Slide 1 / 191 Slide 2 / 191 5th Grade Algebraic Concepts 2015-10-16 www.njctl.org Table of Contents click on the topic to go to that section Slide 3 / 191 What is Algebra? Order of Operations Grouping Symbols Writing Simple Expressions & Interpreting Numerical Expressions Writing & Interpreting Expressions Application Problems Function Tables Graphing Patterns & Relationships in the Coordinate Plane Glossary
Table of Contents click on the topic to go to that section Slide 3 (Answer) / 191 Vocabulary Words are What is Algebra? underlined with a dashed line in the presentation. This is Order of Operations linked to the page at the end Grouping Symbols of the presentation with the Writing Simple Expressions word defined & Interpreting on it. Numerical Expressions [This object is a pull tab] Writing & Interpreting Expressions Application Problems Function Tables Graphing Patterns & Relationships in the Coordinate Plane Glossary Teacher Notes Slide 4 / 191 What is Algebra? Return to Table of Contents Slide 5 / 191 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'Khwarizmi was a Persian mathematician who wrote on Hindu-Arabic numerals (the numeral system we use today). He was also one of the first to use zero as a place holder. al-jebr (from the book title) means "reunion of broken parts". Kind of like puzzle pieces.
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. Slide 6 / 191 What is the missing piece to this number puzzle? Algebra help us tie together many mathematical ideas. (Like hours worked and money earned.) Slide 7 / 191 Sarah earns $10 for every hour she works. (hours) $10 x $40 What is the missing value? Things change. Slide 8 / 191 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)
Slide 9 / 191 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. Slide 10 / 191 Order of Operations Return to Table of Contents Imagine this: You put your shoes on, and then your socks. Slide 11 / 191 Wait... What? What's wrong with this picture? It's out of order. click
Slide 12 / 191 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 13 / 191 In an expression with more than one operation, use the rules called Order of Operations. 1. Do all multiplication and division in order from left to right. 2. Do all addition and subtraction in order from left to right. 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 -. Slide 14 / 191 Name the operation that should be done first. 6 x 3 + 4 click multiplication 3 + 4 x 6 multiplication click 5-3 + 6 subtraction click 9-6 3 division click
1 Do you add or multiply first? 6 + 3 x 2 + 7 Slide 15 / 191 A B add multiply 1 Do you add or multiply first? 6 + 3 x 2 + 7 Slide 15 (Answer) / 191 A B add multiply Answer B [This object is a pull tab] 2 Do you divide or add first? 12 + 3 12 4 Slide 16 / 191 A B add divide
2 Do you divide or add first? 12 + 3 12 4 Slide 16 (Answer) / 191 A B add divide Answer B [This object is a pull tab] 3 10-6 x 6 + 4 x 10 Slide 17 / 191 Which operation do you do... 1st? 2nd? 3rd? A B C + - x D E F + - x + - x G H I Choose an operation for each step. 3 10-6 x 6 + 4 x 10 Slide 17 (Answer) / 191 Which operation do you do... 1st? 2nd? 3rd? A B C + - x D E F + - x + - x Answer C,E,G G H I [This object is a pull tab] Choose an operation for each step.
Slide 18 / 191 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. Evaluate the expression. Slide 19 / 191 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. 4 Evaluate 6 + 3 x 2 + 7 = Slide 20 / 191
4 Evaluate 6 + 3 x 2 + 7 = Slide 20 (Answer) / 191 Answer 19 [This object is a pull tab] 5 Evaluate 12 + 12 4 3 = Slide 21 / 191 5 Evaluate 12 + 12 4 3 = Slide 21 (Answer) / 191 Answer 13 [This object is a pull tab]
6 Evaluate 100-6 x 6 + 4 x 10 = Slide 22 / 191 6 Evaluate 100-6 x 6 + 4 x 10 = Slide 22 (Answer) / 191 Answer 104 [This object is a pull tab] 7 Evaluate 50 10 + 15 Slide 23 / 191
7 Evaluate 50 10 + 15 Slide 23 (Answer) / 191 Answer 20 [This object is a pull tab] Slide 24 / 191 Grouping Symbols Return to Table of Contents Sarah has 8 dollars. Jonathan has 5 dollars more than Sarah. He spends half of his money. Write an expression that represents this scenario. Slide 25 / 191 Sara's $ plus $5 divided in half 8 + 5 2 Think, pair, share: Is this correct? Explain your answer on your paper.
If we follow the order of operations, we would divide the "$5 more" before we added it to Sarah's money. Slide 26 / 191 Sara's $ plus $5 divided in half 8 + 5 2 So how can we represent this scenario? We can use parenthesis. Parentheses ( ) are used to group calculations to be sure that they are done in a certain order. Slide 27 / 191 When you use parentheses ( ), you are saying, "DO THIS FIRST." Sara's $ plus $5 divided in half (8 + 5) 2 The parenthesis tell us to add $5 to Sarah's money before we divide it in half. Sarah has 8 dollars. Jonathan has 5 dollars more than Sarah. He spends half of his money. Slide 28 / 191 Lets look at the results of each expression. Step 1: 8 + 5 2 (8 + 5) 2 8 + 2.5 13 2 Step 2: 10.5 6.5 John has $10.50. John has $6.50. Which makes more sense?
Evaluate each expression using the order of operations. Slide 29 / 191 Remember, to do what is inside the parenthesis first. Step 1 Step 2 10-2 x 4 10-8 2 click click (10-2) x 4 8 x 4 32 What do you notice? Slide 30 / 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 Evaluate the expression. (10 + 6 x 6) - 4 x 10 Slide 31 / 191 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.
8 What is the value of this expression? 5 + 3 x (7-1) Remember to do inside the parentheses( ) first. Slide 32 / 191 A 23 B 25 C 48 D 64 8 What is the value of this expression? 5 + 3 x (7-1) Remember to do inside the parentheses( ) first. Slide 32 (Answer) / 191 A 23 B 25 C 48 D 64 Answer A [This object is a pull tab] 9 What is the value of this expression? (8 + 4) 3 x 6 Slide 33 / 191 A 6 B 9 C 24
9 What is the value of this expression? (8 + 4) 3 x 6 Slide 33 (Answer) / 191 A 6 B 9 C 24 Answer C [This object is a pull tab] 10 Use the Order of Operations. Write each step and evaluate the expression. Slide 34 / 191 5 x (12-5) + 7 11 Evaluate (14-5) + ( 10 2) Slide 35 / 191
12 Which expression equals 72? Slide 36 / 191 A 36 4-3 x 2 B (36 4-3) x 2 C 36 (4-3 x 2) D 36 (4-3) x 2 13 Enter your answer. Slide 37 / 191 From PARCC sample test 14 Evaluate (8 x 9) - (6 x 7) Slide 38 / 191
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) (Problem derived from ( Slide 39 / 191 A B C < > = (Problem derived from 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. ( Slide 40 / 191 A B C D 3 x (6 5) 3 (5 x 6) (3 x 6) 5 3 x 6 5 ( Problem derived from 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. = 24 ( Slide 41 / 191 A15 C+ E3 Gx I2 B8 D- F4 H J5
18 Use what you know about the order of operations to select the correct parenthesis locations to make this equation true. 50 2 + 8-3 = 2 A B C D E F ( ( ( ( ( ( Problem derived from ( ( Slide 42 / 191 19 Which of the following does not equal 134.86 + 52.14? Slide 43 / 191 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) Slide 44 / 191 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 out.
Evaluate the expression Slide 45 / 191 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. Evaluate the expression Slide 46 / 191 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. Let's evaluate an expression together. Remember the Order of Operations, and solve parentheses ( ) first, then brackets [ ]. Slide 47 / 191 5 x [(11-3) - (13-9)] 5 x [8-4] 5 x 4 20
Your turn...evaluate the expression. Write each step. Slide 48 / 191 8 x [(7 + 4) x 2] Step 1 8 click x [11 x 2] Step 2 8 x [22] click Step 3 176 click 20 Evaluate an expression, with multiple grouping symbols, from the inside out. Slide 49 / 191 True False 21 In the following expression, what operation would you do first? Slide 50 / 191 4 x [(15-6) x (7-3)] A B C multiiply add subtract
22 Evaluate the expression. Rewrite each step. Slide 51 / 191 40 - [(8 x 7) - (5 x 6)] 23 Evaluate the expression. Slide 52 / 191 60 [(20-6) + (14-8)] 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. Slide 53 / 191 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.
Let's evaluate an expression together. Remember the Order of Operations and to solve parentheses ( ), brackets [ ] and braces{ } from the inside out. Slide 54 / 191 7 + {32 + [(7 x 2) - (2 x 5)]} 7 + {32 + [14-10]} 7 + {32 + 4} 7 + 36 43 24 Evaluate the expression. Slide 55 / 191 3 x {30 - [(9 x 2) - (3 x 4)]} 25 Evaluate the expression. Slide 56 / 191 10 + {36 [(14-5) - (10-7)]}
26 Which expressions equals 8? Slide 57 / 191 A {5+[6-(3 x 2)] -1} B {[5 + (6-3) x 2] - 1} C {5+ 6 - [3 x (2-1)]} Slide 58 / 191 Writing Simple Expressions & Interpreting Numerical Expressions Return to Table of Contents Word problems use expressions that you can write with symbols. An algebraic expression has at least one variable. 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. Slide 59 / 191 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
Examples: Slide 60 / 191 17 more than 5 "more than" means add "more than x" means add 17 to 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 The words mean multiply 4 by (17 + 5) It is written as 4(17 + 5). Practice writing a simple algebraic expression for these words. Addition Subtraction Slide 61 / 191 9 increased by 12 336 less 24 9 + 12 336-24 click 322 more than 4 129 decreased by 106 322 + 4 129-106 click 8 plus 92 subtract click 13 from 155 8 + 92 155 click- 13 click click Multiplication Division 8 times 6 16 divided by 4 8 6 16 4 click click 9 multiplied by 5 the quotient of 24 and 14 9 x 5 24 14 click click 27 Which phrase is the correct algebraic expression? the sum of 3 and 9 Slide 62 / 191 A 3 + 9 B 9 + 3
28 Which phrase is the correct algebraic expression? 4 more than 5 Slide 63 / 191 A B 5 + 4 4 + 5 29 Which phrase is the correct algebraic expression? 12 decreased by 7 Slide 64 / 191 A 12-7 B 7-12 30 Three students were working on this problem. Slide 65 / 191 Each of them had a different answer. Which answer is correct? A B C From PARCC sample test
31 Which phrase is the correct algebraic expression? 13 less than 15 Slide 66 / 191 A 13-15 B 15-13 32 Which phrase(s) is the correct algebraic expression? product of 2 and 4 Slide 67 / 191 A 4 + 2 B 2 4 33 Which phrase is the correct algebraic expression? 9 divided by 3 A 3 9 B 9 3 Slide 68 / 191
34 Which phrase is the correct algebraic expression? three times the sum of 8 and 7 Slide 69 / 191 A 8 x (3 + 7) B 3 (8 + 7) 35 Which is the correct algebraic expression? 12 divided by the sum of 5 and 2 Slide 70 / 191 A 12 (5 + 2) B (5 + 2) 12 Let's practice writing phrases for these algebraic expressions. Slide 71 / 191 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
Add (+) sum, more than, increased by Examples: Slide 72 / 191 5 + 4 5 and 4 more click 270 + 19 19 added to 270 click 100 + 160 160 increased by 100 click Slide 73 / 191 Subtract (-) difference, less than, decreased by Examples: 400-199 400 decreased by 199 click 65-12 12 less than 65 click 50-31.5 difference of 50 and 31.5 click Slide 74 / 191 Multiply (x) product, times, twice, doubled, of Examples: 9 x 4 9 times 4 click 45 2 product of 45 and 2 click 2 12 twice 12 click
Slide 75 / 191 Divide ( ) quotient, half, per Examples: 12 6 12 divided by six click 8 2 half of 8 click 15 5 5 per 15 click 36 The phrase,16 less than 20, is the same as 20-16. Slide 76 / 191 True False 37 The phrase, 23 subtract from 233, is the same as 23-233. Slide 77 / 191 True False
38 Is the product of a 5 and 12, the same as 5 12? Slide 78 / 191 Yes No 39 Which phrase is correct for the expression 21 7? Slide 79 / 191 A 21 decreased by seven B the quotient of 21 and seven C the quotient of seven and 21 40 Which phrase(s) are correct for the expression 3 2 + 9? Slide 80 / 191 A three times 2 plus nine B three times 9 plus 2 C triple 2 added to nine
41 The sum of 6 and 5 is 11, is the same as 6 + 5 = 11. Slide 81 / 191 True False Slide 82 / 191 Writing & Interpreting Expressions Application Problems Return to Table of Contents Slide 83 / 191 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 made. If Patty bought 20 nuts and placed 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.
Slide 84 / 191 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 Slide 85 / 191 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 Slide 86 / 191 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 box. What number sentence could represent how many he has now? Today = + 5
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? Slide 87 / 191 A 32 x 4 = B 4-32 = C 4 x = 32 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? Slide 88 / 191 A 46 + = 64 B 64 + 46 = C 64 + = 46 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 Slide 89 / 191
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 Slide 90 / 191 46 True/False: The following two scenarios can be represented by the same expression. 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 True Gil has 7 baseballs in False ( Problem derived from bags. days. ( Slide 91 / 191 47 Part I 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 ( Problem derived from ( Slide 92 / 191 Part II 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.
Slide 93 / 191 Function Tables Return to Table of Contents In John's class, the boys are outnumbered 3 to 1. Slide 94 / 191 So, for every boy there are 3 girls. If there is 1 boy, there are 3 girls. (1,3) If there are 2 boys, there are 6 girls. (2,6) If there are 5 boys, there are girls. (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. Slide 95 / 191 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).
They are many ways to show that two different items are related. Slide 96 / 191 a word description For every boy, there are 3 girls an algebraic rule or equation girls = boys x 3 a table boys girls 1 3 2 6 3 9 a graph a list of ordered pairs {(1,3); (2,6); (3,9)} Number of girls Let's practice using tables, equations and graphs to describe a function or relation. Number of boys My Function Machine A function shows the relationship between an Input amount and an Output amount. Slide 97 / 191 Input Output Rule Input The value of the output relies on 1. The value of the input 2. The rule Slide 98 / 191 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.
Let's Practice figuring out the rule. Slide 99 / 191 Step 1. Assign a value to the input. The input and output values will show on this table. Step 2. Hit Enter to see the output. 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. Slide 100 / 191 A function table shows you the relationship between pairs of numbers. The relationship is defined by a rule, and this rule 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). ) x, y ) 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: Slide 101 / 191 input(x) output(y) 0 5 1 6 2 7 3 8 input(x) 0 1 2 3 output(y) 5 6 7 8
The variables x and y are usually used for an unknown value, but other letters can be used. Slide 102 / 191 Examples: m = miles c = cost h = hours l = length Let's apply a rule to the function table. Pull Teacher Note Slide 103 / 191 Let's apply a different rule to the function table. Slide 104 / 191 Pull Teacher Note
48 Rule: Add 4 The missing value is 14. Slide 105 / 191 True False input(x) 6 10 output(y) 7 11 8 12 10 12 16 49 Rule: Multiply 3 The missing value is 12. Slide 106 / 191 True False input(x) 2 6 output(y) 3 9 6 18 8 24 50 Rule: Add 9 What is the missing value? Slide 107 / 191 input(x) 0 1 2 3 4 output(y) 9 10 11 12 13
51 Rule: Divide by 2 What is the missing value? Slide 108 / 191 input(x) output(y) 48 24 16 8 8 4 2 1 52 Rule: Subtract 8 What is the missing value that the arrow is pointing to? Slide 109 / 191 input(x) output(y) 8 10 19 22 40 0 2 11 14 32 53 Rule: Subtract 8 What is the missing value that the arrow is pointing to? Slide 110 / 191 input(x) output(y) 8 10 19 22 40 0 2 11 14 32
Let's find the rule for the function table. Add, subtract, multiply or divide the Input(x) to get the Output(y). Slide 111 / 191 Pull Teacher Note Let's study the pattern between the input and output values. Look at the difference between the numbers. Slide 112 / 191 The rule is y = x 8 or divided by 8. Let's practice finding the rule or function on the tables. Remember to look at the given Input-Output pairs. Slide 113 / 191 Input (x) Output(y) 2 9 3 10 5 12 6 13 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. Input(x) 15 3 20 4 25 5 40 8 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.
54 Is the rule or function y = x 9? Yes No Input(x) 9 1 18 2 Output(y) Slide 114 / 191 27 3 36 4 55 Is the rule y = x - 6? Yes No input(x) output(y) 6 8 10 14 0 2 4 8 Slide 115 / 191 56 Is the rule or function y = 2x - 1? Input(x) True False 10 19 Output(y) Slide 116 / 191 8 15 14 27 6 11
57 What is the rule or function? Slide 117 / 191 A Subtract 2 B Add 3 C Add 2 Input(x) Output(y) 5 8 7 10 6 9 8 11 58 What is the rule or function? Slide 118 / 191 A y = x + 2 B y= 2x C y = x 2 Input(x) Output(y) 2 0 20 10 14 7 10 5 59 What is the rule or function? Slide 119 / 191 A y = 2x + 2 B y = 3x + 2 C y = 2x - 2 Input(x) Output(y) 1 5 2 8 3 11 4 14
A function table can be used to solve Miguel and Mary's problem with their pens. Slide 120 / 191 Pull Teacher Note 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 A function table can be used to solve Sharon's problem with number of miles she will run given any number of hours. Slide 121 / 191 Pull Teacher Note 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 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. Slide 122 / 191 miles traveled, m 1 2 3 4 5 6 7 cost of taxi ride, c $6 $7 $8 $9 $1 0 $11 $12 Pull Teacher Note 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
60 Given the rule used in the function table, will 15 people fit in 3 vans? Slide 123 / 191 Yes No number of people, p 5 10 15 20 number of vans, v 1 2 3 4 Pull Teacher Note 61 Use the function table and equation. How many vans are needed for 35 people? Slide 124 / 191 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 62 If each package contained 2 cookies, what is the tenth number of cookies in the package? Slide 125 / 191 (p) (c) number number of of packages cookies 5 10 7 14 8 16 10 20
63 Use the function table and equation. How many hours will it take for the car to travel 495 miles? Slide 126 / 191 A 7 hours B 9 hours C 11 hours D 15 hours time (hr) 1 2 3 4 distance (miles) 55 110 165 220 64 Use the function table and equation. How much money will you earn in 4 weeks? Slide 127 / 191 Hours Worked (h) Money Earned (m) 1 $6.25 2 $12.50 3 $18.75 4 $25.00 5 $31.25 65 Given the rule add 3 and starting at 0 complete the table below. Given the rule add 6 and starting at 0 complete the table below. Slide 128 / 191 0 0 Look at both of the tables once they are complete, and identify the relationship between the corresponding terms. A Each term in the second table is equal to the corresponding term in the first table. Each term in the second table is twice the value of B the corresponding term in the first table. Each term in the second table is three more than the C value of the corresponding term in the first table. ( Problem derived from (
( Problem derived from 66 Complete the pattern below using the rule 3n-1. (type in the numbers) ( Slide 129 / 191 5,, 11,, ( Problem derived from 67 The output of a function table is 4 less than each input. Angela says the function rule is x - 4 = y. Kara says the rule is 4 - x = y. Who is correct? Explain your answer on your paper. ( Slide 130 / 191 AAngela BKara 68 Use the table below to show the ordered pairs for the pattern in the previous problem. (Select one value for each box.) ( Problem derived from ( A B C D E F G H I J 1 2 3 14 2 4 5 10 6 14 5 8 11 14 17 Slide 131 / 191
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. On your paper, complete the table by writing the first five numbers in the number patterns for the number of boxes and cans collected. Cans Boxes Slide 132 / 191 Then, select the rule for the table that compares the two patterns. A times 2 B times 2, then minus 2 C add 2 D subtract 4 Slide 133 / 191 Graphing Patterns and Relationships in the Coordinate Plane Return to Table of Contents Number patterns, functions tables and equations can be shown in graphs on a coordinate plane. Slide 134 / 191 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.
Slide 135 / 191 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 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. 10 9 8 7 6 5 4 3 2 1 0 1 Origin (0, 0) 2 3 4 5 6 7 8 9 10 Slide 136 / 191 The point at which the x and y axes intersect is called the origin. The coordinates of the origin are (0, 0). 10 9 8 7 6 5 4 3 2 1 0 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. y Slide 137 / 191 1 2 3 4 5 6 7 8 9 10 (x,y) x
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. y 10 9 8 7 6 5 4 3 2 1 0 1 (4,3) 2 3 4 5 6 7 8 9 10 x Slide 138 / 191 (x,y) y 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 x Slide 139 / 191 This point is (3,2). To plot the point, go over 3, then up 2. (x,y) y 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 x Slide 140 / 191 This point is (1,4). To plot the point, go over 1, then up 4.
(x,y) y 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 x Slide 141 / 191 This point is (5,0). To plot the point, go over 5, then up 0. 70 Which point is at the origin? Slide 142 / 191 10 9 8 7 6 5 4 3 2 1 0 B D 1 A C 2 3 4 5 6 7 8 9 10 71 Which point is at (1,3)? Slide 143 / 191 10 9 8 7 6 5 4 3 2 1 0 B A C D 1 2 3 4 5 6 7 8 9 10
72 Which point is at (3,3)? Slide 144 / 191 10 9 8 7 6 5 4 3 2 1 0 B D A C 1 2 3 4 5 6 7 8 9 10 73 Which point is at (0,5)? Slide 145 / 191 10 9 8 7 6 5 4 3 2 1 0 D A 1 C B 2 3 4 5 6 7 8 9 10 74 Which ordered pair is the origin? Slide 146 / 191 A (4,0) B (0,0) C (0,4) D (4,4)
75 Which number in the ordered pair (7,3) is the x-coordinate? Slide 147 / 191 A 7 B 3 C 0 D x 76 Which number in the ordered pair (5,9) is the y-coordinate? Slide 148 / 191 A 0 B 5 C 9 D y 77 Which number in the ordered pair (7,12) is the y-coordinate? Slide 149 / 191 A 7 B 12 C 0 D y
78 Which number in the ordered pair (7,12) is the x-coordinate? Slide 150 / 191 A 7 B 12 C 0 D x 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 column. Then solve for y. Do this for each x value. 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 origin (0); 3 tells you to move three units up (vertical). 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 Teacher Note Slide 151 / 191 Function Table: Equation y = x + 2 y Slide 152 / 191 x y 1 3 2 4 3 5 4 6 Graph: Quadrant I: positive numbers 0 x
Equations that result in straight lines are called linear equations. Slide 153 / 191 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. Equation: y = x - 1 Function Table Graph y Slide 154 / 191 x y 1 0 2 1 Quadrant I - positive numbers 3 2 4 3 Solve for y. 0 x 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. Equation: y = 2x + 3 Function Table Graph y Slide 155 / 191 x y 0 3 1 5 2 7 Quadrant I - positive numbers 3 9 0 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. x
79 Which of the following points is on the line? y A (1, 5) B (4, 10) C (2, 7) D (8, 3) Slide 156 / 191 x 80 Which graph represents the given function? Slide 157 / 191 A Graph A B Graph B 81 Which graph does not represent the given function? Slide 158 / 191 1 2 3 4 5 6 7 1 2 3 4 5 6 7 A Graph A B Graph B
82 Which graph represents the given function? Slide 159 / 191 A Graph A B Graph B Graphing Relations can be used in "real" world problems. Slide 160 / 191 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 dollars earned 42 36 30 24 18 12 6 0 1 2 3 4 5 6 number of hours worked Slide 161 / 191 Using the equation from the table or graph, y = 6x, you can calculate how much you would earn given any amount of hours. Pull Teacher Note If the clerk worked 12 hours, how much would be earned? $72.00 click If the clerk worked 30 hours, how much would be earned? $180.00 click If the cleark worked 40 hours, how much would be earned? $240.00 click
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 x y or y = 3x distanc ordere time e d (t) (d) pairs 0 0 (0, 0) 1 3 (1, 3) 2 6 (2, 6) 3 9 (3, 9) distance Pull Teacher Note Slide 162 / 191 time Slide 163 / 191 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 Pull Teacher Note If you walked 8 hours, how much distance did you travel? 24 miles click We can use graphs to compare two relations. Line a on the graph below shows Blaise's kilometers per hour. (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. Table A Equation: d = 3t or y = 3x Table B Equation: d = 9t or y = 9x x time (t) distanc e (d) 0 0 y 1 3 2 6 3 9 x time (t) y distanc e (d) 0 0 1 6 2 12 3 18 distance 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 a 0 1 2 3 4 5 6 7 8 9 time b Slide 164 / 191
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). 20 b Table A Equation: d = 3t or y = 3x Table B Equation: d = 9t or y = 9x x time (t) y distance (d) 0 0 1 3 2 6 3 9 x time (t) y distance (d) 0 0 1 6 2 12 3 18 distance 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 a Click 0 1 2 3 4 5 6 7 8 9 time 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 Teacher Note Slide 165 / 191 83 Which function table does the graph best represent? Slide 166 / 191 A Table A number of quarts (q) 0 0 1 2 2 4 3 6 number of gallons (g) B Table B number of quarts (q) 0 0 4 1 8 2 12 3 number of gallons (g) 6 gallons 5 4 3 2 1 0 4 8 12 16 quarts 84 Which best describes a graph that shows the relationship between the cost of heating a home and the outside temperature? Slide 167 / 191 A horizontal line B an increasing straight line C a decreasing straight line D a vertical line
85 What is the relationship between the y-coordinates on line a and the y-coordinates with the same x-coordinate on line b? 12 11 10 9 8 7 6 5 4 3 2 1 a b 0 1 2 3 4 5 6 7 8 9 A The coordinate on line b is half the value of the coordinate on line a. The coordinate on line b is twice the value of the B coordinate on line a. The coordinate on line b is one-fourth the value of C the coordinate on line a. Slide 168 / 191 Slide 169 / 191 Glossary Return to Table of Contents Slide 169 (Answer) / 191 Teacher Notes Vocabulary Words are underlined with a dashed line in the Glossary presentation. This is linked to the page at the end of the presentation with the word defined on it. [This object is a pull tab] Return to Table of Contents
Algebra Slide 170 / 191 A type of Math that uses letters (variables) and symbols to represent numbers. y + 7 = 10 y = 3 Algebra is like a Math puzzle, finding the missing pieces and how they fit together. Back to Instruction 4x number: operation: variable: Algebraic Expression An expression that consists of one or more: numbers, variables, and operations. x 2 number: operation: 3 + 2 = y equals sign: operation: variable: Slide 171 / 191 Back to Instruction Calculate Slide 172 / 191 To find a number/answer by adding, subtracting, multiplying, and/or dividing. mental math 4x3 pencil & paper calculator Back to Instruction
Coordinate Plane The two dimensional plane or flat surface that is created when the x-axis intersects with the y-axis. Slide 173 / 191 y (0,0) x Also known as a coordinate graph and a Cartesian plane. Back to Instruction Equation Slide 174 / 191 Two expressions that are equivalent to each other. Equivalence is shown with an equal sign. 4x=8 equivalent expressions 4 =x 3 equivalent expressions x 3 no equivalence Back to Instruction Evaluate Slide 175 / 191 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. Back to Instruction
Expression Slide 176 / 191 Numbers, symbols and operations grouped together that show the value of something. 2 An 3 2 + 1 expression is one side of an equation. 2 x 3 = 6 Expressions DO NOT have equals signs. Back to Instruction Function Slide 177 / 191 A special relationship between x and y values, where each of its input values gives back exactly one output value. (x) (y) 1 1 2 4 3 9 The output (y) depends on the input (x). Not a function. For each input, there can only be one output. (x) (y) 1 1 1 4 3 9 Back to Instruction Input Slide 178 / 191 The input is the independent variable in a function (y). The output (y) depends on the input (x). (x) (y) 1 1 2 4 3 9 y = x 2 Input The value of y is determined by the value of x. 3 Input Rule y = x 2 Back to Instruction
Linear Equation Slide 179 / 191 An equation whose graph is a straight line. 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 Back to Instruction Non-linear Equation Slide 180 / 191 An equation whose graph is not a straight line. Back to Instruction Order of Operations The rules of which calculation comes first in an expression. Parentheses, Exponents, Multiplication or Division, Addition or Subtraction Slide 181 / 191 Please Excuse My Dear Aunt Sally r Back to Instruction
Ordered Pairs Slide 182 / 191 An x and y value that identify a location on a coordinate plane. (x,y) (x,y) (3,2) Back to Instruction Origin Slide 183 / 191 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) origin Back to Instruction Output Slide 184 / 191 The output is the dependent variable in a function (y). The output (y) depends on the input (x). (x) (y) 1 1 2 4 3 9 y = x 2 Output The value of y is determined by the value of x. 3 Input Rule y = x 2 Back to Instruction
Point Slide 185 / 191 A point is not an object, but a location, identified by a set of coordinates. A 10 9 8 7 6 5 4 3 2 1 0 1 A (3,5) 2 3 4 5 6 7 8 9 10 Back to Instruction Relation Slide 186 / 191 A relationship between two sets of data, where one value is taken from each set and represented with a set of ordered pairs. (x) (y) 1 1 2 4 3 9 {(1,1) (2,4) (3,9)} ; ; {(1,3)(2,6)(5, ) ; ; } For every boy, there are 3 girls. {(1,8);(2,16);(3,24)} (Hours worked, $ earned) ( x, y ) Back to Instruction Rule Slide 187 / 191 Defines the relationship between the input and the output, using an algebraic equation. It says what happens to the input in a function. 3 Input Rule: y = x 9 Output Rule: y = 3x 2 (x) (y) 1 3 2 6 3 9 Rule: y = 4x+2 (x) (y) 1 6 2 10 3 14 Back to Instruction
Table Slide 188 / 191 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 Back to Instruction Variable Slide 189 / 191 A letter or symbol that represents a changeable or unknown value. 4x + 2 variable x x x =? Back to Instruction x-axis Slide 190 / 191 Horizontal number line that extends indefinitely in both directions from zero. (Right- positive Left negative) (x,y) 5-4 3 + 2 1 0 1 2 3 4 5 x Back to Instruction
y-axis Slide 191 / 191 Vertical number line that extends indefinitely in both directions from zero. (Up- positive Down- negative) + (x,y) 5 4 3 2 1 y 0 1 2 3 4 5 - Back to Instruction