EE65 Solving Equations with Balances Pages 77 78


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1 EE65 Solving Equations with Balances Pages STANDARDS 6.EE.B.5, 6.EE.B.6 Goals Students will use pictures to model and solve equations. Vocabulary balance equation expression sides (of an equation) variable PRIOR KNOWLEDGE REQUIRED Is familiar with balances Can solve a simple equation to find an unknown value Can substitute numbers for unknowns in an expression Can check whether a number solves an equation MATERIALS paper bags counters identical objects for demonstrations (see below) a pan balance connecting cubes a paper bag for each pair of students a ruler, masking tape, or string NOTE: You will need several identical objects for demonstrations throughout this lesson. The objects you use should be significantly heavier than a paper bag, so that the presence of a paper bag on one of the pans of the balance does not skew the pans. Apples are used in the lesson plan below (to match the pictures in the AP Book), but other objects, such as small fruit of equal size, metal spoons, golf balls, tennis balls, or cereal bars will work well. If a pan balance is not available, refer to a concrete model, such as a seesaw, to explain how a pan balance works, and use pictures or other concrete models during the lesson. Review pan balances. Show students a pan balance. Place the same number of identical (or nearly identical) apples on both pans, and show that the pans balance. Remind students that when the pans, or scales, are balanced, it means there is the same number of apples on each pan. Removing the same number of apples from each pan keeps them balanced. Place some apples in a paper bag and place it on one pan, then add some apples beside the bag. Place the same total number of apples on the other pan. ASK: Are the pans balanced? (yes) What does this mean? (the same number of apples is on each pan) Take one apple off each pan. ASK: Are the pans still balanced? Repeat with two apples. Continue removing the same number of apples from each pan until one pan has only the bag with apples on it. ASK: Are the pans balanced? Can you tell how many apples are in the bag? Show students the contents of the bag to check their answer. Repeat the exercise with a different number of apples in the bag. Expressions and Equations 65 D13
2 Solving addition equations given by a balance model. Divide a desk in half (you can use a ruler, string, or masking tape) and explain that the parts on either side of the dividing line will be the pans. Ask students to imagine that the pans are balanced. As you did above, place a paper bag with apples in it along with some other apples on one side of the line, and place the same number of apples (altogether) on the other side of the line. Ask students how many apples need to be removed from both sides of the balance to find out how many apples are in the bag. Students can signal their answer. Remove the apples, then ask students to tell how many apples are in the bag. Show the contents of the bag to check the answer. Repeat with a different number of apples. ACTIVITY 1 Students can work in pairs to create models of addition equations and solve them. Each pair will need connecting cubes and a paper bag. They can use a ruler, masking tape, string, or the line along which desks meet as the dividing line for their model. Partner 1 places some cubes in a paper bag and some more cubes beside it on one side of the line. Partner 1 then places an equal (total) number of cubes on the other side of the line. Partner 2 has to determine how many cubes are in the bag. Partners switch roles and repeat. x + 2 = 7 The pans balance each other. The numbers on both sides are equal. Writing equations from a balance model. Remind students that we can use variables to represent numbers we do not know. Place a paper bag with 5 apples and 2 more apples on one side of the line, and place 7 apples on the other side of the line. Ask students to write an expression for the number of apples on the side with the paper bag. Explain that an equation is like a pair of balanced pans (or scales), and the equal sign shows that the number of apples on each pan is the same. Remind students that the parts of the equation on either side of the equal sign are called the sides of the equation. Each pan of the balance becomes a side in the equation and the balance on the desk becomes x + 2 = 7. Create more such models and have students write the equation for each one. After you have done a few models that follow this pattern, start placing the bag on different sides of the line, so that students have to write the expressions with unknown numbers on different sides of the equation. Solving addition equations using the balance model. Return to the model that corresponds to the equation x + 2 = 7. ASK: What do you need to do to find out how many apples are in the bag? (remove two apples from each side) Invite a volunteer to remove the apples, then have students write both the old equation and the new one (x = 5), one below the other. Repeat with a few different examples. ASK: What mathematical operation describes taking the apples away? (subtraction) Write the subtraction for each of the equations above vertically. Example: D14 Teacher s Guide for AP Book 6.1
3 x + 2 = ASK: How many apples are left on the right side of the equation? (5) What letter did we use to represent the number of apples in the bag? (x) Remind students that we write this as x = 5. Repeat with the other equations used earlier. Solving addition equations without using the balance model. Present a few equations without a corresponding model. Have students signal how many apples need to be subtracted from both sides of the equation, and then write the vertical subtraction for both sides. Exercises a) x + 5 = 9 b) n + 17 = 23 c) 14 + n = 17 d) p + 15 = 21 Students who have trouble deciding how many apples to subtract without drawing a model can complete the following problems. Exercises: Write the missing number. a) x + 15 b) x + 55 c) x + 91 Bonus: x x x x 38 Finally, give students a few equations and have them work through the whole process of subtracting the same number from both sides to find the unknown number. Exercises: a) x + 5 = 14 b) x + 9 = 21 c) 2 + x = 35 d) x + 28 = 54 Sample solution: a) x + 5 = x = 9 Bonus: The scales in the margin are balanced. Each bag has the same number of apples in it. How many apples are in the bag? Hint: You can cross out whole bags too! (MP.3) Solving multiplication equations given by a model. Divide a desk into two parts and place 3 bags (with 4 cubes in each) on one side of the line and 12 separate cubes on the other side. Tell students that the pans are balanced. What does this say about the number of cubes on both pans? (they are equal) How many cubes are on the pan without the bags? (12) How many cubes are in the bags in total? (12) How many cubes are in each bag? (4) How do you know? (divide 12 into 3 equal groups, 12 3 = 4) Invite a volunteer to group the 12 cubes into 3 equal groups to check the answer. Show students the contents of the bags to confirm the answer. Repeat the exercise with 4 bags and 20 cubes, 5 bags and 10 cubes, 2 bags and 6 cubes. Students can signal the number of cubes in one bag each time. Expressions and Equations 65 D15
4 Writing equations from models. Remind students that the pans of the balance become the sides of an equation, and that the equal sign in the equation shows that the pans are balanced. If students have, say, 3 bags with the same number of cubes in each, they write the total number of cubes in the bags as 3 b. Present a few equations in the form of a model, and have students write the corresponding equations using the letter b for the unknown number. Exercises a) b) Now have students use the models to solve the equations. Show them how to write the solution below the equation and have students record their solutions. Example: 2 b = 12 b = 6 ACTIVITY 2 Students can work in pairs to create models of multiplication equations and solve them. Each pair will need connecting cubes and several paper bags. They can use a ruler, masking tape, string, or the line along which desks meet as the dividing line for their model. Partner 1 places the same number of cubes in several paper bags on one side of the line, and places the number of cubes equal to the total in the bags on the other side. Partner 2 has to write the equation, tell how many cubes are in each bag, and record the solution. Partners switch roles and repeat. (MP.2) Drawing models to solve equations. Tell students that the next task will be the opposite of what they have been doing: now they will start with an equation and draw a model for it. Remind students that when we draw pictures in math class it is most important to draw the correct numbers of objects. Shading, color, and other artistic features or details are not important. Our drawings in math should be simple and we shouldn t spend too much time on them. Demonstrate making a simple drawing of a pan balance, and remind students that they can use circles, squares, or big dots for cubes and boxes for paper bags. Exercises: Draw models and use them to solve the equations. a) 3 b = 15 b) 4 b = 8 c) 9 b = 18 d) 8 b = 24 Using division to find the missing factor. ASK: Which mathematical operation did you use to write an equation for each balance? (multiplication) Which mathematical operation did you use to find the number of apples in each bag? (division) Have students show the division in the models they have drawn by circling equal groups of dots. For example, in Exercise a) above they should circle three equal groups of dots. ASK: What number do you divide by? (the number of bags) D16 Teacher s Guide for AP Book 6.1
5 Rewriting equations so that the missing number is the first factor. Remind students that order does not matter in multiplication. Have students give you a few examples and write them on the board (Example: 3 4 = 12 and 4 3 = 12, so 3 4 = 4 3). Remind students that letters represent numbers that you do not know, so anything that works with numbers will work with letters too. Then write: b 3 = Ask students what this expression will be equal to. (3 b) Repeat with a few other products where the unknown comes first. Write the following equations on the board: a) 3 b = 18 b) 4 b = 16 c) 9 b = 18 d) 8 b = 21 Have students rewrite them so that the variable is the first factor. Ask students how they can solve these equations. Have students make a model to find the answers. Multiplying and dividing by the same number does not change the starting number. Have students solve the following questions: a) (5 2) 2 b) (3 2) 2 c) (8 2) 2 d) (5 4) 4 e) (9 3) 3 f) (10 6) 6 SAY: Look at the questions you solved. How are they all the same? (you start with a number, then multiply and divide by the same number) ASK: Did you get back to the same number you started with? (yes) Does it matter what number you started with? Does it matter what number you multiplied and divided by as long as it was the same number? (no) Have students write their own question of the same type, swap questions with a partner, and check that the answer is the number they started with. Illustrate the same principle using concrete materials: Show students a train of 4 connecting cubes. Tell students that you want to multiply that train by a number, say 3. What will the answer look like? (3 trains of 4 cubes) Make two more trains of 4 cubes. Then say that you want to divide the result by 3. What will the result of the division look like? (1 train) Explain that you can do the same with unknown numbers: Show one paper bag with some cubes and say that you want to multiply it by 3. What will the answer look like? (3 such bags) Now you want to divide the result by 3. What will you get? (1 bag again) Finally, write on the board: ( 3) 3 = ASK: What will we get when we perform the multiplication and the division? (the box) Repeat with equations that include letters as variables: a) (b 3) 3 b) (b 5) 5 c) (b 6) 6 d) (b 10) 10 Expressions and Equations 65 D17
6 Solving equations by dividing both sides by the same number. Write the questions below and have students signal the number they would divide the product by to get back to b. (b 7) = b (b 2) = b (b 4) = b (b 8) = b (b 12) = b (b 9) = b If available, show students a pan balance with 3 bags of 5 apples on one pan and 15 apples on the other pan. Invite a volunteer to write the equation for the balance: 3 5 = 15. ASK: How many apples are in one bag? (5) Have a volunteer make three groups of 5 apples on the side without the bags. Point out that there are three equal groups of apples on both sides of the balance. Remove two of the bags from one side, and two of the groups from the other side. SAY: I have replaced three equal groups on each side with only one of these groups. What operation have I performed? (division by 3) Are the scales still balanced? (yes) Point out that when you perform the same operation on both sides of the balance, the scales remain balanced. What does that mean in terms of the equation? Write the equation that shows the division below the original equation: b 3 3 = 15 3 Have students calculate the result on both sides (b on the left side, 5 on the right side). Write on the board b = 5 (align the equal signs vertically). Demonstrate that the bags indeed contain 5 apples. Repeat with a few more examples. Finally, have students solve equations by dividing both sides of the equation by the same number. Exercises a) b 7 = 21 b) b 2 = 12 c) b 4 = 20 d) b 6 = 42 e) b 3 = 27 f) b 9 = 72 Bonus g) 3 b = 270 h) 8 b = 4,000 i) 7 b = 42,000 j) 6 b = 720,000 Sample solution: a) b 7 = 21 b 7 7 = 21 7 b = 3 Answers: b) 6, c) 5, d) 7, e) 9, f) 8, Bonus: g) 90, h) 500, i) 6,000, j) 120,000 D18 Teacher s Guide for AP Book 6.1
7 EE66 Solving Equations Guess and Check Page 79 STANDARDS 6.EE.B.5 Goals Students will solve equations of the form ax + b = c by guessing small values for x, checking by substitution, and then revising their answer. Vocabulary equation expression solving for a variable PRIOR KNOWLEDGE REQUIRED Can read tables Can substitute numbers for variables in equations Can check whether a number solves an equation MATERIALS paper bags counters x 4x Is equation true? Introduce using a table to solve equations. Draw a model on the board and build the corresponding equation: There are x counters in each bag. There are 4x counters in all the bags because there are 4 bags. There are 4x + 3 counters altogether. Make the same model using 4 bags and 31 counters. Put 3 outside the bags and put 1 counter at a time in each bag until all 31 are used. ASK: What equation can we write for this model? (4x + 3 = 31) Show students how to solve 4x + 3 = 31 by using a table and substituting different values of x in sequence (x = 1, x = 2, x = 3, and so on) into the expression on the left side of the equation (see margin). h 7h Is equation true? Point out the connection between the table and the method using counters: In the table, each time we increase the value of x by 1, it is as though we are adding a counter to each bag (4 counters for 4 bags) and checking how many counters are used in total. We stop when we see that all 31 counters are used. Introduce the guess and check method to solve equations. Show the equation 7h + 2 = 44. Tell students that you are going to solve this equation by guessing and checking. Start by guessing h = 5. ASK: If h = 5, what is 7h + 2? (37) Should h be higher or lower to make 7h + 2 = 44? (higher) What would your next guess be? (6) If h = 6, what is 7h + 2? (44) Is the equation true? (yes) Draw the table at left on the board and SAY: h = 6 makes the equation 7h + 2 = 44 true, so h = 6 is the answer. Compare the two methods of solving equations. ASK: Which method requires less work? Which method is quicker? (the guess and check method is quicker) Which method is more like looking up a word in the dictionary using alphabetical order? (guess and check) Which method is more like Expressions and Equations 66 D19
8 looking up a word in the dictionary without knowing or using alphabetical order? (using the table) Have students explain the connection. (In a dictionary, the words at the top of each page you turn to tell you whether to look to the right or to the left; they tell you if you have gone too far or not far enough.) Exercises a) Replace x with 5 and say whether 5 is too high or too low. i) 4x + 1 = 25 ii) 5x + 3 = 23 iii) 2x + 4 = 16 x 4x + 1 Answer x 5x + 3 Answer x 2x + 4 Answer b) Use the answers in part a) to try a higher or lower number and solve each equation. Answers: a) i) 21, too low, ii) 28, too high, iii) 14, too low, b) x = 6 works, ii) x = 4 works, iii) x = 6 works Extensions (MP.1, MP.7) 1. How many digits does the solution to 3x + 5 = 8,000 have? Explain. Hint: 1digit numbers are between 1 and 9, 2digit numbers are between 10 and 99, and so on. Solution: To determine the number of digits in the solution, we need to determine the first power of 10 (10, 100, 1,000, etc.) that is greater than the solution. We can substitute increasing powers of 10 for the variable until the answer is larger than 8,000: 3(10) + 5 = 35 3(100) + 5 = 305 3(1,000) + 5 = 3,005 3(10,000) + 5 = 30,005 So x is between 1,000 and 10,000, which means that it has 4 digits. (Indeed, x = 2,665.) (MP.7) 2. How many solutions can you find to 2x + 1 = 4y 1 if x and y are whole numbers? Solution: Find 2x + 1 for various values of x: x x Now find 4y  1 for various values of y: y y COPYRIGHT 2012 JUMP MATH: NOT TO BE COPIED. CC EDITION D20 Teacher s Guide for AP Book 6.1
9 Look for numbers that are the same in the second rows: 2x + 1 = 3 = 4y 1 when x = 1 and y = 1 2x + 1 = 7 = 4y 1 when x = 3 and y = 2 2x + 1 = 11 = 4y 1 when x = 5 and y = 3 Students might continue the pattern to find more solutions (x = 7 and y = 4 is the next one). COPYRIGHT 2012 JUMP MATH: NOT TO BE COPIED. CC EDITION Expressions and Equations 66 D21
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