Patterns and RELATIONS Solving Equations Chapter 2 Big Idea Developing and solving equations can help me solve problems. Learning Goals I can use words to show number relationships. I can use equations to show number relationships. I can use objects and drawings to show and explain preservation of equality. I can use symbols to show and explain preservation of equality. I can write and solve an equation using a letter variable to represent a problem. I can write a word problem for a given equation. Essential Question How can equations help me understand the world around me? Important Words equal equivalent equation formula preserve equality solution unknown variable
Simple Symbols Use the =,, <, and > symbols to compare expressions. Use numbers, objects, and pictures to create many ways of showing the same number. Example: Compare the expressions 2 + 3 and 2 3 2 + 3 2 x 3 Two plus 3 is not equal to 2 times 3 because 2 + 3 = 5 and 2 x 3 = 6 and 5 is not the same as 6. 5 6 Five is less than 6. 5 < 6 An expression that is equal to 2 x 3 is 2 + 4 because 2 + 4 = 6 2 + 4 = 2 x 3 = Another expression that is equal to 2 x 3 is 11-5 because 11-5 = 6 11-5 = 2 x 3 = 11. Explain whether each of the following is an example of equal (=) or not equal ( ). Equal means the same or equivalent. a. b. 22 CHAPTER 2: Solving Equations
Simple Symbols (continued) c. d. 12. Compare the two sides of each balance using the words less than or greater than. 1 a. b. c. d. CHAPTER 2: Solving Equations 23
Simple Symbols (continued) 13. Muna and her friends are pooling their money to buy a package of blank CDs for $9.87. Each friend agreed to pay $3.29. Each friend used a different combination of coins. What combinations might they have used? 14. Create as many different numerical representations for the number 20 as you can. 15. Use pennies (1 ), nickels (5 ), dimes (10 ), and quarters (25 ) to answer the following questions. a. Use some of the coins to show one dollar. Record your combination using pictures or symbols. b. Use different coins to show one dollar. Record your combination using pictures or symbols. c. Use a third combination of coins to show one dollar. Record your combination using pictures or symbols. d. Describe how these combinations are similar to, and different from, each other. e. Use two of the combinations to write an equation. Record your equation using pictures or symbols. f. Explain how two different combinations of coins can be equal. An equation is a number sentence with an equal sign. = 24 CHAPTER 2: Solving Equations
Simple Symbols (continued) 16. Copy and complete each statement. Use an equal symbol, =, or a not equal symbol,, to make the statement true. a. 2 + 3 + 5 5 + 3 + 2 c. 3 5 5 3 e. 2 4 4 2 g. 15 3 3 15 b. 15 3 5 d. 16 4 3 + 4 f. 12 4 2 4 h. 10 + 8 20 3 17. Look at each scale. Explain why the pans will not balance. Write a statement using a less than symbol, <, or a greater than symbol, >, for each scale. a. b. c. 12 4 40 d. 45 9 3 5 e. 15 15 6 f. 8 5 15 8. How can a two pan balance show an equation? 9. Write a definition for equal using words, numbers, and pictures. Compare definitions with a partner. I can use words to show number relationships. I can use equations to show number relationships. CHAPTER 2: Solving Equations 25
Equal All the Time Use what you know about preserving equality to correct errors and preserve equality in given equations. Example: Quinn completed this work in her math book, but she knew it was wrong. 3n 6 = 24 + 6 6 3n = 18 Help Quinn find her error and explain how to correct it. She added on one side and subtracted on the other. She should have added six on both sides of the equation. 3n 6 = 24 + 6 + 6 3n = 30 If this were a balance and she added 6 weights to the right side, it would tip, so she would have to add six weights to the other side too, to keep the equation balanced. 11. Four students were asked to change the equation x = 6 into a new equation. Their work is shown below. Anna Belinda Carissa Danika a. x = 6 x 2 x 2 2x = 12 b. x = 6 +2 +2 2x = 8 c. x = 6 + 4 + 4 x + 4 = 10 d. x = 6 x 2 -. 2 2x = 3 Explain which students made equivalent equations and preserved equality in their equations. Explain which students made errors and did not preserve equality in their equations. Equivalent means equal. 26 CHAPTER 2: Solving Equations
Equal All the Time (continued) 12. Four students were solving equations and made errors in their work. Correct the errors in each students work. Albert Bertrand Carlos Decklan a. 2x = 6 -. 2 x 2 x = 12 b. x + 3 = 6 + 3 + 3 x = 9 c. x 4 = 6-4 - 4 x = 2 d. x 5 = 6 + 5 x - 5 = 1 1 13. Preserve equality by performing the given operation on each side of the equation. Write each new equation. You preserve equality when you do the same thing on each side of an equation to keep it equal. a. 2x + 2 18 add 12 to each side x b. 20 3 + 13 subtract 3 from each side c. 12 x 4 multiply each side by 4 d. 15x 60 divide each side by 5 CHAPTER 2: Solving Equations 27
Equal All the Time (continued) 14. Some work has already been done on each of the following equations, but now they are not balanced. Perform the same operation on the other side of the equation and write the new, balanced, equation. a. b. 8 r 5 8 r c. d. q 9 23 q 23 e. 60 12(5) f. p + 2 11 60 5 p 11 28 CHAPTER 2: Solving Equations
Equal All the Time (continued) 15. Draw a picture or build a model to show the equation, then make an equivalent equation. a. q b. r r c. s s 16. Use the operation of your choice to make a new equation while preserving equality. a. g + 18 = 36 c. 75 = h 20 e. 32 = k 8 b. m 3 = 13 d. 4n 9 = 31 f. 15 = (p 4) + 5 7. Why is preserving equality important? 8. What strategies did you use to make sure you preserve equality? I can use equations to show number relationships. I can use objects and drawings to show and explain preservation of equality. I can use symbols to show and explain preservation of equality. CHAPTER 2: Solving Equations 29
Beautiful Balances Solve equations and make conclusions about equations on a two pan balance. Example: The equation 15 = n + 4 is shown on the two pan balance below. n Explain how the weights on this two pan balance show the equation. The ten weight and the five one weights on the left-hand side add up to 15. The four one weights on the right hand side show the 4 and the bag labeled n shows the variable. The two sides are balanced because each side is equal to the other. What is the value of n? Because the two sides are equal and because there are 15 weights on the left side, the right side must equal 15 as well. That means the number of weights in the bag must add to four to make 15. 4 + 11 = 15. There must be 11 weights in the bag. 11. Explain which shape weighs the most. a. b. 30 CHAPTER 2: Solving Equations
Beautiful Balances (continued) c. d. and 12. Explain which shapes will balance two spheres. a. b. c. and CHAPTER 2: Solving Equations 31
Beautiful Balances (continued) 13. Find the unknown mass on each of the following balances. a. b. c. d. e. f. 32 CHAPTER 2: Solving Equations
Beautiful Balances (continued) 14. Solve for the unknown in each equation. An unknown is a letter or a symbol that stands for a number you don t know. a. z + 3 35 b. 5x 80 c. 21 y 14 d. w 4 6 5. How can you show an equation on a two pan balance? 6. Explain how to preserve equality when solving an equation. 7. How can using a two pan balance help you preserve equality? I can use equations to show number relationships. I can use objects and drawings to show and explain preservation of equality. I can use symbols to show and explain preservation of equality. CHAPTER 2: Solving Equations 33
Rank and Solve Use what you know about solving equations to rank the equations by how much effort they will take to solve. Example: Solve the equation 23 = p + 15 using the strategy of your choice. Oliver s strategy: = p I started with 2 rods on the left and 1 rod on the right. Then I crossed off 1 rod on each side. = p I redrew the rod as 10 units, then I crossed off 5 units on each side. = p p must equal 8. I can check by adding 8 + 15 = 23. My solution is correct. I was left with 8 units on the left. Paige s strategy: 1 1 1 10 10 1 1 10 1 1 1 I ll start with 23 on one side and 15 on the other side. 1 1 1 1 1 1 1 10 10 10 1 3 5 I ll add weights to the right side until it s balanced. I added 8 weights. p must be 8. Parker s strategy: 23 = p + 15-15 - 15 8 = p I ll subtract 15 from each side of the equation. p must be 8. That makes sense because 8 + 15 = 23. 34 CHAPTER 2: Solving Equations
Rank and Solve (continued) 11. Rank the following equations on a scale of 1 to 5, where 1 is an equation that takes very little effort to solve and 5 is an equation that takes a great deal of effort to solve. a. a + 5 = 17 b. d 418 = 2223 c. 8 f = 8 d. 10 = m 10 e. 1063 = 642 + c f. 80 + 6 = e 14 g. 120 = 12h k h. = 6 20 i. 6 + b = 8 + 4 j. 3 p = 37 + 8 k. 25 10 = 50n l. 10 + 90 = m. r 40 = 5 5 n. g 7 = 56 o. j 5 = 40 q 10 The solution is the answer to a problem. 12. Solve three of the equations you ranked as 1 or 2. Check your solutions. 13. Show a solution for two of the equations you ranked as 3. Check your solutions. 14. Find a solution for two of the equations you ranked as 4 or 5. Check your solutions. 5. Did the amount of effort required to solve the equations match your predictions? 6. What strategies did you use to solve each type of equation? I can use equations to show number relationships. I can use objects and drawings to show and explain preservation of equality. I can use symbols to show and explain preservation of equality. I can write and solve an equation using a letter variable to represent a problem. CHAPTER 2: Solving Equations 35
Picture it Solved Write an equation and then use a pictorial or concrete representation to solve the equation and preserve equality. Use your solution to answer the question. Example: Lachlan visited the aquarium and saw octopuses that each had eight tentacles. He saw 24 tentacles in all. How many octopuses did Lachlan see? I can write an equation for this problem where x is the number of octopuses. 24 = 8x x x x x x x x x I moved the base ten blocks around until I had eight equal groups. x x x x x x x x = x There must be three octopuses. I can check by multiplying. 3 x 8 = 24 36 CHAPTER 2: Solving Equations
Picture it Solved (continued) 11. Solve the following equations by drawing or building. Show how you could check each answer. a. 9 + r = 13 c. a 4 = 15 b. k 4 = 20 d. p 8 = 12 12. To solve the following problems, write an equation and tell what your variable represents. Then draw or build to solve the equation. Show how you could check each answer. A variable is a letter or a symbol that stands for a number you don t know. a. Maddox collects trains. He has eight more trains this year than he had last year. This year he has a total of 20 trains. How many trains did Maddox have last year? b. Nadia started a game with 20 cards. At the end of the game she had five cards. How many cards did Nadia play during the game? c. Cookies come in a box of 30. The cookies in the box are in rows of ten. How many rows of cookies are in each box? d. Nash divided his baseball cards into three books. Each book holds 45 baseball cards. How many baseball cards does Nash own? e. Ocean chose a secret number. When she divided her secret number by four and then subtracted six she was left with five. What was Ocean s secret number? 3. How are models and pictures of equations similar to equations written with numbers and symbols? 4. What strategies did you use to solve the equations when you represented them pictorially? 5. What strategies did you use to solve the equations when you represented them concretely? 6. How were these strategies the same? different? 7. What strategies did you use to check your answers? I can use objects and drawings to show and explain preservation of equality. I can write and solve an equation using a letter variable to represent a problem. CHAPTER 2: Solving Equations 37
Peculiar Properties Explore number relationships and the properties of shapes using tables and equations. Example: The triangles shown below are equilateral triangles. What do you think makes them equilateral? The sides are all equal in each triangle. These are regular triangles. Complete a table of values to show the perimeter and side length of each triangle. I measured the length of each side of each equilateral triangle and added to find the perimeter. Then I made a table. Side length Perimeter (cm) (cm) 1 3 2 6 3 9 4 12 Describe the relationship between the side length of an equilateral triangle and the perimeter of an equilateral triangle. Multiply the side length by 3. Write an equation that you could use to find the perimeter of an equilateral triangle that has sides that are 10 centimetres long. I can make an equation using p for perimeter and s for side length. P = s x 3 The perimeter of an equilateral triangle with P = (10) x 3 sides that are 10 cm long is 30 cm. P = 30 38 CHAPTER 2: Solving Equations
Peculiar Properties (continued) 11. Macey was looking for patterns in even numbers. She recorded the first four even numbers in the table of values below. Term number Even number 1 2 2 4 3 6 4 8 Macey wrote the pattern rule, Multiply the term number by two to find the even number and the formula e = 2t where e is the even number and t is the term number. a. Compare Macey s rule and formula. b. Explain why the formula for finding even numbers is e = 2t. c. Use two different strategies to find the tenth even number. d. Explain which method you prefer for finding the tenth even number. 12. Kaden was looking for patterns in odd numbers. He recorded the first four odd numbers in the table below. Term number Odd number 1 3 2 5 3 7 4 9 Kaden wrote the pattern rule, Double the term number and add one to find the odd number and the formula n = 2t + 1 where n is the odd number and t is the term number. a. How are Kaden s rule and formula the same? different? b. How are the formulas for even and odd numbers related? c. Use two different strategies to find the tenth odd number. d. Explain which method you prefer for finding the tenth odd number. CHAPTER 2: Solving Equations 39
Peculiar Properties (continued) 13. Quenton knows that there are 10 millimetres in a centimetre. He wants to write a formula to relate centimetres and millimetres. Quenton does not know whether to write 10m = c or 10c = m. a. What might m and c stand for in Quenton s equations? A formula is a rule or equation that works all the time for finding something such as area. b. Try checking Quenton s equations by using some numbers you know, such as 1 cm = 10 mm, 10 cm = 100 mm, or 3 cm = 30 mm. c. Explain which equation Quenton should use. 14. Jenny told her little brother that when you are adding numbers, the sum is always the same, even if you change the order of the numbers. Her proof was this drawing: Three yellow tennis balls and two white tennis balls add up to five tennis balls. Two yellow tennis balls and three white tennis balls add up to five tennis balls. a. Draw a picture or build a model to show another proof for Jenny s rule. b. Use numbers to write three different examples of the rule. Jenny already gave you one example (3 + 2 = 2 + 3). c. Jenny s rule is called the commutative property of addition. Use letter variables and mathematical symbols to write an equation that shows the commutative property of addition. 15. Garrett told his little brother that when you are multiplying two numbers, the product is always the same, even if you change the order of the numbers. His proof was this drawing: Two rows of five columns is ten spaces. Five rows of two columns is ten spaces. 40 CHAPTER 2: Solving Equations
Peculiar Properties (continued) a. Draw a picture or build a model to show another proof for Garrett s rule. b. Use numbers to write three different examples of the rule. Garrett already gave you one example (2 5 = 5 2). c. Garrett s rule is called the commutative property of multiplication. Use letter variables and mathematical symbols to write an equation that shows the commutative property of multiplication. 16. Isack was interested in calculating the area of a rectangle. He drew the rectangles below: A B C a. Copy and complete the table below. Record the width, length, and area of each rectangle. rectangle A B C width 2 3 4 length 3 4 area 6 b. Write a rule you can use to find the area of any rectangle. c. Draw three different rectangles on centimetre grid paper. d. Use the rule you wrote to find the area of each rectangle. e. Check the area of each rectangle using a different strategy. f. Write your rule as a formula using letter variables and mathematical symbols. g. Use the formula you wrote to find the area of each rectangle. CHAPTER 2: Solving Equations 41
Peculiar Properties (continued) 17. Laila drew the rectangles below. A B C a. Copy and complete the table below. Record the width, length, and perimeter of each rectangle. rectangle A B C width 2 2 2 length 3 4 perimeter 10 b. Write a rule you can use to find the perimeter of any rectangle. Hadi drew the rectangles below. D E F c. Create a table to record the width, length, and perimeter of each rectangle. d. Use the rule that you wrote in part b to find the perimeter of each rectangle. e. Write your rule as a formula using letter variables and mathematical symbols. f. Draw three different rectangles on centimetre grid paper. Trade rectangles with a partner. Use the formula to find the perimeter of each rectangle. Compare answers with your partner. 42 CHAPTER 2: Solving Equations
Peculiar Properties (continued) 18. Properties are relationships that are always true and that you can write a formula to represent. For example the number of hours in a certain number of days is always 24 d, where d is the number of days. a. Write a formula to represent the number of fingers on students. b. Write a formula to represent the number of centimetres in metres. c. Write two formulas of your own. Exchange with a partner and see if you can figure out what each other s formulas are for. 9. When might you use a formula instead of a rule? 10. What are some advantages of writing a rule or generalization using numbers and variables? some disadvantages? I can use words to show number relationships. I can use equations to show number relationships. CHAPTER 2: Solving Equations 43
An Age-Old Mystery Match each equation to the related situation, and then use the equations to find the missing age in each question. Example: Gabriela s age is 15 less than the sum of the digits in her telephone number. Gabriela s phone number is 822-8219. Add the phone number digits. 8 + 2 + 2 + 8 + 2 + 1 + 9 = 32. Write an equation using g for Gabriela s age. 32 = g + 15 Hadleigh s strategy: The sum of 15 and 17 is 32. The g must also represent the value of 17. Gabriela s age is 17. Imaan s strategy: Show the equation using base ten blocks and a g for Gabriela s unknown age. = g Take one ten rod from each side of the equation. = g Take five units from each side of the equation. I have to trade one rod on the left for ten units to do this. = g I am left with one rod and seven units on one side of the equation and the unknown on the other side. That means Gabriela s age must be 17. 44 CHAPTER 2: Solving Equations
An Age-Old Mystery (continued) Jada s strategy: Subtract 15 from each side of the equation. 32 = g + 15-15 - 15 17 = g Gabriela s age is 17. 11. Match each question to an equation, then use the equation and the strategy of your choice to find the missing age. a. Seventy-four is the sum when you add 44 to my age. How old am I? b. When you subtract five from my age, the difference is 10. How old am I? c. I am twice as old as my friend. I am 40. How old is my friend? d. When you divide my age by three, the quotient is nine. How old am I? a 3 = 9 40 = 2a 74 = a + 44 a 5 = 10 12. Match each question to an equation, then use the equation and the strategy of your choice to find the missing age. a. Three more than my age is the same as the sum of nine and five. How old am I? b. If you double my age, the result is 10 more than 20. How old am I? c. One quarter of my age is the same as five. How old am I? d. The product of three and five is equal to five less than my age. How old am I? a 3 + a = 9 + 5 = 5 4 2 a = 20 + 10 3 5 = a 5 CHAPTER 2: Solving Equations 45
An Age-Old Mystery (continued) 13. Write you own age-related question for each equation below. Exchange questions with a partner and answer each other s questions. a. (a 6) = 12 b. 18 + 12 = 8 + a c. 6 = a 7 d. a 4 = 3 10 4. Which operations were most common in the age questions? least common? 5. Which words seemed to represent each operation? I can use equations to show number relationships. I can use objects and drawings to show and explain preservation of equality. I can use symbols to show and explain preservation of equality. I can write and solve an equation using a letter variable to represent a problem. I can write a word problem for a given equation. 46 CHAPTER 2: Solving Equations
An Equation is Worth 1000 Words Information is presented in many ways. In this activity, information is presented as a table, a sentence, or a graph. Example: Felix made a graph to show how far he could ride his bike each hour. One day he rode 75 km. How many hours did Felix ride that day? 60 Biking Distance 50 Distance (kilometres) 40 30 20 10 0 0 1 2 3 4 Time (hours) From the graph, I can find the distance by multiplying by 15 the number of hours that Felix rides. I know his total distance is 75 km. I can write an equation to find his time, using t as the time in hours. 75 = 15t 15 15 5 = t It took Felix five hours to ride 75 km. I can check this answer by multiplying 15 5 = 75. CHAPTER 2: Solving Equations 47
An Equation is Worth 1000 Words (continued) Write an equation to represent each situation and tell what your variable represents. Then use the equation to answer the question. Be sure to check your solutions. 11. Cade has 120 trading cards. Some are sports cards and 85 are fantasy character cards. How many sports cards does Cade have? 12. The perimeter of a rectangle is found by adding the two short sides and the two long sides. The perimeter of the rectangle below is 20 cm. How long are the two missing sides? 4 cm 4 cm 13. Dakota s family car has four tires. The tires combined weight is 28 kg. How much does each tire weigh? 14. Easton has marbles for playing a game with his friends. Easton shared the marbles equally between the six players. Each player started the game with seven marbles. How many marbles did Easton have in total? 15. Joe is older than Frank. Their ages are shown in the table below. How old will Frank be when Joe is 20? Joe s age Frank s age 10 5 11 6 12 7 13 8 16. A spider has eight legs. Cadence made the table below to show the number of legs of different numbers of spiders. Her pet spiders have a total of 56 legs. How many pet spiders does Cadence have? Number of spiders Number of legs 1 8 2 16 3 24 4 32 48 CHAPTER 2: Solving Equations
An Equation is Worth 1000 Words (continued) 17. Daisy wanted to rent tandem bicycles for her wilderness group. She received the following table from the rental shop. How many tandem bicycles will she need for the 8 people in her group? Number of people Number of bicycles 2 1 4 2 6 3 18. The animal shelter has room for a total of 33 dogs and cats. The graph below shows the number of dogs and cats that can be housed at the shelter. This week there are 12 dogs at the shelter. How many cats can be at the shelter this week? 35 Cats and Dogs at the Shelter 30 25 Number of dogs 20 15 10 5 0 0 2 4 6 8 10 Number of cats CHAPTER 2: Solving Equations 49
An Equation is Worth 1000 Words (continued) 19. Eden was selling chocolate bars as a fundraiser. She made a graph to show how much she would raise with each chocolate bar sold. How many chocolate bars will Eden have to sell to raise $100? Chocolate Bar Fundraiser 20 15 Profit ($) 10 5 0 0 2 4 6 8 10 Chocolate bars sold (#) 10. Fatimah was playing a magic number game in which she took any starting number and used a magic rule to find a new number. She used a graph like the one below to help her find the new number. What will the new number be if the starting number is 25? New number Magic Number Game 18 16 14 12 10 8 6 4 2 0 0 2 4 6 8 10 Starting number 50 CHAPTER 2: Solving Equations
An Equation is Worth 1000 Words (continued) 11. Compare the equations you wrote for each question with a partner. a. Were they all the same? Why or why not? b. Did both the equations that were different accurately represent the situation? 12. Write your own question for each equation. Present the information in a sentence, a table, or a graph. a. 24 = 8 e b. f + 4 = 12 c. g 4 = 7 d. 20 = h 10 13. How might the data represented in a table, a sentence, and a graph be similar? different? 14. Explain whether the information was easier to interpret in one of these forms than in the others. 15. Why might someone organize information in a table, a sentence, or a graph? I can use equations to show number relationships. I can use symbols to show and explain preservation of equality. I can write and solve an equation using a letter variable to represent a problem. I can write a word problem for a given equation. CHAPTER 2: Solving Equations 51