Unit 3 Operations and Algebraic Thinking: Expressions and Equations

Transcription

1 Unit 3 Operations and Algebraic Thinking: Expressions and Equations Introduction In this unit, students will use the correct order of operations to evaluate expressions involving whole numbers, fractions, and decimals. They will use brackets and verify equations by calculating the expressions on both sides of the equal sign. Students will write simple expressions that record calculations with numbers and interpret numerical expressions without evaluating them. They will also solve word problems with times as many using tape diagrams as models, substitute values for the variables in algebraic expressions, translate simple word problems into algebraic expressions, and solve multiplication word problems by finding the scale factors for small and large quantities. Operations and Algebraic Thinking N-1

2 OA5-6 Order of Operations and Brackets Pages STANDARDS 5.OA.A.1 Vocabulary brackets expression operation order of operations Goals Students will understand the need for assigning an order to the operations in expressions and how to use brackets. Students will evaluate expressions involving two operations by using the correct order of operations. PRIOR KNOWLEDGE REQUIRED Can add, subtract, multiply, and divide 1-digit and small 2-digit numbers Review operation. Remind students that the term operation is a general word describing addition, subtraction, multiplication, and division. The need for an order of operations. Write on the board: Tell students that two people did this problem and got two different answers. Have students predict what the two answers were and why. (To get the answer 5, subtract 8-5 first, then add 2. To get the answer 1, add 5 and 2 first, then subtract 8.) Emphasize that we need a clear way to say what we mean. Sometimes we might mean the operations should be done in the order they appear, but sometimes we need a different order. ASK: What could we do to make it clear which operation to do first? (Students may suggest ideas other than brackets if they are not yet familiar with brackets; accept all answers.) Introduce brackets. Write on the board: (8-5) (5 + 2) SAY: The brackets tell you to do the operations in brackets first. Writing (8-5) + 2 means 3 + 2, which is 5; writing 8 - (5 + 2) means 8-7, which is 1. Exercises: Do the operation in the brackets first. Then do the second operation. a) (8 + 4) - 3 b) 8 + (4-3) c) (8-2) 3 d) 8 - (2 3) e) 12 (2 3) f) (12 2) 3 g) (5 + 3) 4 h) 5 + (3 4) Answers: a) 9, b) 9, c) 18, d) 2, e) 2, f) 18, g) 32, h) 17 Calculate any brackets but otherwise add or subtract from left to right. Point out that which operation you do first often changes the answer. Tell students that mathematicians have come up with shortcuts so that they don t have to write brackets all the time. SAY: When there are no brackets, do addition and subtraction in the order the operations appear from left to right. For example, means If you want the expression to mean 8-7, you have to add brackets: 8 - (5 + 2). N-2 Teacher s Guide for AP Book 5.2

3 Exercises: Add or subtract from left to right. a) b) c) d) Answers: a) 7, b) 5, c) 15, d) 9 Bonus: Do you need to add brackets to get the answer given below? a) = 3 b) = 9 c) = 3 d) = 3 e) = 6 f) = 13 g) = 7 h) = 2 i) = 8 Answers: a) yes, b) no, c) no, d) yes, e) no, f) no, g) yes, h) no, i) yes Multiply and divide from left to right. SAY: When there are no brackets, do multiplication and division from left to right. Exercises: Multiply or divide from left to right. a) b) c) d) Answers: a) 4, b) 8, c) 4, d) 16 Bonus: Do you need to add brackets to get the answer given below? a) = 9 b) = 1 c) = 20 d) = 2 e) = 24 f) = 2 Answers: a) no, b) yes, c) no, d) yes, e) no, f) no Multiply or divide before adding or subtracting. Write on the board: SAY: This one is a bit different. Remember that we do addition and subtraction in the order the operations appear from left to right. And we do multiplication and division in the order that they appear from left to right. However, we haven t yet looked at the order for addition and multiplication or other combinations. We don t do addition and multiplication from left to right. When there are no brackets, always do multiplication or division first, and then do any addition or subtraction. Write on the board: = 3 + (4 5) Exercises: Put in brackets (or circle) the operation you do first. a) b) c) d) Answers: a) 4 + (5 2), b) (3 4) + 5, c) (12 2) + 5, d) 14 - (6 2) Choosing and doing the first operation. Teach students to evaluate expressions involving two operations by deciding which operation to do first, doing that operation only, and then rewriting the expression. Complete the expression on the board: = 3 + (4 5) = Operations and Algebraic Thinking 5-6 N-3

4 Exercises: Choose which operation to do first. Do that operation and then rewrite the rest of the expression. 1. a) b) c) d) e) f) g) h) Bonus i) j) k) Answers: a) , b) , c) 12-10, d) 24-5, e) 3 + 2, f) 4 + 2, g) 3-2, h) 14-3, Bonus: i) , j) 1,300-10, k) a) b) c) d) e) f) g) h) Bonus: Answers: a) 13-12, b) , c) , d) 14-2, e) 12-3, f) 18 3, g) 6 + 3, h) 6 2, Bonus: Doing the second operation. Now ask students to do the second operation in a problem. Complete the expression on the board: = 3 + (4 5) = = 23 Students who are struggling can start by writing brackets around the operation they would do first. Exercises: Finish evaluating the expressions in the previous exercise. Answers: a) 1, b) 16, c) 13, d) 12, e) 9, f) 6, g) 9, h) 12, Bonus: 12 Extensions 1. Use the numbers 1, 2, 3, 4, 5, 6, 7, 8, and 9 once each to make all the equations true. Sample answers: 0 = (4-4) (4 + 4) 1 = = 4 4 (4 + 4) 3 = ( ) 4 4 = (4-4) = ( ) 4 6 = (4 + 4) = = (4 + 4) 9 = = (44-4) 4 a) ( ) = 1 b) - ( + ) = 2 c) ( + ) = 4 Sample answers a) 6 (3 2) = 1 8 (2 4) = 1 b) 8 - (5 + 1) = (1 + 6) = 2 c) (7 + 9) 4 = 4 (5 + 7) 3 = 4 2. Using exactly four 4s each time, make expressions equal to each number from 0 through 10. You may use brackets and any of the four operations. Example: (4 4) (4 + 4) = 16 8 = 2. Hint: You many need to use the 2-digit number 44. Try to come up with many different expressions (sample answers are in the margin). N-4 Teacher s Guide for AP Book 5.2

5 OA5-7 Numerical Expressions Pages STANDARDS 5.OA.A.1 Goals Students will verify equations by calculating the expressions on both sides of the equal sign and verifying that they are equal. Vocabulary brackets equation numerical expression operations order of operations verify PRIOR KNOWLEDGE REQUIRED Can add, subtract, multiply, and divide fractions Knows order of operations Introduce numerical expressions. Explain to students that an expression shows or represents something. For example, a facial expression is when a person s face shows or represents how that person feels. Demonstrate by making one or two facial expressions to show emotions and ask students to guess which emotions you were showing. SAY: A numerical expression represents calculations with numbers. For example, is a numerical expression. Exercises: Calculate the numerical expression. a) b) c) d) Answers: a) 10, b) 30, c) 51, d) 11 Review brackets. Remind students that brackets in an expression tell you to do everything inside them before you do all the other operations. For example, 8 - (2 3) is calculated as 8-6 = 2, but (8-2) 3 = 6 3 = 18. Invite volunteers to solve these problems on the board: (2 3) + 5 Answer: = 11 2 (3 + 5) Answer: 2 8 = 16 Exercises: Evaluate the expression. a) (20-5) 3 b) 5 (3 + 2) c) 7 - (4 + 2) d) 18 (2 + 4) Answers: a) 5, b) 25, c) 1, d) 3 Expressions with fractions and decimals. Explain to students that any number can be in an expression, including, for example, fractions or decimals. Write on the board: ASK: Which operation is done first? (multiplication) Do the multiplication and write on the board: = 3.6 Operations and Algebraic Thinking 5-7 N-5

6 Exercises: Calculate the numerical expression. 3 a) b) c) d) ( ) 2 Answers: a) 0.8, b) 6, c) 1 7, d) 8 5 Introduce equations. Write two expressions beside each other on the board: SAY: Both expressions are equal to 10. When two equal expressions are separated by an equal sign, we call it an equation. Add an equal sign between the expressions on the board = 50 5 SAY: To verify that an equation is true, you can calculate the numerical expressions on both sides of the equal sign and see whether they have the same value. Example: = 7-4 is a true equation because both sides have the same value; 2 3 = is not true because the left side is equal to 6 and the right side is equal to 5. Exercises: Which of the following equations are true? a) 20-5 = 5 3 b) 5 (3 + 2) = c) = 26 d) = 3 3 Answers: a), b), and d) are true equations Exercise: What s the mistake in part c) above? Answer: Multiplication has to be done first: = = 17 Bonus 1. a) Verify that each equation is true = = = 6 6 b) Look for a pattern in part a). Use the pattern to predict and then check: = Answer: a) Verify that each equation is true = (7 + 3) + (6-2) + (10-1) = (8 + 4) + (5 + 3) + (12-7) b) Look for a pattern in part a). Use the pattern to write another similar equation. Evaluate the expressions on both sides of the equal sign to verify that your equation is true. Sample answer: = (12 2) + (31 1) + (14 + 3) N-6 Teacher s Guide for AP Book 5.2

7 Extensions 1. Add brackets where necessary to the equation to make it true. Hint: In some equations, you might need to add one set of brackets inside another. a) = 20 b) = 56 c) = 28 d) = 7 Answers: a) (3 + 1) (7-2), b) (3 + 1) 7 2 = 56, c) (8-4) (2 + 5) = 28, d) (5 (4-3)) + 2 = 7 2. Verify that the equation is true. a) (5-2) (1 + 7) = 24 b) (2 + 5) (8-3) = (6 6) - 1 c) (3 + 2) (9-1) 4 = 10 d) (2 5) + (2 2) = (6 3) (4 + 3) 3. Find the missing digits. a) (2 1,000) + (6 100) + ( 10) + 5 = 2,645 b) 2 7 = ( 100) + (3 10) + c) (3 100,000) + (9 10,000) + ( 1,000) + ( 100) + = 90,7 2 Answers: a) 4; b) 3, 2, 7; c) 0, 7, 2, 3, 0 Operations and Algebraic Thinking 5-7 N-7

8 OA5-8 Unknown Quantities and Equations Pages STANDARDS preparation for 5.OA.A.2 Goals Students will write and solve easy equations and word problems. Vocabulary equation numerical expression unknown PRIOR KNOWLEDGE REQUIRED Can add, subtract, multiply, and divide Knows multiplication and division up to Materials masking tape, string, or a ruler Solving algebraic equations. Divide a desk in half using masking tape, string, or a ruler. Put five counters on one side of the dividing line, and put two counters and a paper bag containing three more counters on the other side. NOTE: Don't let students see how many counters are in the bag. Tell students that the number of counters in total is the same on both sides of the line. Have students guess how many counters are in the bag. (3) Before representing the concrete model with an abstract one, repeat the concrete example with different numbers of counters, but don t use more than one bag. Draw a representation of the concrete model on the board (see example in the margin) = 5 5 ASK: How many counters are on the right side? (5) How many counters can you see on the left side? (2) Write the numbers 2 and 5 under the corresponding number of circles (see example in margin). SAY: The box represents the bag. To find how many circles are in the box, I put a circle in the box. Then draw a circle inside the box. ASK: How many circles can you see on the left side in total? (3) Erase the number 2 and write 3 instead, but write 3 a little to the left, between the box and the two counters (see example in margin). ASK: Is the number of circles on both sides equal? (no) SAY: So, I m going to add another circle in the box. Then draw another circle inside the box. ASK: How many circles can you see on the left side in total? (4) Erase the number 3 and write 4 instead (see example in margin). SAY: The number of circles on both sides is still not equal, so I'm going to put another circle in the box. Then draw another circle inside the box. Erase the number 4 and write 5 instead (see example in margin). ASK: How many circles are on the left side in total? (5) Is the number of circles on both sides equal? (yes) How many circles are in the box? (3) SAY: We can replace the dotted line with an equal sign because the two sides are equal. Erase the dotted line and draw an equal sign in its place, as shown in the margin, and SAY: This shows that the two sides are equal. N-8 Teacher s Guide for AP Book 5.2

9 Have a student open the bag to check that there are three counters in the bag. Students should also see that they can find the number of hidden counters either by counting up from 2 to 5 or by subtracting 2 from 5. Have students check that the equation was solved correctly: Does the number of counters drawn in the box make the equation true? (yes) Exercises: Draw circles in the box until the number of circles is the same on both sides. a) b) = = After students have solved more problems like these, explain that it is inconvenient to draw counters all the time. We can use numbers to represent all the quantities instead. Have students practice writing equations that represent pictures, similar to Question 2 on AP Book 5.2 p. 43. For example, the equation for the first example picture in this lesson is: + 2 = 5 Challenge students to solve several more examples. Students can create models for equations that involve addition using counters. They can also draw models: in this case, ask students to use a box for the unknown (the hidden number or the number we don t know) and use a set of circles to model the numbers in the equation. For example, a model for the equation + 2 = 7 is shown below. = Ask students to make a model with boxes and circles to solve the following problems. They should explain how many circles they would put in each box to make the equation true. Exercises a) 7 + = 11 b) 6 + = 13 c) 4 + = 10 d) 9 + = 12 Answers: a) 4, b) 7, c) 6, d) 3 Equations with addition. Read several word problems to students. Invite volunteers to draw models, write equations using boxes and numbers, and solve the equations. Examples: a) There are 10 trees in the garden. Three of them are apple trees. All the rest are cherry trees. How many cherry trees are in the garden? b) Jane has 12 T-shirts. Three of them are plain. All the rest have designs. How many of Jane's T-shirts have designs? Exercise: Write an equation to solve the problem: There are 15 plants in the flowerbed. Six are lilies. All the rest are peonies. How many peonies grow in the flowerbed? Operations and Algebraic Thinking 5-8 N-9

10 Bonus 1. There are 150 pirates on two ships, a galleon and a schooner. Forty of the pirates are on the schooner. How many are on the galleon? 2. A dragon has 15 heads. A mighty and courageous knight cut off some of the heads. The dragon has seven remaining heads. How many heads did the knight remove? Equations with subtraction. Present this word problem: Sindi has a box of apples. She took two apples from the box. Four were left. How many apples were in the box before she removed any? Draw the box with four apples in it (see example in margin). Draw two more apples in the box and cross them out to show that they have been taken away (see example in margin). - ASK: How many apples were in the box at the beginning? (6) Explain that, when we write a subtraction equation, we draw it a bit differently than the addition equations we drew earlier. We draw a box for the number we don t know (the original number), show a minus sign, and then draw the apples we took out of the box (see example in margin). - = We show the four apples that were left in the box on the other side of the equation (see example in margin). To solve the equation, we have to put all the apples into the box the ones that we took out and the ones that we left there. Remind students that they also learned to write equations using numbers. How could you write this equation using numbers? ( - 2 = 4) Draw several models for subtraction equations (like those in Questions 4 and 5 on AP Book 5.2 pp ) and ask students to write the equations for them. Ask volunteers to present the answers on the board. Exercises: Draw the model to solve the equation. a) - 6 = 9 b) - 7 = 12 c) - 5 = 3 d) - 3 = 10 2 = Equations with multiplication. Tell students that they can also write equations for multiplication problems. Remind students that 2 means that some quantity will be two times larger in other words, it will be doubled. Examples: 2 = and 2 = Present the problem as shown in the margin and ask students to draw the appropriate number of circles in the box. Students should solve the problem by dividing the circles on the right side into two equal groups. ASK: How would you write an equation with numbers for this problem? (2 = 12) Present more problems like this and ask students to write and solve the corresponding numerical equations. Then show students how to write an equation for a word problem involving multiplication using this word problem: N-10 Teacher s Guide for AP Book 5.2

11 Tony has four boxes of pears. Each box holds the same number of pears. He has 12 pears in total. How many pears are in each box? Students might think in the following way: We usually use a box to represent the thing that we do not know (the unknown). So, four times the unknown makes 12, and we have the equation: 4 = 12 Exercises 1. Jenny uses three eggs to bake muffins. Seven eggs remain in the carton. How many eggs were in the carton before Jenny took some out? 2. Bob has nine pets. Three of them are snakes. All the rest are iguanas. How many iguanas does Bob have? 3. Solve the equations. a) 3 + = 8 b) 3 = 15 c) - 4 = 11 d) 2 = Draw models to solve the problems. a) Avi has 12 stamps. Four of them are American and the rest are foreign. How many foreign stamps does she have? b) Joe has 15 stamps. Five of them are French and the rest are German. How many German stamps does Joe have? Answers: 1. 10; 2. 6; 3. a) 5, b) 5, c) 15, d) 7; 4. a) 8, b) 10 Bonus: Solve the equations. a) = 248 b) 8 = 56 c) - 4 = 461 d) 60 = 240 Answers: a) 5, b) 7, c) 465, d) 4 Extensions 1. The same symbol in the equation means the same number. What does each symbol represent? a) + = 12 b) + + = 9 c) = 13 d) = Two birds each laid the same number of eggs. Seven eggs hatched, and three did not. How many eggs did each bird lay? 3. Sixty baby alligators hatched from three alligator nests of the same size. We know that only half of the total number of eggs hatched. How many eggs were in each nest? (Hint: How many eggs were laid in total?) Answers: 1. a) 6, b) 3, c) 4, d) 2; 2. 5; Operations and Algebraic Thinking 5-8 N-11

12 OA5-9 Translating Words into Expressions Pages STANDARDS 5.OA.A.2 Goals Students will write and solve easy addition equations. Vocabulary bracket equation numerical expression operation unknown PRIOR KNOWLEDGE REQUIRED Can add, subtract, multiply, and divide Associating words and phrases with operations. SAY: You can use clues to write expressions. The words give clues to the operations you need to use. On the board, make a table with four columns and these headings: Add, Subtract, Multiply, and Divide. Have students discuss which operation each phrase makes them think of. Based on the class response, create a chart on the board like the following, with each phrase under its correct heading. Add Subtract Multiply Divide increased by sum more than less than difference decreased by reduced by fewer than product times twice as many Exercises: Translate the phrase into an expression. a) 5 more than 7 b) 5 less than 7 divided by divided into share equally c) 5 times 7 d) the product of 7 and 5 e) 7 reduced by 5 f) 7 divided by 5 g) 5 divided into 7 h) 5 divided by 7 i) 7 divided into 5 j) 7 decreased by 5 k) 7 increased by 5 l) the sum of 7 and 5 m) 5 fewer than 7 n) the product of 5 and 7 Bonus: 7 multiplied by 3 then increased by 5 Answers: a) 7 + 5, b) 7-5, c) 5 7, d) 5 7, e) 7-5, f) 7 or 7 5, 5 g) 7 5, h) 5 7, i) 5 7, j) 7-5, k) 7 + 5, l) 7 + 5, m) 7-5, n) 5 7, Bonus: Phrases with decimals and fractions. Explain to students that any number can be in an expression not just whole numbers. Write on the board: twice as many as 3.1 Ask a volunteer to write the numerical expression for the phrase (2 3.1). Have students complete the following exercises. N-12 Teacher s Guide for AP Book 5.2

13 Exercises: Translate the phrase into an expression. a) 2.5 more than 6 b) 5.1 less than c) 3 times 4 d) the product of 1 2 and 3 5 e) 2 reduced by 2 f) 4.5 divided by 9 3 Answers: a) , b) , c) 1 3 4, d) , e) 2-2 3, f) 45. or Associating phrases with expressions with brackets. Proceed to expressions with more than one operation. Write on the board: Multiply 2 and 3. Then subtract 1. Ask a volunteer to write the numerical expression for the phrase (2 3-1). Exercises: Translate the phrase into an expression with more than one operation. Use brackets to indicate which operation has to be done first. a) Divide 6 by 2. Then add 3. b) Add 4 and 6. Then divide by 5. c) Multiply 5 and 4. Then add 2. d) Divide 8 by 4. Then multiply by 3. e) Subtract 2 from 5. Then multiply by 4. Then add 3. Answers: a) (6 2) + 3, b) (4 + 6) 5, c) (5 4) + 2, d) (8 4) 3, e) (5-2) Writing mathematical expressions in words. Start with an easy expression. On the board, write and ask students to read the expression. If some students answer three plus five, say that, rather than using the word plus, we prefer to use the verb add. Then write on the board add 3 and 5. Exercises: Write the operation in words. a) 5 2 b) 7-4 c) d) 15 3 Answers: a) multiply 5 by 2, b) subtract 4 from 7, c) add 4 and 7, d) divide 15 by 3 Teach students how to write a mathematical expression with two or more operations in words. On the board, write: (3 + 2) 4 ASK: Which operation would you do first, addition or multiplication? (addition, because it s in brackets) On the board, write: Add 3 and 2 SAY: The first operation is done and so we have to end the sentence. Put a period to show the sentence is ended. ASK: What operation would you do next? (multiplication) Ask a volunteer to write Multiply by 4. on the board, following your first sentence. (Add 3 and 2. Multiply by 4.) Operations and Algebraic Thinking 5-9 N-13

14 Exercises: Write the mathematical expression in words. a) (5 + 1) 2 b) (9-3) 3 c) (4 2) Bonus: ( ) 9 Answers: a) Add 5 and 1. Then multiply by 2. b) Subtract 3 from 9. Then divide by 3. c) Multiply 4 and 2. Then subtract 3. Then add 7. Bonus: Multiply 3 and 4. Then add 7. Then divide by 9. Interpreting expressions. SAY: A parking lot charges $3 per hour, so you need to pay 2 3 dollars for two hours. ASK: How much do you need to pay for four hours? (4 3) ASK: If the cost of parking is 5 3, how many hours does that pay for? (5 hours) Exercises: To rent skates, you must pay $6 for each hour. Complete the meaning of the expression. a) 3 6 is the cost of renting skates for hours b) 2 6 is the cost of renting skates for hours c) 5 6 is the cost of renting skates for hours Answers: a) 3, b) 2, c) 5 Writing word problems as mathematical expressions. Start with an example. SAY: A movie ticket costs $8 for adults and $5 for students. ASK: How much will it cost for two adults to watch the movie? (2 8) If some students answer 16, say that we don t want to calculate the expressions right now; instead, we just want to write a proper numerical expression that describes the problem. ASK: How much will it cost for three students? (3 5) ASK: How much will it cost for two adults and three students? ( ) Exercises 1. Six people can travel in one van. Sixteen students and two teachers go to the museum. Write an expression to show the number of vans that they need. Answer: (16 + 2) 6 2. Kim wants to buy a new MP3 player that costs $45. Kim has already saved $9. a) Write an expression to show how much money she needs to save. b) Kim decides to save the same amount of money each month for the next four months. Write an expression to show the amount of money that she has to save each month. Bonus: Kim s father agrees to give her $8 per month for the next three months. Write an expression to show the amount of money that she has to save each month. Answer: a) 45-9, b) (45-9) 4, Bonus: ( ) 4 N-14 Teacher s Guide for AP Book 5.2

15 Extensions 1. Movie tickets cost $5 for kids and $8 for adults. Write an expression to represent the cost of tickets for three kids and seven adults. Answer: Fifteen students from each class go on a trip. There are six classes as well as two teachers and three parents for each class. How many buses will be needed if 34 people can ride in each bus? Answer: 6 ( ) 34 or ( ) 34 Operations and Algebraic Thinking 5-9 N-15

16 OA5-10 Tape Diagrams I Pages STANDARDS 5.OA.A.2 Goals Students will solve word problems with times as many using models. Vocabulary bracket difference part total PRIOR KNOWLEDGE REQUIRED Understands the expression times as many Can identify a part, total, and difference in a problem Drawing a model for a times as many situation. Tell students that two people, Kim and Ron, have some stickers. Write on the board: Kim has four times as many stickers as Ron. Ron s stickers Ron s stickers Kim s stickers SAY: I want to draw a model to represent this situation. Who has more stickers, Kim or Ron? (Kim) On the board, draw a bar consisting of a small rectangular block and explain that this block represents Ron s stickers. Label the bar as shown in the margin. ASK: How can we show that Kim has four times as many stickers as Ron? Accept all reasonable answers. Then explain that you are going to use a specific way to draw a model. It is similar to what students did with problems such as Ron has four stickers. Kim has two more stickers than Ron. Draw a second bar that contains the block of Ron s stickers repeated four times. Finish the picture shown in the margin on the board and keep it for future reference. SAY: This is an example of a tape diagram. A tape diagram has two or more strips, or bars, on top of each other and it is made of blocks of equal units. number of dimes number of pennies number of dimes number of pennies number of dimes number of pennies number of dimes number of pennies Present the following situation: Karen has three times as many pennies as dimes. On the board, draw the four pairs of labeled tape diagrams as shown in the margin; ask which of them would fit the situation and which would not. (the first and second models work; the third and fourth do not) Have students explain why the models that do not fit the situation do not work. (The third is incorrect because it shows more dimes than pennies. The fourth is incorrect because it shows four times as many pennies as dimes, not three.) ASK: How do you know that the short bar should be the number of dimes? (Karen has more pennies than dimes) Present this situation: Karen is twice as old as Ariel. ASK: Whose age will be the smaller bar? (Ariel s) Why? (because Karen is older, so her age is shown with the larger bar) Ask students to draw a model for this situation. Repeat with more examples, such as the ones below. As you give each example, ask students to first identify which number is the smaller one, and remind them that this should be the shorter bar. For the last exercise, make sure students understand the meaning of twice. a) Bethany is three times as tall as her baby brother. b) Pria is a nickname. Pria s full name is four times as long as Joshua s. N-16 Teacher s Guide for AP Book 5.2

17 c) There are eight times as many students in the school as in our class. d) A book is twice as thick as a notebook. Ron s stickers 3 Kim s stickers Finding the length of the bars when the smaller part is given. Return to the model with Ron and Kim that was discussed earlier. Tell students that Ron has three stickers. Write 3 in Ron s block. Remind students that each block is an equal unit, so write 3 in each of Kim s blocks as shown in the margin. ASK: Can you tell from the model how many stickers Kim has? (yes, 12) How do you know? (there are 4 blocks of 3) Have students write the multiplication statement for the length of the longer bar. (3 4 = 12) Have students draw a model and find the lengths of the bars for this situation: Ella has three red marbles. She has twice as many green marbles as red marbles. Use volunteers to show the answers. (Check that one block of three marbles is noted as red. There are two blocks, so 2 3 should be noted as green.) Exercises: Draw a model and find the length of the bars. a) A car holds five people. A van holds three times as many people. b) Dan s apartment building is three stories high. Ron s building is five times as high as Dan s. c) Ethan is five years old. David is four times as old as Ethan. d) A sparrow has four eggs in its nest. A duck s nest has three times as many eggs as the sparrow s nest. An ostrich s nest has five times as many eggs as the sparrow s nest. Solving problems when the larger part is given. Present the following situation: Sylvia has 20 stickers. Sylvia has four times as many stickers as George. Invite a student to draw the tape diagram for the situation without writing the numbers. SAY: How many blocks are in Sylvia s bar? (4) Sylvia has 20 stickers. How many stickers does each block represent? (5) How do you know? (20 4 = 5) How many stickers does George have? (5) Have students draw bars in a tape diagram and find the length of each block for the following situations. Work through the first exercise as a class, then have students work individually on the rest. Exercises a) There are six apples on the table. There are twice as many apples as pears. How many pears are there? b) A mini-bus holds 16 people. The mini-bus holds twice as many people as a van. How many people can the van hold? Operations and Algebraic Thinking 5-10 N-17

18 c) Dan s apartment building is 30 stories high. Dan s building is five times as high as Ron s building. How tall is Ron s building? d) Ethan is 14 years old. Ethan is seven times as old as Gregory. How old is Gregory? Bonus: A sugar pine cone is 18 inches long. It is three times as long as an eastern white pine cone. The sugar pine cone is nine times as long as a jack pine cone. How long are the eastern white pine cone and the jack pine cone? Finding the size of a single block when the difference is given. Explain that a student you know drew the model in the margin for a word problem. The problem said that the difference between the parts was 18. ASK: What does this mean? (the longer bar is 18 more than the shorter bar) Show how to mark this on the diagram by adding a bracket below the difference and mark it as 18. ASK: How many blocks is the difference? (3) What is the size of each block? (6) How do you know? (18 3 = 6) Exercises: What is the size of one block? a) b) c) d) Answers: a) 6, b) 9, c) 7, d) 14 Finding the size of a single block when the total is given. Explain that another student you know drew the model in the margin for a different word problem. Again, all the blocks are the same size. The blocks combine to give a total of 18. Show how to mark this on the diagram using a vertical bracket. ASK: How many blocks are there in total? (9) What is the size of each block? (2) How do you know? (18 9 = 2) Exercises: What is the size of one block? 1. a) b) 35 c) d) 24 Answers: a) 7, b) 7, c) 4, d) N-18 Teacher s Guide for AP Book 5.2

19 2. Now mix together the problems with a total given and a difference given. What is the size of the block? a) 56 b) 21 c) d) Answers: a) 8, b) 7, c) 21, d) 10 Extensions 1. Abdul reads the same number of pages every school day. He reads twice as many pages every weekend day. He finished a book of 108 pages in a week. How many pages does he read on Monday? How many pages does he read on Sunday? Answer Week days Step 1: Use five blocks to represent the number of pages he reads each week day (see example in margin). Week days Step 2: He reads twice as many pages on weekend days, so use two blocks for each day (see example in margin). Weekend days Sat Sun Step 3: There are nine blocks in total and = 12. He reads 12 pages on Monday and 24 on Sunday. 2. Choose a model from the block models above, and create a word problem that would fit the model. Have a partner solve the problem. 3. There are three apples and two oranges for each plum in the basket. There are 30 fruits altogether. How many of each fruit are there? Plums Apples 30 Oranges Answer: Use the tape diagram with three bars in the margin = 5, so there are 5 plums, 15 apples, and 10 oranges. Operations and Algebraic Thinking 5-10 N-19

20 OA5-11 Tape Diagrams II Pages STANDARDS 5.OA.A.2 Goals Students will solve times as many and multi-step word problems with fractions using models. Vocabulary difference part total PRIOR KNOWLEDGE REQUIRED Understands the expression times as many Can identify a part, total, and difference in a problem Can do operations with fractions Solving problems with fractions. Explain to students that they can use a tape diagram in problems with fractions. Write on the board: Nests Eggs Nests 10 Eggs 5 5 a) b) c) There are 1 as many nests as eggs. 3 SAY: When there are 1/3 as many nests as eggs, it means that there are three eggs for every nest. Draw the tape diagram in the margin on the board. SAY: Suppose there are 10 more eggs than nests; to find out how many nests there are, you can complete the tape diagram with more information (see example in the margin). ASK: What is the size of each block? (5) SAY: So there are five nests because there is just one block of nests. Exercises: Use a tape diagram to find the number of eggs and the number of nests. a) 2 as many nests as eggs, six more eggs than nests. 5 b) 3 as many nests as eggs, three fewer nests than eggs. 4 c) 3 as many nests as eggs, eight more eggs than nests. 5 d) 2 as many nests as eggs, five less nests than eggs. 3 d) Answers: a) 4 nests and 10 eggs, b) 9 nests and 12 eggs, c) 12 nests and 20 eggs, d) 10 nests and 15 eggs Solving multi-step problems. Tell students that now they will need to draw the models themselves. Present the problem below: Irene is four times as old as Kara. Kara is 15 years younger than Irene. How old is Kara? Ask students to draw a model that fits the first sentence; have a volunteer present the answer. What does the second sentence give us: the difference, the total, or one of the parts? (the difference: 15) Have students mark that on the diagram. ASK: How large is one block? How do you know? (15 3 = 5) How many blocks long is Kara s bar? (1 block) How old is Kara? (5) How long is Irene s bar? (4 blocks) How old is Irene? (20 years old) N-20 Teacher s Guide for AP Book 5.2

21 Work through the first two problems as a class, then have students work individually. Exercises a) Mara saved three times as much pocket money as Shayan. Shayan saved $18 less than Mara. How much money do they have together? b) Robert and Brenda use all their pocket money to buy a shared present for their grandmother. They have $60 together. Robert has twice as much money as Brenda has. How much money does each of them have? c) The number of students in the school who are in Grade 5 is 2 7 the number of students who are not in Grade 5. There are 231 students in the school who are not in Grade 5. How many students are there altogether in the school? d) A number is 3 the size of another number. If you add the two numbers 5 together, you get 64. What are the numbers? Answers a) They have $36. Mara saved $27; Shayan saved $9. b) Brenda has $20; Robert has $40. c) There are 297 students in the school in total. d) 40 and 24 Extensions 1. Abdul reads the same number of pages every school day. He reads half as many pages every weekend day. He finished a book of 84 pages in a week. How many pages does he read on Monday? How many pages does he read on Sunday? Answer: 14 pages on Monday, 7 pages on Sunday 2. Choose a tape diagram from the previous lesson and create a word problem with fractions that would fit the model. Have a partner solve the problem. Operations and Algebraic Thinking 5-11 N-21

22 OA5-12 Variables Page 51 STANDARDS 5.OA.A.1, 5.OA.A.2 Goals Students will substitute values for the variables in algebraic expressions and translate simple word problems into algebraic expressions. Vocabulary algebraic expression bracket equation evaluate formula numerical expression unknown value variable PRIOR KNOWLEDGE REQUIRED Can add, subtract, and multiply Can find rules and formulas for patterns Variables and algebraic expressions. Explain to students that today they will learn to write equations the way mathematicians write them. Instead of drawing a square or a diamond for the unknown, mathematicians usually write letters. They call these letters variables. Remind students that they have used letters in formulas before. On the board, write 2 (3 + 4). SAY: This is a numerical expression. If I replace some numbers with variables, then I have an algebraic expression. Erase the number 3 on the board and write n in its place: 2 (n + 4) Explain to students that when letters are used in an expression, the multiplication sign ( ) is often omitted to avoid confusion with the letter x and to make the notation shorter. In addition we write variables in italics. In this case, instead of writing 2 (n + 4), we can simply write 2(n + 4). Exercises: Rewrite the expression. a) 2 n b) (2 n) + 3 c) 2 (n + 3) Answers: a) 2n, b) (2n) + 3, c) 2(n + 3) evaluate value Substituting numbers for variables and evaluating expressions. On the board, write the expression n + 4. Tell students that we can replace n with a number and get a numerical expression. For example, if we replace n with 3, then the expression becomes 3 + 4, which is 7. Writing 7 is called evaluating the expression, because we are saying the value of the expression. Write the words on the board with underlining, as shown in the margin, to emphasize the connection. Exercises: Replace n with 3 in the expression and evaluate the expression. a) n + 2 b) n 1 c) 5 n d) 7 + n Answers: a) 5, b) 2, c) 2, d) 10 Now tell students that you are going to try to trick them. Write 5n on the board. Have students replace n with 3. Discuss the problem that students run into. The answer looks like the number 53, but 5n really means 5 n, so we mean 5 3, not 53. To avoid this problem, we include brackets if replacing a variable with a number could cause confusion. This confusion could happen whenever a variable is being multiplied by a number. Tell students that 5(3) is another way to write 5 3. N-22 Teacher s Guide for AP Book 5.2

23 Exercises: Evaluate. a) 5(4) b) 7(3) c) 6(2) d) 9(6) Bonus: 9(2,000) Now tell students that, after evaluating an expression, we can add to it or subtract from it. On the board, write the expression 5(4) + 3. Tell students that this means multiply 5 and 4, and then add 3. SAY: We should probably write it like this to show what we mean: (5(4)) + 3 But that s awkward because there are too many brackets, so we ll just write it like this 5(4) + 3 and we ll all understand that it means do 5(4) first. Exercises: Evaluate the expression. a) 3(5) + 4 b) 2(3) + 7 c) 3(4) 5 d) 2(4) 7 Answers: a) 19, b) 13, c) 7, d) 1 Now have students combine the steps: replace the variable with a number and evaluate the resulting expression. Exercises: Replace n with 5 and then evaluate. a) 3n b) 10n c) 10n + 1 d) 10n 2 e) 10n + 4 f) 8n 7 Answers: a) 15, b) 50, c) 51, d) 48, e) 54, f) 33 Interchangeable expressions. Write on the board four different expressions: 2n + 3 2p + 3 2t + 3 2w + 3 Explain to students that using different variables in the same expression doesn t change the meaning of the expression. You can ask your students to verify that all the expressions have the same value for the same number, for example, n = 5, p = 5, t = 5, and w = 5. Extensions 1. Evaluate the expression with the given value. a) 2x + 3y, x = 4, y = 5 b) 3m n, m = 4, n = 10 c) x y + 1, x = 2, y = 0 Answers: a) 23, b) 2, c) 1 Operations and Algebraic Thinking 5-12 N-23

24 2. In the following magic trick, the magician can always predict the result of the sequence of operations performed on any chosen number. Try the trick with students and then encourage them to figure out how it works. Students can use blocks to represent the mystery number and counters to represent the numbers that are added. Give students lots of hints as they manipulate the concrete materials. The Trick Pick any number. Add 4. Multiply by 2. The Algebra Use a square block to represent the mystery number. Use 4 circles to represent the 4 ones that were added. Create 2 sets of blocks to show the doubling. Subtract 2. Take away 2 circles to show the subtraction. Divide by 2. Subtract the mystery number. Remove one set of blocks and circles to show the division. Remove the square. The result is 3. No matter what number you choose, after performing the operations in the magic trick, you will always get the number 3. The model above shows why the trick works. Encourage students to make up their own tricks of the same type. N-24 Teacher s Guide for AP Book 5.2

25 OA5-13 Multiplication and Word Problems Pages STANDARDS 5.NF.B.6 Goals Students will solve multiplication word problems by finding the small and large quantities scale factor. Vocabulary equation equivalent equation operation scale factor unknown variable PRIOR KNOWLEDGE REQUIRED Understands the expression times as many Can identify small and large quantities in a problem NOTE: All explanations and definitions in this lesson are based on positive numbers. If you multiply negative numbers with a fractional scale factor, the result is a greater number. If some students are familiar with operations on negative numbers, you may clarify this difference in the classroom. Identifying the larger and the smaller numbers in a situation. Point out that many problems deal with a situation in which there is a larger number and a smaller number in a multiplication relationship. Write on the board: There are 3 green apples. There are 2 times as many red apples as green apples. ASK: What types of objects are in this situation? (green apples and red apples) On the board, start a table with columns labeled Larger quantity (L) and Smaller quantity (S) ASK: Which piece of information is given: the number of green apples or the number of red apples? (green) ASK: Which sentence tells us which color of apple we have more of? (there are 2 times as many red apples as green apples) ASK: Do I need the number of green apples to find out that there are more red apples than green apples? (no) Write red apples on the board under the Larger quantity (L) column and green apples under the Smaller quantity (S) column. Some students might want to note the quantity of green apples now. You can assure them that, while you will not address the exact quantity now, you will later in the lesson plan (p. N-27). Repeat with the following situation: There are 4 times as many oranges as apples. SAY: In this situation, we only know that there are four times as many oranges as apples. ASK: Which fruit do we have more of? (oranges) Write on the board: There are 1 as many skates as bikes. 3 Explain to students that, in this situation, the number of skates is 1/3 of the number of bikes. ASK: Which are there more of, skates or bikes? (bikes) Provide the following situations one at a time, and have students signal what goes in the Larger quantity column (with thumbs up) and the Smaller quantity column (with thumbs down). Operations and Algebraic Thinking 5-13 N-25

26 Exercises: Determine the larger quantity in each situation. a) three times as many albums as books b) two times as many boys as girls 3 c) as many vans as cars 5 Answers: a) albums, b) boys, c) cars Review scale factor. Present a situation: Laura is three times as old as Sam. ASK: Who is older, Laura or Sam? (Laura) How many times? (3 times) Write the equation L = 3 S on the board. Remind students that the number that tells us how many times larger or smaller one part is than the other is called the scale factor. Remind students that multiplying by a scale factor larger than one makes the number larger, and multiplying by a scale factor smaller than one makes the number smaller. Write on the board: S = 1 3 L Explain to students that in the equation L = 3 S, the scale factor is 3, but in the equation S = 1 3 L, the scale factor is 1 3. Large is always greater than small. Write the equation on the board: L = scale factor S SAY: Large is greater than small, so to enlarge the smaller quantity, you can multiply it by a scale factor greater than one. Explain to students that the equations L = 2 S and L = 3.5 S are possible equations because the scale factor is greater than one. In contrast, L = 1 S is not possible 2 because half of a small quantity is even smaller and cannot be equal to the large part; for example, half of 3 cannot be equal to 12. Exercises: L is the larger quantity, and S is the smaller quantity. Determine which equation is possible and which is not. a) S = 3 L b) S = 1 5 L c) L = 2 S d) S = 4 3 L Answers: b) and c) are possible Exercises: Determine if the scale factor is greater than one or less than one. a) 6 = scale factor 3 b) 4 = scale factor 12 1 c) 2 = scale factor 1 Bonus: 2 5 = scale factor 1 5 Answers: a) greater than 1, b) less than 1, c) less than 1, Bonus: greater than 1 Writing equations using a scale factor. When solving word problems with scale factors, start by determining which quantity in the problem is larger. Write on the board: There are 2 times as many red apples as green apples. N-26 Teacher s Guide for AP Book 5.2

27 ASK: Which one is the larger quantity, red apples or green apples? (red apples) Write the letter L on top of the red apples and SAY: So the smaller quantity must be the green apples. Write the letter S on top of the green apples. ASK: How many times as many red apples as green apples are there? (2 times) Then write the equation L = 2 S on the board. Remind students that we usually say twice instead of two times. Exercises: Write L above the larger quantity and S above the smaller quantity. Then write the equation. a) three times as many albums as books b) twice as many boys as girls 3 c) as many vans as cars 5 Answers: a) L = 3 S, b) L = 2 S, c) S = 3 5 L In the next step, ask students to replace variables with the given number. Return to the earlier example of red and green apples (p. N-25) and write on the board: There are twice as many red apples as green apples. There are 3 green apples. Ask a volunteer to write the equation for the first line on the board. (L = 2 S) Point to the equation L = 2 S and ASK: In this equation, what variable shows the number of green apples? (S) How many green apples are there? (3) Then replace S by 3 and write L = 2 3 under the existing equation. L = 2 S L = 2 3 Exercises: Write the equation. Then replace the correct letter with the given number. a) Three times as many albums as books. There are five books. b) Two times as many boys as girls. There are 10 boys. 3 c) as many vans as cars. There are 15 cars. 5 Answers: a) L = 3 S, L = 3 5; b) L = 2 S, 10 = 2 S; c) S = 3 5 L, S = Equivalent equations. Write on the board: There are 2 times as many boys as girls. SAY: I would like to write an equation that represents the situation. Write two equations on the board: L = 2 S and S = 1 2 L Explain to students that they are the same because when there are two times as many boys as girls, there are 1/2 as many girls as boys. SAY: We call these two equations equivalent equations. Operations and Algebraic Thinking 5-13 N-27

28 Exercises: Write the equation that means the same thing. a) L = 5 S b) S = 1 4 L Answers: a) S = 1 5 L, b) L = 4 S Write on the board: There are 2 times as many boys as girls. There are 10 boys. SAY: If I write the equation in the L = scale factor S form, then the equation will be 10 = 2 S because there are 10 boys. Write the equation 10 = 2 S on the board. SAY: On the other hand, if I write the equation in the S = scale factor L form because there are 1/2 as many girls as boys, then the equation will be S = Write this equation beside the other 2 equation on the board. Point at the equations and explain that solving the equation S = 1 10 is easier because the unknown is by itself. SAY: Half 2 of 10 is equal to 5. Write on the board: S = 5 and SAY: So there are five girls. Repeat with this problem: There are 4 times as many eggs as nests. There are 8 nests. Exercises: Write the equation. If the unknown is not by itself, write the equivalent equation and then solve the equation. a) Three times as many albums as books. There are five books. b) Two times as many boys as girls. There are 10 boys. 3 c) as many vans as cars. There are 15 cars. 5 Answers: a) L = 3 5, L = 15; b) 10 = 2 S, S = 1 10, S = 5; 2 c) S = , S = 3 (15 5) = 3 3 = 9 Extensions 1. Rewrite the equation so the unknown is by itself. Then, solve the equation. a) 10 = 4 S b) 5 = 1 3 L Bonus: c) 1 2 = 4 S d) 8 = 2 3 L Answers: a) S = = 10, b) L = 5 3 = 15, 4 Bonus: c) S = = 1 8, d) L = =12 N-28 Teacher s Guide for AP Book 5.2