# Lesson 9: Using If-Then Moves in Solving Equations

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Student Outcomes Students understand and use the addition, subtraction, multiplication, division, and substitution properties of equality to solve word problems leading to equations of the form pppp + qq = rr and pp(xx + qq) = rr, where pp, qq, and rr are specific rational numbers. Students understand that any equation can be rewritten as an equivalent equation with expressions that involve only integer coefficients by multiplying both sides by the correct number. Lesson Notes This lesson is a continuation from Lesson 8. Students examine and interpret the structure between pppp + qq = rr and pp(xx + qq) = rr. Students will continue to write equations from word problems including distance and age problems. Also, students will play a game during this lesson which requires students to solve 1 2 problems and then arrange the answers in correct numerical order. This game can be played many different times as long as students receive different problems each time. Classwork Opening Exercise (10 minutes) Have students work in small groups to write and solve an equation for each problem, followed by a whole group discussion. Opening Exercise Heather practices soccer and piano. Each day she practices piano for hours. After 55 days, she practiced both piano and soccer for a total of hours. Assuming that she practiced soccer the same amount of time each day, how many hours per day, hh, did Heather practice soccer? hh: hours per day that soccer was practiced 55(hh + ) = = = = (5555) = (1111) Heather practiced soccer for hours each day. hh = 130

2 Over 55 days, Jake practices piano for a total of hours. Jake practices soccer for the same amount of time each day. If he practiced piano and soccer for a total of hours, how many hours, hh, per day did Jake practice soccer? hh: hours per day that soccer was practiced = = 5555 = (5555) = (1111) Jake practiced soccer. 66 hours each day. hh =. 66 MP.2 MP.7 MP.2 Examine both equations. How are they similar, and how are they different? Both equations have the same numbers and deal with the same word problem. They are different in the set-up of the equations. The first problem includes parentheses where the second does not. This is because in the first problem, both soccer and piano were being practiced every day, so the total for each day had to be multiplied by the total number of days, five. Whereas in the second problem, only soccer was being practiced every day, and piano was only practiced a total of two hours for that time frame. Therefore, only the number of hours of soccer practice had to be multiplied by five, and not the piano time. Do the different structures of the equations affect the answer? Explain why or why not. Yes, the first problem requires students to use the distributive property, so the number of hours of soccer and piano practice are included every day. Using the distributive property changes the 2 in the equation to 10, which is the total hours of piano practice over the entire 5 days. An if-then move of dividing both sides by 5 first could have also been used to solve the problem. The second equation does not use parentheses since piano is not practiced every day. Therefore, the 5 days are only multiplied by the number of hours of soccer practice, and not the piano time. This changes the end result. Which if-then moves were used in solving the equations? In the first equation, students may have used division of a number on both sides, subtracting a number on both sides, and multiplying a number on both sides. If the student distributed first, then only the if-then moves of subtracting a number on both sides and multiplying a non-zero number on both sides were used. In the second equation, the if-then moves of subtracting a number on both sides and multiplying a non-zero number on both sides were used. Interpret what 3.6 hours means in hours and minutes? Describe how to determine this. The solution 3.6 hours means 3 hours 36 minutes. Since there are 60 minutes in an hour and 0.6 is part of an hour, multiply 0.6 by 60 to get the part of the hour that 0.6 represents. 131

3 Example 1 (8 minutes) Lead students through the following problem. Example 1 Fred and Sam are a team in the local mile bike-run-athon. Fred will compete in the bike race, and Sam will compete in the run. Fred bikes at an average speed of 88 miles per hour and Sam runs at an average speed of 44 miles per hour. The bike race begins at 6:00 a.m., followed by the run. Sam predicts he will finish the run at 2: a.m. the next morning. a. How many hours will it take them to complete the entire bike-run-athon? From 6:00 a.m. to 2:00 a.m. the following day is hours. minutes, in hours, is 6666 = 1111 = hours. Therefore, the total time it will take to complete the entire bike-run-athon is hours. Scaffolding: Refer to a clock when determining the total amount of time. Teachers may need to review the formula dd = rrrr from Grade 6 and Module 1. b. If tt is how long it takes Fred to complete the bike race, in hours, write an expression to find Fred s total distance. dd = rrrr dd = 8888 The expression of Fred s total distance is c. Write an expression, in terms of tt to express Sam s time. Since tt is Fred s time and is the total time, then Sam s time would be the difference between the total time and Fred s time. The expression would be tt. d. Write an expression, in terms of tt, that represents Sam s total distance. dd = rrrr dd = 44( tt) The expressions 44( tt) or is Sam s total distance. e. Write and solve an equation using the total distance both Fred and Sam will travel ( tt) = = = = = = (4444) = (5555) tt = 1111 Fred s time: Sam s time: tt = 1111 hours tt = = hours f. How far will Fred bike, and how much time will it take him to complete his leg of the race? 88(1111) = miles, and Fred will complete the bike race in 1111 hours.. 132

4 g. How far will Sam run, and how much time will it take him to complete his leg of the race? 44( tt) 44( ) 44( ) =. miles Sam will run. miles, and it will take him hours. Discussion (5 minutes) Why isn t the total time from 6:00 a.m. to 2: a.m. written as 20. hours? Time is based on 60 minutes. If the time in minutes just became the decimal, then time would have to be out of 100 because 20. represents 20 and hundredths. To help determine the expression for Sam s time, work through the following chart. (This will lead to subtracting Fred s time from the total time.) Total Time (hours) Fred s Time (hours) Sam s Time (hours) = = = = tt tt How do you find the distance traveled? Multiply the rate of speed by the amount of time. Model how to organize the problem in a distance, rate, and time chart. Rate (mph) Time (hours) Distance (miles) Fred 88 tt 8888 Sam tt 44( tt) Explain how to write the equation to have only integers and no decimals. Write the equation. Since the decimal terminates in the tenths place, if we multiply every term by 10, the equation would result with only integer coefficients. The equation would be 40tt + 8 =

6 Name Date Exit Ticket 1. Brand A scooter has a top speed that goes 2 miles per hour faster than Brand B. If after 3 hours, Brand A scooter traveled 24 miles at its top speed, at what rate did Brand B scooter travel at its top speed if it traveled the same distance? Write an equation to determine the solution. Identify the if-then moves used in your solution. 2. At each scooter s top speed, Brand A scooter goes 2 miles per hour faster than Brand B. If after traveling at its top speed for 3 hours, Brand A scooter traveled 40.2 miles, at what rate did Brand B scooter travel if it traveled the same distance as Brand A? Write an equation to determine the solution and then write an equivalent equation using only integers. 135

7 Exit Ticket Sample Solutions 1. Brand A scooter has a top speed that goes miles per hour faster than Brand B. If after hours, Brand A scooter traveled miles at its top speed, at what rate did Brand B scooter travel at its top speed if it traveled the same distance? Write an equation to determine the solution. Identify the if-then moves used in your solution. xx: speed, in mph, of Brand B scooter xx + : speed, in mph, of Brand A scooter dd = rrrr = (xx + )() = (xx + ) Possible solution 1: Possible solution 2: = (xx + ) 88 = xx + 88 = xx + 66 = xx = (xx + ) = = = (1111) = 11 () 66 = xx If-then Moves: Divide both sides by. If-then Moves: Subtract 66 from both sides. Subtract from both sides. Multiply both sides by At each scooter s top speed, Brand A scooter goes miles per hour faster than Brand B. If after traveling at its top speed for 3 hours, Brand A scooter traveled miles, at what rate did Brand B scooter travel if it traveled the same distance as Brand A? Write an equation to determine the solution and then write an equivalent equation using only integers. xx: xx + : speed, in mph, of Brand B scooter speed, in mph, of Brand A scooter dd = rrrr = (xx + )() = (xx + ) Possible solution 1: Possible solution 2: = (xx + ) = xx = = = (111111) = (111111) = xx = (xx + ) = = = = () = () = xx Brand B's scooter travels at miles per hour. 136

8 Problem Set Sample Solutions 1. A company buys a digital scanner for \$1111, The value of the scanner is 1111, nn after nn years. The 55 company has budgeted to replace the scanner when the trade-in value is \$, After how many years should the company plan to replace the machine in order to receive this trade-in value? They will replace the scanner after 44 years. 1111, nn =, , , =, , , , =, , , = 99, nn = Michael is 1111 years older than John. In 44 years, the sum of their ages will be Find Michael s present age. xx represents Michael s age now in years. Michael s present age is years old. Now 44 years later Michael xx xx + 44 John xx 1111 (xx 1111) + 44 xx xx = 4499 xx xx 1111 = = = = () = 11 (5555) xx = 3. Brady rode his bike 7777 miles in 44 hours. He rode at an average speed of 1111 mph for tt hours and at an average rate of speed of mph for rest of the time. How long did Brady ride at the slower speed? Use the variable tt to represent the time, in hours, Brady rode at 1111 mph. Rate (mph) Time (hours) Distance (miles) The total distance he rode: The total distance equals 7777 miles. The time he rode at 1111 mph is. 66 hours. Brady speed tt Brady speed 2 44 tt (44 tt) (44 tt) (44 tt) = (44 tt) = = = tt = = 1111 tt =. 66 Total distance 137

9 4. Caitlan went to the store to buy school clothes. She had a store credit from a previous return in the amount of \$ If she bought 44 of the same style shirt in different colors and spent a total of \$5555. after the store credit was taken off her total, what was the price of each shirt she bought? Write and solve an equation with integer coefficients. tt: the price of one shirt The price of one shirt was \$ = = = (4444) = 11 ( ) 44 tt = A young boy is growing at a rate of. 55 cccc per month. He is currently 9999 cccc tall. At that rate, in how many months will the boy grow to a height of cccc? Let mm represent the number of months = = = (. 5555) = (4444) mm = 1111 The boy will grow to be cccc tall 1111 months from now. 6. The sum of a number, of that number, 11 of that number, and 77 is 11. Find the number. Let nn represent the given number. nn nn + 11 nn + 77 = nn = nn = nn = nn = nn + 00 = nn = nn = = nn = The number is

10 7. The sum of two numbers is and their difference is. Find the numbers. If I let xx represent the first number, then xx represents the other number since their sum is. xx ( xx) = xx + ( xx) = xx + ( ) + xx = + ( ) = + ( ) + = = = 11 = = xx xx = = , Aiden refills three token machines in an arcade. He puts twice the number of tokens in machine A as in machine B, and in machine C, he puts of what he put in machine A. The three machines took a total of 1111, tokens. How 44 many did each machine take? Let AA represent the number of tokens in machine A. Then 11 AA represents the number of tokens in machine B, and AA represents the number of tokens in machine C. 44 AA + 11 AA + AA = 1111, AA = 1111, 44 AA = 88, Machine A took 88, tokens, machine B took 44, tokens, and machine C took 66, tokens. 9. Paulie ordered pens and pencils to sell for a theatre club fundraiser. The pens cost 1111 cents more than the pencils. If Paulie s total order costs \$ , find the cost of each pen and pencil. Let ll represent the cost of a pencil in dollars. Then, the cost of a pen in dollars is ll (ll + ll ) = ( ) = = ( ) + (. 55) = (. 55) [ (. 55)] = = = = ll = \$ A pencil costs \$ , and a pen costs \$

11 10. A family left their house in two cars at the same time. One car traveled an average of 77 miles per hour faster than the other. When the first car arrived at the destination after hours of driving, both cars had driven a total of miles. If the second car continues at the same average speed, how much time, to the nearest minute, will it take before the second car arrives? If I let rr represent the speed in miles per hour of the faster car, then rr 77 represents the speed in miles per hour of the slower car (rr) (rr 77) = (rr + rr 77) = ( 77) = ( 77) = ( 77) = ( 77) = = = = = = = 5555 rr = 5555 The average speed of the faster car is 5555 miles per hour, so the average speed of the slower car is 5555 miles per hour. dddddddddddddddd = rrrrrrrr ddddccdd dd = The slower car traveled. 55 miles in hours. dd = dd =. 55 dd = dd = dd = The faster car traveled miles in hours. OR = The slower car traveled. 55 miles in hours. The remainder of their trip is. 55 miles because. 55 =. 55. dddddddddddddddd = rrrrrrrr ddddccdd. 55 = 5555 (tt) (. 55) = (5555)(tt) = 11tt = tt This time is in hours. To convert to minutes, multiply by 6666 minutes per hour = = 4444 minutes The slower car will arrive approximately 4444 minutes after the first. 140

12 11. Emily counts the triangles and parallelograms in an art piece and determines that all together, there are 4444 triangles and parallelograms. If there are total sides, how many triangles and parallelograms are there? If tt represents the number of triangles that Emily counted, then 4444 tt represents the number of parallelograms that she counted. There are 1111 triangles and parallelograms. + 44(4444 tt) = ( tt) = (4444) + 44( tt) = ( 4444) = ( 4444) = tt = tt = tt + 00 = 1111 tt = ( tt) = 11 ( 1111) 1111 = 1111 tt = 1111 Note to the Teacher: Problems are more difficult and may not be suitable to assign to all students to solve independently. 12. Stefan is three years younger than his sister Katie. The sum of Stefan s age years ago and of Katie s age at that time is How old is Katie now? If I let ss represent Stefan s age in years, then ss + represents Katie s current age, ss represents Stefan s age years ago, and ss also represents Katie s age years ago. (ss ) + ss = 1111 ss + ( ) + ss = 1111 ss + ss + ( ) = 1111 ss + ( ) = 1111 ss + 55 ss + ( ) = ss + ( ) + = ss + 00 = ss = ss = = ss = 99 Stefan s current age is 99 years, so Katie is currently 1111 years old. 141

13 13. Lucas bought a certain weight of oats for his horse at a unit price of \$00. per pound. The total cost of the oats left him with \$11. He wanted to buy the same weight of enriched oats instead, but at \$00. per pound, he would have been \$ short of the total amount due. How much money did Lucas have to buy oats? The difference in the costs is \$ for the same weight in feed. Let ww represent the weight in pounds of feed. Lucas bought pounds of oats. CCCCdddd = uudddddd pprrddccdd wwwwwwwwwwww (ppccuudddddd) CCCCdddd = (\$00. pppppp ppccuudddd) ( pppppppppppp) CCCCdddd = \$ = = 11 ww = ww = = ww = Lucas paid \$66 for pounds of oats. Lucas had \$11 left after his purchase, so he started with \$

14 Group 1: Where can you buy a ruler that is feet long? What value(s) of zz makes the equation 7 6 zz = 5 6 true; zz = 1, zz = 2, zz = 1, or zz = 36 63? DD Find the smaller of 2 consecutive integers if the sum of the smaller and twice the larger is 4. SS Twice the sum of a number and 6 is 6. Find the number. YY Logan is 2 years older than Lindsey. Five years ago, the sum of their ages was 30. Find Lindsey s current age. AA The total charge for a taxi ride in NYC includes an initial fee of \$3.75 plus \$1.25 for every 1 mile traveled. 2 Jodi took a taxi, and the ride cost her exactly \$ How many miles did she travel in the taxi? RR The perimeter of a triangular garden with 3 equal sides is feet. What is the length of each side of the garden? EE A car travelling at 60 mph leaves Ithaca and travels west. Two hours later, a truck travelling at 55 mph leaves Elmira and travels east. All together, the car and truck travel miles. How many hours does the car travel? AA The Cozo family has 5 children. While on vacation, they went to a play. They bought 5 tickets at the child s price of \$10.25 and 2 tickets at the adult s price. If they spent a total of \$89.15, how much was the price of each adult ticket? LL 143

15 Group 1 Sample Solutions YARD SALE What value(s) of zz makes the equation 7 6 zz = 5 36 true; zz = 1, zz = 2, zz = 1, or zz = 6 63? 11 Find the smaller of 2 consecutive integers if the sum of the smaller and twice the larger is 4. Twice the sum of a number and 6 is 6. Find the number. Logan is 2 years older than Lindsey. Five years ago, the sum of their ages was 30. Find Lindsey s current age The total charge for a taxi ride in NYC includes an initial fee of \$3.75 plus \$1.25 for every 1 mile traveled. 2 Jodi took a taxi, and the ride cost her exactly \$ How many miles did she travel in the taxi?. 55 mmmm. The perimeter of a triangular garden with 3 equal sides is feet. What is the length of each side of the garden? 44. ffffffff A car travelling at 60 mph leaves Ithaca and travels west. Two hours later, a truck travelling at 55 mph leaves Elmira and travels East. All together, the car and truck travel miles. How many hours does the car travel? hhhhhhhhhh The Cozo family has 5 children. While on vacation, they went to a play. They bought 5 tickets at the child s price of \$10.25 and 2 tickets at the adult s price. If they spent a total of \$89.15, how much was the price of each adult ticket? \$

16 Group 2: Where do fish keep their money? What value of zz makes the equation 2 3 zz 1 2 = 5 12 true; zz = 1, zz = 2, zz = 1 8, or zz = 1 8? KK Find the smaller of 2 consecutive even integers if the sum of twice the smaller integer and the larger integers is 16. BB Twice the difference of a number and 3 is 4. Find the number. II Brooke is 3 years younger than Samantha. In five years, the sum of their ages will be 29. Find Brooke s age. EE Which of the following equations is equivalent to 4.12xx = 8.23? (1) 412xx + 52 = 823 (2) 412xx = 823 (3) 9.32xx = 8.23 (4) 0.412xx = 8.23 RR The length of a rectangle is twice the width. If the perimeter of the rectangle is 30 units, find the area of the garden. NN A car traveling at 70 miles per hour traveled one hour longer than a truck traveling at 60 miles per hour. If the car and truck traveled a total of 0 miles, for how many hours did the car and truck travel all together? AA Jeff sold half of his baseball cards then bought sixteen more. He now has 21 baseball cards. How many cards did he begin with? VV 145

17 Group 2 Sample Solutions RIVER BANK What value of zz makes the equation 2 3 zz 1 2 = 5 12 true; zz = 1, zz = 2, zz = 1 8, or zz = 1 8? Find the smaller of 2 consecutive even integers if the sum of twice the smaller integer and the larger integer is Twice the difference of a number and 3 is 4. Find the number. 11 Brooke is 3 years younger than Samantha. In 5 years, the sum of their ages will be 29. Find Brooke s age. 88 Which of the following equations is equivalent to 4.12xx = 8.23? (1) 412xx + 52 = 823 (2) 412xx = 823 (3) 9.32xx = 8.23 (4) 0.412xx = 8.23 The length of a rectangle is twice the width. If the perimeter of the rectangle is 30 units, find the area of the garden uuuuuuuuuu A car traveling at 70 miles per hour traveled one hour longer than a truck traveling at 60 miles per hour. If the car and truck traveled a total of 0 miles, for how many hours did the car and truck travel all together? 55 hhhhhhhhhh Jeff sold half of his baseball cards then bought 16 more. He now has 21 baseball cards. How many cards did he begin with?

18 Group 3: The more you take, the more you leave behind. What are they? An apple has 80 calories. This is 12 less than 1 the number of calories in a package of candy. How many 4 calories are in the candy? The ages of 3 brothers are represented by consecutive integers. If the oldest brother s age is decreased by twice the youngest brother s age, the result is 19. How old is the youngest brother? A carpenter uses 3 hinges on every door he hangs. He hangs 4 doors on the first floor and xx doors on the second floor. If he uses 36 hinges total, how many doors did he hang on the second floor? OO PP FF Kate has pounds of chocolate. She gives each of her 5 friends xx pounds each and has 3 1 pounds left 3 over. How much did she give each of her friends? TT A room is 20 feet long. If a couch that is 12 1 feet long is to be centered in the room, how big of a table can 3 be placed on either side of the couch? EE Which equation is equivalent to 1 4 xx = 2? (1) 4xx + 5 = 1 2 (2) 2 9 xx = 2 (3) 5xx + 4 = 18 (4) 5xx + 4 = 40 During a recent sale, the first movie purchased cost \$29, and each additional movie purchased costs mm dollars. If Jose buys 4 movies and spends a total of \$64.80, how much did each additional movie cost? The Hipster Dance company purchases 5 bus tickets that cost \$150 each, and they have 7 bags that cost bb dollars each. If the total bill is \$823.50, how much does each bag cost? SS OO SS The weekend before final exams, Travis studied 1.5 hours for his science exam, 2 1 hours for his math exam, 4 and h hours each for Spanish, English, and social studies. If he spent a total of 11 1 hours studying, how 4 much time did he spend studying for Spanish? TT 147

19 Group 3 Sample Solutions FOOTSTEPS An apple has 80 calories. This is 12 less than 1 the number of calories in a package of candy. How 4 many calories are in the candy? cccccccccccccccc The ages of 3 brothers are represented by consecutive integers. If the oldest brother s age is decreased by twice the youngest brother s age, the result is 19. How old is the youngest brother? A carpenter uses 3 hinges on every door he hangs. He hangs 4 doors on the first floor and xx doors on the second floor. If he uses 36 hinges total, how many doors did he hang on the second floor? 88 Kate has pounds of chocolate. She gives each of her 5 friends xx pounds each and has pounds left over. How much did she give each of her friends? pppppppppppp A room is 20 feet long. If a couch that is 12 1 feet long is to be centered in the room, how big of a table 3 can be placed on either side of the couch? ffffffff Which equation is equivalent to 1 4 xx = 2? (1) 4xx + 5 = 1 2 (2) 2 9 xx = 2 (3) 5xx + 4 = 18 (4) 5xx + 4 = During a recent sale, the first movie purchased cost \$29, and each additional movie purchased costs mm dollars. If Jose buys 4 movies and spends a total of \$64.80, how much did each additional movie cost? \$ The Hipster Dance company purchases 5 bus tickets that cost \$150 each, and they have 7 bags that cost bb dollars each. If the total bill is \$823.50, how much does each bag cost? \$ The weekend before final exams, Travis studied 1.5 hours for his science exam, 2 1 hours for his math 4 exam, and h hours each for Spanish, English, and social studies. If he spent a total of hours studying, how much time did he spend studying for Spanish? 11 wwccuurrdd 148

### Writing & Solving Equations Bell Work 1. Name: Period:

1) The perimeter of a rectangle is 0 inches. If its length is three times its width, find the dimensions. Writing & Solving Equations Bell Work 1 Tuesday: ) You are designing a rectangular pet pen for

### SECTION 2 Time 25 minutes 20 Questions

2 2 123456789 123456789 SECTION 2 Time 25 minutes 20 Questions Turn to Section 2 of your answer sheet to answer the questions in this section. Notes 1. Choose the best answer choice of those provided.

### Free Animal Alphabet Match. Halloween Version & Anytime version. October and November Calendar pieces

Free Animal Alphabet Match Halloween Version & Anytime version October and November Calendar pieces Aa alligator Recognize and name all upper- and lowercase letters of the alphabet. Table of Contents ALL

### Letter Detectives Uppercase & Lowercase Letter Recognition

Uppercase & Lowercase Letter Recognition Created By Pam Ballingall 2012 Pocketful of Centers http://pocketfulofcenters.blogspot.com Graphics provided by Scrappin Doodles Uppercase & Lowercase Letter Recognition

### Veterans Upward Bound Algebra I Concepts - Honors

Veterans Upward Bound Algebra I Concepts - Honors Brenda Meery Kaitlyn Spong Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) www.ck12.org Chapter 6. Factoring CHAPTER

### MM 6-1 MM What is the largest number you can make using 8, 4, 6 and 2? (8,642) 1. Divide 810 by 9. (90) 2. What is ?

MM 6-1 1. What is the largest number you can make using 8, 4, 6 and 2? (8,642) 2. What is 500 20? (480) 3. 6,000 2,000 400 = (3,600). 4. Start with 6 and follow me; add 7, add 2, add 4, add 3. (22) 5.

### Word Problems Involving Systems of Linear Equations

Word Problems Involving Systems of Linear Equations Many word problems will give rise to systems of equations that is, a pair of equations like this: 2x+3y = 10 x 6y = 5 You can solve a system of equations

### Mental Math Grade 3 MM 3-3 MM 3-1. Count by tens. What comes next? 1. Start at 30... (40) 1. Write the number that is 5 tens and 2 ones.

MM 3-1 Count by tens. What comes next? 1. Start at 30... (40) 2. Start at 60... (70) 3. Start at 20... (30) 4. What is 40 + 30 = (70) 5. What is 30 + 60 = (90) 6. What are the next 3 numbers? 2, 4, 6,

### Excel Math Placement Tests A grade-level evaluation tool

Excel Math Placement Tests A grade-level evaluation tool Attached are six tests that can be used to evaluate a student s preparedness for Excel Math. The tests are labeled A - F, which correspond to first

### Lesson 19: Comparison Shopping Unit Price and Related Measurement Conversions

Comparison Shopping Unit Price and Related Measurement Conversions Student Outcomes Students solve problems by analyzing different unit rates given in tables, equations, and graphs. Materials Matching

What five coins add up to a nickel? five pennies (1 + 1 + 1 + 1 + 1 = 5) Which is longest: a foot, a yard or an inch? a yard (3 feet = 1 yard; 12 inches = 1 foot) What do you call the answer to a multiplication

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION INTEGRATED ALGEBRA. Tuesday, January 27, 2015 1:15 to 4:15 p.m.

INTEGRATED ALGEBRA The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION INTEGRATED ALGEBRA Tuesday, January 27, 2015 1:15 to 4:15 p.m., only Student Name: School Name: The possession

### FSCJ PERT. Florida State College at Jacksonville. assessment. and Certification Centers

FSCJ Florida State College at Jacksonville Assessment and Certification Centers PERT Postsecondary Education Readiness Test Study Guide for Mathematics Note: Pages through are a basic review. Pages forward

### Math Ready Unit 1. Algebraic Expressions Student Manual

SREB Readiness Courses Transitioning to college and careers Math Ready Unit 1. Algebraic Expressions Student Manual Name 1 SREB Readiness Courses Transitioning to college and careers Math Ready. Unit 1.

### 7-2 Factoring by GCF. Warm Up Lesson Presentation Lesson Quiz. Holt McDougal Algebra 1

7-2 Factoring by GCF Warm Up Lesson Presentation Lesson Quiz Algebra 1 Warm Up Simplify. 1. 2(w + 1) 2. 3x(x 2 4) 2w + 2 3x 3 12x Find the GCF of each pair of monomials. 3. 4h 2 and 6h 2h 4. 13p and 26p

### B. Franklin, Printer. LESSON 3: Learning the Printing Trade

B. Franklin, Printer Elementary School (Grades K-2) LESSON 3: Learning the Printing Trade OVERVIEW Benjamin Franklin was born January 17, 1706, to Josiah and Abiah Franklin. He was the ninth of eleven

### DISTRICT Math Olympics-MATH BY MAIL-REVISED

AREAS OF COMPUTATION 3RD GRADE + - x whole numbers 4TH GRADE + - x whole numbers +- fractions +- decimals Express remainders as fractions (lowest simplest terms), or r. 5TH GRADE + - x whole numbers +

### Lesson 9: Radicals and Conjugates

Student Outcomes Students understand that the sum of two square roots (or two cube roots) is not equal to the square root (or cube root) of their sum. Students convert expressions to simplest radical form.

### Printing Letters Correctly

Printing Letters Correctly The ball and stick method of teaching beginners to print has been proven to be the best. Letters formed this way are easier for small children to print, and this print is similar

### APPLICATION REVIEW. 5. If a board is (2x + 6) feet long and a piece (x 5) feet long is cut off, what is the length of the remaining piece?

APPLICATION REVIEW 1. Find the cost of the carpet needed to cover the floor of a room that is 14 yards long and 10 yards wide if the carpet costs \$13.00 per square yard. 2. A can of soup is in the shape

### To Use: You can find more printables online at

Included in this download are printable a-z flashcards. Included are cards containing manuscript, numbers, and cursive fonts. There are smaller 2x3 cards and larger 4x6 cards to hang on the wall in your

### Lesson 1: Introduction to Equations

Solving Equations Lesson 1: Introduction to Equations Solving Equations Mentally Key to Solving Equations Solving Equations Lesson 2: Solving Addition Equations Golden Rule for Solving Equations: In order

### DIAMOND PROBLEMS 1.1.1

DIAMOND PROBLEMS 1.1.1 In every Diamond Problem, the product of the two side numbers (left and right) is the top number and their sum is the bottom number. Diamond Problems are an excellent way of practicing

### Basic Pre Algebra Intervention Program

Basic Pre Algebra Intervention Program This 9 lesson Intervention Plan is designed to provide extra practice lessons and activities for students in Pre Algebra. The skills covered are basics that must

### ALGEBRA I (Common Core) Thursday, January 28, 2016 1:15 to 4:15 p.m., only

ALGEBRA I (COMMON CORE) The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA I (Common Core) Thursday, January 28, 2016 1:15 to 4:15 p.m., only Student Name: School Name: The

### Use your TI-84 graphing calculator to check as many problems as possible.

Name: Date: Period: Dear Future Algebra Honors student, We hope that you enjoy your summer vacation to the fullest. We look forward to working with you next year. As you enter your new math class, you

### ALGEBRA I (Common Core) Monday, January 26, 2015 1:15 to 4:15 p.m., only

ALGEBRA I (COMMON CORE) The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA I (Common Core) Monday, January 26, 2015 1:15 to 4:15 p.m., only Student Name: School Name: The possession

### Factoring, Solving. Equations, and Problem Solving REVISED PAGES

05-W4801-AM1.qxd 8/19/08 8:45 PM Page 241 Factoring, Solving Equations, and Problem Solving 5 5.1 Factoring by Using the Distributive Property 5.2 Factoring the Difference of Two Squares 5.3 Factoring

### Algebra EOC Practice Test #4

Class: Date: Algebra EOC Practice Test #4 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. For f(x) = 3x + 4, find f(2) and find x such that f(x) = 17.

### Lesson 19: Unknown Area Problems on the Coordinate Plane

Unknown Area Problems on the Coordinate Plane Student Outcomes Students find the areas of triangles and simple polygonal regions in the coordinate plane with vertices at grid points by composing into rectangles

### Apply and extend previous understandings of arithmetic to algebraic expressions. Draft

6th Grade - Expressions and Equations 1 Standard: 6.EE.1 No Calculator: Write and evaluate numerical expressions involving whole-number exponents The population of fruit flies increases by a factor of

### Mathematics Common Core Sample Questions

New York State Testing Program Mathematics Common Core Sample Questions Grade7 The materials contained herein are intended for use by New York State teachers. Permission is hereby granted to teachers and

### Score 4.0. The student: Score 3.0

Strand: 5.NS.1 (Decimal Place Value) Topic: Use a number line to compare and order decimals In addition to, in-depth inferences and applications that go beyond what was taught. 3.5 In addition to score

### Principles of Mathematics MPM1D

Principles of Mathematics MPM1D Grade 9 Academic Mathematics Version A MPM1D Principles of Mathematics Introduction Grade 9 Mathematics (Academic) Welcome to the Grade 9 Principals of Mathematics, MPM

### STUDENT NAME: GRADE 8 MATHEMATICS

STUDENT NAME: GRADE 8 MATHEMATICS Administered October 2009 Name: Cla ss: 8th Grade TAKS Practice Test 1 1 A librarian arranged some books on the shelf using the Dewey decimal system. Choose the group

### Lesson 1: Positive and Negative Numbers on the Number Line Opposite Direction and Value

Positive and Negative Numbers on the Number Line Opposite Direction and Value Student Outcomes Students extend their understanding of the number line, which includes zero and numbers to the right or above

### Simplifying Expressions

UNIT 1 Math 621 Simplifying Expressions Description: This unit focuses on using quantities to model and analyze situations, interpret expressions, and create expressions to describe situations. Students

### Algebra Word Problems

WORKPLACE LINK: Nancy works at a clothing store. A customer wants to know the original price of a pair of slacks that are now on sale for 40% off. The sale price is \$6.50. Nancy knows that 40% of the original

### bello words how to build them? tt tt

99 practical stuff Reproducing and using this material in any form without a written permission from Underware is probhibited. p 1/8 bello words how to build them? tt tt tt 11 Rep input the same character

### The Distributive Property

The Distributive Property Objectives To recognize the general patterns used to write the distributive property; and to mentally compute products using distributive strategies. www.everydaymathonline.com

### 4. What could be the rule for the pattern in the table? n 1 2 3 4 5 Rule 3 5 7 9 11

5 th Grade Practice Test Objective 1.1 1. John has two fewer marbles than Kay. If Kay has marbles, how many marbles does John have? 2 2 2 2 2. What is if + 17 = 26? 43 19 11 9 3. ll the cakes at the bake

### ALGEBRA I (Common Core) Wednesday, August 12, :30 to 11:30 a.m., only

ALGEBRA I (COMMON CORE) The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA I (Common Core) Wednesday, August 12, 2015 8:30 to 11:30 a.m., only Student Name: School Name: The

### TEKS TAKS 2010 STAAR RELEASED ITEM STAAR MODIFIED RELEASED ITEM

7 th Grade Math TAKS-STAAR-STAAR-M Comparison Spacing has been deleted and graphics minimized to fit table. (1) Number, operation, and quantitative reasoning. The student represents and uses numbers in

### Algebra 1 Unit Students will be able to solve word problems by substituting data into a given equation. Worksheet

Algebra 1 Unit 2 1. Students will be able to solve two-step equations. (Section 2.1) Worksheet 1 1-24 Page 72 1 54 (optional) 2. Students will be able to solve word problems by substituting data into a

### Math and FUNDRAISING. Ex. 73, p. 111 1.3 0. 7

Standards Preparation Connect 2.7 KEY VOCABULARY leading digit compatible numbers For an interactive example of multiplying decimals go to classzone.com. Multiplying and Dividing Decimals Gr. 5 NS 2.1

### Algebra: Equations and formulae

Algebra: Equations and formulae. Basic algebra HOMEWORK A 6 7 8 0 FM AU Find the value of x + when a x = b x = 6 c x = Find the value of k when a k = b k = c k = 0 Find the value of + t when a t = b t

### Mean, Median, Mode, and Range

Mean, Median, Mode, and Range Practice to review I can find the mean, median, mode, and range of a set of data! HW 14.3A Sue rode her bike each afternoon after school. One week she recorded the number

### PowerScore Test Preparation (800) 545-1750

Question 1 Test 1, Second QR Section (version 2) Two triangles QA: x QB: y Geometry: Triangles Answer: Quantity A is greater 1. The astute student might recognize the 0:60:90 and 45:45:90 triangle right

### Algebra I Module 1 Lessons 1 28

Eureka Math 2015 2016 Algebra I Module 1 Lessons 1 28 Eureka Math, Published by the non-profit Great Minds. Copyright 2015 Great Minds. No part of this work may be reproduced, distributed, modified, sold,

### Possible Stage Two Mathematics Test Topics

Possible Stage Two Mathematics Test Topics The Stage Two Mathematics Test questions are designed to be answerable by a good problem-solver with a strong mathematics background. It is based mainly on material

### Sect 6.7 - Solving Equations Using the Zero Product Rule

Sect 6.7 - Solving Equations Using the Zero Product Rule 116 Concept #1: Definition of a Quadratic Equation A quadratic equation is an equation that can be written in the form ax 2 + bx + c = 0 (referred

### ( ) ( ) Math 0310 Final Exam Review. # Problem Section Answer. 1. Factor completely: 2. 2. Factor completely: 3. Factor completely:

Math 00 Final Eam Review # Problem Section Answer. Factor completely: 6y+. ( y+ ). Factor completely: y+ + y+ ( ) ( ). ( + )( y+ ). Factor completely: a b 6ay + by. ( a b)( y). Factor completely: 6. (

### Unit 10: Quadratic Equations Chapter Test

Unit 10: Quadratic Equations Chapter Test Part 1: Multiple Choice. Circle the correct answer. 1. The area of a square is 169 cm 2. What is the length of one side of the square? A. 84.5 cm C. 42.25 cm B.

### 2-2 Solving One-Step Equations. Solve each equation. Check your solution. g + 5 = 33. Check: 104 = y. Check: Check: Page 1

Solve each equation. Check your solution. g + 5 = 33 104 = y 67 4 +Manual t = 7- Powered by Cognero esolutions Page 1 4+t= 7 a + 26 = 35 6 + c = 32 1.5 = y ( 5.6) Page 2 1.5 = y ( 5.6) Page 3 Page 4 Page

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION MATHEMATICS A. Monday, January 26, 2004 1:15 to 4:15 p.m.

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION MATHEMATICS A Monday, January 26, 2004 1:15 to 4:15 p.m., only Print Your Name: Print Your School s Name: Print your name and the

### NMC Sample Problems: Grade 5

NMC Sample Problems: Grade 5 1. What is the value of 5 6 3 4 + 2 3 1 2? 1 3 (a) (b) (c) (d) 1 5 1 4 4 4 12 12 2. What is the value of 23, 456 + 15, 743 3, 894 expressed to the nearest thousand? (a) 34,

### Algebra 1 End-of-Course Exam Practice Test with Solutions

Algebra 1 End-of-Course Exam Practice Test with Solutions For Multiple Choice Items, circle the correct response. For Fill-in Response Items, write your answer in the box provided, placing one digit in

### 4.5 Some Applications of Algebraic Equations

4.5 Some Applications of Algebraic Equations One of the primary uses of equations in algebra is to model and solve application problems. In fact, much of the remainder of this book is based on the application

### IOWA End-of-Course Assessment Programs. Released Items ALGEBRA I. Copyright 2010 by The University of Iowa.

IOWA End-of-Course Assessment Programs Released Items Copyright 2010 by The University of Iowa. ALGEBRA I 1 Sally works as a car salesperson and earns a monthly salary of \$2,000. She also earns \$500 for

### Eureka Lessons for 7th Grade Unit ONE ~ ADD/SUBTRACT Rational Numbers Concept 1-1

Eureka Lessons for 7th Grade Unit ONE ~ ADD/SUBTRACT Rational Numbers Concept 1-1 LESSONS 7, 8, & 9 are the FOURTH set of Eureka Lessons for Unit 1. Also Includes four fluency drills, 2 in lesson 8 and

### Copperplate Victorian Handwriting. Victorian. Exploring your History. Created by Causeway Museum Service

Victorian Coleraine Exploring your History Copperplate Victorian Handwriting Postcards courtesy of Coleraine Museum Collection Created by Causeway Museum Service In Victorian times hand writing was very

### 2.4 Two-Step Equations

www.ck12.org Chapter 2. Expressions and Equations 2.4 Two-Step Equations Learning Objectives Solve a two-step equation using addition, subtraction, multiplication, and division. Solve a two-step equation

### SAINT JOHN PAUL II CATHOLIC ACADEMY. Entering Grade 6 Summer Math

SAINT JOHN PAUL II CATHOLIC ACADEMY Summer Math 2016 In Grade 5 You Learned To: Operations and Algebraic Thinking Write and interpret numerical expressions. Analyze patterns and relationships. Number and

### Lesson 24: Surface Area

Student Outcomes Students determine the surface area of three-dimensional figures, those that are composite figures and those that have missing sections. Lesson Notes This lesson is a continuation of Lesson

### Algebra 1. Practice Workbook with Examples. McDougal Littell. Concepts and Skills

McDougal Littell Algebra 1 Concepts and Skills Larson Boswell Kanold Stiff Practice Workbook with Examples The Practice Workbook provides additional practice with worked-out examples for every lesson.

### 2.5. section. tip. study. Writing Algebraic Expressions

94 (2-30) Chapter 2 Linear Equations and Inequalities in One Variable b) Use the formula to predict the global investment in 2001. \$197.5 billion c) Find the year in which the global investment will reach

### Grade 5 Module 2 Lessons 1 29

Eureka Math Grade 5 Module 2 Lessons 29 Eureka Math, Published by the non-profit Great Minds. Copyright Great Minds. No part of this work may be reproduced, distributed, modified, sold, or commercialized,

### Example 1: If the sum of seven and a number is multiplied by four, the result is 76. Find the number.

EXERCISE SET 2.3 DUE DATE: STUDENT INSTRUCTOR: 2.3 MORE APPLICATIONS OF LINEAR EQUATIONS Here are a couple of reminders you may need for this section: perimeter is the distance around the outside of the

### Summer Math Exercises. For students who are entering. Pre-Calculus

Summer Math Eercises For students who are entering Pre-Calculus It has been discovered that idle students lose learning over the summer months. To help you succeed net fall and perhaps to help you learn

### Integrated Algebra Ratios, Proportions and Rates \$ =

% ½ ⅘ Integrated Algebra Ratios, Proportions and Rates \$ = ⅓ Ratio A ratio is a comparison of two quantities. Ex: A teacher graded 180 bonus quizzes during the school year. The number of quizzes receiving

### American Diploma Project

Student Name: American Diploma Project ALGEBRA l End-of-Course Eam PRACTICE TEST General Directions Today you will be taking an ADP Algebra I End-of-Course Practice Test. To complete this test, you will

### SUMMER MATH PACKET 2016 FOR RISING EIGHTH GRADERS WHO WERE IN PRE-ALGEBRA

SUMMER MATH PACKET 2016 FOR RISING EIGHTH GRADERS WHO WERE IN PRE-ALGEBRA 1 DIRECTIONS FOR COMPLETING THE PACKET The following page contains the list of help topics available on the Glencoe website. Each

### 5.1 Introduction to Rational Expressions. Technique: 1. Reduce: 2. Simplify: 6. Simplify:

5.1 Introduction to Rational Expressions The Technique: 1. Reduce: 2. Simplify: When simplifying rational expressions, you are actually dividing both the numerator and the denominaror by common factors.

### Georgia Online Formative Assessment Resource (GOFAR) 7th Grade VMath Algebra Review

7th Grade VMath Algebra Review Name: Date: Copyright 2014 by Georgia Department of Education. Items shall not be used in a third party system or displayed publicly. Page: (1 of 17 ) 1. If =, what is the

### Applications of Algebraic Fractions

5.6 Applications of Algebraic Fractions 5.6 OBJECTIVES. Solve a word problem that leads to a fractional equation 2. Apply proportions to the solution of a word problem Many word problems will lead to fractional

### MTH 092 College Algebra Essex County College Division of Mathematics Sample Review Questions 1 Created January 17, 2006

MTH 092 College Algebra Essex County College Division of Mathematics Sample Review Questions Created January 7, 2006 Math 092, Elementary Algebra, covers the mathematical content listed below. In order

### NMC Sample Problems: Grade 5

NMC Sample Problems: Grade 5 1. Wonjung is riding a bike at the speed of 300 m/min. This speed equals to cm/sec. What number should be in the box? (a) 50 (b) 300 (c) 500 (d) 3000 5000 2. Which of the following

### Grade 4. Operations and Algebraic Thinking 4.OA COMMON CORE STATE STANDARDS ALIGNED MODULES 2012 COMMON CORE STATE STANDARDS ALIGNED MODULES

THE NEWARK PUBLIC SCHOOLS THE OFFICE OF MATHEMATICS Grade 4 Operations and Algebraic Thinking 4.OA.1-2 2012 COMMON CORE STATE STANDARDS ALIGNED MODULES 2012 COMMON CORE STATE STANDARDS ALIGNED MODULES

### 2.3 Solving Equations - Mathematical & Word

2.3 Solving Equations - Mathematical & Word Problems 7.EE.4a Students will be able to use properties of numbers to correctly solve mathematical and word problems in one variable using fraction bars, word

### Lesson 1: Posing Statistical Questions

Student Outcomes Students distinguish between statistical questions and those that are not statistical. Students formulate a statistical question and explain what data could be collected to answer the

### Solving Systems of Equations Algebraically Examples

Solving Systems of Equations Algebraically Examples 1. Graphing a system of equations is a good way to determine their solution if the intersection is an integer. However, if the solution is not an integer,

### 12. [Place Value] A Q. What is the largest odd, 4 digit number, that contains the digits 0, 4, 5 and 7?

12. [Place Value] Skill 12.1 Understanding the place value of a digit in a number (1). When writing numbers the following is true: Each digit in a number occupies a special place or column. Larger numbers

### Math BINGO MOST POPULAR. Do you have the lucky card? B I N G O

MOST POPULAR Math BINGO Do you have the lucky card? Your club members will love this MATHCOUNTS reboot of a classic game. With the perfect mix of luck and skill, this is a game that can be enjoyed by students

### Solving Equations With Fractional Coefficients

Solving Equations With Fractional Coefficients Some equations include a variable with a fractional coefficient. Solve this kind of equation by multiplying both sides of the equation by the reciprocal of

### Add Decimal Numbers LESSON 4

LESSON 4 Add Decimal Numbers In this lesson, we will begin using the algebra-decimal inserts to represent decimals. Turn a red hundred square upside down so the hollow side is showing, and snap the flat

### Student Outcomes. Lesson Notes. Classwork. Exercises 1 3 (4 minutes)

Student Outcomes Students give an informal derivation of the relationship between the circumference and area of a circle. Students know the formula for the area of a circle and use it to solve problems.

### ALGEBRA I (Common Core) Wednesday, June 17, :15 to 4:15 p.m., only

ALGEBRA I (COMMON CORE) The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA I (Common Core) Wednesday, June 17, 2015 1:15 to 4:15 p.m., only Student Name: School Name: The possession

### MATH COMPUTATION. Part 1. TIME : 15 Minutes

MATH COMPUTATION Part 1 TIME : 15 Minutes This is a practice test - the results are not valid for certificate requirements. A calculator may not be used for this test. MATH COMPUTATION 1. 182 7 = A. 20

### Contents. 3 Introduction. 4 Basic Operations. 6 Order of Operations. 8 Basic operations with fractions. 16 Introduction to decimals

Judyth Hayne 0 Contents Introduction Basic Operations 6 Order of Operations 8 Basic operations with fractions 6 Introduction to decimals 0 Basic operations with decimals Significant figures and rounding

### USING THE PROPERTIES

8 (-8) Chapter The Real Numbers 6. (xy) (x)y Associative property of multiplication 66. (x ) x Distributive property 67. 4(0.) Multiplicative inverse property 68. 0. 9 9 0. Commutative property of addition

### The graphs coincide. Therefore, the trinomial has been factored correctly. The factors are (x + 2)(x + 12).

8-6 Solving x^ + bx + c = 0 Factor each polynomial. Confirm your answers using a graphing calculator. 1. x + 14x + 4 In this trinomial, b = 14 and c = 4, so m + p is positive and mp is positive. Therefore,

### Benchmark Assessment Test 2005 Math Grade 5 Test 1 Interpretive Guide

Benchmark Assessment Test 2005 Math Grade 5 Test 1 Interpretive Guide STRAND A MAA122 Compare relative size and order of whole numbers, fractions, decimals and percents 1 2 Comparing common fractions with

### PERCENT PROBLEMS USING DIAGRAMS 5.1.1 and 5.1.2

PERCENT PROBLEMS USING DIAGRAMS 5.1.1 and 5.1.2 A variety of percent problems described in words involve the relationship between the percent, the part and the whole. When this is represented using a number

### Grade 8 Linear Equations in One Variable 8.EE.7a-b

THE NEWARK PUBLIC SCHOOLS THE OFFICE OF MATHEMATICS Grade 8 Linear Equations in One Variable 8.EE.7a-b 2012 COMMON CORE STATE STANDARDS ALIGNED MODULES THE NEWARK PUBLIC SCHOOLS Office of Mathematics MATH

### SPECIAL PRODUCTS AND FACTORS

CHAPTER 442 11 CHAPTER TABLE OF CONTENTS 11-1 Factors and Factoring 11-2 Common Monomial Factors 11-3 The Square of a Monomial 11-4 Multiplying the Sum and the Difference of Two Terms 11-5 Factoring the

### Number Models for Area

Number Models for Area Objectives To guide children as they develop the concept of area by measuring with identical squares; and to demonstrate how to calculate the area of rectangles using number models.