Contents Domain 1 Operations and Algebraic Thinking........ 4 Lesson 1 Evaluating Numerical Expressions.................... 6 Lesson 2 Writing and Interpreting Numerical Expressions........ 10 Lesson 3 Analyzing and Generating Numerical Patterns......... 14 Common Core State Standards 5.OA.1 5.OA.2 5.OA.3 Domain 1 Review........................................ 20 Domain 2 Number and Operations in Base Ten...... 24 Lesson 4 Multiplying and Dividing by Powers of Ten............ 26 Lesson 5 Using Place Value to Read and Write Decimals......... 32 Lesson 6 Comparing Decimals.............................. 38 Lesson 7 Rounding Decimals Using Place Value............. 42 Lesson 8 Multiplying Whole Numbers................. 48 Lesson 9 Dividing Whole Numbers....................... 54 Lesson 10 Adding and Subtracting Decimals................... 62 Lesson 11 Multiplying Decimals.............................. 70 Lesson 12 Dividing Decimals............................. 78 5.NBT.1, 5.NBT.2 5.NBT.3.a 5.NBT.3.b 5.NBT.4 5.NBT.5 5.NBT.6 5.NBT.7 5.NBT.7 5.NBT.7 Domain 2 Review........................................ 86 Domain 3 Number and Operations Fractions....... 90 Lesson 13 Lesson 14 Adding and Subtracting Fractions and Mixed Numbers............................... 92 Problem Solving: Adding and Subtracting Fractions and Mixed Numbers.................. 100 5.NF.1 5.NF.2 Lesson 15 Problem Solving: Interpreting Fractions as Division.................................. 106 Lesson 16 Multiplying Fractions............................. 112 Lesson 17 Interpreting Multiplication of Fractions.............. 120 Lesson 18 Problem Solving: Multiplying Fractions and Mixed Numbers.............................. 126 5.NF.3 5.NF.4.a, 5.NF.4.b 5.NF.5.a, 5.NF.5.b 5.NF.6 2 Problem Solving Fluency Lesson Performance Task
Common Core State Standards Lesson 19 Dividing with Unit Fractions and Whole Numbers...... 132 Lesson 20 Problem Solving: Dividing with Unit Fractions..... 138 5.NF.7.a, 5.NF.7.b 5.NF.7.c Domain 3 Review....................................... 142 Domain 4 Measurement and Data................... 146 Lesson 21 Converting Units of Measure to Solve Problems....... 148 Lesson 22 Line Plots....................................... 154 Lesson 23 Understanding and Measuring Volume.............. 160 5.MD.1 5.MD.2 5.MD.3.a, 5.MD.3.b, 5.MD.4 Lesson 24 Finding Volume of Rectangular Prisms...................................... 164 5.MD.5.a, 5.MD.5.b Lesson 25 Recognizing Volume as Additive................ 170 5.MD.5.c Domain 4 Review....................................... 174 Domain 5 Geometry.................................. 178 Lesson 26 Graphing Points on the Coordinate Plane............ 180 Lesson 27 The Coordinate Plane......................... 186 5.G.1 5.G.2 Lesson 28 Extending Classification of Two-Dimensional Figures.......................... 190 5.G.3, 5.G.4 Domain 5 Review....................................... 196 Glossary................................................. 200 Math Tools................................................ 203 3
1 LESSON Evaluating Numerical Expressions EXAMPLE A Evaluate this numerical expression. (15 2 5) 1 (4 3 3) 1 Do the operation inside the first set of parentheses ( ). Subtract. (15 2 5) 5 10 10 1 (4 3 3) 2 Do the operation inside the second set of parentheses. Multiply. 4 3 3 5 12 3 Evaluate the remaining expression. Add. 10 1 12 5 22 10 1 12 22 The value of the expression is 22. DISCUSS Explain how you would evaluate (3 3 9) 2 (8 1 8). What is the value of the expression? 6 Domain 1: Operations and Algebraic Thinking
EXAMPLE B Evaluate this numerical expression. 30 2 [(5 3 4) 4 2] 1 17 1 Do the operation inside the parentheses. Multiply. (5 3 4) 5 20 30 2 [20 4 2] 1 17 2 Do the operation inside the brackets [ ]. Divide. [20 4 2] 5 10 3 Evaluate the remaining expression. The subtraction comes first in the expression. Subtract first. 30 2 10 5 20 30 2 10 1 17 20 1 17 Then add. 20 1 17 5 37 37 The value of the expression is 37. TRY Insert brackets in 36 4 (12 2 9) 3 3 so that the value of the expression is 4. Lesson 1: Evaluating Numerical Expressions 7
Practice Use the following expression to answer questions 1 through 3. 13 1 (8 3 3) 1. What operation should you do first? HINT Which expression is inside the parentheses? 2. What operation should you do next? 3. What is the answer? Use the order of operations to evaluate each expression. 4. [8 1 (12 2 6)] 3 5 5. 7 3 (6 2 2) 4 2 The value of the expression is. The value of the expression is. REMEMBER Start with the operation inside the parentheses. Then work inside the brackets. 6. 2 3 (3 3 4) 2 (5 1 13) 7. [8 3 (5 1 5)] 4 4 The value of the expression is. 8. (6 1 8) 4 2 1 1 The value of the expression is. The value of the expression is. 9. 55 4 5 2 [(3 3 2) 1 4] The value of the expression is. 8 Domain 1: Operations and Algebraic Thinking
Insert parentheses so that the value of the expression matches the value given. 10. 10 1 7 3 6 2 2 The value of the expression is 50. 11. 9 3 4 1 8 4 2 The value of the expression is 40. 12. 25 2 10 1 5 3 4 The value of the expression is 35. 13. 4 1 6 4 2 2 5 The value of the expression is 2. Choose the best answer. 14. Which operation should you do first to evaluate 5 1 [(3 3 4) 2 7]? A. addition B. subtraction C. multiplication D. division 15. Which expression does not have a value of 6? A. 3 1 (7 3 2) 2 11 B. 2 3 [(2 1 5) 2 3] C. (8 4 4) 1 (2 3 2) D. [4 2 (8 2 7)] 3 2 Solve. 16. Denzel bought a large pizza for $14 and 2 drinks for $3 each. He had a coupon for $1 off the total amount of his purchase. The following expression represents how much Denzel spent. 14 1 (2 3 3) 2 1 How much did Denzel spend on the pizza and the 2 drinks? 17. DECIDE Marcus says the value of 1 1 [4 3 (9 2 6)] is 31. Erica says the value of the expression is 13. Who is correct? Explain. Lesson 1: Evaluating Numerical Expressions 9
Contents Domain Assessment Operations and Algebraic Thinking.......................... 4 Domain Assessment Number and Operations in Base Ten........................ 14 Domain Assessment Number and Operations Fractions........................ 22 Domain Assessment Measurement and Data.................................. 32 Domain Assessment Geometry............................................ 42 Summative Assessment.................................................... 52 3
Domain Assessment Operations and Algebraic Thinking 1. What does the expression 75 2 5 mean? 4 A. 75 divided by 5 B. 5 less than 75 C. 5 more than 75 D. 75 times 5 2. Which expression means the same as 9 more than 18 divided by 3? A. (18 1 3) 4 9 B. (9 1 3) 4 18 C. (18 4 3) 1 9 D. (9 4 3) 1 18 3. What is the value of 21 1 4 3 8 2 2? A. 51 B. 98 C. 150 D. 198 4. Which word phrase is equivalent to 35 1 28 2 10? A. 35 more than the quotient of 28 divided by 10 B. 10 less than the sum of 35 and 28 C. 10 more than the sum of 35 and 28 D. the sum of 35 and 28 and 10 5. Which of the following is a true statement? A. 76 2 (6 3 2) is 76 less than (6 3 2). B. (100 2 25) 4 3 is 3 times as great as (100 2 25). C. 5 3 (5 1 10) is 5 more than (5 1 10). D. 10 3 (43 2 3) is 10 times as great as (43 2 3). 6. How would you use numbers and symbols to express 28 divided by the product of 1 2 and 8? A. 28 4 ( 1 8 3 8 ) B. 28 4 ( 1 2 3 8 ) C. 28 2 ( 1 2 3 8 ) D. 1 2 3 8 3 28 7. Use the order of operations to evaluate 1 4 3 (40 1 16). A. 14 B. 26 C. 44 D. 176 Go On
8. Look at the patterns below. Pattern 1: 2, 5, 8, 11, 14, Pattern 2: 2, 14, 26, 38, 50, What are the rules for the patterns? A. Pattern 1: add 3; Pattern 2: add 6 10. Khalia s teacher wrote two rules on the board. Start at 3, add 5 Start at 6, add 4 Operations and Algebraic Thinking B. Pattern 1: add 3; Pattern 2: add 12 C. Pattern 1: add 3; Pattern 2: add 14 D. Pattern 1: add 3; Pattern 2: multiply by 7 9. Use the order of operations to evaluate 5 3 [36 4 (4 3 3)]. A. 15 B. 29 C. 90 D. 135 If Khalia were to write out the sequences based on these two rules, what would they look like? A. 3, 6, 9, 12, 15, 18 and 6, 10, 14, 18, 22, 26 B. 3, 8, 11, 14, 17, 20 and 6, 12, 18, 24, 30, 36 C. 3, 8, 11, 14, 17, 20 and 6, 12, 18, 24, 30, 36 D. 3, 8, 13, 18, 23, 28 and 6, 10, 14, 18, 22, 26 Go On 5
11. Jamie and Marcus created two different patterns. The rule for Jamie s pattern was start at 2 and add 2. The rule for Marcus s pattern was start at 2 and multiply by 2. Which of the following tables shows the ordered pairs made from the corresponding terms of the sequences for Jamie and Marcus s patterns? A. Jamie s Pattern Marcus s Pattern Ordered Pair C. Jamie s Pattern Marcus s Pattern Ordered Pair 2 2 (2, 2) 4 4 (4, 4) 6 8 (6, 8) 8 16 (8, 16) 10 32 (10, 32) 2 4 (2, 4) 2 4 (2, 4) 6 8 (6, 8) 8 16 (8, 16) 8 10 (8, 10) B. Jamie s Pattern Marcus s Pattern Ordered Pair D. Jamie s Pattern Marcus s Pattern Ordered Pair 2 4 (2, 4) 6 8 (6, 8) 10 2 (10, 2) 4 8 (4, 8) 16 32 (16, 32) 2 4 (2, 4) 4 2 (4, 2) 6 8 (6, 8) 8 6 (8, 6) 8 16 (8, 16) Go On 6
12. Which of the following shows how to find the value of the expression 7 3 (18 2 9)? A. Find the quotient of 18 divided by 9, then multiply by 7. B. Find the difference of 18 and 9, then multiply by 7. 14. Which expression has a value of 33? A. 5 1 (10 3 2) 1 8 4 4 1 1 B. 5 1 10 3 2 1 (8 4 4) 1 1 C. 5 1 10 3 (2 1 8) 4 4 1 1 D. (5 1 10) 3 2 1 8 4 4 1 1 Operations and Algebraic Thinking C. Find the sum of 18 and 9, then multiply by 7. D. Multiply 7 and 18, then subtract 9. 13. Use the order of operations to evaluate 2 5 8 1 1 4 2 1 1 2 A. 1 1 4 B. 1 3 8 C. 2 D. 2 3 8 Go On 7
15. Zack created a table of ordered pairs from the sequences for two patterns. The rules for the two patterns are shown below. Pattern A: Start at 0 and add 4 Pattern B: Start at 0 and add 6 Pattern A gives the x-value and Pattern B gives the y-value. Zack graphed the ordered pairs, as shown below. y 32 28 24 20 16 12 8 4 0 2 4 6 8 10 12 14 16 18 x Which of these tables best reflects the patterns displayed in Zack s graph? A. Pattern A Pattern B Ordered Pair 0 1 (0, 1) 4 12 (4, 12) 8 18 (8, 18) 12 22 (12, 22) 18 36 (18, 36) C. Pattern A Pattern B Ordered Pair 0 0 (0, 0) 4 6 (4, 6) 8 12 (8, 12) 12 18 (12, 18) 16 24 (16, 24) B. Pattern A Pattern B Ordered Pair 1 0 (1, 0) 4 6 (4, 6) 8 12 (8, 12) 12 18 (12, 18) 16 24 (16, 24) D. Pattern A Pattern B Ordered Pair 0 12 (0, 12) 4 24 (4, 24) 8 36 (8, 36) 12 48 (12, 48) 16 60 (16, 60) Go On 8
16. What is the value of the expression [(7 3 8) 4 2] 2 11? Show your work. Operations and Algebraic Thinking 17. Xenia told her teacher that she had found a mistake in her math textbook. In one of the lessons, it said that (222 1 11) 4 3 is three times as great as the sum of 222 and 11. What mistake did Xenia find? Go On 9
18. Ms. Solis wrote the following equation on the blackboard for her fifth-grade class. 1 61-7 7 + 17 = 47 2 Where in the equation do you need to add parentheses and brackets so that the equation is correct? Explain your answer using an example of what would happen if parentheses and brackets are added incorrectly. Go On 10
19. A fancy pastry shop charges $2 for each macaron and $6 for each petit four. macarons, $2 petit fours, $6 Operations and Algebraic Thinking A. Write the pattern that shows the cost of 0 to 6 macarons. Write the pattern that shows the cost of 0 to 6 petit fours. B. Use the corresponding terms from the patterns you created in Part A to complete the table below. Then describe the relationship between corresponding terms. Macarons Pattern (in $) Petit Fours Pattern (in $) Ordered Pair Go On 11
20. Will chose 2 cards from a pile. The cards are shown below. Card A Card B Start with 0 and add 2. Start with 0 and add 3. Will used the rules on the cards to write a sequence of numbers for each pattern. He then created a table of ordered pairs using the corresponding terms from the patterns. A. Use the blank table below to show how Will completed his table of ordered pairs. Card A Pattern Card B Pattern Ordered Pair Go On 12
B. Graph the ordered pairs on the graph below. Be sure to label each axis and scale. Then describe the relationship between the terms that make up the ordered pairs. y Operations and Algebraic Thinking x STOP 13
Contents Instructional Overview......................................... 4 Common Core State Standards Correlation Chart................. 12 Domain 1 Operations and Algebraic Thinking.......... 16 Lesson 1 Evaluating Numerical Expressions..................... 18 Lesson 2 Writing and Interpreting Numerical Expressions.......... 20 Lesson 3 Analyzing and Generating Numerical Patterns........... 22 Common Core State Standards 5.OA.1 5.OA.2 5.OA.3 Domain 2 Number and Operations in Base Ten......... 24 Lesson 4 Multiplying and Dividing by Powers of Ten.............. 26 Lesson 5 Using Place Value to Read and Write Decimals........... 28 Lesson 6 Comparing Decimals................................ 30 Lesson 7 Rounding Decimals Using Place Value............... 32 Lesson 8 Multiplying Whole Numbers................... 34 Lesson 9 Dividing Whole Numbers......................... 36 Lesson 10 Adding and Subtracting Decimals..................... 38 Lesson 11 Multiplying Decimals................................ 40 Lesson 12 Dividing Decimals............................... 42 5.NBT.1, 5.NBT.2 5.NBT.3.a 5.NBT.3.b 5.NBT.4 5.NBT.5 5.NBT.6 5.NBT.7 5.NBT.7 5.NBT.7 Domain 3 Number and Operations Fractions.......... 44 Lesson 13 Adding and Subtracting Fractions and Mixed Numbers.... 46 5.NF.1 Lesson 14 Problem Solving: Adding and Subtracting Fractions and Mixed Numbers............................. 48 5.NF.2 Lesson 15 Problem Solving: Interpreting Fractions as Division.... 50 5.NF.3 Lesson 16 Multiplying Fractions................................ 52 Lesson 17 Interpreting Multiplication of Fractions................. 54 Lesson 18 Problem Solving: Multiplying Fractions and Mixed Numbers................................. 56 Lesson 19 Dividing with Unit Fractions and Whole Numbers........ 58 Lesson 20 Problem Solving: Dividing with Unit Fractions........ 60 5.NF.4.a, 5.NF.4.b 5.NF.5.a, 5.NF.5.b 5.NF.6 5.NF.7.a, 5.NF.7.b 5.NF.7.c 2 Problem Solving Fluency Lesson Performance Task
Common Core State Standards Domain 4 Measurement and Data....................... 62 Lesson 21 Converting Units of Measure to Solve Problems.......... 64 Lesson 22 Line Plots.......................................... 66 Lesson 23 Understanding and Measuring Volume................. 68 Lesson 24 Finding Volume of Rectangular Prisms............... 70 Lesson 25 Recognizing Volume as Additive................... 72 5.MD.1 5.MD.2 5.MD.3.a, 5.MD.3.b, 5.MD.4 5.MD.5.a, 5.MD.5.b 5.MD.5.c Domain 5 Geometry...................................... 74 Lesson 26 Graphing Points on the Coordinate Plane............... 76 Lesson 27 The Coordinate Plane............................ 78 Lesson 28 Extending Classification of Two-Dimensional Figures...... 80 5.G.1 5.G.2 5.G.3, 5.G.4 Answer Key................................................... 82 Math Tools................................................... 104 Appendix A: Fluency Practice.................................... A Appendix B: Standards for Mathematical Practice.................. B 3
LESSON 1 Evaluating Numerical Expressions Learning Objective Students will evaluate numerical expressions using the correct order of operations. Vocabulary numerical expression a combination of numbers and operation signs Common Core State Standard 5.OA.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. Before the Lesson Write a numerical expression that includes parentheses, such as (12 1 8) 2 (3 3 5), and ask for a volunteer to evaluate it. Then write an expression such as the following: 18 1 [(2 3 3) 4 3] 2 2. Ask another student to evaluate the expression. Ask the students to explain what is different about the two expressions, aside from the numbers. Encourage them to notice that the first expression includes two sets of parentheses, while the second expression includes both parentheses and brackets. They should notice, too, that the parentheses are inside the brackets. Be sure students understand that they should work from the inside to the outside, so that they start with the innermost set of parentheses and do the work inside. Then they work on what is inside the brackets. After all the grouping symbols have been removed, they use the remaining operation symbols to finish evaluating the expression. Examples EXAMPLE A This example illustrates how to apply the order of operations to a numerical expression that includes parentheses. Emphasize that the operations that are inside parentheses must be performed first. Ask: Which two operations must be performed first? As students look over their work, point out how after each operation is performed, the entire expression is rewritten with the result of that operation substituted into it. Stress the importance of rewriting the expression to help students keep track of their work. DISCUSS MP7 Use this feature to encourage students to talk about the procedures involved in evaluating a numerical expression. First perform the operations in parentheses: 3 3 9 5 27 and 8 1 8 5 16. Then subtract: 27 2 16 5 11; the answer is 11. EXAMPLE B This example shows how to evaluate a numerical expression that includes two types of grouping symbols: parentheses and brackets. Use this example to demonstrate that students must evaluate the entire expression inside the brackets, (5 3 4) 4 2, before they evaluate anything else. Ask: Which operation must be performed first? Which must be performed second? Point out how the entire expression is rewritten with 10 replacing the entire expression inside the brackets. Reinforce the importance of rewriting the entire expression to help students keep track of their work. To ensure that students understand this content, rewrite the problem on the board as [30 2 (5 3 4)] 4 2 1 17. Ask: How would the value of the expression change if the brackets were moved like this? Use the fact that the value is now 22 to show students how important it is to pay attention to the positioning of the parentheses and brackets. 18
TRY MP1 Encourage students to perform operations mentally or use trial and error to determine where to place the brackets to get the correct result. If students use trial and error, ask students how many tries it took to find the correct answer. 36 4 [(12 2 9) 3 3] Practice As students are working, pay special attention to problem 16, which provides an opportunity for students to see how a numerical expression that includes parentheses can be used to represent a real-world problem. For answers, see page 82. Common Errors When evaluating numerical expressions with parentheses inside brackets, some students may not do the work inside the parentheses first before evaluating the rest of the expression inside the brackets. Remind students to treat the entire expression inside the brackets as a numerical expression and do what is inside the parentheses first. Domain 1 19
LESSON 2 Writing and Interpreting Numerical Expressions Learning Objectives Students will write numerical expressions to represent verbal phrases or statements using grouping symbols to indicate which operations are performed first. Students will write verbal phrases or statements to represent numerical expressions. Before the Lesson Before class, get a bottle of water, with the cap screwed on, and a cup. Tell students to imagine that you are an alien from outer space. Tell them that the alien wants to know how to pour a cup of water for itself. Ask: How would you tell the alien to do this, using simple steps? Write students responses on the board. Using bullets, list each step on the board or on a sheet of paper. You can list them one above the other or next to one another. For example, students might suggest these steps: Unscrew the cap of the bottle. Remove the cap. Tilt the bottle over the glass so that its top is over the glass. Then follow the directions out of order, stating aloud which step you are currently following. For example, tilt the bottle with the cap still screwed on over the glass. Then unscrew the cap and remove it. Ask: Why didn t it work? The alien followed all your steps. Use this to explain that sometimes, when you write directions, the order of the steps matters and sometimes it does not. Ask students how you could have made this clearer to the alien. They may suggest replacing the bullets with ordinal numbers. Explain that sometimes students will need to write a numerical expression to represent a phrase or a statement. If the order in which the operations must be performed matters, students should use parentheses or other grouping symbols to indicate the correct order. Common Core State Standard 5.OA.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation add 8 and 7, then multiply by 2 as 2 3 (8 1 7). Recognize that 3 3 (18,932 1 921) is three times as large as 18,932 1 921 without having to calculate the indicated sum or product. Examples EXAMPLE A This example introduces how to write a numerical expression to represent a phrase. When they are considering the first part of the expression, be sure students understand that subtract 6 from 10 means 10 2 6, not 6 2 10. If necessary, review the fact that subtraction is not commutative to help students understand why it is important to pay attention to which number is being subtracted. Emphasize that parentheses must be included in the expression to clarify which operation should be performed first. Since 6 must be subtracted from 10 and only then multiplied by 5, parentheses must be placed around the subtraction expression 10 2 6. 20
TRY MP2 Encourage students to include parentheses in the numerical expressions they write. Emphasize that including the parentheses makes it very clear which operation is to be performed first. 12 1 8; (6 2 5) 1 3; (18 4 6) 3 9 EXAMPLE B This example is the reverse of Example A. It shows students how to write a word phrase to represent a numerical expression. Stress that there may be more than one way to do this. For example, after working through Example B, ask: Can you think of another way to use words to represent this expression? Students may suggest add 2 and 7, then divide by 3 or divide the sum of 7 and 2 by 3. DISCUSS MP7 MP8 Emphasize that students should not evaluate (9 1 1) 3 6 but should instead apply their understanding of the work they performed for (10 2 6) 3 5 to reason the answer. If some students are not convinced, you may need to evaluate (9 1 1) 3 6 to show that it is six times as large as (9 1 1). (10 2 6) 3 5 5 20; (10 2 6) 5 4; 20 is five times as great as 4. Answers may vary. Possible answer: Without calculating, you can tell that the value of (9 1 1) 3 6 is six times as great as the value of (9 1 1). Practice As students are working, pay special attention to problems 21 and 22, which require students to translate real-world problem situations into numerical expressions. These are less straightforward than other translations in this lesson. For example, help any students who are confused by problem 21 see that having 12 crackers and eating 2 can be represented as the subtraction expression 12 2 2. Then help them see why John s dividing the crackers equally between himself and his friend is a division by 2. For answers, see page 82. Common Errors A common error may be not understanding that the work inside parentheses is done first, no matter where parentheses fall in the expression. If students do not understand this, they may write multiply 24 by 709 and then add 54 for question 13. In addition, they may believe that problem 19 has no correct answer and that only the expression (593 1 88) 3 12 is a correct way to represent the sentence. Review the fact that multiplication is commutative to help students see that 12 3 (593 1 88) 5 (593 1 88) 3 12, so either is the correct answer for problem 19. Domain 1 21
LESSON 3 Analyzing and Generating Numerical Patterns Learning Objective Students will find the rule for a numerical pattern, identify the relationship between corresponding terms in two different numerical patterns, write ordered pairs to represent corresponding terms in two different numerical patterns, and plot ordered pairs on the coordinate plane. Vocabulary coordinate plane a grid formed by a horizontal line, called the x-axis, and a vertical line, called the y-axis ordered pair two numbers that give a location on a coordinate plane pattern a series of numbers or figures that follows a rule rule tells how the numbers in a pattern are related term a number or figure in a pattern x-axis the left-right or horizontal axis on a coordinate plane x-coordinate the first number in an ordered pair y-axis the up-down or vertical axis on a coordinate plane y-coordinate the second number in an ordered pair Common Core State Standard(s) 5.OA.3 Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule Add 3 and the starting number 0, and given the rule Add 6 and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so. Before the Lesson Write the following on the board: 1, 3,,. Ask: What number goes in the blank to continue the pattern? Some students may say that the next number is 5. Others may say it is 9. Still others may correctly state that not enough information is given to determine the next number. Whatever answers students give, use them to illustrate the fact that the numbers, or terms, in a pattern follow a rule. Students who believe the next number is 5 believe that the rule is add 2. Students who believe that the next number is 9 believe that the rule is multiply by 3. Additional information is needed to determine the rule, but if these numbers form a pattern, they must follow a rule. Examples EXAMPLE A This example introduces how to find a rule for a numerical pattern. Emphasize the fact that since the numbers are increasing, the pattern could involve either addition or multiplication. TRY MP8 Explain that students should start with the term 57 and then apply the rule add 9 to find the next three consecutive terms. 66, 75, 84 EXAMPLE B This example requires students to relate the corresponding terms of two patterns. Be sure to stress to students that even though the first terms in both patterns are the same, 0, the other consecutive terms are not the same, so they should try to determine how the terms are related. 22
TRY MP7 Encourage students to use the rule for each pattern to find its next term. Emphasize that if they do this correctly, the relationship between the two terms they find will be the same as the relationship between Pattern 1 and Pattern 2 corresponding terms in Example B. 25; 75; Answers may vary. Possible answer: The next term in Pattern 2 is three times the corresponding term in Pattern 1. DISCUSS MP6 To benefit visual learners, show how to plot (3, 5) and (5, 3) on the same grid. This will help students understand that the two ordered pairs represent different locations on the coordinate plane. Answers may vary. Possible answer: To graph (3, 5), move 3 units to the right and then move 5 units up. To graph (5, 3), move 5 units to the right and then move 3 units up. Domain 1 EXAMPLE C This example is similar to Example B, except that the rules and first terms are given and students must generate the first several terms in each pattern. Explain that the ellipsis ( ) is placed after the last term found for each pattern to show that the established pattern will continue. TRY MP7 Encourage students to find at least five terms in each pattern and to organize the terms in a table so it is easier for them to compare corresponding terms. The first pattern is 0, 2, 4, 6, 8,. The second pattern is 0, 6, 12, 18, 24,. Answers may vary. Possible answer: The terms in the second pattern are three times the corresponding terms in the first pattern. This is true because adding 6 to each term in the second pattern is three times as much as adding 2 to each term in the first pattern. EXAMPLE D This example shows how corresponding terms in two patterns can be considered as ordered pairs and how those ordered pairs can be plotted on a grid to help students see the relationship between them. If necessary, review how to plot points on a coordinate plane. For example, show students that (2, 4) can be located by starting at the origin and moving 2 units to the right and 4 units up. Show that the next point, (4, 8), can be found by starting at (2, 4) and moving 2 units to the right and 4 units up. Emphasize that for each unit you move to the right, you move twice as many units up. This shows that each term in Pattern 2 is twice that of its corresponding term in Pattern 1. Practice As students are working, pay special attention to problems 6 through 8, which provide an opportunity for students to generate patterns and then explore the relationship between them by writing and plotting ordered pairs. While reviewing the answers, ask: What is the relationship between the corresponding terms in the pattern? After establishing that each term in the second pattern is 3 times its corresponding term in the first pattern, have students describe all the different ways they could determine this by comparing the rules for the patterns, by comparing the x- and y-coordinates of the ordered pairs, and by examining the graph. For answers, see pages 82 and 83. Common Errors One error that students may make when determining the rule for a pattern is not considering all of the terms. If students consider only the first and second terms, 1 and 4, for problem 2 on page 18, they may mistakenly believe that the rule is add 3. Reinforce the importance of considering more than three terms in a pattern to be sure the correct rule has been determined. 23