163A Chapter 3Chapter 3
About the MathProfessional Development
About the MathProfessional Development
LESSON AT A GLANCE
Progressto AlgebraLESSON 3.4
Professional Development Videos
Interactive Student Edition
Personal Math Trainer
Math on the Spot Video
iTools: Base-Ten Blocks
Progress to AlgebraTeaching for DepthIn this lesson, students learn how to record partial products in vertical form. Sometimes partial products are recorded beginning with the greatest place value because:
• the expanded form of a number is written starting with the greatest place value.
• the Distributive Property, when written with the numbers broken apart (expanded form), starts with the greatest place value.
• the area model that represents the partial products starts with the greatest place value.
In middle and high school, when students multiply binomials, they are often taught a mnemonic FOIL (fi rst, outer, inner, last). If this method is applied to recording partial products, students record the greatest place value fi rst.
Multiply Using Partial Products
Learning ObjectiveUse place value and partial products to multiply 2-digit numbers.
Language ObjectiveStudent pairs explain how to use place value to multiply 2-digit numbers, and then ask a partner to explain the partial products strategy.
MaterialsMathBoard
F C R Focus:Common Core State Standards
4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
MATHEMATICAL PRACTICES (See Mathematical Practices in GO Math! in the Planning Guide for full text.)MP4 Model with mathematics. MP7 Look for and make use of structure. MP8 Look for and express regularity in repeated reasoning.
F C R Coherence:Standards Across the GradesBefore3.OA.B.5
Grade 44.NBT.B.5
After5.NBT.B.5
F C R Rigor:Level 1: Understand Concepts....................Share and Show ( Checked Items)Level 2: Procedural Skills and Fluency...........On Your Own, Practice and HomeworkLevel 3: Applications..................................Think Smarter and Go Deeper
F C R For more about how GO Math! fosters Coherence within the Content Standards and Mathematical Progressions for this chapter, see page 143J.
FOCUS COHERENCE RIGOR
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ENGAGE1
Lesson 3.4 163B
Daily RoutinesCommon Core
Daily RoutinesCommon Core
How can you use place value and partial
products to multiply 2-digit numbers?
with the Interactive Student Edition
Essential QuestionHow can you use place value and partial products to multiply 2-digit numbers?
Making ConnectionsInvite students to tell you what they know about multiplying two-digit numbers.
When can you multiply two two-digit numbers in real life? Possible answer: finding the number of students in a school that has at least 10 classrooms
Learning ActivityConnect the story to the problem.
• What are you trying to find in this problem? the total weight of the rocks
• How many bags of rocks does the owner of the diner need for the rock garden? 24 bags
• How much does each bag of rocks weigh? 45 pounds
• What mathematical operation could you use to solve the problem? multiplication
Have students think about how they might use partial products to find the total weight of the rocks.
Literacy and Mathematics• Have one partner act out weighing each bag of rocks. Have the
other partner record the weight of each bag as it is weighed.
• Have students write their own word problems about the bags of rocks.
1 23 4 Fluency BuilderMultiply 2-Digit Numbers by 1-Digit Numbers Write the following examples on the board. Have students practice multiplying 2-digit numbers by 1-digit numbers. Have students check answers with a partner.
24 × 4 = 96
51 × 3 = 153
36 × 7 = 252
87 × 2 = 174
92 × 5 = 460
Problem of the Day 3.4Use rounding to estimate the product 3 × 2,810. 9,000
Vocabulary
Common Core Fluency Standard 4.NBT.B.5
• Interactive Student Edition• Multimedia Glossary e
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DifferentiatedInstruction1
2
3 DifferentiatedDifferentiatedInstruction
EXPLORE2
Name
tens ones
43
325
__
8 0 0
6 0
2 0 0
1 1 5
1, 0 7 5
Record the product.
1. 2. 3.
4. 5. 6.
Multiply Using Partial Products
Multiply 25 3 43. Record the product.
Think: I can use partial products to find 25 3 43.
Step 1 Multiply the tens by the tens. 20 × 4 tens = 80 tens, or 800.
Step 2 Multiply the ones by the tens. 20 × 3 ones = 60 ones, or 60.
Step 3 Multiply the tens by the ones. 5 × 4 tens = 20 tens, or 200.
Step 4 Multiply the ones by the ones. 5 × 3 ones = 15 ones, or 15.
Step 5 Add the partial products. 800 + 60 + 200 + 15 = 1,075.
So, 25 3 43 5 1,075 .
38 3 95
_
85 3 72
_
76 3 23
_
46 3 52
_
59 3 38
_
25 3 62
_
Lesson 3.4Reteach
3,610
2,700720150
1 40
6,120
5,600350160
1 10
2,392 1,748
1,400120210
1 18
2,000300
801 12
2,242
1,500270400
1 721,550
1,200300
401 10
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3-11 ReteachChapter Resources© Houghton Mifflin Harcourt Publishing Company
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Name
35250210
+ 1,500
1,995
Reverse and RecordWhen Nestor records 2-digit by 2-digit multiplication, he always records the partial products in the following order:
(1) Multiply the tens by the tens.(2) Multiply the ones by the tens.(3) Multiply the tens by the ones.(4) Multiply the ones by the ones.
One day he decides to reverse the order. He starts by multiplying the ones by the ones and works backward to multiplying the tens by the tens. He uses this strategy to multiply some numbers. Will Nestor’s strategy give him the correct products?
Use Nestor’s new strategy to find the products below. The first problem has been completed for you.
1. 2. 3.
4. 5. 6.
7. Stretch Your Thinking How can Nestor check that his products are correct?
75 3 27
_
59 3 95
_
83 3 19
_
44 3 63
_
31 3 22
_
57 3 35
_
Lesson 3.4Enrich
45250810
1 4,5005,605
27720
301 8001,577
35490100
1 1,4002,025
12120240
1 2,4002,772
Possible answer: Nestor can multiply again in the original
order of the tens by the tens, the ones by the tens, the tens by
the ones, and the ones by the ones to check the products.
26020
1 600682
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EnrichChapter Resources© Houghton Mifflin Harcourt Publishing Company
3-12
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1
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3 DifferentiatedInstruction
Progressto Algebra
30
4
50 7
30
4
50 7
30
4
50 7
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Unlock the ProblemUnlock the Problem
MATHEMATICAL PRACTICES 8MathTalk
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Multiply Using Partial ProductsEssential Question How can you use place value and partial products to multiply 2-digit numbers?
Lesson 3.4
Chapter 3 163
connect You know how to break apart a model to find partial products. How can you use what you know to find and record a product?
Multiply. 34 × 57 Estimate. 30 × 60 = __
SHADE THE MODEL THINK AND RECORD
So, 34 × 57 = 1,938. Since 1,938 is close to the estimate of 1,800, it is reasonable.
57 × 34
__
57 ×34
__
1,500
57 ×34
__
1,500
210
__
57 ×34
__
1,500
210
200
+
← Multiply the tens by the tens.
30 × 5 tens = 150 tens
← Multiply the ones by the tens. 30 × 7 ones = 210 ones
← Multiply the tens by the ones. 4 × 5 tens = 20 tens
← Multiply the ones by the ones. 4 × 7 ones = 28 ones← Add the partial products.
STEP 1
STEP 2
STEP 3
STEP 4
Use Repeated Reasoning You can write 10 × 4 ones = 40 ones as 10 × 4 = 40. What is another way to write 10 × 3 tens = 30 tens?
Number and Operations in Base Ten—4.NBT.B.5
MATHEMATICAL PRACTICESMP1, MP2, MP4, MP8
1,800
10 × 30 = 300
1,500
210
200
28
1,938
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Problem Type: Area • Unknown Product
163 Chapter 3
Enrich 3.4Reteach 3.4
LESSON 3.4
Unlock the Problem MATHEMATICAL PRACTICES
MP6 Attend to precision. The model in this lesson connects to the symbolic form of the Distributive Property.34 × 57 = (30 + 4) × (50 + 7) = (30 × 50) + (30 × 7) + (4 × 50) + (4 × 7)Since students work from left to right on the model, they fi rst multiply and record the product of tens times tens in the vertical form. Ask students to describe each step in the model and the corresponding recording.
• How did you use place value to help you multiply two 2-digit numbers? Possible answer: I fi rst multiplied the tens in 57 by the tens in 34.Then I multiplied the ones in 57 by the tens in 34.Then I multiplied the tens in 57 by the ones in 34. Then I multiplied the ones in 57 by the ones in 34.
• How did partial products help you to multiply two 2-digit numbers? Possible answer: it was easy to multiply the factors in place-value parts. Then I could add the smaller products together.
• How could you estimate to check for reasonableness? Possible answer: 30 × 60 = 1,800; 1,800 is close to 1,938.
MathTalk Use Math Talk to focus on
students’ understanding of renaming numbers.
• How do you write 3 tens and 30 tens in standard form? Possible answer: Write each number followed by a zero in the ones place.
ELL Strategy: Model Concepts
Write on the board 14 × 36.
• Describe and model the steps you take as you solve the problem.
• Have students apply the same process to another problem. Students can use your language and steps as a model.
36× 14 300 10 times 3 tens 60 10 times 6 ones 120 4 times 3 tens+ 24 4 times 6 ones 504 Add the partial products.
4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
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120 16
600
4
20 80
30 4
MATHEMATICAL PRACTICES 1MathTalk
Share and ShowShare and ShowShare and ShowShare and ShowShare and ShowShare and ShowShare and ShowShare and Show MATHBOARDMATHBOARDMATHBOARDMATHBOARDMATHMATHMATHMATHBOARDBOARDBOARDBOARD
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164
The apples from each tree in an orchard can fill 23 bushel baskets. If 1 row of the orchard has 48 trees, how many baskets of apples can be filled?
Multiply. 48 × 23 Estimate. 50 × 20 = ______
So, 1,104 baskets can be filled.
1. Find 24 × 34.
Example
3 4
× 2 4
__
THINK RECORD
STEP 1
Multiply the tens by the tens.
STEP 2
Multiply the onesby the tens.
STEP 3
Multiply the tensby the ones.
STEP 4
Multiply the ones bythe ones. Then addthe partial products.
23 × 48
__
← 40 × ______ tens = ______ tens
23 × 48
__
800
← 40 × ______ ones = ______ ones
23 × 48
_
800
120
← 8 × ______ tens = ______ tens
23 × 48
_
800
120
160
+
__ ← 8 × ______ ones = ______ ones
Evaluate Reasonableness How do you know your answer is reasonable?
Possible answer: since 1,104 is close to the estimate of 1,000, it is reasonable.
1,000
6 0 0
8 0 1 2 0
2 80
3 120
2 16
3 24
800
120
160
1,104
24
+ 1 6
__
8 1 6
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COMMON ERRORS COMMON ERRORS
EXPLAIN3
Advanced LearnersAdvanced Learners
Problem Type: Equal Groups • Unknown Product
Lesson 3.4 164
ExampleA bushel is a U.S. customary unit of dry volume. A bushel of apples weighs about 42 to 48 pounds.Explain to students that estimating before computing an exact answer tells them what a reasonable answer might be.
• How can you use rounding to the greatest place value to estimate the product? Round 48 to 50 and 23 to 20. 50 × 20 = 1,000
• In Step 1, explain why 80 tens is the same as 800. Possible answer: 80 tens means 80 groups of 10. If you added 10 eighty times you would get 800.
• How do you line up the partial products when you record them? I line up the ones places.
DEEPER
MP8 Look for and express regularity in repeated reasoning. After students determine that the product 48 × 23 = 1,104, give them an opportunity to represent 1,104 in different ways. Ask volunteers to write their ways on the board. Possible answers: 1,000 + 100 + 4; 11 hundreds + 4 ones; 23 × 2 × 24; 552 × 2; 1,105 − 1 Students should generalize that there are many ways to represent any number.
Error Students may not align the partial products on the ones place.
Example
Springboard to Learning Give students ruled paper turned sideways or grid paper to help them align the partial products correctly.
Share and Show MATHBOARDMATHBOARDMBOARDMMMMBOARDBOARDBOARDBOARDMATHATHABOARDMMMMAAAATHATHATHTHTHATHATHATHAATHAAAATHAAATHATHTHTHATHATHAAATHATHATHAAATHABOARDBOARDBOARDBOARD
The fi rst problem connects to the learning model.
1 2 × 1 2 ____
100 20 20 + 4 ______
540
Kinesthetic / VisualIndividual
Materials 2 number cubes, 1-Centimeter Grid Paper (see eTeacher Resources)
• Challenge students to fi nd products by using either a break-apart place-value model or no model with place value to fi nd partial products. Give students grid paper to record their work.
• Have students toss two number cubes to make a 2-digit number, and record. Have them toss the number cubes again to make another 2-digit number, and record.
• Challenge students to fi nd the product using the method they choose. Have them compare methods and solutions.
HandsOn
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Quick Check
If
Rt I RR1
2
3
Quick Check
If
Rt I 1
2
3
Then
On Your OwnOn Your Own
MATHEMATICAL PRACTICES 4MathTalk
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Chapter 3 • Lesson 4 165
2. 12
× 12
__
3. 31
× 24
__
4. 25
× 43
__
5. 37
× 26
__
Record the product.
Record the product.
MATHEMATICALPRACTICE 2 Reason Abstractly Algebra Find the unknown digits. Complete the problem.
6. 54
× 15
__
7. 87
× 16
__
8. 62
× 56
__
9. 49
× 63
__
10. 38 × 47
14. 53 × 68
11. 46 × 27
15. 76 × 84
12. 72 × 53
16. 92 × 48
13. 98 × 69
17. 37 × 79
Practice: Copy and Solve Record the product.
18. 6
× 4
__
1,4 0 0
1 2 0
2 8 0
+ 2 4
___
19. 2
× 7
__
7,2 0 0
1 8 0
5 6 0
+ 1 4
___
20. 6
× 5
__
1,5 0 0
3 0 0
9 0
+ 1 8
___
21. 3
× 8
__
6 0 0
8 0
2 4 0
+ 3 2
__
Model Mathematics How would you model and record 74 × 25?
144 744 1,075 962
1,786 1,242 3,816 6,762
7
810 1,392 3,472 3,087
3,604 6,384 4,416 2,923
2
1, 8 2 4
89
7, 9 5 4
33
1, 9 0 8
4
2
9 5 2
See Additional Answers, TE p. 165.
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PROBLEM TYPE SITUATIONS
165 Chapter 3
On Your Own If students complete the checked exercises correctly, they may continue.MP2 Reason abstractly and quantitatively. Exercises 18–21 require that students use logical reasoning to use the partial products to determine the unknown digits in the factors.
Additional answers:
Math Talk, p. 165
Possible explanation: fi rst draw a rectangle to represent 74 × 25. Then break apart the factors into 70 + 4 and 20 + 5 and draw the smaller rectangles to represent the four partial products. Write the problem in vertical form:
25 × 74 _____
Then multiply the tens by the tens (70 × 2 tens) and record the partial product, 1,400 under the problem. Next, multiply the tens by the ones (70 × 5 ones) and record the partial product, 350 under the problem. Then, multiply the ones by the tens (4 × 2 tens) and record the partial product 80 under the problem. Next, multiply the ones by the ones (4 × 5 ones) and record the partial product 20 under the problem. Finally, add the partial products: 1,400 + 350 + 80 + 20 = 1,850.`
a student misses the checked exercises
Differentiate Instruction with • Reteach 3.4
• Personal Math Trainer 4.NBT.B.5
• RtI Tier 1 Activity (online)
Use the checked exercises for Quick Check.
MathTalk Use Math Talk to focus students'
understanding of lesson concepts.
• How do you model 74 × 25? Possible answer: draw an area model of 74 × 25 and break it apart into 4 parts by tens and ones.
• What is the next step in fi nding the product of 74 × 25? Possible answer: fi nd the area of each part of the model. These are partial products.
• What is the fi nal step? Possible answer: add all the partial products.
Addition and Subtraction
Put Together/Take Apart • Total Unknown Exercise: 13
Multiplication and Division
Equal Groups • Unknown Product Exercises: 12, 13, 14, 15
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Differentiated Centers Kit
DIFFERENTIATED INSTRUCTION INDEPENDENT ACTIVITIES
ELABORATE4
Games
EVALUATE5 Formative Assessment
Differentiated Centers Kit
DIFFERENTIATED INSTRUCTION INDEPENDENT ACTIVITIES
Math on the Spot videos are in the Interactive Student Edition and at www.thinkcentral.com.
Citrus Fruit Weight per Box (in pounds)
Pounds of Citrus Fruit per Box
Key: Each = 10 pounds. Key: Each = 10 pounds.
Orange
Tangelo
Tangerine
Grapefruit
Key: Each = 10 pounds.
Tangerine
WRITE Math • Show Your Work
MATHEMATICAL PRACTICES MODEL • REASON • MAKE SENSE
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Use the picture graph for 22–24.
22. MATHEMATICALPRACTICE 4 Use Graphs A fruit-packing
warehouse is shipping 15 boxes of grapefruit to a store in Santa Rosa, California. What is the total weight of the shipment?
23. DEEPER How much less do 13 boxes
of tangelos weigh than 18 boxes of tangerines?
24. What is the weight of 12 boxes of oranges?
25. SMARTER Each person in the United States eats about 65 fresh apples each year. Based on this estimate, how many apples do 3 families of 4 eat each year?
26. DEEPER The product 26 × 93 is greater than 25 × 93. How much greater? Explain how you know without multiplying.
27. SMARTER Margot wants to use partial products to find 22 × 17.
Write the numbers in the boxes to show 22 × 17.
( × ) + ( × ) + ( × ) + ( × )
Problem Solving • ApplicationsProblem Solving • Applications
1,275 pounds
540 pounds
780 apples
1,080 pounds
The difference is 93. Possible explanation:
26 × 93 is one more group of 93 than
25 × 93.
20 10 20 7 2 10 2 7
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Lesson 3.4 166
Students take turns using number cards to make and solve 2-digit by 1-digit multiplication problems.
Students read about how Julia uses multiplication to decide how to arrange the stamps in a collection.
Students complete orange Activity Card 5 by using multiplication to find numbers that match given products.
GamesMultiplication Marathon
LiteraturePutting the World on a Page
ActivitiesFirst One Out
Essential QuestionUsing the Language Objective:Reflect Have student partners take turns explaining to answer the Essential Question.How can you use place value and partial products to multiply 2-digit numbers? Possible answer: I can break apart the numbers into tens and ones and multiply to find partial products, and then add the partial products.
Math Journal WRITE Math
Explain why it works to break apart a number by place values to multiply.
MATHEMATICAL PRACTICES
MP4 Model with mathematics. Discuss the picture graph. Students need to find the total weight in one box of each kind of fruit used in Exercises 22–24 before finding the weight of a number of boxes.
SMARTER
Problem 25 requires students to multiply twice to solve this multistep problem. Because multiplication is associative, it does not matter in which order students multiply. Many will multiply 3 × 4 first to find the total people. Others may multiply 4 × 65 first to find the total apples one family eats.
Exercise 27 assesses students’ ability to find the product of two 2-digit numbers using partial products. Students who incorrectly fill in the boxes may have difficulty using the Distributive Property to represent partial products. They may be able to multiply using partial products but not be able to write how to find the partial products in this form.
Math on the Spot Video TutorUse this video to help students model and solve this type of Think Smarter problem.
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Meeting Individual Needs
Problem Solving • ThinkingProblem Solving • Applications
Problem SolvingProblem Solving
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Multiply Using Partial Products
Chapter 3 167
Record the product.
1. 23 × 79
__
2. 56 × 32
__
3. 87 × 64
__
4. 33 × 25
__
8. Evelyn drinks 8 glasses of water a day, which is 56 glasses of water a week. How many glasses of water does she drink in a year? (1 year = 52 weeks)
9. Joe wants to use the Hiking Club’s funds to purchase new walking sticks for each of its 19 members. The sticks cost $26 each. The club has $480. Is this enough money to buy each member a new walking stick? If not, how much more money is needed?
5. 94 × 12
__
6. 51 × 77
__
7. 69 × 49
__
1,400
210
180
+ 27
1,817
10. WRITE Math Explain why it works to break apart a
number by place values to multiply.
COMMON CORE STANDARD—4.NBT.B.5 Use place value understanding and properties of operations to perform multi-digit arithmetic.
Practice and HomeworkLesson 3.4
1,792 5,568 825
1,128 3,927 3,381
2,912 glasses No; $14 more is needed.
Check students’ work.
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167 Chapter 3
Practice and HomeworkUse the Practice and Homework pages to provide students with more practice of the concepts and skills presented in this lesson. Students master their understanding as they complete practice items and then challenge their critical thinking skills with Problem Solving. Use the Write Math section to determine student’s understanding of content for this lesson. Encourage students to use their Math Journals to record their answers.
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Personal Math Trainer
FOR MORE PRACTICE GO TO THE
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Spiral Review (4.OA.A.1, 4.OA.A.3, 4.NBT.B.5)
168
Lesson Check (4.NBT.B.5)
1. A carnival snack booth made $76 selling popcorn in one day. It made 22 times as much selling cotton candy. How much money did the snack booth make selling cotton candy?
2. List the partial products of 42 × 28.
3. Last year, the city library collected 117 used books for its shelves. This year, it collected 3 times as many books. How many books did it collect this year?
4. Washington Elementary has 232 students. Washington High has 6 times as many students. How many students does Washington High have?
5. List the partial products of 35 × 7. 6. Shelby has ten $5 bills and thirteen $10 bills. How much money does Shelby have in all?
$1,672
351 books
210, 35
800, 40, 320, 16
1,392 students
$180
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Monitoring Common Core Success
Maintaining Focus on the Major WorkThe major work in Grade 4 includes using the four operations with whole numbers to solve problems, as well as using place-value understanding and properties of operations to perform multi-digit arithmetic (4.NBT.B). Chapter 3 expands students’ ability to multiply whole numbers to include multiplying 2-digit numbers by multi-digit numbers. The topics covered often connect to the major work. For example, in Lesson 3.1, students apply place value understanding to round the numbers in a multiplication problem to produce an estimate for the product. In Lessons 3.1–3.4, students learn about the patterns when multiplying with multiples of 10. Students also expand their knowledge of the partial products method of multiplication, which requires addition of whole numbers as well as multiplication.
Connecting Content Across Domains and ClustersIn Lessons 3.1–3.4, a connection is made between Clusters 4.NBT.A and 4.OA.A. Lessons 3.2–3.4 (4.NBT.B) contain several Think Smarter and Go Deeper problems that require students to solve multi-step word problems (4.OA.A). For example, in Lesson 3.4,
problem 25 on page 166 requires the student to first multiply 3 by 4, and then multiply the result by 65 to find the answer.
Building FluencyIn Lessons 3.1–3.4, students enhance their understanding
of 4.NBT.B.4, which requires students to be fluent with addition and subtraction of multi-digit whole numbers. In Lessons 3.3 and 3.4, students use partial products to multiply, first with models and then without. In the partial products process, students are required to add several smaller products of varying numbers of digits to find the overall product.
Build fluency with HMH Mega Math’s fun and engaging activities. Use HMH Mega Math: The Number Games; Up, Up, and Array; Level C and The Number Game; Up, Up, and Array; Level I to strengthen students’ mastery of adding and subtracting within 1,000,000.
1 23 4
Lesson 3.4 168
Continue concepts and skills practice with Lesson Check. Use Spiral Review to engage students in previously taught concepts and to promote content retention. Common Core standards are correlated to each section.
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