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Place Value Learning Goal: I will understand place value. Model: Shade 1 =1 Shade 1 10 __________ tenths = 1 Shade 1 100 ___________hundredths = 1 Show the following numbers by shading rectangles. 1. .3 means __________________________ 2. .09 means __________________________ 1 3. .3 + .09 equal_____ which means __________________________ 4. 2 means __________________________ 5. Why is it not possible to show 4 below?_________________________ 6. 1.43 means____________ + _____________ + ______________ + + 7. 1.43 means __________________________ What is interesting about adding .7 to .3?__________________________ 2 + = .7 + .3 means____________+ ______________ = __________________ Show why .08 is less than .5 Explain____________________________________________________ Show why .4 is greater than .35 Explain___________________________________________________ Moving to Symbols Why don’t we always use models to show numbers? Why do we use numerals? Define the word sum. _____________________________________________________ Think of the values of the numbers to find the sums. 3 1. .3 .2 equals __________ which means _________________________ 2. .04 .01 equals ________ which means ________________________ 3. .3 .02 equals ________ which means_________________________ 4. 2 .6 equals ____________ which means_______________________ Line up the numbers in order. .35, .08 1.02 .90 .37 .9 Make two sentences from your order. ________________________ is less than ________________________ _______________________ is greater than _____________________ _______________________ is the same as ______________________ 4 Homework Practice for Place Value Parent Signature________________ Name______________ Goal: Understand place value. Show the following numbers by shading rectangles if equals the number 1. 1. .5 means __________________________ 2. .02 means __________________________ 3. .5 + .02 equals_____ which means __________________________ 4. 2.5 means __________________________ 5. 1.37 means _________ + ______________ + ___________ 7. 1.37 means __________________________. Show its value below. 5 Show why .18 is less than .09 Explain____________________________________________________ Show why .7 is greater than .30 Explain____________________________________________________ Think of the values of the numbers to find the sums. 1. .5 .1 equals __________ which means ________________________ 2. .07 .02 equals ________ which means _________________________ 3. .7 .01 equals ________ which means_________________________ 4. 2 .6 equals ____________ which means________________________ Line up the numbers in order. 1.05, 1.08 .92 .90 2 Make two sentences from your order. ________________________ is less than ________________________ _________________________ is greater than ____________________ Rate your self on the goal of understanding place value_______ 0 Don’t understand place value at all 1 Understand it a little 2 I still make a lot of mistakes, but I think I get it. 3 I make little mistakes only, but I get it. 4 I could do tougher place value problems than I have seen in class. 6 Line up the following numbers. 1.45, 4.15, 8.30, 1.95, .85 Neighbor check _________ What are these numbers? 1.45 _______________________________________ 4.15 _______________________________________ 8.30 _______________________________________ 1.95 _______________________________________ .85 _______________________________________ Make sentences. _______________________ is smaller than ______________________. _______________________ is bigger than ______________________. Eighty-five hundredths is _________________________ eight and three tenths. Eight and three tenths is _________________________ eighty-five hundredths. Less than Greater than Make sentences. _______________________ is less than ______________________. _______________________ is greater than _____________________. What is an inequality? 7 Inequality symbols Practice. Ex. 1.92 ______ .67 A. .75 ______ 2.3 B. .89 ______ .90 C. 3.1 ______ 2.74 D. .65 ______ 5.60 Write two sentences for each inequality. Ex. 4.52 > .36 Four and fifty-two hundredths is greater than thirty-six hundredths. Thirty-six hundredths is less than four and fifty-two hundredths. A. .59 < .81 ____________________ is greater than ___________________. ____________________ is less than ______________________. B. 1.35 > 1.34 C. .47 < 2.25 8 Write the sentences using symbols. Ex. Six tenths is greater than thirty-one hundredths. .6 > .31 OR .31 < .6 A. Fifty-four hundredths is less than seventy-seven hundredths. B. Three and five tenths is greater than three and twenty-nine hundredths. C. Nine hundredths is less than seventeen hundredths. Line up these numbers in order. A. .85, .90, .78 What are these numbers? .85 _________________________________________ .90 _________________________________________ .78 ________________________________________ Make a sentence. Use the term “less than” or “greater than”. Write your sentence with an inequality symbol. 9 Homework Name __________________ What are these numbers? .7 ________________________________ .08 ________________________________ .70 ________________________________ .78 ________________________________ Line up the numbers in order. Make two sentences from your order. _____________________ is less than _________________________. _____________________ is greater than ______________________. What is an inequality? Use an inequality symbol in each problem. A. .65 ______ .64 B. 7.55 ______ 5.7 C. 4.9 ______ 4.92 D. 5.88 ______ 6.1 Write two sentences for each inequality.. Ex. 4.52 > .36 Four and fifty-two hundredths is greater than thirty-six hundredths. Thirty-six hundredths is less than four and fifty-two hundredths. A. 6.1 < 7.28 B. 4.91 > 4.77 Parent Signature: 10 What are these numbers? .8 ___________________________ 3.4___________________________ 2.5___________________________ 4.7 ___________________________ Line up the numbers in order. Number Line (Horizontal) (Vertical) Put these prices on a number line. $6.50 for two boxes of Lucky Charms cereal, $3.00 for a plastic bucket, $.80 for a KitKat candy bar, $1.50 for a yard of fabric, $1.75 for a bunch of bananas, $0 for a high-five 11 In the last example, a plastic bucket was $3.00. How much would two cost? _______________ How much would three cost? _______________ How much would four cost? _______________ How much would none cost? _______________ Line up these prices on a number line. In the last example, a plastic bucket was $3.00. How much would two cost? _______________ How much would three cost? _______________ How much would four cost? _______________ How much would none cost? _______________ Line up these prices on a number line. 12 Homework Name ______________________ Line up these numbers in order. $.42, $.83, $.05, $.17, $.35 Put these numbers on a number line. A salad costs $2.50. How much would two cost? _______________ How much would three cost? _______________ How much would four cost? _______________ How much would none cost? _______________ Put these numbers on a number line. Parent Signature _________________________ 13 A Better Way to Organize Who is Rene’ Descartes? Cartesian coordinate system I can put my data on the ____________________________________. A blue pen costs $1.50. What is the cost of two pens? __________________________ What is the cost of three pens? ________________________ What is the cost of four pens? _________________________ What is the cost of five pens? __________________________ What is the cost of zero pens? __________________________ Label a coordinate system for this data. Put the cost on the vertical number line. Write a sentence to describe the graph. 14 A frozen pizza costs $5.05. What is the cost of two pizzas? __________________________ What is the cost of three pizzas? ________________________ What is the cost of four pizzas? _________________________ What is the cost of five pizzas? __________________________ What is the cost of zero pizzas? __________________________ Label a coordinate system for this data. Put the cost on the vertical number line. Write a sentence to describe the graph. 15 Data collected: 4 corn on the cob for $1.00. What is the cost of 8 cobs? __________________________ What is the cost of 12 cobs? ________________________ What is the cost of 16 cobs? _________________________ What is the cost of 20 cobs? __________________________ What is the cost of 0 cobs? __________________________ Label a coordinate system for this data. Put the cost on the vertical number line. How should we label the horizontal number line? Write a sentence to describe the graph. How much does one cob cost? 16 Homework Name ___________________ Data collected: Parent Signature________________ What is the cost of ? __________________________ What is the cost of ? ________________________ What is the cost of ? _________________________ What is the cost of ? __________________________ What is the cost of ? __________________________ Label a coordinate system for this data. Put the cost on the vertical number line. How should we label the horizontal number line? Write a sentence to describe the graph. Create this same graph for display in class. Use a fresh sheet of graph paper. 17 A blue pen costs $1.80. If we start with $10.00, how much money will we have left if we buy: One pen?___________________________ Two pens? __________________________ Three pens? ________________________ Four pens? _________________________ Five pens? __________________________ Zero pens? __________________________ What is the greatest number of pens that we could buy with $10.00? Label a coordinate system for this data. Put the money left on the vertical number line. Write a sentence to describe the graph. 18 We have been given $50 to buy pizzas for a party. If A frozen pizza costs $4.95, how much money would we have left if we buy: one pizza? __________________________ two pizzas? ________________________ three pizzas? _________________________ four pizzas? __________________________ five pizzas? __________________________ zero pizzas? __________________________ How many pizzas could we buy for $50? Label a coordinate system for this data. Put the money left on the vertical number line. Write a sentence to describe the graph. 19 Data collected: 3 cans of Sprite for $1.29. Given a soda budget of $20, how much money will be left if we buy: 3 cans? __________________________ 6 cans? ________________________ 9 cans? _________________________ 12 cans? __________________________ What is the cost of 0 cans? __________________________ Label a coordinate system for this data. Put the money left on the vertical number line. How should we label the horizontal number line? Write a sentence to describe the graph. How much money will be left if we buy one can? How many cans of soda could be purchased before going over the $20 budget? 20 Homework Name ___________________ Parent Signature________________ Collect data Find a price that is less than $2______________________ Starting with $15.00, how much money will I have left if : I buy ? __________________________ I buy ? __________________________ I buy ? __________________________ I buy ? __________________________ I buy ? __________________________ I buy zero ?___________________________ Label a coordinate system for this data. Put the money left over on the vertical number line. What should be the label for the horizontal number line? Write a sentence to describe the graph. Create this same graph for display in class. Use a fresh sheet of graph paper. 21 How to Write a Check Class Exercise Our electricity budget for the year is $600.00. Here is a copy of our bill: Practice writing a check to Heber Light and Power Company 22 Create a ledger of your budget for twelve months. Starting Amount Month 0 $____________ Month 1 $____________ Month 2 $____________ Month 3 $____________ Month 4 $____________ Month 5 $____________ Month 6 $____________ Month 7 $____________ Month 8 $____________ Month 9 $____________ Month 10 $____________ Month 11 $____________ Month 12 $____________ Use graph paper to put your data on a Cartesian coordinate system. Include labels and even scale on your graph. Write sentences to describe your data. _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ 23 Given: Communication budget for the year: $1,400.00 Your bill will be the same for each month. Create a ledger of your budget for twelve months. Starting Amount Month 0 $____________ Month 1 $____________ Month 2 $____________ Month 3 $____________ Month 4 $____________ Month 5 $____________ Month 6 $____________ Month 7 $____________ Month 8 $____________ Month 9 $____________ Month 10 $____________ Month 11 $____________ Month 12 $____________ Write checks for each payment. You will write twelve checks and attach them to your work. 24 Use graph paper to put your data on a Cartesian coordinate system. Include labels and even scale on your graph. Write sentences to describe your data. _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ 25 26 27 Given: Power budget for the year: $500.00 Your bill will be the same for each month. Create a ledger of your budget for twelve months. Starting Amount Month 0 $____________ Month 1 $____________ Month 2 $____________ Month 3 $____________ Month 4 $____________ Month 5 $____________ Month 6 $____________ Month 7 $____________ Month 8 $____________ Month 9 $____________ Month 10 $____________ Month 11 $____________ Month 12 $____________ Write checks for each payment. You will write twelve checks and attach them to your work. 28 Use graph paper to put your data on a Cartesian coordinate system. Include labels and even scale on your graph. Write sentences to describe your data. _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ 29 30 31 Given: Cell phone budget for the year: $1,300.00 Your bill will be the same for each month. Create a ledger of your budget for twelve months. Starting Amount Month 0 $____________ Month 1 $____________ Month 2 $____________ Month 3 $____________ Month 4 $____________ Month 5 $____________ Month 6 $____________ Month 7 $____________ Month 8 $____________ Month 9 $____________ Month 10 $____________ Month 11 $____________ Month 12 $____________ Write checks for each payment. You will write twelve checks and attach them to your work. 32 Use graph paper to put your data on a Cartesian coordinate system. Include labels and even scale on your graph. Write sentences to describe your data. _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ 33 34 35 Given: Car budget for the year: $6,000.00 Your bill will be the same for each month. Create a ledger of your budget for twelve months. Starting Amount Month 0 $____________ Month 1 $____________ Month 2 $____________ Month 3 $____________ Month 4 $____________ Month 5 $____________ Month 6 $____________ Month 7 $____________ Month 8 $____________ Month 9 $____________ Month 10 $____________ Month 11 $____________ Month 12 $____________ Write checks for each payment. You will write twelve checks and attach them to your work. 36 Use graph paper to put your data on a Cartesian coordinate system. Include labels and even scale on your graph. Write sentences to describe your data. _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ 37 38 39 Homework. Name ________________________________ Parent Signature ________________________ Given: Student loan budget for the year: $900.00 Your bill will be the same for each month. Create a ledger of your budget for twelve months. Starting Amount Month 0 $____________ Month 1 $____________ Month 2 $____________ Month 3 $____________ Month 4 $____________ Month 5 $____________ Month 6 $____________ Month 7 $____________ Month 8 $____________ Month 9 $____________ Month 10 $____________ Month 11 $____________ Month 12 $____________ Write checks for each payment. You will write twelve checks and attach them to your work. 40 Use graph paper to put your data on a Cartesian coordinate system. Include labels and even scale on your graph. Write sentences to describe your data. _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ 41 42 43 Interpreting Graphs Use the following graph to answer the questions below. Total price $16.00 $14.00 $13.50 $12.00 $11.25 $10.00 $9.00 $8.00 Total price $6.75 $6.00 $4.00 $4.50 $2.00 $2.25 $0.00 $0.00 0 2 4 6 8 number of 3-ring binders 1. What is the price of a 3-ring binder? _______________________ 2. What is the total price of 4 binders? __________________________ 3. How many binders could we buy for $11.25_______________________ 4. How many binders could we buy for $10.00_______________________ 5. Could a straight line be drawn exactly through all of the points that are plotted on the graph? ___________ 6. Why is the line not shown?___________________________________ 7. Finish the statement: As the number of binders increases by one, the total price ______________________________________________. 44 Use the following graph to answer the questions below. Money left in my pocket $12.00 $10.00 $10.00 $8.00 $8.40 $6.80 Money left in my $6.00 $5.20 pocket $4.00 $3.60 $2.00 $2.00 $0.00 $0.40 0 2 4 6 8 Snicker bars 1. How much was in my pocket when I went into the store? 2. What is the price of a Snicker bar? 3. Do I have enough money for 7 Snicker bars? 4. How much money will be left in my pocket if I buy 5 bars? 5. What is the maximum number of bars that I can buy? How much will be left if in my pocket if I buy as many as possible? 6. Fill in the blanks to complete the statement: Starting with __________, the amount of money in my pocket decreases by ________________ for every additional _________________________________. 45 Use the graph below to estimate the answers of the questions below. Money left in my pocket $8.00 $7.00 $6.00 $5.00 Money left in my $4.00 pocket $3.00 $2.00 $1.00 $0.00 0 2 4 6 8 Sharpies 1. What is a good estimate of the money I started with?_______________ 2. Estimate the price of a Sharpie._________________________ 3. Estimate how much money that I will have left if I buy 5 sharpies.______________ 4. If I walked out of the store with just over $4 in my pocket, how many sharpies did I buy?_________________________________________ 5. Estimate the largest number of sharpies that I can buy. Show your work. 46 Homework Interpreting Graphs Name_________________ Parent Signature_________________ Use the graph to estimate the answers to the questions below. Total cost $8.00 $7.00 $6.00 $5.00 $4.00 Total cost $3.00 $2.00 $1.00 $0.00 0 2 4 6 8 Trading Cards 1. Estimate the cost of one package of trading cards._________________ 2. How many packages can you buy for $7.00?______________________ 3. Estimate the total cost of 5 packages.__________________________ 4. Can a straight line be drawn exactly through the points?__________________ 5. Why does the graph not have a line drawn?___________________________________________________ 6. Finish the statement: Starting with a minimum total cost of ________ for __________________________, the total cost _______________ for every additional _________________________________________. 47 Graphing Continuous Data Name____________________ Parent Signature____________ The cost of denim fabric $6.00 per yard. What is the cost of one yard? __________________________ What is the cost of two yard? ________________________ What is the cost of three yards? _________________________ What is the cost of five yards? __________________________ What is the cost of zero yards? __________________________ What is the cost of a half of a yard? ______________________ What is the cost of two and one half yards?_________________ What is the cost of one foot (0ne-third of a yard)?______________ What is the cost of three and one-third yards?_________________ Label a coordinate system for this data. Put the cost on the vertical number line. Write a sentence to describe the graph. 48 The cost of bananas is $1.20 per pound. What is the cost of one pound of banana? ____________________ What is the cost of three pounds of bananas? _________________ What is the cost of five pounds of bananas? ___________________ What is the cost of zero pounds of bananas? _________________ What is the cost of a half of a pound of bananas?____________ What is the cost of four and a half pounds of bananas?__________ What is the cost of a fourth of a pound of bananas? _____________ What is the cost of two and one fourth pounds of bananas?________ Label a coordinate system for this data. Put the cost on the vertical number line. Write a sentence to describe the graph. 49 Data collected: The cost of diesel fuel is $3.00 per gallon. What is the cost of 8 gallons of diesel fuel? ___________________ What is the cost of 12 gallons of diesel fuel? _________________ What is the cost of 16 gallons of diesel fuel? _________________ What is the cost of 20 gallons of diesel fuel? _________________ What is the cost of 0 gallons of diesel fuel? ___________________ What is the cost of 0.1 gallons?____________________ What is the cost of 10.1 gallons?___________________ What is the cost of 0.4 gallons?____________________ What is the cost of 5.4 gallons?____________________ Label a coordinate system for this data. Put the cost on the vertical number line. How should we label the horizontal number line? Write a sentence to describe the graph. 50 Homework Name ___________________ Parent Signature________________ Data collection: Find the cost of a product that is sold by weight, volume, or length. Product Name______________________ Product Price___________Per_________ (Unit of measure) The cost of one ______________ of _____________ is _____________ (Unit of measure) (Product) (dollar amount) The cost of three ____________ of _____________ is _____________ The cost of five ______________ of _____________ is _____________ The cost of zero ______________ of ____________ is _____________ The cost of a half ____________ of ___________ is _____________ The cost of four and a half_____________ of __________ is _________ Label a coordinate system for this data. Put the cost on the vertical number line. How should we label the horizontal number line? Write a sentence to describe the graph. 51 Homework Name _______________________ Parent Signature ____________________________ Look at an advertisement. Choose three items that are priced by the pound. Find the price of 0,1,2,3,4,5 pounds of each item. Graph the total costs of each item on the same Cartesian coordinate system. Be sure to label the vertical and horizontal number lines. Use different colors to label each graph. List what each color represents. Is this data continuous? Is a line graph appropriate? Why or why not? Which item has the steepest curve? ______________________ data has the steepest curve because _______ _________________________________________________________. Compare the slopes of the graphs. Use complete sentences. 52 Graphing From a Starting Cost Name____________________ Parent Signature____________ The total cost of going to a movie depends on two costs: the cost of an $8.00 ticket and the cost of $1.00/bag for bags of popcorn that we buy. The total cost of going to the movie if we buy zero bags of popcorn is__________. The total cost of going to the movie if we buy one bag of popcorn is__________. The total cost of going to the movie if we buy two bags of popcorn is__________. The total cost of going to the movie if we buy five bags of popcorn is__________. Label a coordinate system for this data. Put the cost on the vertical number line. Write a sentence to describe the graph. 53 The total cost of renting a car depends on two costs: a rental fee of $30 and the cost of $.50/mile for every mile we drive the car. The total cost of renting a car if we drive zero miles is__________. The total cost of renting a car if we drive 50 miles is__________. The total cost of renting a car if we drive 100 miles is__________. The total cost of renting a car if we drive 200 miles is__________. The total cost of renting a car if we drive 350 miles is __________. Label a coordinate system for this data. Put the cost on the vertical number line. Write a sentence to describe the graph. 54 The total cost of a certain weight-loss program depends on two costs: A membership fee of $30 and $10/lb. for every pound we lose. What is the total cost if we lose zero pounds?__________ What is the total cost if we lose one pound?____________ What is the total cost if we lose 3 pounds?_____________ What is the total cost if we lose a half of a pound?_______ What is the total cost if we lose six and a half pounds?_______ Label a coordinate system for this data. Put the cost on the vertical number line. How should we label the horizontal number line? Write a sentence to describe the graph. 55 Homework Name ___________________ Parent Signature________________ Data collection: Estimate the cost of one small jar of peanut butter________________ Estimate the cost of one dinner roll_______________ Use your estimates that you listed to find the following total costs: One jar of peanut butter and zero rolls_________________________ One jar of peanut butter and one roll___________________________ One jar of peanut butter and two rolls___________________________ One jar of peanut butter and three rolls_________________________ One jar of peanut butter and four rolls__________________________ One jar of peanut butter and eight rolls___________________________ Label a coordinate system for this data. Put the total cost on the vertical number line. How should we label the horizontal number line? Write a sentence to describe the graph. 56 Rules Name____________________ Parent Signature____________ Use each sentence to create a rule for calculating the total cost. 1. The total cost of a number of plastic buckets, if each bucket costs $3.00 2. The total cost of a number of frozen pizzas, if each costs $5.05 3. The total cost of a number of cobs of corn if 4 corn on the cob sell for $1.00. 4. The total cost of going to a movie depends on two costs: the cost of an $8.00 ticket and the cost of $1.00/bag for bags of popcorn that we buy. 5. The total cost of renting a car depends on two costs: a rental fee of $30 and the cost of $.50/mile for every mile we drive the car. 57 6. The total cost of a certain weight-loss program depends on two costs: A membership fee of $30 and $10/lb. for every pound we lose. 7. The cost of denim fabric $6.00 per yard. Find the total cost of denim for a certain length of fabric. 8. The cost of bananas is $1.20 per pound. Find the total cost for a certain weight (in pounds) of bananas. 9. The cost of diesel fuel is $3.00 per gallon. Find the total cost given a certain quantity (in gallons) of diesel fuel. 58 Estimate the cost of one small jar of peanut butter________________ Estimate the cost of one dinner roll_______________ Use the estimates to make a rule for the total cost based on number of rolls. Use each sentence to create a rule that finds the money left over. 1. A blue pen costs $1.80. If we start with $10.00, how much money will we have left if we buy a certain number of pens. 2. We have been given $50 to buy pizzas for a party. If A frozen pizza costs $4.95, how much money would we have left if we buy a certain number of pizzas? 3. 3 cans of Sprite for $1.29. Given a soda budget of $20, how much money will be left if we buy a certain number of cans? 4. Choose the price of a particular item to complete the following sentence: The price of a ___________ is _____________ a piece. Starting with $15.00, how much money will I have left if I buy a certain number? 59 Write a rule for the following graphs. 1. Total price $16.00 $14.00 $13.50 $12.00 $11.25 $10.00 $9.00 $8.00 Total price $6.75 $6.00 $4.00 $4.50 $2.00 $2.25 $0.00 $0.00 0 2 4 6 8 number of 3-ring binders 2. Money left in my pocket $12.00 $10.00 $10.00 $8.00 $8.40 $6.80 Money left in my $6.00 $5.20 pocket $4.00 $3.60 $2.00 $2.00 $0.00 $0.40 0 2 4 6 8 Snicker bars 60 Homework: Rules Name ___________________ Parent Signature______________ 1. Data collection: Find the cost of a product that is sold by weight, volume, or length. Product Name______________________ Product Price___________Per_________ (Unit of measure) Write a rule that calculates the total cost for any amount that you could buy. 2. Write a rule for the following graph. Total cost $8.00 $7.00 $6.00 $5.00 $4.00 Total cost $3.00 $2.00 $1.00 $0.00 0 2 4 6 8 Trading Cards 61 Ordered Pairs Name____________________ Parent Signature____________ The cost of a notebook is $2.25. Write a sentence that describes the total cost in terms of the number of notebooks that we buy. Fill out the chart below using the description that we wrote. Number of notebooks Total cost 0 1 2 3 4 5 An ordered pair is a combination of two numbers. For example 1, 2.25 is an ordered pair from the table above. The first number 1 represents 1 notebook and the second number 2.25 represents $2.25, the cost of 1 notebook. Ordered pairs can be drawn as points on a graph in the same way that we have placed points on a graph before. Label a coordinate system for this data. Put the total cost on the vertical number line. Label each point with an ordered pair. 62 Continuous Data: The cost of apples is $1.60 per pound. Write a sentence that describes the total cost in terms of the pounds of apples that we buy. Fill out the chart below using the description that we wrote. Pounds of apples Total cost 0 1 .5 3 4.5 5 Label a coordinate system for this data. Put the total cost on the vertical number line. Label each point with an ordered pair. 63 Starting Cost: The cost of diesel fuel is $3.00 per gallon. If we buy a bottle of additive for $5.20 before we fill up, write a sentence that describes the total cost in terms of the gallons we buy. Fill out the chart below using the description that we wrote. Gallons of Diesel fuel Total cost 0 .5 10.5 20 30 50 Label a coordinate system for this data. Put the total cost on the vertical number line. Label each point with an ordered pair. 64 Homework Name ___________________ Parent Signature________________ Data collection: Find the cost of a product that is sold by weight, volume, or length. Product Name______________________ Product Price___________Per_________ (Unit of measure) Write a sentence that describes the total cost in term of the amount that is bought. Fill out the chart below using the description that we wrote. Quantity of: Total cost 0 .5 3 5 10.5 12 Label a coordinate system for this data. Put the cost on the vertical number line. Label the points with ordered pairs. 65 Variables Name____________________ Parent Signature____________ For each rule: Underline the words that represent numbers that can change. Write the words that you underlined, and assign a each variable. Write a formula for the rule. Try the rule out. Example: The total distance (in miles) travelled can be found by multiplying 45 by the number of hours. D= total distance travelled (miles) H= number of hours D = 45 X H 45 X 2 = 90 D = 90 miles 1. Since I start with $50 in my pocket, the amount of money I have left will be 50 minus 4.50 times the number of anchor bolts that I buy. Underline the words that represent numbers that can change. Write the words that you underlined, and assign each a variable. Write a formula for the rule. Try the rule out. 66 2. The amount of money that Marie spends at the carnival can be found by multiplying the number of tickets that she buys by $1.25, then adding $5.00 for the entrance fee. Underline the words that represent numbers that can change. Write the words that you underlined, and assign each a variable. Write a formula for the rule. Try the rule out. 3. The remaining distance on a trip from Salt Lake City to Heber City can be found by subtracting the minutes travelled from 44. (Assume an average speed of 60 mph.) Underline the words that represent numbers that can change. Write the words that you underlined, and assign each a variable. Write a formula for the rule. Try the rule out. 4. The class collected and analyzed hand span and height data to create a model that suggested that the height of a person (in inches) could be estimated by multiplying the span of their hand (in centimeters) by 3. Underline the words that represent numbers that can change. Write the words that you underlined, and assign each a variable. Write a formula for the rule. Try the rule out. 67 5. Between 6:00 and 7:00pm, the number of fans in The Nest can be approximated by multiplying 12 by the number of minutes past 6:00pm plus 253. Underline the words that represent numbers that can change. Write the words that you underlined, and assign each a variable. Write a formula for the rule. Try the rule out. 6. The number of fans who remain in The Nest can be estimated by subtracting from 989 the number of minutes after the final buzzer times 57. Underline the words that represent numbers that can change. Write the words that you underlined, and assign each a variable. Write a formula for the rule. Try the rule out. 7. Assign variables then write a formula for the following graph. $30.50 $30.00 0, $30.00 Money left in the wallet $29.50 1, $29.35 $29.00 2, $28.70 $28.50 $28.00 3, $28.05 $27.50 4, $27.40 $27.00 5, $26.75 $26.50 $26.00 6, $26.10 $25.50 0 1 2 3 4 5 6 7 Cans of soda 68 Homework: Variables Name______________ Parent Signature___________________ The final price of a shirt (which includes tax) can be found by multiplying the price on the shirt’s tag by 1.07. Underline the words that represent numbers that can change. Write the words that you underlined, and assign each a variable. Write a formula for the rule. Try the rule out. Write a rule for the graph below. Assign variables then write a formula. Try the formula out to see if it works. total cost of going to the movie $18.00 $16.00 $15.50 $14.00 $14.25 $13.00 $12.00 $11.75 $10.00 $10.50 $9.25 $8.00 $8.00 $6.00 $4.00 $2.00 $0.00 0 1 2 3 4 5 6 7 bags of popcorn 69 Slope and Intercept Name __________________ Parent Signature _____________________ Cost of Trip to State Tournament $12.00 $10.00 5 , 10.50 4 , 9.00 Cost of Trip $8.00 3 , 7.50 $6.00 2 , 6.00 Cost $4.00 1 , 4.50 0 , $3.00 $2.00 $0.00 0 1 2 3 4 5 6 Number of Trips to the Snack Bar Describe the data. 1. My data starts at ______________________________________. This starting amount is known as the ________________________. Write a sentence using this vocabulary and the starting amount. 2. My data is ________________________________ for every __________________________________________________. This change is known as the ___________________. Write a sentence using this vocabulary and this change. Write a rule for the data. 3. The cost of the trip _______________________________________ ____________________________________________________. Circle the words that represent numbers that can change. Write the words that you underlined, and assign each a variable. Write a formula for the rule. Try the rule out. 70 Fundraiser $120.00 0, $110.00 $100.00 2, $95.00 Sports Fee $80.00 4, $80 $60.00 6, $65 Sports Fee 8, $50 $40.00 10, $35 $20.00 $0.00 0 2 4 6 8 10 12 Number of Cookie Dough Packs Sold Describe the data. 4. My data starts at ______________________________________. This starting amount is known as the ________________________. Write a sentence using this vocabulary and the starting amount. 5. My data is ________________________________ for every __________________________________________________. This change is known as the ___________________. Write a sentence using this vocabulary and this change. Write a rule for the data. 6. The sports fee _______________________________________ ____________________________________________________. Circle the words that represent numbers that can change. Write the words that you underlined, and assign each a variable. Write a formula for the rule. Try the rule out. 71 Loan Balance 2500 2000 0, 2000 Loan Balance 1, 1750 1500 2, 1500 3, 1250 1000 4, 1000 5, 750 500 6, 500 7, 250 0 8, 0 0 1 2 3 4 5 6 7 8 9 Months Describe the data. 6. My data starts at ______________________________________. This starting amount is known as the ________________________. Write a sentence using this vocabulary and the starting amount. 7. My data is ________________________________ for every __________________________________________________. This change is known as the ___________________. Write a sentence using this vocabulary and this change. Write a rule for the data. 6. The loan balance _______________________________________ ____________________________________________________. Circle the words that represent numbers that can change. Write the words that you underlined, and assign each a variable. Write a formula for the rule. Try the rule out. 72 Homework: Slope and Intercept Name __________________ Parent Signature _____________________ 1. A refrigerator is set at 41 degrees. When the power goes out, the temperature increases 3.8 degrees every ten minutes. Time (minutes) Refrigerator Temperature 0 41 degrees 10 __________ 20 __________ 30 __________ 40 __________ 50 __________ 2. How much does the refrigerator temperature increase every minute? This change is known as the ________________. Write a sentence using this vocabulary and the change. 3. What is the starting temperature? This starting temperature is known as the __________________. Write a sentence using this vocabulary and the starting temperature. 4. Write a rule for the data. The refrigerator temperature ____________________________ ____________________________________________________ ____________________________________________________ Circle the words that represent numbers that can change. Write the words that you underlined, and assign each a variable. Write a formula for the rule. Try the rule out. 73 Slope-Intercept Form Name __________________ Parent Signature _____________________ When a formula is in slope-intercept form, we can easily find the slope and intercept of the graph. For Example: A = 8 + 3 X P is in slope-intercept form because we can see that the slope is 3 and the intercept is 8. For each formula, find the slope and the intercept. 1. For the formula D = 400 + 50 X H: The slope is _____ and the intercept is ______. 2. For the formula M = 50 – 2 X B: The slope is _____ and the intercept is ______. 3. For the formula C = 25 X R + 650: The slope is _____ and the intercept is ______. 4. For the formula T = 5.60 X P The slope is _____ and the intercept is ______. How do you know which number is the slope? How do you know which number is the intercept? Which formula’s graph has the steeper slope? Why? A. T = 40 + 3 X P Or B. T = 20 + 4 X P 74 Weight of the dump truck 25000 21000 21750 20000 19500 20250 Total weight 18000 18750 15000 10000 5000 0 0 1 2 3 4 5 6 scoops of gravel 5. The intercept is _________ and the slope is __________. Assign variables: _____ = Total weight, ______ = Scoops of gravel. The formula for the total weight is ___________________________. 6. The Shyster Glass Company deducts $50 for every vase that Roger breaks from his $3000 earnings. A) Write a rule that finds what Roger has left of his earnings when he breaks vases. B) Underline the words that represent numbers that can change. C) Assign variables to the words you underlined. D) Write a formula for the rule. E) The slope is ____________ because___________________________________________. F) The intercept is __________ because___________________________________________. 75 Homework: Slope-Intercept Form Name______________ Parent Signature_______________ Gallons that have leaked from a faulty kitchen faucet 2030 2027.5 2025 2022 wasted water 2020 2015 2016.5 2010 2011 2005 2005.5 2000 2000 1995 0 1 2 3 4 5 6 days 7. The intercept is _________ and the slope is __________. Assign variables: _____ = Total weight, ______ = Scoops of gravel. The formula for the gallons of wasted water is __________________. 8. Jerry’s grandmother gave him $500 to start a savings account for college and his mother promises to deposits $40 to the account every month until he goes to college. A) Write a rule that finds the balance in Jerry’s college fund for a given number of months. B) Underline the words that represent numbers that can change. C) Assign variables to the words you underlined. D) Write a formula for the rule. E) The slope is ____________ because___________________________________________. F) The intercept is __________ because___________________________________________. 76 Make Formulas Mean Something Name __________________ Parent Signature _____________________ Often in math classes we practice with formulas that have no specific meaning. For example, the formula J =70 – 3 X B has no particular meaning because we don’t know what the variables J and B represent. Let’s see if we can make up a rule for the formula. We can see two things for sure: 1. The intercept is 70 2. The slope is -3. The formula suggest that something called J starts at 70 then goes down in steps of 3 for every one of something called B. Think about something represented by the variable J that could start at 70 and decrease in steps of 3. Example: J = The temperature in Juan’s room. 1. ____________________________________________ 2. ____________________________________________ 3. ____________________________________________ For each of the J’s we listed let’s find something represented by the variable B that relates to the decrease in J. Example: B = The hours after Juan turns off the heat. 1. ____________________________________________ 2. ____________________________________________ 3. ____________________________________________ Write a rule for each situation. Example: The temperature in Juan’s room started at 700 then decreased by 30 every hour after he turned off the heat. 1. 2. 3. 77 Graph the formula J = 70 – 3 X B. Label the axes with the words from one of the rules on the previous page. Use an appropriate scale and consistent spacing of the numbers on each axis. 14 12 10 8 6 4 2 5 10 15 20 Useful skills are learned and practiced using formulas or equations that have no specific meaning. Find the slope and intercept of each equation. 1. A = 7 + 2 X B Slope = ____________ Intercept = ___________ 2. F = 5 X T + 4 Slope = ____________ Intercept = ___________ 3. C = 9 – 3 X D Slope = ____________ Intercept = ___________ Since the letter X could be used as a variable, we avoid using X for a multiplication symbol. For example: 5 X B is written 5B or 5 B. Find the slope and intercept of each equation. 4. A = 2 + 3B Slope = ____________ Intercept = ___________ 5. Y = 7 – 5X Slope = ____________ Intercept = ___________ 78 Group Activity For each formula: (a) find the slope, (b) find the intercept, (c) assign meaning to each variable, and (d) write a rule that makes sense. 1. D = 500 – 28 X P (a) The slope = _____________. (b) The intercept is ________________. (c) D = _________________________________________ P = __________________________________________ (d) Rule: __________________________________________ _____________________________________________ _____________________________________________ 2. T = 8.25 + 3 X B (a) The slope = _____________. (b) The intercept is ________________. (c) T = _________________________________________ B = __________________________________________ (d) Rule: __________________________________________ _____________________________________________ _____________________________________________ 3. W = 150 + 2 X G (a) The slope = _____________. (b) The intercept is ________________. (c) W = _________________________________________ G = __________________________________________ (d) Rule: __________________________________________ _____________________________________________ _____________________________________________ 4. P = 1.25 X C (a) The slope = _____________. (b) The intercept is ________________. (c) P = _________________________________________ C = __________________________________________ (d) Rule: __________________________________________ _____________________________________________ 79 Homework: Make Formulas Mean Something Name____________________ Parent Signature_______________ For the formula P = 35 – 3T, find the slope, find the intercept, assign meaning to each variable, and write a rule that makes sense. 1. The slope = _____________. Increasing or Decreasing? 2. The intercept = ________________. 3. Assign Variables P = _________________________________________ T = __________________________________________ 4. Rule: __________________________________________ _____________________________________________ _____________________________________________ Graph the formula J = 35 – 3T. Label the axes with the words from one of the rule you wrote above. Use an appropriate scale and consistent spacing of the numbers on each axis. 14 12 10 8 6 4 2 5 10 15 20 80 Rational Numbers Name ________________________ Parent Signature _____________________ A rational number is any number that ________________________________________________________. Draw a figure that represents each rational number. 1. 4/5 2. 2 1/4 Ex. 3. 1/9 4. 7/12 5. 27/50 6. 3 7/11 7. 8/15 8. 1 7/8 81 Prime Factorization Ex. 32 2 2 2 2 2 Practice. Write the prime factorization for each number. 45 120 81 Reducing Fractions- Dividing out common factors. 18 2 3 3 2 1 2 Ex. 63 3 3 7 1 7 7 Practice. Reduce the following fractions. You must show factorization for each. 12 15 9 14 42 65 45 42 20 21 72 3a 3b 2c 36 36 30 12abc 3 82 To compare rational numbers, we need to look at pieces that are the same size. This is called finding a common denominator. Compare and put the numbers in order. , , , , We can also compare by changing fractions into their decimal equivalent. How? Terminating Repeating 83 Write the fraction or mixed number as a decimal. Tell whether the fraction is a terminating decimal or a repeating decimal. 1. 3/5 2. 2 2/5 3. 1/6 4. 5/12 5. 33/50 6. 14 7/16 7. 2/3 8. 1 4/5 Order the numbers from exercises 1-8. 84 Homework: Rational Numbers Name________________________ Parent Signature _______________ Draw a figure representing each number. Write the fraction or mixed number as a decimal. Tell whether the fraction is a terminating decimal or a repeating decimal. 1. 3/5 2. 2 2/5 3. 1/6 4. 5/12 5. 2/3 6. 1 4/5 Compare and order the numbers from least to greatest. 9. 1 1/8, 1 3/7, 1.1, 1.43, 1 4/15 10. 1/8, 0.3, 1/3, 4/9, 0.7 85 Rational Numbers on a Number Line Name ________________________ Reduce each fraction. Then compare and order the numbers. 11 19 21 41 , , , 14 21 28 49 To place these numbers on a number line, we need to think about good landmarks (scale). Compare and order these numbers. Place the numbers on a number line. 2 1 9 2 7 3 , , , 2 , 1 5 10 20 5 10 86 Compare and order these numbers. Place the numbers on a number line. 2 1 9 2 7 1 , , , 2 , 1 7 7 21 7 14 Compare and order these numbers. Place the numbers on a number line. 1 1 9 4 7 , , , , 9 3 12 6 12 What is an inequality? Compare the two numbers. Write two inequalities for the comparison. 1 3 8 16 87 Compare the two numbers. Write two inequalities for the comparison. 1 2 15 10 Compare the two numbers. Place the numbers on a number line. Write two inequalities for the comparison. 8 5 12 14 Compare the first two numbers. Place the numbers on a horizontal number line. The compare the second two numbers and place on a vertical number line. 5 4 Horizontal: 11 12 1 2 Vertical: 2 3 3 3 88 Homework: Rational numbers on Name _____________________ a number line Parent Signature _____________ Compare and order these numbers. Place the numbers on a number line. 3 4 7 2 7 2 , , , 2 , 1 5 10 20 5 10 Compare the two numbers. Write two inequalities for the comparison. 3 5 8 10 Compare the two numbers. Place the numbers on a number line. Write two inequalities for the comparison. 6 7 18 15 89 On the SECRET coordinate plane, you should plot the locations of five ships. The five ships will follow these specifications: Aircraft carrier -- 5 points long Battleship -- 4 points long Submarine -- 3 points long Destroyer -- 3 points long PT-Boat -- 2 points long The boats must be either horizontal or vertical. They may not overlap. Draw a rectangle around each “boat”. SECRET Record the coordinates of each boat below. AC B Sub D PT 90 Fold this paper in half so your opponent does not see the contents. Battle Station You will take turns guessing the location of your opponent’s boats. Record the coordinate pair of each guess. On your Battle Station coordinate system, place an O for a “miss” and an X for a “hit”. If you guess all the points on your opponent’s ship, they are required to say “You sunk my battleship” or whatever type of boat it is. (The same goes for your battleships) Your guesses: 91 Decreasing to Zero Horizontal Intercept Name____________________ Parent Signature____________ You start with $42 on your lunch balance, and are charged $2.50 everyday you eat at the cafeteria. Write a rule that gives the remaining balance for a given number of meals. Underline the words that represent numbers that can change then define variables for them. Write a formula. What is the slope?_________ What is the vertical intercept? ______ Graph the formula and label the axes. 8 6 4 2 5 10 15 How many meals can you get before you run out of money? How much money will be left in the balance? 92 Anna has 20 gallons in the tank of her Toyota Tercel, which uses 1/35 gallon per mile (35 mpg). Write a rule that gives the remaining gallons of fuel in the tank of Anna’s Tercel after she drives so many miles. Underline the words that represent numbers that can change then define variables for them. Write a formula. What is the slope?______ What is the vertical intercept? ______ Graph the formula and label the axes. 8 6 4 2 5 10 15 How many miles will she be able to travel on the 20 gallons of fuel? Find the ordered pair at the horizontal intercept of the graph. Find the ordered pair at the vertical intercept of the graph. 93 Phillip rides an elevator that starts on the 17th floor and goes downward 1 floor in 3 seconds (or at the rate of 1/3 floor per second). Write a rule that gives the number of floors that Phillip is above ground level after a given number of seconds. Underline the words that represent numbers that can change then define variables for them. Write a formula. What is the slope?______ What is the vertical intercept? ______ Graph the formula and label the axes. 8 6 4 2 5 10 15 How long will it take Phillip to reach the ground level? Find the ordered pair at the horizontal intercept of the graph. Find the ordered pair at the vertical intercept of the graph. 94 Rebecca knows that she can average 43 mph (.724 miles per minute) from her grandma’s house in Mapleton to her home 38 miles away in Heber City. Write a rule that gives the number of miles Rebecca is away from home given the number of minutes travelling there. Underline the words that represent numbers that can change then define variables for them. Write a formula. What is the slope?______ What is the vertical intercept? ______ Graph the formula and label the axes. 8 6 4 2 5 10 15 How long will it take Rebecca to get home? Find the ordered pair at the horizontal intercept of the graph. Find the ordered pair at the vertical intercept of the graph. 95 Homework: Decreasing to zero Name______________ Parent Signature___________________ You know that you can mow 20 square yards of lawn in 1 minute, and that your whole yard has 1600 square yards of lawn. Write a rule that gives the number yards left to mow given the number of minutes that you have been mowing the lawn. Underline the words that represent numbers that can change then define variables for them. Write a formula. What is the slope?______ What is the vertical intercept? ______ Graph the formula and label the axes. 8 6 4 2 5 10 15 How long will it take to mow the whole lawn? Find the ordered pair at the horizontal intercept of the graph. Find the ordered pair at the vertical intercept of the graph. 96 Reaching a Target Value Graphing a Solution Name____________________ Parent Signature____________ Andy wants to drop weight for wrestling. He weighs 187 now, and plans to lose 1.5 lbs. per week. Write a rule that gives Andy’s weight for a given number of weeks of dieting. Underline the words that represent numbers that can change then define variables for them. Write a formula. Graph the formula and label the axes. 8 6 4 2 5 10 15 Draw a horizontal line at 170 lbs. How many weeks until Andy weighs 170 lbs? 97 A hot-air balloon starts at 1000 feet above Heber City and rises at a rate of 5 feet per second. Write a rule that gives the balloon’s height at a given number of seconds. Underline the words that represent numbers that can change then define variables for them. Write a formula. Graph the formula and label the axes. 8 6 4 2 5 10 15 Draw a horizontal line at 2000 feet. How many seconds until the balloon is 2000 feet above Heber City? Write an ordered pair where the two lines cross. Write a sentence for the meaning of the ordered pair. 98 8 6 Math can be used to save Lives. 4 2 5 10 15 On August 5, 2010, 33 miners were trapped 2300 feet underground. After 69 days the men were brought to the surface one at a time in the pod shown above. The pod could be pulled up as fast as 92 feet per minute. Write a rule that gives the pod’s height from the bottom of the shaft at a given number of minutes of a miner’s rescue. Underline the words that represent numbers that can change then define variables for them. Write a formula. Graph the formula and label the axes. Graph a horizontal line at 2300 feet. Write an ordered pair where the two lines cross. Write a sentence for the meaning of the ordered pair. 99 Homework: Reaching a Target Value Name____________________ Parent Signature________________ Miguel knows that it will take $5000 dollars to pay for his first year of college, so he set aside the $1200 that he earned this summer in a savings account then plans to deposit $200 per month to the account. Write a rule that gives Miguel’s college savings for a given number of months of deposits Underline the words that represent numbers that can change then define variables for them. Write a formula. Graph the formula and label the axes. 8 6 4 2 5 10 15 Draw a horizontal line at $5000. How many months of saving until Miguel can pay for one year of college? Label the ordered pair where the lines intersect. Write what the point means. 100 Inverse Machines Name____________________ Parent Signature____________ A) In the top machine write what has been done to the variable N. B) In the bottom machine write how to reverse what has been done to N. C) Try out a number for N to see if the bottom machine reverses what the top machine does to the number. 7 4 N-4 N+7 ______ ______ 18 8 N 2 3N ______ ______ 101 10 5 3N - 4 2N + 3 15 8 N 3-4 N 2 + 5 _____ ______ _ ____ try a number. ____ try a number. (N + 1) 2 5(N – 1) ______ ______ 102 Homework: Inverse Machines Name_____________________ Parent Signature___________________ A) In the top machine write what has been done to the variable N. B) In the bottom machine write how to reverse what has been done to N. C) Try out a number for N to see if the bottom machine reverses what the top machine does to the number. 5N N-8 ______ ______ (N – 1) 3 2N - 3 ______ ______ 103 Putting Inverse Machines in Reverse Name____________________ Parent Signature____________ A) In the top machine write what has been done to the variable. B) In the bottom machine write how to reverse what has been done. C) Try the number in the bottom machine. Try the result in the top machine. ______ ______ J+5 V-8 7 4 ______ ______ N 7 P 5 14 8 104 _____ _____ _ _ 3N - 4 2N + 3 17 15 _____ _ N 4 - 8 N 3 + 1 32 8 _____ _____ _ _ (N + 1) 2 5(N – 1) 5 20 105 Homework: Putting Inverse Machines in Reverse Name_____________________ Parent Signature___________________ A) In the top machine write what has been done to the variable. B) In the bottom machine write how to reverse what has been done. C) Try the number in the bottom machine. Try the result in the top machine. ______ ______ H - 14 G+7 5 23 ______ ______ (N + 1) 5 3N + 1 3 19 106 Solving Equations with Inverse Machines Name____________________ Parent Signature____________ The equation 3x - 5 = 22 can be shown as a machine. If we make the inverse machine of 3x – 5 and put 22 into it, we can find the x that makes the equation true. Check this out! 3x – 5 Times by 3 then reduce by 5 is 22 Increase by 5 then divide by 3 ( 22 + 5) 3 27 3 9 How can we see that x = 9 makes the equation true? Solve the equations using these three steps. 1. Write what happened to the variable. 2. Write the inverse by reversing what happened to the variable. 3. Use the inverse to solve the equation. (You solve an equation when you find out what makes it true.) 4. Always check your answer. Example: Solve 7x – 4 = 38 What happened Inverse Solve Check x was multiplied by 7 Add 4 then ( 38 + 4 ) 7 7 6-4 then decreased by 4 divide by 7 42 7 42 – 4 6 38 Solve 4B + 3 = 11 What happened Inverse Solve Check 107 Solve the equations using these three steps. 1. Write what happened to the variable. 2. Write the inverse by reversing what happened to the variable. 3. Use the inverse to solve the equation. (You solve an equation when you find out what makes it true.) 4. Always check your answer. Equation What happened Inverse Solve Check N – 5 = 11 J + 17 = 25 3T = 21 L 7=9 5V + 2 = 27 Y 3 + 2 = 11 (D – 4)/3 = 7 (G + 2) 5 = 15 3X – 11 = 10 7(R – 3) = 49 D4 11 3 108 Homework: Solving Equations with Inverse Machines Name_______________________ Parent Signature_________________ Solve the equations using these three steps. 1. Write what happened to the variable. 2. Reverse what happened to the variable to write the inverse. 3. Use the inverse to solve the equation. (You solve an equation when you find out what makes it true.) 4. Always check your answer. Equation What happened Inverse Solve Check Q + 3 = 12 Increase by 3 Decrease by 3 12 – 3 9 + 3 = 12 9 12 = 12 J - 17 = 2 6T = 54 L 4=7 4V + 3 = 27 K 3 + 2 = 23 (M + 4)/2 = 7 (G -7) 2 = 10 3P – 17 = 10 8(R + 3) = 40 109 Putting it all Together Name ___________________ http://www.youtube.com/watch?v=LcWyyyms07w In 1940, about 40 Siberian tigers existed in the wild. Russian anti-poaching controls have helped to increase the population over the last 70 years. On average, the tiger population increased by 5 tigers each year. Write a rule that represents the number of Siberian tigers in the wild for any year after 1940. Identify variables. Write a formula for the Siberian tiger population. Gather data. In 1940, the population was _______________. After 1 year, the population was ___________. After 2 years, the population was ___________. After 10 years, the population was ___________. After 20 years, the population was ___________. After 50 years, the population was ___________. After 70 years, the population was ___________. Graph the data. Label your number lines and use an appropriate scale. 8 6 4 2 5 10 15 110 Write the formula for the Siberian tiger population again. Is the data increasing or decreasing? What is the slope? What is the intercept? How can we determine when the population will be at 500 tigers? Write a specific equation. Describe the equation in words. Write the inverse by reversing what happened to the variable. Use the inverse to solve the problem. Check your solution. 111 Homework: Putting it all together Name ___________________ Write the formula for the Siberian tiger population again. How can we determine when the population will be at 450 tigers? Write a specific equation. Describe the equation in words. Write the inverse by reversing what happened to the variable. Use the inverse to solve the problem. Check your solution. 112 Positive and Negative Integers Name____________________ Parent Signature____________ Solve the equation using the inverse. B + 9= 5 What happened to B Write the inverse Solve Is it possible to solve the equation B + 9 = 5? If so what kind of number is the solution? Can two numbers add to zero? Give some examples. We are going to model adding and subtracting Integers with blocks. You and a partner will be given two different colored stacks of 8 blocks. Decide which stack will represent positives and which stack will represent negatives. Positive color___________________ Negative color________________ + - Positive negative What number is shown by - ? + 113 Start with + - to represent zero, then model each + - + - problem with blocks. Draw a picture to show each answer. 1. 5+3 2. 5–3 3. 3–5 4. 5 + -3 5. -2 + 3 6. -4 + 3 7. -2 + -3 8. -2 – 4 9. 4 – (-2) 10. -4 – (-3) 11. 2 + (-5) 12. 0 – (-2) 114 Solve the equations using these steps. 1. Write what happened to the variable. 2. Write the inverse by reversing what happened to the variable. 3. Use the inverse to solve the equation. (You solve an equation when you find out what makes it true.) 4. Always check your answer. Equation What happened Inverse Solve Check N – 5 = -3 J – 17 = -8 T - 10 = -21 L+9=7 5V + (-2) = 23 Y + (-5) = -11 D 2 +(-1) = 7 G – (-5) = 2 H – (-6) = -3 7T – (-4) = 11 D 5 2 3 115 Putting It All Together (B) Name ________________________ Cal scoops ice cream at the Mooseum. He looks in the tip jar at 6:30 and there are $4.50. Every 20 minutes, he looks and there are $2.00 more in the jar. 1. On average, how much money gets added per minute? 2. Find a rule that gives the amount of money in the tip jar for a certain number of minutes. 3. Define variables for the numbers that change. 4. Write a formula. 5. Use your formula to fill out the chart below. Minutes after 6:30 Amount in the Tip Jar 0 10 20 30 60 116 6. Graph the data. Use an appropriate scale, and label the axes. 8 6 The slope is ______________ The intercept is ___________ 4 2 5 10 15 20 7. Write an equation for when the amount in the tip jar is $10.00. _____________________________________ 8. Write the equation in words ______________________________________________ 9. Write the inverse in words _______________________________________________ 10. Use the inverse to solve it. 11. Check your answer. 117 The Distributive Property Name _______________________ I want to give each of the 28 students in class 6 M&Ms. There are 20 boys and 8 girls. How many M&Ms do I need? Ideas from Partner Discussion: Ideas from Class Discussion: Which idea you like best? One way to represent the product of two numbers is to draw a rectangular array. For example the product 3 2 can be illustrated with the following array: 3 2 This array shows that 3 groups of 2 are the same as 2 groups of 3. The Product is 6. How could we use an array to solve the M&M problem? 118 Some people do two-digit multiplication using a similar method. For example, consider the product: 15 38 38 30 8 10 15 5 Fill in the chart with the appropriate products. What is the total area? Use the diagram to find the product 15 38 . Partner Practice. Use the arrays to find the products. 1. 27(15) 2. 82x17 119 3. 61(41) 4. 6(132) Class Practice. 5. 4(y) 6. 3h(4) 7. 9(7m) 8. 12(6v) 9. d(3d) 10. 5g(5g) (w+9) Sometimes we will need to multiply groups. For example, 4(w+9). w 9 4 Product: 120 Practice: 11. 3(x-10) 12. .5(y+8) Partner Practice. 13. (h+7)11 14. 6(9+c) 15. 2v(v+12) 16. 5z(z-3+w) 17. a(a + b + c) 18. 10(x + 2y +z) 121 Homework: The Distributive Property Name ___________________ Parent Signature _______________ Use an array to multiply. 1. 19(73) 2. 56(13) 3. 85(14) 4. 910(32) 5. 8(x+7) 6. 12(y-5) 7. 7(b+4+c) 8. 3(x+y+5z) 122 Match Game Match Game Name: Name: 3 (5 2) 2 11-20 1 2 2 [16+3(4-1)] [14 2(7 1) 3(5 1)] 9 6 5 1 2 12 1 13 0(2 19) 23 3 2 160 2 1 3 4 Reduce Reduce 64 96 24 36 40 56 55 77 Descrive the equation. Describe the equation. Write the inverse. Then use it to solve. Write the inverse. Then use it to solve. 3 x 9 21 10 x 5 35 123 Decomposing Numbers Name____________________ A sum is the result of addition. For example the sum of 3 and 4 is written ( 3 + 4 ), which can be written as 7. The sum of Q and 8 can only be written as ( Q + 8 ) if we do not know the value of Q. Write each number as 3 different sums. 1. 45 a) b) c) 2. 73 a) b) c) 3. -5 a) b) c) 4. 0 a) b) d) A product is the result of multiplication. For example the product of 3 and 7 is written 3 7 , which can be written as 21. The product of 5 and B can only be written as 5B or 5 B unless you know the value of B. Write each number as 3 different products. 5. 24 a) b) d) 6. 18 a) b) d) 7. 20 a) b) d) 8. 72 a) b) d) Some products are easy to find. Explain why the following numbers are easy to multiply. 9. ( 300 X 5000) 10. ( 13 X 1,000) 124 When you multiply with arrays, make sure you write each number as a sum of easy numbers to multiply. If you have to use a calculator for any step, you probably need to use easier numbers. For example the product 47 X 58 can be found in the following way 58 50 8 40 47 7 Fill in the array with the appropriate products. What is the total area? What is the product 47 X 58? Use arrays to find the products. 11. 35 X 18 12. 52(33) 13. 17 . 83 125 Use arrays to find the products. 14. 62 X 35 15. 19(63) 16. 24 . 95 17. 120 . 37 18. 537(8) 19. 7 X 214 20. 325 X 271 126 Starting Blocks Charge Model Integer Addition Model each problem by drawing the appropriate positive and negative blocks above the starting blocks. Strike out blocks to show your result. 1. 5 – 7 2. 4 – (-2) 3. – 2 – 5 4. 5 – 3 5. -4 + -3 6. 1 – (-5) 7. -6 + 5 8. -4 – (-5) 9. -5 – (-2) 127 Homework: Multiplying with Arrays Name______________ Starting Blocks Parent Signature_____________ Use arrays to find the products. 1. 38 X 37 2. 53(41) 3. 23 . 75 Model each problem by drawing the appropriate positive and negative blocks above the starting blocks. Strike out blocks to show your result. 4. -4 – (-2) 5. -3 – (-5) 6. -3 - 4 128 Match Game 2 Match Game 2 Name: Name: 7 2 22 30 9 2 4 2 2 40 8 3 3 1 16 4 20 2 5 3 2 8 6 2 6 5 1 23 1 1 35 2 3 2 3 1 6 2 3 7 1 5 16+5 4 1 2 2 2 8-5 6 3 6 Reduce Reduce 81 48 54 60 Describe the equation. Describe the equation. Write the inverse. Then use it to solve. Write the inverse. Then use it to solve. 7 x 4 32 8x 3 35 129 Decomposing Numbers B Name ______________ Write each number as 3 different sums. 1. 56 a) b) c) 2. 61 a) b) c) 3. -3 a) b) c) 4. 0 a) b) c) Write each number as 3 different products. 5. 32 a) b) c) 6. 16 a) b) c) 7. 40 a) b) c) 8. 64 a) b) c) Use arrays to find the products. 9. 64 X 37 10. 17(43) 11. 44 . 75 12. 130 . 27 13. 526(419) 14. 8 X 317 130 Algebra 1 Name ________________________ Vocabulary Write the words for each number. 1 1 1. 2. 6 7 1 1 3. 4. 3 5 1 1 5. 6. 2 4 1 1 7. 8. 8 9 1 9. 10. .1 10 1 11. 12. .01 100 Write a symbol for each term. 13. 4 cards 14. 2 kings 15. 5 sweaters 16. 3 numbers 131 Write using symbols and simplify if you can. 17. 2 cards plus 4 sweaters plus 5 numbers plus 3 cards plus 10 kings plus 2 sweaters plus 4 numbers plus 8 cards. Draw a picture and use symbols for each situation. 18. 2 groups of 3 chickens 19. 4 groups of 2 camels 20. 2 groups of 3chickens and 7 eggs 21. 3 groups of 2 camels and 1 tent Write a situation for the given symbols. 22. 4m 23. 5d 24. 7a + 8r 25. 6(2d + 3e) 26. 3(2y + 4f) 132 Write the symbol for each word. 27. is 28. plus 29. times 30. of 31. decreased by 32. quotient 33. added to Write a sentence for each equation. 34. 2x + 3h = 14 35. 4(5a + 7) = 13 1 1 36. y z 20 3 5 133 Match Game 3: Combining Like terms Goal: Be able to combine like terms. Steps to get there: A) Practice order of operation. B) Practice adding and subtracting fractions. C) Practice adding and subtracting decimals. D) Practice adding and subtracting integers. E) Practice using the distributive property. F) Learn basic factoring, change the order with the distributive property. G) Combine like terms. Name____________________ Name____________________ Match Game (3) Match Game (3) A) Order of operations 3 23 1 0 10 20 6 1. 7 1. 6 32 4 3 2. 5 + 2(3 - (7 - 5) + 1) 2. 3 + 3( 8 – (14 – 7) + 1) 3. 5 3 1 3. 20 7 5 2 2 4 2[3 5 12 11] 7 1 2 3 18 17 4. 4. 5 3 134 Name___________________ Name_____________________ B) Adding and subtracting fractions (no calculators – show every step) 3 2 1 4 5. 5. 11 11 11 11 3 2 4 3 6. 6. 5 5 5 5 1 3 1 1 7. 7. 5 10 3 6 2 1 3 1 8. 8. 3 6 5 10 1 1 2 2 9. 9. 5 3 3 15 5 1 1 1 10. 10. 8 4 4 8 135 C) Adding and subtracting decimals (no calculators) Name_______________________ Name_______________________ 11. 5.03 + 7.204 11. 2.102 + 10.132 12. 11.037 + 2.142 12. 7.153 + 6.026 13. 8.876 – 2.451 13. 9.766 – 3.341 14. 3.5 + 5.8 14. 2.4 + 6.9 15. 12.05 – 3.75 15. 13.15 – 4.85 136 D) Adding and subtracting integers. Model each problem by drawing the appropriate positive and negative blocks above the starting blocks. Strike out blocks to show your result. Name____________________ Name_____________________ 16. -5 + -2 16. -6 + -1 17. 5 – (-3) 17. 2 – (-6) 18. -5 – (-7) 18. -1 – (-3) 137 E) The distributive property Use arrays to find the products Name______________________ Name___________________ 19. 52(64) 19. 128(26) 20. 12(x + 3) 20. 4(3x + 9) 21. G(5 + 2) use an array 21. G(3 + 4) use an array 22. ( 7 + 5)T 22. ( 2 + 10)T 23. Show that the arrays for 23. Show that the arrays for (2 + 9)x and 11x are equal. (4 + 7)x and 11x are equal. 138 F) Basic Factoring: The distributive property Rewrite each product using the distributive property. Show that both ways are equal. Example: 5(2) + 7(2) can be written (5 + 7)(2) 10 + 14 (12)(2) 24 24 Name____________________ Name_______________________ 24. 7(3) + 5(3) 24. 3(3) + 9(3) 25. 5(10) + 6(10) 25. 8(10) + 3(10) 26. 3(x) + 11(x) 26. 7(x) + 7(x) Check with x = 2 Check with x = 2 27. 13(y) - 11(y) 27. 28(y) - 26(y) Check with x = 7 Check with x = 7 28. 5(x2) + 4(x2) 28. 2(x2) + 7(x2) Check with x = 3 Check with x = 3 139 G) Now you are ready to combine like terms! Name________________________ Name________________________ Simplify each sum or difference by combining like terms. 29. 3x + 2x 29. x + 4x 30. 5w + 8w 30. 2w + 11w 31. 11R – 5R 31. 8R – 2R 5 1 3 3 32. x x 32. x x 8 8 8 8 1 3 1 2 33. y y 33. y y 7 14 14 7 34. 1.5K + 3.4K 34. 4.2K + .7K 35. 9.102x + 1.310x 35. 3.211x + 7.201x 36. 5x2 + 3x2 36. 2x2 + 6x2 37. -3x + 7x 37. -8x + 12x 38. 5x – (-2x) 38. 3x – (-4x) 39. 2x + 5x + 3y + 7 y 39. x + 6x + 4y + 6y 140 Homework: Combining Like Terms (1) Name___________________ Parent Signature_______________ No Calculator on any problem. A) Order of operations 7 32 2 3 2 3 13 10 1. 20 2. 4 2 B) Adding and subtracting fractions (show every step) 5 2 1 1 3. 4. 13 13 4 3 C) Adding and subtracting decimals 5. 5.03 + 7.204 6. 2.102 + 10.132 D) Adding and subtracting integers. Model each problem by drawing the appropriate positive and negative blocks above the starting blocks. Strike out blocks to show your result. 7. -2 – (-3) 8. 6 - 8 141 E) The distributive property Use arrays to find the products 9. 32(63) 10. 141(25) F) Basic Factoring: The distributive property Rewrite each product using the distributive property. Show that both ways are equal. Example: 5(2) + 7(2) can be written (5 + 7)(2) 10 + 14 (12)(2) 24 24 11. 2(8) + 5(8) 12. 3(x) + 9(x) Check with x = 2 G) Combine like terms. Simplify each sum or difference by combining like terms. 13. 2x + 8x 14. 6x - x 15. -5w + 8w 16. -2w + -11w 3 1 17. T T 18. 3.5R – 2.1R 5 5 142 Combining Like-Terms to Solve Equations The Commutative Property of Addition: Generalizaton: a + b = Show that the commutative property can make addition easier. 1. 7 + 38 + 3 2. 25 + 157 + 5 3. 21 + 1229 + 29 The Inverse Property of Addition says that every number but zero has an opposite, and when you add a number to its opposite you get 0. Use n’s for negative counters and p’s for positive counters to show examples that Subtracting is adding the opposite. a) use subtraction b) use addition. 4. 4 – 3 5. 3 – (-2) 6. 2 – 5 7. -3 – (-1) a) a) a) a) b) b) b) b) 143 8. Explain why addition is commutative but subtraction is not commutative. Show how subtraction by adding the opposite and the commutative property make these problems easier. 9. 23 + 88 – 13 10. 58 + 275 – 48 11. 1255 + 2749 - 1254 More on combining like terms Remember 5x means x+x+x+x+x and 2x means x+x so 5x + 2x means (x+x+x+x+x) + (x+x) = 7x Can you think of another way to show that 5x + 2x = 7x? Explain why 3x + 2y cannot be simplified. Explain why 3ab + 8ab equals 11ab. 144 Multiplying integers Explain how 4(5) can be written as an addition problem using only 5s. What does 4(-3) mean in terms of groups of -3? What does -3(2) mean in terms of groups of 2? Explain what -4(-5) means? Finish the following generalizations a(b) = a(-b) = -a(b) = -a(-b) = Examples 3(4)= 3(-4)= -3(4)= -3(-4)= What does the product (1/2)(2) equal? How about (5) (1/5)? ½ and 2 are reciprocals. Likewise 5 and 1/5 are reciprocals. What is the reciprocal of 17? What happens when you multiply a number by its reciprocal? 145 The Inverse Property of Multiplication says that every number but zero has a reciprocal, and that when you multiply a number by its reciprocal you get 1. b 1 Finish the generalization: 1, b ________ b b This means that dividing by 2 is the same as multiply by ½. It also means that dividing by ½ is the same as multiplying by 2. The same goes for any number and its reciprocal. Big Idea: “Dividing is the same as multiplying by the reciprocal.” Generalization: a b Change each division problems to a multiplication problem. 1 12. 5 7 13. 15 3 14. 12 2 Complete the following generalizations. a b a b (a) b a b The Commutative Property of Multiplication means: Generalization: ab = Show that the commutative property can make multiplication easier. 1 1 15. 15 11 16. 40 12 3 6 146 Explain why multiplication is commutative but division is not commutative. Change the division problems to multiplication problems then evaluate. 1 2 3 17. 5 18. 19. 3 7 1 5 3 2 5 20. 1 1 3 20. 21. 5 3 2 8 4 9 Solving Equations by Balancing Solving equations by balancing is very similar to solving with inverses. Solve 4x + 3 = 11 Inverses Balancing x was multiplied by 4 then increased Undo but keep the equation by 3 to get 11 balanced at all times. 11 can be decreased by 3 then divided by 4 to get x equation balanced at all times. (11 – 3)/4 4x + 3 = 11 8/4 -3 -3 subtract 3 from both sides 2 4x + 0 = 8 (identity property of +) 147 4x = 8 4x = 8 divide by sides by 4 4 4 1x = 2 (identity property of X) x = 2 the variable is isolated To isolate the variable means: Combine like terms then solve equation by balancing. Show every step. Check your answer by substituting it back into the equation to see if it makes a true statement. 21. 3x 9 4 2x 3 22. 5 x 3 x 3 23. 5x 2 3x 11 4x 1 148 Homework: Combining Like Terms to Solve Equations Name____________________ Parent Signature____________________ Combine like terms then solve equation by balancing. Show every step. 1. x + 5 + 2x = 14 2. 5 + 3x – 3 + 2x = 22 3. 3 – 4x + 11 + x = -1 4. 3x + 1 = 4x 5. 3(2x + 1) – 4x = 17 6. 6 + 2(x – 1) + x = 12 - x 149 Introduction to Factoring (The Distributive Property) Name____________________ Simplify. 1. aaaaaa ________ 2. a a a a a a ________ 3. sstttt _________ 4. a 2 a 4 ________ 5. 2a 4a __________ 6. st st 3 _________ 7. b b b bbb ________ 8. aaa aaa _________ 9. ccc cccc _______ Find the greatest common factor. 1. 5, 35 _________ 2. 14, 42 _______ 3. 33, 55 _________ 4. x, 5x _________ 5. t 2 , t 3 ________ 6. 4r , rb _________ 7. 7 x,14 x5 _______ 8. 12tw,9w2 _________ 9. 18c 2 d , 24cd 4 _________ Use an array to multiply. 1. 5 1 7 2. 7 2 6 3. 11 3 5 4. x 1 5 5. t 2 1 t 6. r 4 b 7. 7 x 1 2 x 4 8. 3w 4t 3w 9. 6cd 3c 4d 3 150 Factor (rewrite using the distributive property) 1. 6t 12 2. 5x 7 x2 3. 12a 16b 4. 28x 7 xy 5. 54a3b 63a 2b2 6. 13x 26 x 2 Solve by balancing. Check your answer. 1. 3x 11 8 2. 2x 3 x 2 x 5 3. 2 3 x 2 x 1 Check: Check: Check: Sylvia has made 3 birthday invitations in 2 minutes. Write a formula that she could use to find the total number of invitations given the number of minutes she continues to work. (hint: What is the intercept?______ What is the slope? ____________) Graph the function. Write an equation then solve it by balancing to find the number of minutes that it will take Sylvia to make a total of 16 invitations. 151 Factoring Revisited (The Distributive Property) Name____________________ Simplify. 1. ggggg 2. g g g g g 3. mmmvv 4. g 2 g 3 5. 2 g 3g 6. mv 2 mv 4 7. c c c c cccc 8. nn nn nn 9. bbb bb b Find the greatest common factor. 1. 6, 30 2. 18, 45 3. 22, 77 4. a, 8a 5. h 2 , h 3 6. 5k , 2kb 7. 6 x,12 x3 8. 21hy ,14h 2 9. 20 w2 x,30 wx 3 152 Use an array to multiply. 1. y 2 4 2. w2 3 w 3. a 9 c 4. 5x 1 4 x3 5. 2w 7d 5w 6. 3cd 2c 6d 4 Factor (rewrite using the distributive property) 1. 7t 14 2. 7 x 6 x2 3. 16a 32b 4. 27 x 9 xy 5. 56a 2b 64a3b2 6. 15 x 30 x3 153 Solve by balancing. Check your answer. 1. 2 x 13 17 2. 4x 6 x 3 x 15 3. 1 4 x 3 2x 1 Check: Check: Check: Matt’s company can produce 5 snowmobiles in 2 hours. How many snowmobiles can the company produce in 1 hour? __________ What is the slope of the data? __________ Matt starts the work day with 16 snowmobiles in storage. What is the intercept of the data? _________ Write a formula that he could use to find the total number of snowmobiles given the number of hours worked. How many snowmobiles in the shop for each time given? 0 hours? 2 hours? 6 hours? Graph the function. Label your axes and scale. Write an equation and solve to find the number of hours that it will take Matt’s company to have a total of 42 snowmobiles in storage. 154 Homework: Introduction to factoring Name___________________________ Parent Signature___________________ Factor (rewrite using the distributive property) 1. 2t 8 2. 5u 7u 2 3. 24a 36c 4. 8 y 32 xy 5. 50a3b 45a 4b3 6. 17 x3 34 x2 Solve by balancing. Check your answer. 1. 7 x 1 16 2. 3x 3 2x 2 x 9 3. 2 5 x 3 x 8 Check: Check: Check: Rose has made 5 pinwheels in 20 minutes. Write a formula that she could use to find the total number of pinwheels given the number of minutes she continues to work. (hint: What is the intercept?______ What is the slope? ____________) Graph the function. Write an equation then solve it by balancing to find the number of minutes that it will take Rose to make a total of 23 pinwheels. 155 Perimeter and Area Name ___________________ Perimeter: Use a ruler to find the dimensions of each rectangle. Write the dimensions on the rectangle. Then calculate the perimeter. Practice worksheet Given the algebraic dimensions on each rectangle, calculate the perimeter. 2x x+3 3w + 7 9 - 2w 156 Area: Write the dimensions of each rectangle below. (Hint: you already measured these). Use the dimensions to calculate the area. Practice Worksheet. Use the algebraic dimensions to calculate the area of each rectangle. 2x x+3 3w + 7 9 - 2w What is the difference between perimeter and area? 157 Homework: Perimeter and Area Name ___________________ Parent Signature ______________________ Given the dimensions, find the perimeter and the area of each rectangle. 16 12 11 5 5x 2x + 4 4w + 3 10 - 4w w + 3z 3x + 4y 2y w + 3z 158 Slope-Intercept Form Name ___________________ Exponents, Factoring Carl graduates from college with student loans totaling $20,000. He sets up a payment plan where each month he pays $250. What is the intercept? ________ Is this amount increasing or decreasing? What is the slope? ________ Define variables. Write a formula to determine how much Carl owes. Collect some data. Graph From a different story, we might find the formula Q = 14 + 3B What is the intercept? ________ What does that mean? What is the slope? ________ What does that mean? Collect data and graph. 159 From a different story, we might find the formula A = 315 – 22C What is the intercept? ________ What does that mean? What is the slope? ________ What does that mean? Collect data and graph. From a different story, we might find the formula Y = 20x + 10 What is the intercept? ________ What does that mean? What is the slope? ________ What does that mean? Collect data and graph. Quick Practice 1. M = 90 -11K 2. Y = 14x + 35 Intercept _________ Intercept __________ Slope ________ Slope _________ Graph Graph 160 Exponents What does x5 mean? What does 7y mean? Simplify 1. m3(m2) 2. (4ab)3 3. (2x)(5x)2 4. 5y(12xy) 5. 120 6. (22a)(a2b) Combine Like Terms 7. 5g + 3g + 19 8. 4m + -3y + 2m + -8y 9. 3x + 5x2 10. -11a2 + 9a2 +3a + 8a 11. 16rg + -2r + 3g + -5r 12. 10x2 – 5xx Multiply (Distribute) 13. 3(x + 7) 14. 8k(2k – 5) 15. 3xy(y2 +2) 161 Factoring Factor the following. 1. 32 2. 60 3. 88x2 Find the greatest common factor for each. 4. 16, 30 5. 20ab2 , 35abc Factor the following. 6. 12x + 36 7. 3a2 + 2a 8. 5d2k – 20dk 9. 18w3 + 12w2 + 24w Graphing coordinate practice worksheet. ( , ) 162 Homework: slope-intercept form, Name __________________ Exponents, Factoring Parent Signature __________ 1. From a story, we might find the formula H = 25 + 4M What is the intercept? ________ What does that mean? What is the slope? ________ What does that mean? Collect data and graph. 2. 3y(7xy) 3. 80 4. (15a3)(a2b) 5. -8a2 + 3a2 +15a + 6a 6. 10rg + -3r + 2g + -7r 7. 9x2 – 3xx Multiply (Distribute) 8. 4(x + 11) 9. 3a(2a + 3y) 10. 5fb(b + 3f) Factor 11. 4x + 24 12. 3a2 + 5ab 163 Slope-Intercept Form Name ___________________ Exponents, Factoring B The Fillmans bought a new hot tub and filled it with water from the garden hose. Initially, the temperature of the water was 56 degrees. When they turned on the tub, the temperature rose about 4 degrees every 40 minutes. What is the intercept? ________ Is this amount increasing or decreasing? How many degrees does the temperature change every minute? What is the slope? ________ Define variables. Write a formula to determine the temperature of the tub. Collect some data. Graph From a different story, we might find the formula Q = 12 + 7B What is the intercept? ________ What does that mean? What is the slope? ________ What does that mean? Collect data and graph. 164 From a different story, we might find the formula A = 285 – 15C What is the intercept? ________ What does that mean? What is the slope? ________ What does that mean? Collect data and graph. From a different story, we might find the formula Y = 15x + 22 What is the intercept? ________ What does that mean? What is the slope? ________ What does that mean? Collect data and graph. Quick Practice 1. M = 130 - 9K 2. Y = 8x + 16 Intercept _________ Intercept __________ Slope ________ Slope _________ Graph Graph 165 Exponents What does x6 mean? What does 4y mean? Simplify 1. m5(m3) 2. (2abc)3 3. (3x)(5x)2 4. 4y(10xy) 5. 880 6. (10a)(a2b3) Combine Like Terms 7. 4g + 9g + 14 8. 2m + -5y + 8m + -7y 9. 2x + 11x2 10. -8a2 + 4a2 +7a + 5a 11. 15rg + -3r + 2g + -8r 12. 31x2 – 6xx Multiply (Distribute) 13. 4(x + 5) 14. 6k(3k – 7) 15. 2xy(y2 + 5) 166 Factoring Factor the following. 1. 54 2. 40 3. 30y2 Find the greatest common factor for each. 4. 14, 98 5. 12ab2 , 32a3bc Factor the following. 6. 10x + 24 7. 5a2 + 30a 8. 6d2k – 21dk 9. 15w3 + 35w2 + 20w4 Find the Area of each Rectangle 3x +8 2a+3b x+9 a-5 167 Homework: slope-intercept form, Name __________________ Exponents, Factoring B Parent Signature __________ 1. From a story, we might find the formula H = 27 + 3M What is the intercept? ________ What does that mean? What is the slope? ________ What does that mean? Collect data and graph. 2. 2y(8x2y) 3. 90 4. (4a3)(7a2b) 5. -3a2 + 8a2 +11a + 7a 6. 11rg + -2r + 6g + -9r 7. 5x2 – 2xx Multiply (Distribute) 8. 3(x + 10) 9. 5a(7a + 6y) 10. 3fb(b + 2f2) Factor 11. 7x + 21 12. 8a2 + 12ab 168 Slope-Intercept, General Form Name ___________________ Exponents, Factoring Elmo lost some extra weight to look really good for his 5-year class reunion. Tipping the scales at 195 pounds, he successfully lost 2 pound per month until he reached his target weight. What is the intercept? ________ Is this amount increasing or decreasing? What is the slope? ________ Define variables. Write a formula to find how much Elmo weighed. Collect some data. Graph From a different story, we might find the formula Q = 23 + 2B What is the intercept? ________ What does that mean? What is the slope? ________ What does that mean? Collect data and graph. 169 Quick Practice 1. M = 83 + 5K 2. Y = -3x + 18 Intercept _________ Intercept __________ Slope ________ Slope _________ Graph Graph General Form: AX + BY = C Tanya has $20 to buy food at a roadside stand. She can buy F pounds of fruit at $2 per pound or N pounds of nuts at $5 per pound. If Tanya spends all $20, the equation 2F + 5N = 20 describes all of the possible weight combinations of nuts and fruit that she could buy. Use the equation 2F + 5N = 20 to answer the following questions. 1. How many pounds of nuts can Tanya buy if she buys 10 pounds of fruit? 2. How many pounds of fruit can she get if she buys 2 pounds of nuts? 3. How many pounds of nuts can Tanya buy if she buys 2 pounds of fruit? 4. How many pounds of nuts can she buy if she buys 0 pounds of fruit? Use the data you collected in problems 1-4 to graph 2F + 5N = 20. Lbs of Nuts Lbs. of Fruit 170 Exponents 1. What is the difference between 7x and x7? Simplify 2. m3(m+m+m) 3. (3a2)3 4. (3x3)(2x)2 5. (5y(7xy))0 6. Why is 3x2 not the same as (3x)2? Combine Like Terms 7. 5g + 3t + 7g 8. 4y + -y + 2m + -8y + m 9. 17x3 + x(x2) Multiply (Distribute) 10. 3(x + 7) 11. 8k(2k – 5) 12. 3xy(y2 +2) 13. Distribute then combine like terms. 14. Distribute then solve. 5( x + 3) + 7x + 8 8( b – 3) = 3( b + 2) 171 Factoring Factor the following. 1. 80x 2. 16x2 3. 8xy Find the greatest common factor for each. 4. 16x2, 80x 5. 80x, 8xy Factor the following. 6. 15x + 25 7. 16x2 + 8xy + 80 Use the given arrays to factor. 8. x2 + 5x + 6 = ( )( ) 9. x 2 + 8x + 15 = ( )( ) x2 3x x2 3x 2x 6 5x 15 172 Due (Wednesday, January 20) Homework: slope-intercept form, General Form Name __________________ Exponents, Factoring Parent Signature __________ 1. T = 8 + 3K 2. y = -2x + 15 Intercept _________ Intercept __________ Slope ________ Slope _________ Graph Graph Simplify 3. (114)0 4. (3a3)(a2b)2 5. -a2 + 3a2 + 4a + 6a2 6. 5 + 3gx + 2 + 4xg 7. Distribute then simplify. 8. Distribute then solve. 4(x + 1) + 2 5(2a + 3) = 4a + 3 Factor 11. 4x + 24 12. a2 + 7a + 12 = ( )( ) a2 3a 4a 12 173 Factoring, General Form, Name _____________________ Solving Equations Use arrays to factor each of the following. x2 + 9x + 20 x2 + 12x + 32 x2 5x x2 8x 4x 20 4x 32 x2 + 12x + 27 x2 + 13x +40 x2 x2 27 40 Exponent Practice x y 4 3 (3ab 2 )(4a) 2 174 General Form: AX + BY = C For her book club, Courtney wants to have crackers and cheese on hand. She has $45 to buy mozzarella and gouda cheese. Courtney already has crackers. She can buy M pounds of mozzarella at $5 per pound or G pounds of gouda at $8 per pound. If Courtney spends all $45, the equation 5M + 8G = 45 describes all of the possible weight combinations of mozzarella and gouda that she could buy. Use the equation 5M + 8G = 45 to answer the following questions. 1. How many pounds of gouda can Courtney buy if she buys 9 pounds of mozzarella? 2. How many pounds of mozzarella can she get if she buys 3 pounds of gouda? 3. How many pounds of gouda can Courtney buy if she buys 2 pounds of mozzarella? 4. How many pounds of gouda can she buy if she buys 0 pounds of mozzarella? Use the data you collected in problems 1-4 to graph 5M + 8G = 45 . Lbs of Mozzarella Lbs. of Gouda Exponent Practice 5 gh(2 g 9h) 4m(3mn 2 )(2m) 175 Homework Due Friday January 21st Name ___________________ Factoring, General Form, Solving Equations 1. Use arrays to factor the following. x2 + 16x + 64 x2 + 15x +56 x2 8x x2 8x 64 56 2. From a story, we get an equation 4x + 3y = 12. Use the equation to answer the following questions. a. If we buy 3 of the x’s, how many y’s can we buy? b. If we buy 4 of the y’s, how many x’s can we buy? c. If we buy 2 of the x’s, how many y’s can we buy? d. Use your data from a-c to graph 4x + 3y = 12. y X 3. Solve the equation. 7(2x + 1) = 2x +4 176 Linear graphs, solving equations, Name ___________________ Factoring For their wedding, Elvis and Diana created their own blend of M&Ms. Diana’s choice of purple with the message “Celebrate” costs $19 per pound. Elvis’s choice of silver with the message “Love” costs $15 per pound. They have a total budget of $1,900 for the candy. The equation 19D + 15E = 1900 can be used to help determine different combinations to buy. http://www.youtube.com/watch?v=mlyM_janCSE 1. If they decide to buy 100 pounds of Diana’s choice, how many pounds of Elvis’s choice can be purchased? 2. If they decide to buy 48 pounds of Elvis’s choice, how many pounds of Diana’s choice can be purchased? 3. If they decide to buy 30 pounds of Diana’s choice, how many pounds of Elvis’s choice can be purchased? Use the data you collected from problems 1-3 to graph the different combinations. Elvis’s Choice (pounds) Diana’s Choice (pounds) 4. Is the data increasing or decreasing? 5. Can you determine the slope? 6. Using your graph, what is the intercept? 177 Solving Equations Solve. 1. 3x + 5 = 22 2. 8x + 10 = -14 3. 5(2x + 1) = 12 4. 3(5x – 8) = 2(4x + 12) 5. 7x + 9 + 3x = 12 – 8 - 2x 6. 18 + 2x – 9 = 8x + 3x + 1 7. 9(2x – 4) + 8 = 17 8. 19 - 4x = 3(8x + 11) + 2 Exponent Practice. 1. (5 xy ) 2. (2ab)(3a ) 3 2 178 Factoring Practice. Use arrays to factor. 1. 8ab + 6b 2. 9x2 + 12x Use arrays to factor the trinomials. 3. x 17 x 72 4. a 3a 2 2 2 5. d 6d 8 6. k 11k 30 2 2 Paste here for the sorting activity. Explain why you sorted the way you did. Explanation: Explanation: 179 Cut Up the Following Equations and Graphs. Sort everything into two categories and tape them on the previous page. Explain how you sorted. 4 M 17 3H g 5x2 12 xy 2 D 8W 17 y 80 14 x a 2 b 2 22 (3af ) 4 2 19 9q 12b 20 r 3 x 11 Homework Name ________________________ Linear graphs, solving equations, Factoring Solve. 1. 4x + 2 + 9x = 12 – 4 + x 2. 3(2d + 7) = 25 Factor. 3. x 2 6 x 9 4. x2 14 x 49 180 Linear Equations, Graphing, Name ___________________ Solving Equations, Factoring Label each equation as LINEAR or NONLINEAR. As a class, choose one LINEAR equation to graph. ___________________ If we know the graph is a line, we need _______ ordered pairs. Choose ANY value for x: _______ Find the y coordinate that completes the ordered pair. ( , ) Choose ANY value for y: ________ Find the x coordinate that completes the ordered pair. ( , ) 181 As a class, choose another LINEAR equation to graph. ___________________ If we know the graph is a line, we need _______ ordered pairs. Choose ANY value for x: _______ Find the y coordinate that completes the ordered pair. ( , ) Choose ANY value for y: ________ Find the x coordinate that completes the ordered pair. ( , ) For the LINEAR equation 3x + 7y = 21, find two ordered pairs and graph. Choose any value for x: ________ Find the y coordinate that completes the ordered pair. ( , ) Choose ANY value for y: ________ Find the x coordinate that completes the ordered pair. ( , ) 182 The equations have been collected from different stories. Solve each equation. 1. 5x + 7 = 23 2. 4 = 3a – 14 3. 2y + 5 = 19 4. 6 + 5c = -29 5. 5x – 3 = 13- 3x 6. -4c – 11 = 4c + 21 7. 9(4b – 1) = 2(9b + 3) 8. 3(6+5y) = 2(-5 + 4y) 9. 3(d – 8) – 5 = 9(d + 2) + 1 10. 2(a – 8) + 7 = 5(a + 2) – 3a – 19 183 Factor each trinomial using an array. 184 Homework: Linear Equations and graphs, Name ________________________ Solving equations, Factoring 1. For the LINEAR equation 2x + 5y = 20, find two ordered pairs and graph. Choose any value for x: ________ Find the y coordinate that completes the ordered pair. ( , ) Choose ANY value for y: ________ Find the x coordinate that completes the ordered pair. ( , ) Solve each equation. 2. 7x + 18 = 22 3. 8x + 2 + 4x = 27 4. 3(2x + 8) -5 = 14 + 2x 5. 5(x – 7) = 9x + -3 Factor the trinomial using an array. 6. x 13x 42 7. x 5 x 84 2 2 185 Introduction to Radicals Vocabulary: a is called a radical. The number a is called the radicand. Evaluate the radicals 1. 36 2. 81 3. 49 4. Graph the radicals on a number line. 8 48 24 3 1 2 3 4 5 6 7 8 9 Factor then split the factors into equal groups (if possible). 5. 100 6. 18 7. 64 8. 24 9. 36 10. 42 Why is 4 9 the same as 36 ? What does a = c mean? What does b d mean? Use c’s and d’s to show why a b always equals ab ? 186 Simplify. 11. 2 5 12. 3 7 13. 2 16 14. 5 5 15. 27 3 16. 5 15 Put the numbers from problems 11-16 into two groups. How are the numbers different? A radical is simplified if the radicand has no square factors. 15 is simplified, 12 is not simplified. Why? 26 is simplified, 75 is not simplified. Why? Simplify the radicals. 1. 8 2. 27 3. 32 187 Like Radicals Not Like Radicals 2 1 2 5 2 -7 2 .3 2 2 5 3 -2 2 -2 7 11 5 7 Numbers are like radicals if:_________________________________________________ Just like 5x + 3x = 8x, 5 3 7 3 12 3 Simplify by combining like radicals. 1. 3 5 4 5 2. 4 3 3 3. 2 34 2 4. 8 3 2 5. 12 27 6. 2 18 3 8 Solve the equations. 1. 3x 7 11 2. 5x 2 3x 7 x 3. 7 x 3 5 4 2 x Factor 1. 4 x2 12 x 2. 15ax 12x 3. 14a 2b 28ab2 Factor using arrays 1. a 2 12a 32 2. c 2 10c 16 3. x 2 3x 54 188 Find the intercepts of the linear equation then sketch the graph 1. 3x + 2y = 12 2. x – 7y = 14 3. 3x + 5y = 10 Find any two points that make the linear equation true then sketch the graph 1. 3x – y = 4 2. y = 3 – x 3. 3y = x Solve each equation for b. 1. b + 3 = x 2. 8b = v 3. b – 1 = g b b5 4. y 5. 7b + r = a 6. t 5 2 3a b 4x y 2x 3 3 2 3 2 Simplify 7. 8. 189 Homework: Introduction to Radicals Name___________________ Simplify 1. 50 2. 20 3. 48 Add like radicals 4. 3 7 5 7 5. 2 3 5 3 3 5 Simplify 6. 84 2 7. 18 7 3 Find two points that make the equation true then graph 8. 3x 5 y 15 9. x 2 y 8 Solve for g 10. g 5 h 11. 4 g y 15 190 Linear equations, graphing, solving Name ___________________ Perimeter and Area From a story about tollbooths, we get the equation T = 250 + 50C. What is the intercept? ___________ What is the slope? __________ Is the data increasing or decreasing? Collect some data C 250+50(C) T (C,T) 0 Graph your data. From a story about pizza, we get the equation 6w + 5h = 30. Is it easy to see the intercept? The slope? Collect some data. W h (w,h) Graph your data. 191 Solving Equations. 7 x 8 100 4 x 3 2 x 9 8x 5 1 4(2 x 1) 15 2(9w 3) 10 3( w 1) Which equations are linear? Write Linear or Non-linear next to each. G 3 4b 3x 7 f 14 7 x2 y 4 y 12 xy 8 8 15d y 7 x .3 y 12 Simplify the radicals. 50 80 242 Use a calculator to approximate each radical above. 192 Like Radicals. Circle the like radicals in each expression and add when possible. 4 2 3 7 5 7 8 3 9 3 48 3 Factoring. Use an array to factor each expression. h2 5h 6 x2 18x 81 3dg 6 g 2 r 3 q 14rq Simplify. 4w(3w)(2w) 10 x 2 (3x) 2 (132 xyz )0 193 Perimeter: How do we calculate perimeter? Find the perimeter of each figure. 8.3cm 6.3cm 2w 14 11.2cm 5.5cm 4x +7 11+4w 6.3cm 7.4cm 9x Perimeter= Perimeter= Area: When we calculate the area of a rectangle, we _______________ _______________________________________________________. Find the area of each rectangle. 5.8in d+7 1.4in 3d 16cm g+3 16cm g+3 194 Finding the dimensions of squares. For each square, the area is given. Find the dimensions. 144 484 Area = 144 Area = 144 122 12 Each side length = 12 Each side length = 289 361 Area = Area = Each side length = Each side length = 81 12 Area = Area = Each side length = Each side length = 195 Homework: Solving equations, Name _________________ Simplifying radicals Solve. 1. 8x + 10 = 18 2. 5x + 3 -2x = 14 + 1 + x 3. 4(2x + 7) = 19 4. -3(x + 9) + 1 = 5(4x – 8) Simplify. 5. 60 6. 125 Given the area of the square, find the dimensions. Area = 576 Side length = 196 Pythagorean Theorem, Like Radicals, Name ______________ Solving Equations Draw square areas to help find the missing dimensions of the right triangles. 9 7 c 12 24 c 8 6 11 c 5 c 10 15 a 14 b 25 197 Radicals Simplify. Then use a calculator to get an approximation. 1. 121 2. 36 3. 40 4. 28 5. 64 6. 45 Like radicals have ______________________________________. Ex. Simplify. Do not use a calculator approximation. 7. 3 2 5 7 2 8. 8 5 2 11 11 9. 5 3 12 10. 20 45 Distribute (Multiply) 11. 5(3x + 7) 12. 4(2x – 3) 198 Solve each equation. 13. 14 = 8x + 9 14. 22 = 7 + 3x 15. x + 7x + 8 = 4x + 2 16. 10x – 5 – 3x = 12x + 3 + 2 17. 4(x + 3) = 25 18. 2(6x + 13) = 5(x – 1) Clearing Fractions. 3x 4 3x Ex. 5 Ex. 2 7 5 5 5x 2 x4 19. 1 20. 2 3 3 6 199 Radicals, Pythagorean Theorem, Name ___________________ Slope-Intercept Form Simplify. Then use a calculator to approximate. 1. 24 2. 75 3. 45 4. 2 7 63 5. 20 72 Find the missing side of each right triangle. 6. 7. 10 15 a 14 b 25 8. Find the perimeter of a triangle whose side lengths are 3x, x + 8, x – 1. 9. Find the perimeter of a square whose side length is 2a+b. 200 10. The area of a hexagon can be calculated using the formula Area = .5ap Find the area if a = 3.7 and p=27.4. 11. Find the value of 9gk + d2 if g=4, k=3.8, and d=5. Slope-Intercept Form Ex. *Make sure one variable is alone For each, find the intercept and slope. Then create a graph. 12. y = 16 + 3x 13. y = 22 – 4x Intercept _______ Intercept _______ Slope ______ Slope _______ 14. y = -(3/4)x + 5 15. y = 4x + -3 Intercept _______ Intercept _______ Slope ______ Slope _______ 201 Given the following graphs, find the intercept and the slope. Write an equation. 8 Intercept: Increasing or Decreasing? 6 How much? Slope: 4 2 Equation: y= -15 -10 -5 5 10 15 20 -2 -4 -6 -8 -10 8 Intercept: 6 Increasing or Decreasing? 4 How much? Slope: Equation: 2 -15 -10 -5 5 10 15y= 20 -2 -4 -6 -8 -10 8 Intercept: Increasing or Decreasing? How much? 6 4 Slope: 2 Equation: y= -15 -10 -5 5 10 15 20 -2 -4 -6 -8 -10 202 Factor 16. w2 6w 8 17. b2 15b 56 18. 9ab 12bc 19. 2d 18d 2 Simplify. 20. 4x(3x + 7g) 21. 7f(2 – 3r) 22. (x + 8)(x +4) 23. (k + 3)(k – 2) Put each equation in slope-intercept form. Find the intercept and slope. 24. y + 3 = 5x 25. y + 2x = 4 26. 8y + x = 24 27. 3x + 2y = 12 203 Homework: CRT B concepts #1-5,7,10 Name___________________ Estimate the square roots. Check your estimate with a calculator. 1. 18 2. 98 3. 51 Guess__________ Guess__________ Guess__________ Check__________ Check__________ Check__________ Classify each number as rational or irrational. Explain 4. 4.0087 Rational Irrational Why?______________________________________ 5. 3 49 Rational Irrational Why?______________________________________ 6. .3232… Rational Irrational Why?______________________________________ 7. 14 Rational Irrational Why?______________________________________ Simplify 8. 3 50 6 2 9. 54 24 x 10. Find the perimeter of a triangle whose sides measure: 3x, , and x 3 if x = 6. 3 Write the equation in slope-intercept form by solving for y then identify the slope and the y-intercept. 11. 3x + 2y = 5 12. x – 4y = 6 13. 2 x y 4 Slope_______ Slope_______ Slope_______ y-intercept________ y-intercept_________ y-intercept_______ 204 Slope, Intercepts Name __________________ Solving Equations, Linear and Quadratic Mikayla sells Girl Scout cookies for a fundraiser. Her grandma gave her $25 to begin. Each additional box that she sells will raise $3.50. How much money did Mikayla have to start? _________ What is the intercept? _______ Is the data increasing or decreasing? How much? ______ What is the slope? _______ Write a formula to describe how much money Mikayla has: How much money did Mikayla have after selling one box? ________ How much money did Mikayla have after selling two boxes? ________ How much money did Mikayla have after selling ten boxes? ________ Graph your data. From another story, we produced the following graph: 15 10 $ in Pat's 5 account -10 10 Weeks 20 30 40 50 60 -5 What is the intercept?______ Is the data increasing or -10 decreasing? How much? ______ -15 What is the slope? _______ Write an equation describing the data. 205 I-tunes collected some data from a new release. This data is shown in the table below. 0 .99 1 1.98 2 2.97 3 3.96 4 4.95 5 5.94 What is the intercept? _________ Is the data increasing or decreasing? How much? __________ What is the slope? _________ Write an equation describing the data. Avalanche Control suggests staying off terrain with a slope greater than 1. Would the following pitch be considered safe? Find the slope to explain your answer. Is this slope safe? Find the slope to explain your answer. 206 Solve each equation. 1. 5x + 9 = 32 2. 4(x + 1) = 8.3 3. 5 + 3x + 1 = 2x + 4x + 10 4. 12(x + 1) = 3(2x – 1) x 1 x 1 5 5. 3 6. 4 5 5 2 2 Is the data closer to linear? Is the data close to linear? Is it increasing or decreasing? Is it increasing or decreasing? Correlation: Correlation: 6 6 4 4 2 2 5 10 15 5 20 10 25 15 -2 -2 -4 -4 -6 -6 207 -8 Which equations are linear? G 3 4b 3x 7 f 14 7 x2 y 4 y 12 xy 8 8 15d y 7 x .3 y 12 Factor x 2 7 x 12 a 2 11a 30 Solving quadratics by factoring and using the zero product property. x 2 7 x 12 0 a 2 11a 30 Find the missing side of the right triangle. 24 7 208 Homework: Slope, solving equations Name __________________ 1. Given the following data: 0 17 1 15 2 13 3 11 4 9 What is the intercept? _________ Is the data increasing or decreasing? 22 How much? What is the slope? 20 18 2. Given the following data:16 14 12 10 8 6 4 2 -5 5 10 15 20 25 30 35 40 45 50 -2 What is the intercept? _________ -4 Is the data increasing or decreasing? How much? What is the slope? 3. Solve each equation. 5x + -3 + 8x = 2 – 9 + 11x -2(x +7) = 3x -8 4. Factor and solve each equation. x 2 7 x 10 0 a 2 9a 20 209 Zero Product, Inequalities, Slope, and Intercept Name________________________ Write each equation in slope-intercept form (Solve for y) to identify the slope and intercept. 1. y 5 x 4 2. 7 x y 2 Slope________ Slope_________ Intercept_____ Intercept______ 3. 3x 2 y 6 4. x 2 y 4 Slope________ Slope_________ Intercept_____ Intercept______ Write a generalization with the variables a and b to show: “If we get a zero by multiplying two numbers then one of the numbers has to be zero.” If p 6 0 then what is the value of p ? If q r 0, and r 7, then what is the value of q? If x 2 8 0, then x 2 must equal what value? What is the value of x ? If x 3 y 0, and y 5 , then what is the value of x ? 210 Solve using the zero product rule. 1. a 3b 1 0 2. x 9 x 4 0 Solve the quadratic equations by factoring then using the zero product rule. 3. x2 3x 40 0 4. x 2 7 x 12 0 5. x 2 8x 12 0 6. x 2 12 x 36 0 7. x2 2 x 35 0 8. x2 x 56 0 211 Station 1: Simplifying radicals Expert_________________ Simplify 1. 75 initials______ 2. 32 18 initials______ 3. 12 4 27 initials______ 4. 3 2 20 7 5 initials______ Station 2: Pythagorean Theorem Expert_________________ Solve for the unknown side of the right triangle. ? 13 6 5 8 ? initials______ initials______ 212 Station 3: Solving Equations Expert__________________ Solve. 1. 3x 4 11 initials______ 2. 4 2 x 7 initials______ x 3 3. 3 x 4 2 x initials______ 4. 3 initials______ 2 2 Station 4: Slopes and Intercepts from graphs Expert________________ Find the slope and intercept. 1. 2. initials______ initials______ Slope________ Slope_______ Intercept______ Intercept______ Equation_____________________ Equation___________________________ 213 Homework: Stations A Name__________________ Solve using the zero product rule. 1. a 8 b 4 0 2. x 2 x 5 0 Solve the quadratic equations by factoring then using the zero product rule. 3. x2 2 x 63 0 4. x 2 9 x 18 0 5. x 2 10 x 16 0 6. x 2 10 x 25 0 7. Simplify 3 5 45 7 2 8. Solve 3x 2( x 1) 7 9. Find the slope and y-intercept of 2 x 5 y 200 214 Systems, Name ___________________ Solving Equations System of Equations: When we solve a system, we are looking for ________________________ ________________________________________________________. For example, Practice. Find the solution to the system. 1. 2. Solution: Solution: 3. 4. Solution: Solution: 215 Systems with algebra These two equations describe two sets of linear data. We want to know which data point they have in common. 3x + y = 12 and y = 7 Solution: Try this system: x + 4y = 8 and y = 2 Solution: Practice: 1. Solve the system: 2x + y = 11 and y = 4 Solution: 2.Solve the system: x + 3y = 9 and x =2 Solution: 3. Solve the system: 2x + y = 10 and y = 3x Solution: 216 Station 1: Linear Equations Expert__________________ 1. Explain which of the following equations are NOT linear. Initial ____ a. A 3 7B b. y 4 x 2 7 c. 3xw 9 w 5 d. y 4 x 7 e. .4 x .9 y 13 f. g 12 f 2. Which equations are in slope-intercept form? Initial ____ a. 8w+3z=10 b. y = 13+2x c. x = -15 -2y d. y = 11x +5 e. 8x + y = 2 f. y = 15 – x/2 3. Put each equation in slope-intercept form. Initial ____ a. 3x + y =12 b. 4y = 12 + 24 c. 2x + 3y = 6 Initial _____ Initial ______ Initial ______ Station 2: Solving Quadratic Equations Expert __________________ Factor and solve using the zero product property. 1. x 2 12 x 35 0 2. w2 12w 36 0 Initial _____ Initial _____ 3. d 12d 11 0 2 4. x 2 3x 40 0 Initial _____ Initial _____ 217 Station 3: Pythagorean Theorem Expert ________________ Find the missing side of the right triangle. ? 6 10 12 ? 7 Initial _____ Initial _____ Station 4: Simplifying Radicals Expert ________________ 1. 40 Initial _____ 2. 45 80 Initial ____ 3. 18 Initial _____ 4. 12 75 5 Initial ____ Use a calculator to approximate: 5. 17 Initial _____ 6. 22 Initial ____ 218 Homework: Station B Name _________________ 1. Solve the system. 2. Solve the system. 3x + y = 8 and y = 1x 3. Put in slope-intercept form. Then find the intercept and slope. 7x + y = 9 Intercept ______ Slope _____ 4. Factor. Then use the zero product property to solve. x 2 9 x 18 0 219

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