Unit 3 Number and Operations in Base Ten: Multiplication

Transcription

1 Unit 3 Number and Operations in Base Ten: Multiplication Introduction In this unit, students will learn how to perform multiplication using the standard algorithm. Students will build on their understanding of how to compute products of one-digit numbers and multiples of 10,100, and 1,000. Students will display an understanding of the role played by the distributive property in the multiplication process. They will also connect diagrams of areas or arrays to numerical work to develop understanding of general base ten multiplication methods. The unit begins with multiplication of single-digit numbers and progresses to multiplication of multi-digit numbers by 2-digit numbers. Methods include expanded form, base ten materials, and the standard algorithm for multiplication. NOTE: The term product in this unit is used to refer to both an expression involving multiplication and to the evaluation of that expression. So, for example, the product of 3 and 4 is both 3 4 and 12. The use of grid paper for multiplication using the standard algorithm helps student organize their work and line up digits in the appropriate columns. If students do not use grid paper notebooks in general, you will need to have lots of grid paper on hand throughout the unit. Number and Operations in Base Ten D-1

2 NBT5-13 Introduction to Multiplication Pages STANDARDS preparation for 5.NBT.B.5 Vocabulary addition array column commutative multiplication product row Goals Students will write a product as repeated addition. Students will recognize the commutative property of multiplication, and write a product for a given array. PRIOR KNOWLEDGE REQUIRED Can add single-digit numbers MATERIALS BLM Multiplication Charts (pp. D-55 57) BLM Using the Multiplication Chart to Multiply (p. D-58) calculators BLM Multiplication Facts Commutative Property (p. D-59) NOTE: The Grade 3 Common Core State Standards require that students know all one-digit multiplication facts. You can use the exercises in How to Learn Your Times Tables in a Week (p. A-47), and in the BLM Multiplication Charts and BLM Using the Multiplication Chart to Multiply to help students learn their facts. We recommend you review basic facts regularly with students who are struggling. Review the concept of multiplication as repeated addition. On the board, write: 5 4 =? Ask students for the answer. Allow time for the class to agree on an answer. ASK: How can we draw a diagram to show that 5 4 = 20? Although a diagram such as: that has 4 groups with 5 items in each group is correct, try to direct students toward a diagram with 5 groups of 4 items in each group, such as: Students should think of 5 4 as adding five 4s. The first number is the number of times we are adding the second number. ASK: What addition statement could we write for this diagram? ( = 20) Tell students that 5 4 is really just a short form for adding five 4s. D-2 Teacher s Guide for AP Book 5.1

3 Draw the following diagram on the board (which students may have already suggested for the previous question): ASK: What addition statement could we write for this diagram? ( = 20) Remind students that this is a way of adding four 5s to get 20. ASK: What multiplication question can you write for = 20? Although 5 4 = 20 is correct, direct them toward 4 5 = 20 instead. Students should think of 4 5 as adding four 5s. Exercises: Write each multiplication question as an addition question. a) 5 3 = 15 b) 4 7 = 28 c) 5 1 = 10 d) 3 0 = 0 Answers: a) = 15, b) = 28, c) = 5, d) = 0 Exercises: Write each addition question as a multiplication question. a) = 12 b) = 16 c) = 36 Answers: a) 3 4 = 12, b) 2 8 = 16, c) 4 9 = 36 Writing a product for a given array. Draw the following diagrams on the board: Using dots: Using squares: ASK: What product is represented by each diagram? If students are having difficulty, have them count the number of rows and number of columns. Remind them that rows are horizontal, while columns are vertical. row row row col col col col row row row col col col col Tell students that the product is created by writing the number of rows first, then the number of columns. Here, the number of rows is 3 and the number of columns is 4, so we write the product 3 4. Number and Operations in Base Ten 5-13 D-3

4 Exercises: Write a product for each diagram. a) b) c) d) Answers: a) 4 6, b) 2 5, c) 3 2, d) 4 1 Recognizing the commutative property of multiplication. The commutative property of multiplication states that you can reverse the numbers in a product, and the answers will be equivalent. For example, 5 4 = 4 5. Draw on the board: ASK: What multiplication is represented by each array? (3 4, 4 3) ASK: What can we do to the diagram on the left to make it look like the diagram on the right? (rotate the diagram 90 degrees) SAY: Notice that when we rotate the diagram on the left, the total number of dots doesn t change! So, 3 4 = 4 3. Have students draw an array diagram for 4 5, then for 5 4. ASK: Does rotating the diagram on the left give you the diagram on the right? (yes) ASK: What can we say about 4 5 and 5 4? (they are equivalent) ASK: What do you notice about the answer to a multiplication equation when the numbers are reversed? (the answers are the same) Ask students for other examples that show this is true. (sample answers: 7 4 = 28 and 4 7 = 28, 9 3 = 27 and 3 9 = 27) Bonus: If = 195, what is the answer for 15 13? (195) Ask students to use their calculators to create similar bonus questions with greater numbers. (sample answers: = 19,890 and =?) Using the commutative property to review multiplication facts. Have students complete the multiplication facts in BLM Multiplication Facts Commutative Property. Make sure they notice that each fact they know automatically gives them another fact. D-4 Teacher s Guide for AP Book 5.1

5 Review the perfect squares: 1 1 = 1, 2 2 = 4, 3 3 = 9, and so on up to 9 9 = 81. The perfect squares form a diagonal in the table diagonal of perfect squares SAY: If you know 7 3 = 21, fill in the answer 21. Then travel diagonally through the line of perfect squares to find the spot for 3 7 and fill in the answer 21. Have students complete the multiplication table using the facts they know, and using the commutative property to find another fact. Extensions (MP.2) 1. Can we reverse the order of the numbers in an addition statement and still get the same answer? Provide examples. Answers: yes; = 7 + 5, = 6 + 9, = (MP.2) 2. Can we reverse the order of the numbers in a subtraction statement and still get the same answer? If not, provide a counter example. Answers: no; , (MP.1) 3. The rows and columns of the times tables have been mixed up. Fill in the missing numbers. a) b) Answers a) b) Number and Operations in Base Ten 5-13 D-5

6 (MP.1) 4. Use the numbers 1 to 5 to fill in the missing numbers: Answers: 2 5 = 10, 3 4 =12 (MP.1) 5. Use all the numbers from 0 to 9 to fill in the missing numbers Answers: 8 3 = 24 or 3 8 = 24, 9 6 = 54, 7 5 = 35 or 5 7 = 35, 6 5 = 30, 6 2 = 12 D-6 Teacher s Guide for AP Book 5.1

7 NBT5-14 Multiplication by Adding On Pages STANDARDS preparation for 5.NBT.B.5 Goals Students will find products by adding onto smaller products. Vocabulary array column equation product row sum PRIOR KNOWLEDGE REQUIRED Can add and multiply Can represent multiplication in different ways Finding a product for a given array. Draw the following diagram on the board. For each diagram, ask for volunteers to count the number of rows and number of columns. Remind students that rows are horizontal and columns are vertical a) b) rows columns rows columns Answers: a) 3 rows, 4 columns; b) 4 rows, 6 columns Ask students to write a product for each diagram. Answers: a) 3 4, b) 4 6 Finding a product by adding to a smaller product using arrays. Draw the following diagram on the board: Ask for three volunteers to write a product for each of the circled arrays. Answers: 5 3, 4 3, 1 3 SAY: But the number of dots in the diagrams is the same. So the product on the left must be the same as the sum of the product on the right! So, 5 3 = (4 3) + (1 3). + Number and Operations in Base Ten 5-14 D-7

8 Draw the following diagram on the board. SAY: We can put both diagrams into one diagram. Note that 1 3 = = (4 3) + 3 Exercises: Fill in the missing products and number. Then write an equation. a) b) + + Answers a) 5 4 = (4 4) + 4 b) 3 5 = (2 5) + 5 Finding a product by adding on to a smaller product without using arrays. Write the following on the board: 3 6 = ( 2 6) = ( 4 7) + 7 ASK: In each equation, what is the pattern in the underlined numbers? (the second underlined number is one less than the first underlined number) Exercises: Turn each product into a lesser product and a sum. a) 7 4 = ( 4 ) + 4 b) 9 5 = ( 5) + 5 Answers: a) 7 4 = (6 4) + 4; b) 9 5 = (8 5) + 5 (MP.2) Exercises: Find the product by using a lesser product and sum. a) 8 6 = b) 9 4 = (MP.2) (MP.2) Answers a) 8 6 = (7 6) + 6 b) 9 4 = (8 4) + 4 = = = 48 = 36 Extensions 1. Have students use BLM Finding Easier Ways to Multiply (pp. D-60 62) to learn to turn products into sums of easier products. 2. For extra practice with mental math, have students follow BLM Using Doubles to Multiply (pp. D-63 64) and BLM Using Triples to Multiply (p. D-65). D-8 Teacher s Guide for AP Book 5.1

9 NBT5-15 Multiplying by Tens, Hundreds, Pages and Thousands STANDARDS 5.NBT.A.2, 5.NBT.B.5 Goals Students will multiply by multiples of 10 and 100. Vocabulary multiple power PRIOR KNOWLEDGE REQUIRED Can use base ten materials to represent numbers Can multiply single-digit numbers MATERIALS base ten materials Multiplying a single digit number by multiples of 10. Give each student 9 ones blocks, 9 tens blocks, and 9 hundreds blocks. ASK: Which block is equal in value to 10 ones blocks? (1 tens block) Remind students that = 1 = 10 = 100 Ask students to take 4 ones blocks = 10 SAY: Since each ones block gets multiplied by 10, replace each ones block by a tens block = 10 = = 40 Exercises: Use base ten blocks to model multiplication. a) 10 5 b) 7 10 c) 10 3 Answers: a) 5 tens blocks, b) 7 tens blocks, c) 3 tens blocks For question b), you may have to remind students that 7 10 = ASK: What is a shortcut to multiplying by 10? (add a zero to the end of the number being multiplied by 10) If students are struggling for the answer, write on the board: 10 5 = = = 30 Number and Operations in Base Ten 5-15 D-9

10 4 50 ASK: How many tens are in 50? (5) Draw on the board: 4 50 = 4 = 4 5 tens ASK: How many tens are there after we multiply by 4? (20) What number is the same as 20 tens? (200) Exercises: Multiply. a) 5 30 b) 6 40 c) 7 30 Answers: a) 150, b) 240, c) 210 ASK: What is a shortcut to multiplying a one-digit number by multiples of 10? (multiply the non-zero digits, then add a zero) If students are struggling for the answer, write on the board: 5 30 = = = 210 Multiplying a multi-digit number by multiples of ASK: How many tens are in 30? (3) Which block is the same as ten tens blocks? (a hundreds block) = 10 = = 300 (MP.5) Exercises: Model using base ten blocks as in the example above. a) b) c) Answers: a) 5 hundreds blocks, b) 4 hundreds blocks, c) 2 hundreds blocks For question b), you may have to remind students that = ASK: What is a shortcut to multiplying a multiple of 10 by 10? (multiply the non-zero digits, then add two zeros) If students are struggling for the answer, write on the board: = = = 200 D-10 Teacher s Guide for AP Book 5.1

11 Multiplying a multi-digit number by multiples of ASK: How many hundreds are in 200? (2) What block is the same as 10 hundreds blocks? (a thousands block) Draw on the board: = 10 = = 2,000 (MP.5) Exercises: Solve using base ten materials. a) b) c) Answers: a) 3,000, b) 4,000, c) 6,000 Multiplying a multi-digit number by multiples of 100 without blocks ASK: How many tens are in 30? (3) How many tens are in 40? (4) = (3 tens) (4 tens) = (3 10) (4 10) SAY: With multiplication, we can change the order of the numbers. Write only the first line of the following problem on the board: = (3 4) (10 10) = = 1,200 (MP.2) Point to the first part of the problem and ASK: How much is 3 4? (12) Write 12 on the board. Point to the second part and ASK: How much is 10 10? (100) Finish writing the second line of the problem. ASK: How much is ? (1,200) Write the answer. Exercises: Multiply. a) b) c) Answers: a) 1,400, b) 1,800, c) 20,000 ASK: What is a shortcut for this multiplication? (multiply the non-zero digits, and write all the zeros after) If students need a hint, write on the board: = 1, = 1, = 20,000 Number and Operations in Base Ten 5-15 D-11

12 Identifying patterns when multiplying powers of = = 1, ,000 = 10,000 ASK: What do you notice about the total number of zeros in the answer of each equation compared to the total number of zeros in the question? (they are the same) 100 1,000 ASK: How many zeros are in 100? (2) How many zeros are in 1,000? (3) How many zeros are there in total? (5) What number starting with 1 has 5 zeros at the end? (100,000) SAY: So, 100 1,000 = 100,000. Exercises: Multiply. a) b) 1, c) 10, d) ,000 e) 1,000 1,000 f) 1,000, ,000 Answers: a) 10,000, b) 100,000, c) 1,000,000, d) 10,000,000, e) 1,000,000, f) 100,000,000,000 Extensions (MP.2) 1. Find the missing number: a) 300 = 60,000 b) 200 = 80,000 c) 1,000 = 500,000 d) 400 = 1,200,000 (MP.7) 2. Find as many answers as you can using multiples of 10 for each question. a) = 40,000 b) = 120,000 (MP.7) (MP.1) (MP.1) 3. How many dimes are in these dollar amounts? a) $5 b) $1,000 c) $100, The total weight of all the termites on the planet is approximately 10 times as great as the total weight of humans. The average human weighs 50 kg. The world population is approximately 7,000,000,000 people. What is the approximate total weight of all the termites in the world? 5. Insects first appeared on Earth 350,000,000 years ago. Humans appeared approximately 130,000 years ago. About how many times as many years have insects been on Earth? Explain your answer. D-12 Teacher s Guide for AP Book 5.1

13 Answers: 1. a) 200, b) 400, c) 500, d) 3,000, 2. a) 1 40,000, 5 8,000, 8 5,000, 4 10,000, 10 4,000, , , , ; b) sample answers: 1 120,000, 2 60,000, 3 40,000, 4 30,000, 5 24,000, 6 20,000, 8 15,000, 10 12,000, 12 10,000, 15 8,000, 20 6,000, 24 5,000, 25 4,800, 30 4,000, 40 3,000, 48 2,500, 50 2,400, 60 2,000, 80 1,500, 100 1,200, 120 1,000, , , , , ; 3. a) 50, b) 10,000, c) 1,000,000; 4. 3,500,000,000,000 kg = 3,500,000,000 tonnes; 5. about 300 times as many years Number and Operations in Base Ten 5-15 D-13

14 NBT5-16 Patterns in the 6, 7, 8, and 9 Times Tables Pages STANDARDS preparation for 5.OA.B.3, 5.NBT.B.5 Goals Students will be fluent with multiplication by 6, 7, 8, or 9. PRIOR KNOWLEDGE REQUIRED Vocabulary multiple pattern times table Can skip count by 6, 7, 8, and 9 Can identify multiples of 6, 7, 8, and 9 Recognizing patterns in the ones digits of the 6 times table. (MP.7) 6 1 = 6 2 = 6 3 = 6 4 = 6 5 = 6 6 = 6 7 = 6 8 = 6 9 = Ask students for answers and fill in. ASK: Do you see a pattern in the ones digits of the answers? If students have difficulty seeing the pattern, write on the board: Ask a student to come to the board and circle the ones digits from the answers on the row that counts down repeatedly from ASK: What is the pattern? (ones digits are all four numbers apart) SAY: If you have trouble remembering the numbers, picture the pattern and count four spots each time. Exercises: Multiply the numbers. a) 6 5 b) 6 7 c) 6 9 Bonus d) 6 80 e) Answers: a) 30, b) 42, c) 54, Bonus: d) 480, e) 24,000 Recognizing patterns in the ones digits of the 7 times table. 7 1 = 7 2 = 7 3 = 7 4 = 7 5 = 7 6 = 7 7 = 7 8 = 7 9 = Ask students for answers and fill in. D-14 Teacher s Guide for AP Book 5.1

15 ASK: Do you see a pattern in the ones digits of the answers? If students have difficulty seeing the pattern, write on the board: Ask a student to come to the board and circle the ones digits from the answers on the row that counts down repeatedly from ASK: What is the pattern? (ones digits are all three numbers apart) SAY: If you have trouble remembering the numbers, picture the pattern and count three spots each time. Exercises: Multiply. a) 7 4 b) 7 6 c) 7 9 Bonus d) 7 80 e) Answers: a) 28, b) 42, c) 63, Bonus: d) 560, e) 21,000 Recognizing patterns in the ones digits of the 8 times table. 8 1 = 8 2 = 8 3 = 8 4 = 8 5 = 8 6 = 8 7 = 8 8 = 8 9 = ASK: Do you see a pattern in the ones digits of the answers? If students have difficulty seeing the pattern, write on the board: Ask a student to come to the board and circle the ones digits from the answers on the row that counts down repeatedly from ASK: What is the pattern? (ones digits are all two numbers apart) SAY: If you have trouble remembering the numbers, picture the pattern and count two spots each time. Exercises: Multiply. a) 8 5 b) 8 8 c) 8 6 d) 8 90 e) Answers: a) 40, b) 64, c) 48, d) 720, e) 24,000 Number and Operations in Base Ten 5-16 D-15

16 Recognizing patterns in the ones digits of the 9 times table. 9 1 = 9 2 = 9 3 = 9 4 = 9 5 = 9 6 = 9 7 = 9 8 = 9 9 = ASK: Do you see a pattern in the ones digits of the answers? If students have difficulty seeing the pattern, write on the board: Ask a student to come to the board and circle the ones digits from the answers on the row that counts down repeatedly from ASK: What is the pattern? (ones digits are all one number apart) SAY: If you have trouble remembering the numbers, picture the pattern and count one spot each time. Exercises: Multiply. a) 9 4 b) 9 7 c) 9 5 d) 9 80 Bonus: Answers: a) 36, b) 63, c) 45, d) 720, Bonus: 5,400 NOTE: For other patterns in the times tables see How to Learn Your Times Tables in a Week in the Mental Math section of this guide. Extensions (MP.2, MP.8) 1. Have students complete BLM Circle Charts (p. D-66). Answers multiples of 4 multiples of 4 D-16 Teacher s Guide for AP Book 5.1

17 2. Have students complete BLM Divisibility Rules (pp. D-67 68). multiples of 8 multiples of 3 (MP.2) Answers 1. A number is a multiple of 2 if it ends in 2, 4, 6, 8, or A number is a multiple of 5 if it ends in 5 or A number is a multiple of 3 if the sum of its digits is a multiple of A number is a multiple of 9 if the sum of its digits is a multiple of divisible by 2, 3; 36 divisible by 2, 3, 9; 60 divisible by 2, 3, 5; 135 divisible by 3, 5,9; 525 divisible by 3, 5; 143,172 divisible by 2, 3, 9 3. Ask students to describe any patterns they see in the following products: 2 6 = = = = 48 Answer: Students should notice that when you multiply 6 by an even number, the ones digit of the answer is the same as the number you are multiplying by. The tens digit is half the number you are multiplying by. Number and Operations in Base Ten 5-16 D-17

18 NBT5-17 Multiplying by Powers of 10 Pages STANDARDS 5.NBT.A.1, 5.NBT.A.2, 5.NBT.B.5 Goals Students will write a power as a product and a product as a power. PRIOR KNOWLEDGE REQUIRED Vocabulary base exponent power product Can multiply multi-digit numbers by 2-digit numbers Knows the commutative property of multiplication Knows multiplication is a short form for repeated addition Introduce the parts of a power. ASK: How can we check that 5 4 = 20? (5 4 = = 20) What does the 5 tell us? (how many of the second number we should add) SAY: There is a similar short form for multiplying. 4 5 = SAY: Instead of writing , we can write 4 5. ASK: What does the 5 tell us this time? (how many times to multiply the number by itself) SAY: It is important to learn the correct vocabulary for this short form. Write on the board: base 4 5 exponent power SAY: 4 is the base, 5 is the exponent, and 4 5 is a power of 4. Exercises: State the base, exponent, and power. a) 6 3 b) 8 4 c) 2 10 Answers: a) base = 6, exponent = 3, power = 6 3 ; b) base = 8, exponent = 4, power = 8 4 ; c) base = 2, exponent = 10, power = 2 10 Writing a power as a product. SAY: The exponent tells us how many times to multiply the base. 4 5 = multiply the base 5 times Exercises: Write each power as a product. a) 2 4 b) 6 3 c) 7 2 d) 8 1 Answers: a) 2 4 = , b) 6 3 = 6 6 6, c) 7 2 = 7 7, d) 8 1 = 8 Evaluating a power. 3 4 = D-18 Teacher s Guide for AP Book 5.1

19 SAY: To evaluate the power, we need to start multiplying and keeping track of the product as we go along. What is 3 3? (9) Write the answer in the first box. What is 9 3? (27) Write the answer in the second box. What is 27 3? (81) Write the final answer. Finish the equation on the board: = = 81 Exercises: Evaluate. a) 2 4 b) 5 3 c) 8 2 d) 10 3 Answers: a) 16, b) 125, c) 64, d) 1,000 (MP.7) Evaluating products involving a power of = 2 (10 10) = = = 2 ( ) = 2 = = 2 ( ) = 2 = (2,000, 20,000) ASK: What pattern do you see in the power and the answer? (the number of zeros added to the digit is the same as the exponent) If a hint is needed, write on the board: = = 2, = 20,000 Exercises: Evaluate without writing a product. a) b) c) d) Answers: a) 3,000, b) 700,000, c) 9,000,000, d) 3,200,000 Extensions 1. Write as a power of 10. a) 100 b) 10,000 c) 1,000,000,000 (MP.2) (MP.1, MP.8) 2. Write as a power of 2. a) 4 b) 8 c) Evaluate. a) b) c) How can you predict the number of zeros when evaluating ? 5. Have students complete BLM The Power of Doubling (pp. D-69 70). Answers: 1. a) 10 2, b) 10 4, c) 10 9 ; 2. a) 2 2, b) 2 3, c) 2 5 ; 3. a) 100,000, b) 1,000,000, c) 1,000,000,000; 4. add the exponents; 5. the total after 30 days is $2,147,483,647 Number and Operations in Base Ten 5-17 D-19

20 NBT5-18 Arrays and Multiplication Pages STANDARDS 5.OA.B.3, 5.NBT.B.5 Vocabulary array distributive property expanded place value product Goals Students will write a product for an array using the distributive property. Students will write a product using the distributive property, without using an array. PRIOR KNOWLEDGE REQUIRED Can write a product for a given array MATERIALS grid paper Use the distributive property to write a product for an array. Draw the following diagram on the board: ASK: What is a product that represents this array of squares? (3 14) On the board, shade in the last 4 columns of the array. Draw on the board: 3 14 Ask students to come to the board to write a product for the shaded squares and the unshaded squares. (3 10 and 3 4) SAY: But the total number of squares hasn t changed just because we shaded in some of the rectangles! So what can we say about the expressions 3 14 and (3 10) + (3 4)? (they are equal) Exercises: Write products for the entire diagram, the unshaded squares, and the shaded squares. Then write an equation for the diagram. a) b) D-20 Teacher s Guide for AP Book 5.1

21 Answers: a) 4 13, 4 10, 4 3, 4 13 = (4 10) + (4 3); b) 5 26, 5 20, 5 6, 5 26 = (5 20) + (5 6) SAY: Instead of drawing all the squares, we can draw a diagram and write down how many squares there would have been. Draw on the board: Instead of We draw (MP.7) Exercises: Write products for the entire diagram, the unshaded squares, and the shaded squares. Then write an equation for the diagram. a) b) Answers: a) 6 12, 6 10, 6 2, 6 12 = (6 10) + (6 2); b) 5 23, 5 20, 5 3, 5 23 = (5 20) + (5 3) (MP.8) Using the distributive property to write a product for an array without using an array. NOTE: Students don t need to know the term distributive property, but we recommend you introduce them to the term. Distributive Property: A property that allows multiplication of a sum by multiplying each term, then adding the products. Example: 3 (2 + 4) = (3 2) + (3 4) 3 6 = SAY: Let s look at the answers from our last two questions. Write on the board: 6 12 = (6 10) + (6 2) 5 23 = (5 20) + (5 3) Number and Operations in Base Ten 5-18 D-21

22 What pattern do you see in how the right side of the equation was created? (the first number is shared in each bracket; the second number is broken into a multiple of 10 and the remainder) If students have difficulty seeing the pattern, write on the board: 6 12 = (6 10) + (6 2) 5 23 = (5 20) + (5 3) (MP.8) Exercises: Rewrite each product in expanded form. a) 5 63 b) 7 84 c) 9 48 Answers: a) 5 63 = (5 60) + (5 3); b) 7 84 = (7 80) + (7 4); c) 9 48 = (9 40) + (9 8) Bonus: Write each product as a sum of three or more products. a) b) c) 4 1,375 d) 9 2,486 e) 3 23,756 Answers: a) = (7 100) + (7 20) + (7 5); b) = (5 300) + (5 40) + (5 8); c) 4 1,375 = (4 1,000) + (4 300) + (4 70) + (4 5); d) 9 2,486 = (9 2,000) + (9 400) + (9 80) + (9 6); e) 3 23,756 = (3 20,000) + (3 3,000) + (3 700) + (3 50) + (3 6) Performing the standard algorithm for multiplication in two steps = (5 60) + (5 3) SAY: We can use the right side of the equation to multiply in two steps: 5 3 and 5 60 Write on the board, without the answers: Have students copy the equations into their notebooks, on grid paper. ASK: What is 5 3? (15) SAY: Write the answer in the grid on the left underneath the first line. ASK: What is 6 5? (30) Then what is 60 5? (300) Remind students that to multiply by multiples of 10, we can just add the zero at the end. SAY: Write the answer in the grid on the right = (5 60) + (5 3) D-22 Teacher s Guide for AP Book 5.1

23 SAY: We ve done each of the multiplications, but we haven t done the addition SAY: Copy the answer from 60 5 underneath the answer from 3 5, being careful to line up the numbers under the correct place value. Now we can finish the final step by adding! Have students finish the question. Ask a volunteer to come up and write the answer on the board. Exercises: Use two steps to calculate each product. a) 74 2 b) 83 2 c) 46 3 d) 57 4 Answers: a) 148, b) 166, c) 138, d) 228 Encourage students to do simple calculations mentally. Exercises: Multiply using mental math. a) 14 2 b) 23 3 c) 43 2 d) 72 3 e) 61 4 Answers: a) 28, b) 46, c) 86, d) 216, e) 244 Extension Fill in the missing numbers. (MP.2) a) 0 4 b) 0 7 c) Answers a) b) c) Number and Operations in Base Ten 5-18 D-23

24 NBT5-19 Standard Method for Multiplication Pages STANDARDS 5.NBT.B.5 Goals Students will multiply a 2-digit number by a 1-digit number using the standard algorithm. Vocabulary algorithm area distributive property multiple regrouping PRIOR KNOWLEDGE REQUIRED Can multiply a 2-digit number by a 1-digit number using the distributive property MATERIALS grid paper Use 2 separate grids to multiply a 2-digit number by a 1-digit number. Draw on the board: 3 45 ASK: What multiplication statement represents the area of the rectangle? (45 3) How can we use a multiple of 10 to break up 45? (40 + 5) Add a line to the diagram so it looks like this: ASK: What multiplication statements give the area of each rectangle? (40 3 and 5 3) SAY: But the area is the same! So: 45 3 = (40 + 5) 3 = (40 3) + (5 3) We can perform these multiplications using two grids and adding the results: D-24 Teacher s Guide for AP Book 5.1

25 Exercises (MP.7) 1. Use a multiple of 10 and the distributive property to break up the first factor. a) 36 4 b) 54 7 c) 73 5 d) 87 3 Answers: a) 36 4 = (30 + 6) 4 = (30 4) + (6 4); b) 54 7 = (50 + 4) 7 = (50 7) + (4 7); c) 87 3 = (80 + 7) 3 = (80 3) + (7 3) 2. Now use two separate grids to find the products for the questions above. Selected solution a) Answers: b) 378, c) 365, d) 261 Using the standard algorithm to multiply a 2-digit number by a 1-digit number with no regrouping. SAY: We can combine our work into one grid Step 1: Multiply 2 3 = 6. Place the 6 in the ones column in the bottom row of the grid Step 2: Multiply 4 tens by 3 = 12 tens. Place the ones digit of 12 in the tens column in the bottom row of the grid. Place the tens digit of 12 in the hundreds column in the bottom row of the grid. SAY: It s important to place the digits in the correct columns in the bottom row, since we are really multiplying 40 3, not = 12 4 tens 3 = 12 tens Exercises: Use the standard algorithm to multiply. a) 72 3 b) 54 2 c) 32 4 Answers: a) 216, b) 108, c) 128 Number and Operations in Base Ten 5-19 D-25

26 Using the standard algorithm to multiply a 2-digit number by a 1-digit number with regrouping. SAY: Sometimes the first step may involve a product greater than 9. This will require regrouping Step 1: SAY: Notice that the first step will give us 5 ones 3 = 15 ones. ASK: How many tens are in 15 ones? (1) How many ones are left? (5) We write the 5 remaining ones in the ones column in the bottom row. We write the tens digit of 15 in the tens column in the row above the grid tens digit from 5 3 = 15 ones digit from 5 3 = 15 Step 2: SAY: Multiply 4 tens by 3 = 12 tens. Add the 1 ten from the regrouping. ASK: How many tens do we have? (13) How many hundreds are in 13 tens? (1) How many tens are left? (3) SAY: Write the 13 tens so that the 1 is in the hundreds column in the bottom row, and the 3 is in the tens column in the bottom row = 13 Exercises: Multiply using the standard algorithm. a) 37 2 b) 54 7 c) 93 4 Answers: a) 74, b) 378, c) 372 Extensions 1. Find the missing numbers. a) b) c) D-26 Teacher s Guide for AP Book 5.1

27 d) e) f) Use the digits from 0 to 5 to complete the multiplications Answers: 1. a) 42 3 = 126, b) 61 7 = 427, c) 54 3 = 162, d) 62 9 = 558, e) 94 8 = 752, f) 99 3 = 297; = 252, 51 4 = 204, 92 6 = 552 Number and Operations in Base Ten 5-19 D-27

28 NBT5-20 Multiplying a Multi-Digit Number by a Pages Digit Number STANDARDS 5.NBT.B.5 Goals Students will multiply a multi-digit number by a 1-digit number using the standard algorithm. Vocabulary expanded form regrouping PRIOR KNOWLEDGE REQUIRED Can multiply a 2-digit number by a 1-digit number using the standard algorithm MATERIALS base ten materials Review multiplying a 2-digit number by a 1-digit number in 3 ways: expanded form, base ten materials, and the standard algorithm. (MP.5) a) Using expanded form ASK: How can we write 47 in expanded form? (4 tens + 7 ones) 4 tens + 2 ones 3 12 tens + 6 ones = 1 hundred + 2 tens + 6 ones = 126 ASK: What is 4 tens 3? (12 tens) What is 2 ones 3? (6 ones) How many hundreds are in 12 tens? (1) How many tens are left? (2) b) Using base ten materials Draw: D-28 Teacher s Guide for AP Book 5.1

29 ASK: What do you get when you put all the tens blocks together? (12 tens blocks) What can we exchange 10 tens blocks for? (a hundreds block) Draw: to get the answer 126 c) Using the standard algorithm ASK: What is 2 3? (6) What is 4 tens 3? (12 tens) How many hundreds are there in 12 tens? (1) How many tens are left? (2) So 42 3 = 126. Using base ten materials, the standard algorithm, and expanded form, multiply a 3-digit number by a 2-digit number without regrouping. a) Using expanded form. Tell students you want to multiply ASK: How can we write 423 in expanded form? (4 hundreds + 2 ten + 3 ones) 4 hundreds + 2 ten + 3 ones 2 8 hundreds + 4 tens + 6 ones ASK: What is 4 hundreds 2? (8 hundreds) What is 2 tens 2? (4 tens) What is 3 ones 2? (6 ones) What is the product? (846) Number and Operations in Base Ten 5-20 D-29

30 b) Using base ten materials: Draw on the board: ASK: How many hundreds are there altogether? (8) How many tens are there altogether? (4) How many ones are there altogether? (6) What is the product? (846) c) Using the standard algorithm ASK: What is 3 2? (6) What is 2 2? (4) What is 4 2? (8) So = 826. Using expanded form, base ten materials, and the standard algorithm, multiply a 3-digit number by a 2-digit number with regrouping. SAY: Sometimes regrouping may be involved a) Using expanded form. ASK: How can we write 456 in expanded form? (4 hundreds + 6 tens + 7 ones) 4 hundreds + 6 tens + 7 ones 2 8 hundreds + 12 tens + 14 ones ASK: What is 4 hundreds 2? (8 hundreds) What is 6 tens 2? (12 tens) What is 7 ones 2? (14 ones) What is the product? (846) 8 hundreds + 12 tens + 14 ones = 8 hundreds + ( hundreds + tens) + ( tens + ones) D-30 Teacher s Guide for AP Book 5.1

31 Ask a volunteer to come up to fill in the blanks. (1, 2, 1, 4) Write on the board: = hundreds + tens + ones Ask another volunteer to gather up the hundreds, tens, and ones to fill in the blanks. (9, 3, 4) Write the answer: = 934 b) Using base ten materials. Draw on the board: ASK: How many hundreds are there altogether? (8) How many tens are there altogether? (12) How many ones are there altogether? (14) If we exchange 10 ones for a tens block, and 10 tens for a hundreds block, what will the diagram look like? 12 tens become 1 hundred + 2 tens 14 ones become 1 ten + 4 ones ASK: How many hundreds are there altogether? (9) How many tens are there altogether? (3) How many ones are there altogether? (4) What is the final product? (934) c) Using the standard algorithm. Write on the board, without the top row filled in: ASK: What is 7 2? (14) How do we write 14 in expanded form? (1 ten and 4 ones) SAY: We write the 1 ten in the row above the grid, and we write the 4 in the ones column in the bottom row of the grid. As you fill in the numbers, describe to students what each number represents. Number and Operations in Base Ten 5-20 D-31

32 ASK: What are 6 tens 2? (12 tens) SAY: But we have 1 ten from multiplying 7 2, so we really have 13 tens altogether. ASK: How do we write 13 tens in expanded form? (13 tens = 1 hundred + 3 tens) SAY: So we write the 3 in the tens column in the bottom row, and we write the 1 in the hundreds column in the row above the grid ASK: What is 4 hundreds 2? (8 hundreds) SAY: But we have 1 hundred from multiplying 6 tens 2, so really we have 9 hundreds. Write the 9 in the hundreds column in the bottom row of the grid. Together, solve: problems that require regrouping ones to tens (examples: 219 3, 312 8, 827 2) problems that require regrouping tens to hundreds (examples: 391 4, 282 4, 172 3) problems that require regrouping both ones and tens (examples: 479 2, 164 5, 129 4) Have students solve additional problems in their notebooks. Exercises: Use base ten materials, expanded form, and the standard algorithm to solve each problem. a) b) c) d) Answers: a) 560, b) 2,568, c) 1,505, d) 2,808 Tell students to be sure that they get the same answer all three ways. If they do not, they should check their work to find the mistake. Bonus: Find the products. a) 2,456 3 b) 5,234,562 7 Answers: a) 7,368, b) 36,641,934 D-32 Teacher s Guide for AP Book 5.1

33 Exploring the special case in which the 3-digit number has a 0 digit Describe each step of the process, pointing to each digit as you say it: 6 ones 9 is 54 ones, so that s 5 tens and 4 ones 0 tens 9 is 0 tens, then add the 5 tens 3 hundreds 9 is 27 hundreds, so that s 2 thousands and 7 hundreds Exercises: Find the product. a) b) c) d) e) Bonus: 12,009 7 Answers: a) 3,654, b) 3,680, c) 4,842, d) 4,350, e) 2,124, Bonus: 84,063 Extensions (MP.1) 1. Using only the digits 2, 3, 4, and 6, find the greatest product that can be made by multiplying a 3-digit number by a 1-digit number. Answer: 2,592 (MP.2) 2. Using only the digits 4, 5, 6, and 9, find the least product that can be made by multiplying a 3-digit number by a 1-digit number. Answer: 2,276 (MP.2) 3. What is the greatest product possible when multiplying a 3-digit number by a 1-digit number? Answer: = 8,991 (MP.1) 4. Try the following number trick with a friend. a) Pick a number from 1 to 9. b) Multiply your number by 100. c) Add 3 to your answer. d) Multiply your answer by 6. e) Subtract 18. f) Ask for the answer. To guess the number, remove the zeros at the end of the number, then divide by 6. That will be the number your friend started with. Try it with your friend, then have your friend try it with you. Can you figure out why it works? Number and Operations in Base Ten 5-20 D-33

34 Answer: Multiplying by 100 moves the digits to the left two places. Adding 3, multiplying by 6, then subtracting 18, gives zeros at the end of the number. When you remove the zeros, all that is left is the original number multiplied by 6. (MP.7) 5. Complete BLM Circle Magic (p. D-71). Answers: a) 142,857, b) 285,714, c) 428,571, d) 571,428, e) 714,285, f) 857,142; all the products contain the digits (MP.7) 6. Use mental math to multiply by 9. To multiply by 10, add a zero. To multiply by 9, multiply by 10, then subtract. Example: 3 9 = (3 10) (3 1) 3 10 To calculate 457 9: Step 1: Calculate = 4,570 Step 2: Calculate = 457 Step 3: Subtract: 4,113 Use mental math to calculate. a) b) c) 1,234 9 Answers: a) 1,143, b) 2,232, c) 11,106 D-34 Teacher s Guide for AP Book 5.1

35 NBT5-21 Word Problems I Page 53 STANDARDS 5.NBT.B.5 Goals Students will solve word problems involving multiplication that may require multiplying multi-digit numbers by 1-digit numbers. PRIOR KNOWLEDGE REQUIRED Can use the standard algorithm to multiply multi-digit numbers by 1-digit numbers Extensions (MP.1) 1. Tim earns $9 per hour at his summer job. a) How much does Tim earn in a 40-hour week? b) How much will he earn in 8 weeks? c) How many more weeks will he need to work to buy computer equipment that costs $3,600? Answers: a) $360, b) $2,880, c) 80 hours (MP.1) 2. A T-shirt company sold 1,250 T-shirts at $8 each. a) How much money did the company earn? b) For each T-shirt, the company paid $2 for heat, lighting, and electricity. What are the costs for 1,250 shirts? c) What profit did the company make on the T-shirts? Answers: a) $10,000, b) 2,500, c) $7,500 Number and Operations in Base Ten 5-21 D-35

36 NBT5-22 Multiplying 2-Digit Numbers by Pages Multiples of 10 STANDARDS 5.NBT.A.1, 5.NBT.A.2, 5.NBT.B.5 Goals Students will use the standard algorithm to multiply 2-digit numbers by 2-digit multiples of 10 (10, 20, 30,, 90). Vocabulary array associative property commutative property multiples product regrouping rounding standard algorithm PRIOR KNOWLEDGE REQUIRED Can multiply using arrays Can apply the distributive property Can use the standard algorithm to multiply 2-digit numbers by 1-digit numbers Can use mental math to multiply 2-digit numbers by 2-digit multiples of 10 without regrouping MATERIALS BLM Cutting an Array into Ten Strips (p. D-72) Review mentally multiplying 1-digit numbers by multi-digit numbers. Choose examples that require no regrouping (examples: 2 324, 3 132, 4 201) or only regrouping at the greatest place value. (examples: 4 612, 2 804, 3 430) Using arrays to multiply by multiples of 10. Draw on the board: Tell students that the picture represents an array that is 20 squares high and 32 squares long. If they have trouble visualizing the individual squares, give them BLM Cutting an Array into Ten Strips. ASK: What multiplication gives the total number of squares? (20 32 because the width is 20 squares and the length is 32 squares) Ask students to imagine dividing the array into smaller strips that are 2 squares high and 32 squares long. D-36 Teacher s Guide for AP Book 5.1

37 32 each strip is 2 squares high and 32 squares long 20 ASK: What multiplication gives the number of squares in each strip? (2 32 because each strip is 32 squares long and 2 squares high) ASK: How many strips are in the array? (10; each strip is 2 squares high and the array is 20 squares high) SAY: There are 10 strips each containing 2 32 squares, so you can rewrite as: 10 (2 32) number of strips number of squares in each strip Using the same diagram on the board, write a new height and length on your diagram but do not erase the 10 strips. Make sure the height is a multiple of ten. Ask students to write a multiplication statement for the new dimensions. Students must first determine the height of each strip. Remind them that there are 10 strips. Exercises: Write a multiplication statement for the array. a) 57 b) c) Answers: a) 10 (3 57), b) 10 (4 77), c) 10 (5 42) Number and Operations in Base Ten 5-22 D-37

38 Review why using arrays works. Tell students they can prove that = 10 (2 32) without using a diagram. They can use the commutative and associative properties of multiplication = (10 2) 32 (20 = 2 10 or 10 2, commutative property) = 10 (2 32) (associative property) Remind students that the associative property allows us to multiply three numbers by either multiplying the first and second numbers together, or by multiplying the second and third numbers together. Example: = (2 3) 4, or = 2 (3 4). SAY: Now rewrite the product to multiply a 2-digit number by a multiple of 10: = 10 (2 32) = = 640 Exercises: Calculate the product. a) b) c) Answers: a) = (10 3) 12 = 10 (3 12) = = 360; b) = (10 4) 22 = 10 (4 22) = = 880; c) = (10 5) 31 = 10 (5 31) = = 1,550 Multiplying by multiples of 10 to estimate products ASK: Is 32 closer to 30 or to 40? (30) Is 79 closer to 70 or to 80? (80) (3 80) ,400 Exercises: Estimate by rounding each number to the nearest ten. a) b) c) Answers: a) 1,000, b) 1,600, c) 4,500 Use a chart to find products involving multiples of 10. Tell students they can use a chart to find products such as = 10 (4 57) by following these steps: Step 1: When you multiply a number by 10, you add a 0. So write a 0 in the ones place because you will multiply 10 by (4 57). D-38 Teacher s Guide for AP Book 5.1

39 Write a 0 here because you will multiply 4 57 by 10. Step 2: Now multiply 57 4 using the standard algorithm Multiply 4 7. Write the 2 for the regrouping in the hundreds column. Multiply 4 5. Add the 2 from the regrouping in the hundreds column. Exercises: Multiply. a) b) c) Answers: a) 1,280, b) 2,700, c) 1,840 Extension (MP.1) The strongest animal on Earth is the rhinoceros beetle. It weighs 80 grams and can lift 850 times its own weight. How much can it lift? Use properties of numbers to break the product into simpler products. Explain how you found your answer: Answer: 68,000 g = 68 kg; multiply 85 80, then multiply by 10 by adding zero Number and Operations in Base Ten 5-22 D-39

40 NBT5-23 Multiplying 2-Digit Numbers by Pages Digit Numbers STANDARDS 5.NBT.B.5 Goals Students will use the standard algorithm to multiply 2-digit numbers by 2-digit numbers. Vocabulary algorithm array distributive property double multiples regrouping rounding standard algorithm PRIOR KNOWLEDGE REQUIRED Can multiply using arrays Can apply the distributive property Can use the standard algorithm to multiply 2-digit numbers by 1-digit numbers Can multiply 2-digit numbers by 2-digit multiples of 10 MATERIALS BLM 1 cm Grid Paper (p. I-1) Introduce multiplying 2-digit by 2-digit numbers ASK: How is this multiplication different from any we have done so far? (neither number is a multiple of 10 we have only estimated the product in such cases) Splitting a problem into easier problems. Tell students that you would like to find a way to split the problem into two easier problems, both of which they already know how to do. Have students list all the types of problems they know how to do that might be helpful: Multiply a 1-digit number by a 1-digit number. Multiply a 2-digit number by a 1-digit number. Multiply a 2-digit number by a 2-digit multiple of 10. ASK: How can we break up one of the numbers so that it contains a 2-digit multiple of 10? (45 = ; 32 = ) SAY: Let s pick 32 = (although it would also work with 45 = ). Using the distribute property to multiply 2-digit numbers by 2-digit numbers in separate steps. Draw on the board: D-40 Teacher s Guide for AP Book 5.1

41 ASK: What multiplication statement gives the area? (45 32) SAY: Let s break up 32 into 30 and 2. Draw on the board: ASK: What multiplication statement gives the area of the rectangle on the left? (45 30) What multiplication statement gives the area of the rectangle on the right? (45 2) Write on the board and SAY: Since the total area of the two small rectangles is the same as the large rectangle, we have: = 45 (30 + 2) = (45 30) + (45 2) SAY: We know how to use a grid to multiply each of these multiplication statements. We can add the separate parts in a grid as well Exercises Use BLM 1 cm Grid Paper to multiply using the three separate steps. a) b) c) Answers: a) 777, b) 3,504, c) 1, Check the reasonableness of your answers by rounding each factor to the nearest 10. Selected solution: a) Using the standard algorithm to multiply a 2-digit number by a 2-digit number Number and Operations in Base Ten 5-23 D-41

42 Step 1: Multiply regrouping from 8 4 = = = 112 Step 2: Multiply regrouping from 8 3 = = add 0 because we are multiplying by 30 Step 3: Add the two products product from 28 4 product from Mentally finding the product of 2-digit numbers multiplied by 2-digit numbers. Encourage students to mentally find simple products of pairs of 2-digit numbers. Examples:13 11, 21 12, (143, 252, 682) ACTIVITY Write the product on the board. Ask the class to find the product mentally. After students have had time to do this, ask volunteers to tell you what strategy they used to find the product. Write their answers on the board in abbreviated form. For instance, if a student says I thought of 12 as 10 plus 2. I multiplied 14 by 10 and got 140, then I multiplied 14 by 2 and got 28. I added 140 and 28 and got 168. You could write on the board: = (10 + 2) 14 = = = 168 D-42 Teacher s Guide for AP Book 5.1

43 Encourage students to share unusual strategies. For example, I know 14 is double 7. So is double I know 12 7 is 84. And double 84 is 168. (NOTE: This exercise was inspired by a talk by Marilyn Burns at the NCTM National Conference in 2013.) Exercises: Use the standard algorithm to multiply. a) b) c) d) Answers: a) 1,081, b) 2,790, c) 2,088, d) 1,044 Extensions (MP.5) 1. Instead of dividing the rectangle for into two rectangles, we could divide it into four rectangles. We use the idea that 45 = and that 32 = Write a multiplication statement for each rectangle A B 5 C D Answers: A: 40 30, B: 40 2, C: 5 30, D: 5 2 To calculate 45 32, add up the areas of the individual rectangles: = (40 5) (30 + 2) = (40 30) + (40 2) + (5 30) + (5 2) = 1, = 1,440 Have students use this technique to find the products. a) b) c) Answers: a) 2,128, b) 3,362, c) 2,847 Have students check the reasonableness of their answers by rounding each factor to the nearest 10. Selected solution: a) Use the four-rectangle method in a grid to multiply 2-digit numbers by 2-digit numbers. Number and Operations in Base Ten 5-23 D-43

44 , ,440 Tell the students that the 2 in 32 is multiplied by the 5 and the 40, and the 30 in 32 is multiplied by the 5 and the 40. To find the answer, add the four products. Have students practice this grid method to find the products. a) b) c) Answers: a) 3,286, b) 392, c) 3,441 (MP.7) 3. Distribute BLM Patterns in Multiplication (p. D-73). Have students discover an easy way to find the product of a 2-digit number ending in 5 that is multiplied by itself (examples: 15 15, 25 25, 35 35). After students complete the BLM, summarize their answers. ASK: What are the digits in the tens column and ones column in every case? (25) ASK: Using the tens digit of the original number, what multiplication statement will give the remaining digits in the answer? (multiply the tens digit by 1 more than the tens digit) = ASK: What is the tens digit? (6) What is one more than the tens digit? (7) What is the product of 6 7? (42) SAY: These are the first two digits of the answer. The last two digits will always be 5 5 = 25. So = 4,225 Exercises: Use this technique to find the product. a) b) c) d) Answers: a) 5,625, b) 2,025, c) 1,225, d) 7,225 Challenge: Find the product. a) (Hint: Calculate to find the first 3 digits.) b) c) d) 1,005 1,005 e) 9,995 9,995 Answers: a) 30,625, b) 11,025, c) 990,025, d) 1,010,025, e) 99,900,025 D-44 Teacher s Guide for AP Book 5.1