 complete the table below. A Each term in the second table is equal to the corresponding term in the first table. BEach term in the second table is twice the value of the corresponding term in the first table. CEach term in the second table is three more than the value of the corresponding term in the first table. Given the rule add 6 and starting at 0 complete the table below.
Look at both of the tables once they are complete, and identify the relationship between the corresponding terms.
Problem derived from
( (
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66 Complete the pattern below using the rule 3n-1.
(type in the numbers)
Problem derived from
( ( 5, _____, 11, _____, _____ Slide 129 / 191
67 The output of a function table is 4 less than each
says the rule is 4 - x = y. Who is correct? Explain your answer on your paper. AAngela BKara
Problem derived from
( ( Slide 130 / 191
68 Use the table below to show the
the pattern in the previous problem. (Select one value for each box.)
1 2 5 3 14 8 2 4 11 5 10 14 6 14 17
A B C D E F G H I J
Problem derived from
( (
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69 A fifth-grade class collected boxed and canned foods
for the school's annual food drive. The first day, 2 boxes and 2 cans were collected. Each day after the first, the number of boxes collected went up by 4 and the number of cans collected went up by 2. A times 2 B times 2, then minus 2 C add 2 D subtract 4 On your paper, complete the table by writing the first five numbers in the number patterns for the number of boxes and cans collected. Then, select the rule for the table that compares the two patterns.
Cans Boxes
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Return to Table
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Number patterns, functions tables and equations can be shown in graphs on a coordinate plane. The graph gives an easy visual way to solve problems, and to make predictions based on the patterns seen in the graph.
Lets explore graphing coordinates first.
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1 2 3 4 5 6 8 9 10 2 3 4 5 6 7 8 9 10 1 7
The coordinate plane is formed by perpendicular number lines called axes. The horizontal line is the x-axis. The vertical line is the y-axis.
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1 2 3 4 5 6 8 9 10 2 3 4 5 6 7 8 9 10 1 7
The point at which the x and y axes intersect is called the origin. The coordinates of the origin are (0, 0).
Origin (0, 0)
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Points can be plotted on the plane using one coordinate from each of the axes. These sets are called ordered pairs. The x coordinate always appears first in these pairs. The y coordinate appears second.
(x,y)
1 2 3 4 5 6 8 9 10 2 3 4 5 6 7 8 9 10 1 7
x y Slide 137 / 191
1 2 3 4 5 6 8 9 10 2 3 4 5 6 7 8 9 10 1 7
To graph an ordered pair, such as (4,3): · Start at the origin (0,0). · Move right on the x-axis, since the first number is positive. · Then, move up since the second number is positive. · Plot the point. (4,3)
x y Slide 138 / 191
1 2 3 4 5 6 8 9 10 2 3 4 5 6 7 8 9 10 1 7
(x,y)
This point is (3,2). To plot the point, go over 3, then up 2.
x y Slide 139 / 191
1 2 3 4 5 6 8 9 10 2 3 4 5 6 7 8 9 10 1 7
x y (x,y)
This point is (1,4). To plot the point, go over 1, then up 4.
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1 2 3 4 5 6 8 9 10 2 3 4 5 6 7 8 9 10 1 7
(x,y)
This point is (5,0). To plot the point, go over 5, then up 0.
x y Slide 141 / 191
1 2 3 4 5 6 8 9 10 2 3 4 5 6 7 8 9 10 1 7 70 Which point is at the origin?
A B C D
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1 2 3 4 5 6 8 9 10 2 3 4 5 6 7 8 9 10 1 7 71 Which point is at (1,3)?
A B C D
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1 2 3 4 5 6 8 9 10 2 3 4 5 6 7 8 9 10 1 7 72 Which point is at (3,3)?
A B C D
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1 2 3 4 5 6 8 9 10 2 3 4 5 6 7 8 9 10 1 7 73 Which point is at (0,5)?
A B C D
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74 Which ordered pair is the origin? A (4,0) B
(0,0)
C
(0,4)
D
(4,4)
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75 Which number in the ordered pair (7,3) is
the x-coordinate?
A 7 B
3
C D
x
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76 Which number in the ordered pair (5,9) is
the y-coordinate?
A B
5
C
9
D
y
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77 Which number in the ordered pair (7,12)
is the y-coordinate?
A
7
B
12
C D
y
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78 Which number in the ordered pair (7,12) is
the x-coordinate?
A
7
B
12
C D
x
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Follow the steps to graph the function y = x + 2. Step 1 Complete the function table. Replace the x in the equation with a number from the x
Step 2 Graph each ordered pair (x,y) on the coordinate grid. Look at the first pair (1,3). The 1 tells you to go one unit to the right (horizontal) of the
Step 3 Use the same method to graph (2,4), (3,5), (4,6) Step 4 Connect all the points with a line. You should end up with a straight line that shows the solution for y = x + 2.
Pull Pull
Teacher Note
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Equation y = x + 2
x y 1 3 2 4 3 5 4 6
Quadrant I: positive numbers
Graph: Function Table:
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Increasing line: A line that slants upward from left to right. Decreasing line: A line that slants downward from left to right. Equations that result in curved lines are called nonlinear equations. Equations that result in straight lines are called linear equations.
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Equation: y = x - 1 Function Table Graph
x y 1 2 1 3 2 4 3
Quadrant I - positive numbers
x y Solve for y. Start with x = 1 y = 1 - 1 y = 0 Repeat the above steps to find the value of y when x = 2. Repeat x = 3 and x = 4. Graph the order pairs and connect the points with a line.
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Equation: y = 2x + 3 Function Table Graph
x y 3 1 5 2 7 3 9
Quadrant I - positive numbers
x y
Solve for y. Start with x = 2 y = 2x + 3 y = (2 x 2) + 3 y = 7 Repeat the above steps to find the value of y when x = 3. Graph the order pairs and connect the points with a line.
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79 Which of the following points is on the
line?
A (1, 5) B (4, 10) C (2, 7) D (8, 3) x
y
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80 Which graph represents the given function?
A Graph A B Graph B
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81 Which graph does not represent the given function?
A Graph A B Graph B
1 2 3 4 5 6 7 1 2 3 4 5 6 7
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82 Which graph represents the given function?
A Graph A B Graph B
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Graphing Relations can be used in "real" world problems. The clerk at the video store earns $6.00 per hour. Here is how you would graph the relation between hours worked and amount earned, up to six hours.
First, use a table to show this one-to-one relation.
Hours Worked (x) 0
1 2
3
4 5 6
Dollars Earned (y) 0
$6 $1 2
$1 8
$2 4 $30 $3 6 Second, graph the ordered pairs: (0, 0), (1, 6), (2, 12), (3, 18), (4, 24), (5, 30), (6, 36).
number of hours worked
1 2 3 4 5 6
612
18
24 30 36
42
number of dollars earned
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Using the equation from the table or graph, y = 6x, you can calculate how much you would earn given any amount
If the clerk worked 12 hours, how much would be earned? $72.00 If the clerk worked 30 hours, how much would be earned? $180.00 If the cleark worked 40 hours, how much would be earned? $240.00
Pull Pull
Teacher Note click click click
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Blaise walks home from school at the rate of 3 kilometers per hour. Complete the function table that shows the relationship between d, the distance he walks, and t, the time it takes him to walk this distance. Graph the ordered pairs and connect with a line. Equation: d = 3t or y = 3x time (t) distanc e (d)
d pairs (0, 0) 1 3 (1, 3) 2 6 (2, 6) 3 9 (3, 9)
x
y
time distance
Pull Pull
Teacher Note
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Using the equation from the table or graph, d = 3t, you can calculate how much distance traveled given any amount of time. If you walked 5 hours, how much distance did you travel? 15 miles If you walked 8 hours, how much distance did you travel? 24 miles
click
Pull Pull
Teacher Note
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We can use graphs to compare two relations. Line a
(The ordered pairs from table A) When Blaise rides his bike home from school,
he travels at the rate of 6 kilometers per hour. Complete function table B to show the kilometers he bikes per hour. Then, graph the ordered pairs and connect with a line. Label the line b.
time (t) distanc e (d) 1 3 2 6 3 9
x
y
time distance
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 20 19 2 3 4 5 6 7 8 9 1
Table A
Equation: d = 3t
time (t) distanc e (d) 1 6 2 12 3 18
x
y
Table B
Equation: d = 9t
a b
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We can use graphs to compare two relations. Compare the y-coordinates on line a (and in Table A) to the y-coordinates on line b (and in Table B).
time (t) distance (d) 1 3 2 6 3 9
x
y
time distance
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 20 19 2 3 4 5 6 7 8 9 1
Table A
Equation: d = 3t
time (t) distance (d) 1 6 2 12 3 18
x
y
Table B
Equation: d = 9t
a b
Click
Explain the relationship between the y-coordinates on line a to the y-coordinates on the same x-axis on line b. What does this tell you about his walking and biking speed?
Pull Pull
Teacher Note
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83 Which function table does the graph best represent?
A Table A B Table B
number of quarts (q) number of gallons (g)
1 2 2 4 3 6
number of quarts (q) number of gallons (g)
4 1 8 2 12 3
quarts gallons
4 8
12 16
1 2 3 4 5 6
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84 Which best describes a graph that shows the relationship
between the cost of heating a home and the outside temperature?
A horizontal line B an increasing straight line C a decreasing straight line D a vertical line
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85 What is the relationship between the y-coordinates on
line a and the y-coordinates with the same x-coordinate
A The coordinate on line b is half the value of the coordinate on line a. B The coordinate on line b is twice the value of the coordinate on line a. CThe coordinate on line b is one-fourth the value of the coordinate on line a.
1 2 3 4 5 6 7 8 9 10 11 12 2 3 4 5 6 7 8 9 1
a b
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[This object is a pull tab]
Teacher Notes
Vocabulary Words are underlined with a dashed line in the presentation. This is linked to the page at the end
word defined on it.
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y = 3
Algebra is like a Math puzzle, finding the missing pieces and how they fit together.
A type of Math that uses letters (variables) and symbols to represent numbers. Slide 170 / 191
Algebraic Expression
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variable:
number: number:
variable:
equals sign:
3 + 2 = y An expression that consists of
variables, and operations. Slide 171 / 191
To find a number/answer by adding, subtracting, multiplying, and/or dividing.
4 x 3
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pencil & paper calculator mental math
Slide 172 / 191
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Coordinate Plane
The two dimensional plane or flat surface that is created when the x-axis intersects with the y-axis.
Also known as a coordinate graph and a Cartesian plane.
(0,0) x
y Slide 173 / 191
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Two expressions that are equivalent to each other. Equivalence is shown with an equal sign.
equivalent expressions
equivalent expressions
no equivalence
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To find the value/amount of something.
If x=3, then 2x is 2 times 3; so 2x = 6
You cannot "evaluate" an algebraic expression if you do not know the value of the variable.
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Numbers, symbols and
show the value of something.
2 x 3 = 6
Expressions DO NOT have equals signs.
2
An expression is
an equation.
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A special relationship between x and y values, where each of its input values gives back exactly one output value. The
depends
input (x).
(x) (y)
1 1 2 4 3 9
(x) (y) 1 1 1 4
3 9
Not a function. For each input, there can only be
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The input is the independent variable in a function (y).
The
depends
input (x).
(x) (y) 1 1 2 4
3 9
The value of y is determined by the value of x. Input
Input
Rule
y = x2
3 Slide 178 / 191
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An equation whose graph is a straight line.
1 2 3 4 5 6 8 9 10 2 3 4 5 6 7 8 9 10 1 7
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An equation whose graph is not a straight line.
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Order of Operations
Please Excuse My Dear Aunt Sally The rules of which calculation comes first in an expression.
Parentheses, Exponents, Multiplication or Division, Addition or Subtraction r
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(x,y) (x,y)
An x and y value that identify a location on a coordinate plane.
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The point where zero on the x-axis intersects zero on the y-axis. The coordinates of the origin are (0,0). (0,0) (0,0)
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The output is the dependent variable in a function (y).
The
depends
input (x).
(x) (y) 1 1 2 4
3 9
The value of y is determined by the value of x. Output
Input Rule
y = x2
3 Slide 184 / 191
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A point is not an object, but a location, identified by a set of coordinates. A
1 2 3 4 5 6 8 9 10 2 3 4 5 6 7 8 9 10 1 7
A (3,5)
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A relationship between two sets of data, where
represented with a set of ordered pairs.
(1,3) (2,6) (5,___) { } ; ;
For every boy, there are 3 girls.
(x) (y) 1 1 2 4
3 9
(1,1) (2,4) (3,9)
{ } ; ;
(Hours worked, $ earned)
{(1,8);(2,16);(3,24)} ( x , y )
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Defines the relationship between the input and the output, using an algebraic equation. It says what happens to the input in a function.
Input
3
Output
9
Rule:
(x) (y) 1 3 2 6
3 9
Rule:
(x) (y) 1 6 2 10
3 14
Rule:
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A visual display that organizes the special relationship between two sets of data. (x) (y)
1 1 2 4 3 9
in. ft.
12 1 24 2 36 3
hours worked
3 $ earned24
4 5 32 40
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A letter or symbol that represents a changeable or unknown value.
variable
x x
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Horizontal number line that extends indefinitely in both directions from zero. (Right- positive Left negative)
1 2 3 4 5
x Slide 190 / 191
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Vertical number line that extends indefinitely in both directions from
1 2 3 4 5
y
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