SOLVING EQUATIONS. Judo Math Inc.

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1 SOLVING EQUATIONS Judo Math Inc.

2 6 th grade Problem Solving Discipline: Yellow Belt Training Order of Mastery: Solving Equations 1. From expressions to equations 2. What is an equation? (Answering a question) 3. Solve equations using substitution 4. Looking for structure in equations 5. Intro to inequality Yellow Belt Training Equations Look at the words: Equation Equals Notice anything? Hopefully you noticed that they have the same 4 letters to start the words! This is often a good indication that the words have something to do with each other In this case, in order for an equation to be an equation, it has to contain an equals sign! An equation could be as simple as: Or as complex as X+2=4 In this belt you re going to practice writing equations and even doing a little bit of solving! These are some of your first steps into the world of algebra. I hope you enjoy it! Good Luck Grasshopper. Standards Included: 6.EE.5 Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. 6.EE.6 Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. Judo Math Inc.

3 1. From expressions to equations Write an expression that represents the following statement. Then ask a question about the statement that turns the expression into an equation. 1. Kendra picked 91 pears. D andre picked p fewer pears than Kendra. Expression: 91-p Question: How many fewer pears did D andre pick if he picked a total of 65 pears? Equation: 91-p=65 2. Alana has 46 DVD s. Her mother bought her d more DVD s. 3. Cassandra earned b bonus points. Frank earned 10 more bonus points than Cassandra. 4. Reid has 13 trading cards. Violet has t more trading cards than Reid. 1

4 5. Lucy has r red peppers and 31 green peppers. 6. Gwendolyn has b books. Holly has 56 more books than Gwendolyn. 7. Russ has 57 buttons. He gives away b buttons. 8. Connor s dog weighs 85 pounds, and Isabella s dog weights p pounds. 9. Mrs. Strong ran x miles last week. Mr. Strong ran 25 miles more than she did. 2

5 10. Fiona has m dollars. Since she is such great friends with Sierra, Tannia, and Juanita she decided to divide her money up amongst each of them. 11. James is 9 years older than his sister. 12. Paul has x dollars which is 10 times as much as his best friend Sally. 13. Hannah had $25. Over the weekend she bought some clothes that cost n dollars. How much money does she have left? 3

6 2. Definition of equation Based on your work above and what you read on the first page of this packet, write your own definition of an equation here: Now, trade your definition with another person. Edit each other s definitions and try to find examples of places where their definition breaks. Then write your definition on the board in your classroom and your teacher will work with everyone to make a class definition of equations. Write your class definition here: Important to remember: Every time you see a variable in an equation has the same value! 5N+2N=12+2N 5M+2M=14+2M 5a-2=7+9a But in different equations, the same variable could represent different things. 4

7 Equation A statement that the values of two mathematical expressions are equal This is indicated by the sign = Use the definition of equation above and the one that you generated with the class to check to see if the given values are solutions for the equations. M+12=20 Is M=12 a solution? How do you know? N+7=14 Is N=7 a solution? How do you know? 2M+1=3 Is M= -1 a solution? How do you know? 5

8 J+1=4J Is J=4 a solution? How do you know? -4x+1=1 Is x=0 a solution? How do you know? -5x-1=5 Is there an integer that is a solution to this problem? Explain why or why not. 6

9 3. Solve equations with substitution In the last problems in part 2, you were most likely substituting the values that I gave you in for the variables to see if each side of the equation equaled the other, right? In this section, we are going to continue this strategy for solving equations only I m not going to give you any ideas this time. You will have to guess and check. First, what does guess and check mean? Write your own definition below For the following problems, guess and check solutions for each variable until you come upon one that works. 7

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11 In the last 24 problems, you were guessing and checking for values that would make the equation true. What was the most number of guesses you had to make before you got one right? What was the least number of guesses you had to make before getting one right? Did you notice any patterns or shortcuts that might be easier than guessing and checking for solving equations like this? Discuss your answer to this question with other classmates or in a classroom discussion with your teacher. 9

12 4. Looking for structure in equations Find the value of x that would make this equation true. Explain in writing how you know your solution is correct. 1) 4x+3x=3x+20 2) 7y-3y=14-3y 3) 5N+2N=12+2N 4) 20+3M=3M+20 10

13 5) 40+5z=5z+20 (be careful in this one!) For the following scenarios, explain how the equation relates to the problem? 6) How many 44 cent stamps can you buy with $11? Equation: 0.44n=11 a) What does the n represent in this situation? b) How does this table help you to solve the problem? n n ) Jay bought three packs of balloons. He opened them and counted 48 balloons. How many balloons are in a pack. Equation: 3b=48 a) What does b represent in this situation? b) Would a table like the one above come in handy for this situation? Make one below if so. 11

14 8) You spent $140 for you and 5 of your friends siblings to go to the fair. How much did it cost per person? a) Equation: b) What does your variable represent in this situation? c) Would a table like the one above come in handy for this situation? Create one here: 12

15 5. Intro to inequality So far we have done a lot of talking about equations. In equations, two expressions are EQUAL to each other and by now you are probably becoming quite the expert on working with them! Good job! Now we are going to take a little sidetrack and in this section, instead of talking about things that are equal, we are going to talk about things that AREN T equal! Example: You have to be MORE THAN 42 tall to ride a roller coaster. For this scenario, you don t have to be h=42, instead you have to be h 42, or anything equal to or taller than 42. Inequalities use the following symbols (Under each symbol, write anything you already know about that symbol) < > Now write the following inequalities in words (you don t have to do any solving) 1) X<5 2) 3x 12 3) 2N 7+N 13

16 Equation or Inequality? For each of the scenarios below, explain whether it would be modeled by an equation or inequality. Then try to write an equation or inequality to model it! 4) A number increased by nine is 20. 5) Twice a number is less than eighteen. 6) Four less than a number is 16. 7) A number divided by six is greater than or equal to 10. 8) You have to be shorter than 40 to go in the kiddie jump house at Sea World 9) Jessie needs to set the freezer temperature so the ice cubes do not melt. What temperature should the freezer be set at? 14

17 10) To ride the Manta roller coaster you must be at least 54 inches. Write an inequality to show how tall can you be to ride the roller coaster? 11) Write a scenario that could be modeled by each of these inequalities. a. x > -5 b. 7 < x c. 12 x d. x -2 15

18 12) The city s budget to pay firefighter's wages and benefits is $600,000. If wages are calculated at $40,000 per firefighter and benefits at $20,000 per firefighter, write an equation or inequality whose solution is the number of firefighters the town can employ. Solve the equation or inequality by substituting or looking for patterns 13) Final Inequality Challenge: The FAIR has a log ride that can hold 12 people. They also have a weight limit of 1500 lbs per log for safety reasons. If the average adult weighs 150 lbs, the average child weighs 80 lbs and the log itself weights 200. Write an inequality for this situation and see how many solutions you can find for children and adults on the ride! 16