Assessments

Automaticity

3.11 Connecting the Area Model to the AlgorithmCOMMON CORE STATE STANDARDS4.NBT.B.5 – Number and Operations in Base TenMultiply 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.BIG IDEAStudents will connect the area model and the partial products method to the standard algorithm.

Standards of Mathematical Practice

□ Make sense of problems and persevere in solving them

Reason abstractly and quantitatively□ Construct viable arguments and critique the

reasoning of others Model with mathematics Use appropriate tools strategically□ Attend to precision□ Look for and make use of structure□ Look for and express regularity in repeated

reasoning

Informal Assessments:

□ Math journal□ Cruising clipboard□ Foldable□ Checklist Exit ticket Response

Boards Problem Set Class Discussion

PREPARING FOR THE ACTIVITY MATERIALS Personal response

boards Problem set 3.11 Exit ticket 3.11 Additional Practice 3.11

VOCABULARY Area model

AUTOMATICITY TEACHER NOTESMultiply mentally:

1. Repeat the process from Block 10, expanding to four-digits for possible sequence: 4,312 x 2, 2,032 x 4, and 3,203 x 4.

Multiply in three different ways:1. Write 43 x 2. Say the multiplication sentence in unit

form. (4 tens 3 ones x 2.)2. Show the multiplication sentence using partial

products. Give the students time to respond.3. Show the multiplication sentence using number disks.

Give students time to respond.4. Show the multiplication sentence using the standard

algorithm. Allow students to respond.

Select appropriate activities depending on the time allotted for automaticity.

Multiply Mentally: Reviewing these mental multiplication strategies will provide a foundation for students to succeed during the concept development.

Multiply 3 different Ways: This drill will review content from Blocks 7-10.

Source: https://www.engageny.org/resource/grade-4-mathematicsGrade 4 Unit 3: Block 11

TeachingNew

Concepts

5. Repeat the process using the following possible sequence: 54 x 2, 63 x 3.

SETTING THE STAGE TEACHER NOTESApplication Problem

1. Allow students to work in partners or small groups using the RDW strategy to solve. Students should use their response boards to record work.

Write an equation for each area of the rectangle. Then find the sum of the two areas. Bonus: find a faster method for finding the area of the combined rectangles.

Connection to Big IdeaToday, we will continue working with adding different kinds of measurements using the algorithm. Through our work today we will learn how the algorithm works when we have to add more than two numbers.

Note: This problem is designed to bridge learning from Topic A, in which students solved for the area, to this block where they will learn to model multiplication problems using the area model. The placement of the small rectangle to the right of the larger rectangle is intentional for showing the tens and ones of the area model. It is recommended that this problem be taught immediately prior to Exploring the Concept.

EXPLORE THE CONCEPT TEACHER NOTESProblem 1: Multiply a three-digit number by a one-digit number using the area model.

1. Draw a rectangle with a length of 8 and a width of 200.

2. Tell your neighbor how o find the area. (Multiply 8 times 200. That equals 1,600)Write the area inside your rectangle.

3. Think back to the application problem we just completed. We had two rectangles that also had a length of 8. Let’s combine all three rectangles: this one and the two from the application problem. Model drawing them for students.

4. With your partner, discuss how to find the area of all three rectangles put together. (In the application problem, I multiplied 8 x 4 and 8 x 30. So then I can just multiply a times 200 and add all the sums together.

UDL- Multiple Means of Action and Expression: Scaffold student use of the area model to solve with the following options: Provide a blank area

model template for students to slip into their response boards.

Review expanded form with hide zero cards or number disks during OPDI.

Simplify the multiplication, for example, use 4 as a factor rather than 8.

Source: https://www.engageny.org/resource/grade-4-mathematicsGrade 4 Unit 3: Block 11

5. Record that as one continuous addition problem with your partner.

6. Guide students to record (8 x 200) + (8 x 30) + (8 x 4).

7. You are saying to multiply each section of the width times 8? Record 8(200 + 30 +4) on the board.

8. Solve to find the area of the entire rectangle. Let’s begin with the largest rectangle. 200 times 8? (1,600.)

9. Record 1,600 as a partial product in the area model and in the written method.

10. 30 times 8? (240) Record as a partial product in the area model and in the written method.

11. Show your partner where to record 4 times 8. Tell your partner the multiplication sentence represented by the area model. (234 times 8 equals 1,872.)

12. Compare the partial products to the regular area model. (The area inside each smaller rectangle is the same as each of the partial products.)

13. We recorded the partial products starting with the largest unit, the hundreds. Does the order of partial products change the final product? Work with your partner to solve 234 times 8 using partial products, beginning with the smallest unit, the ones. (The answer is the same. I can multiply in any order using partial products. That’s the commutative property of addition. I can record partial products using the smallest or the largest unit first.

14. Yes, the rectangle, or area model is another way to represent the partial products in multiplication.

Problem 2: Multiply a three-digit number by a one-digit number, connecting the area model to the standard algorithm.

1. Display 316 x 4.2. How many hundred, tens, and ones are in 316? (3

hundreds, 1 ten, 4 ones.)3. Draw an area model with a width of 3 hundreds 1 ten

and 6 ones and a length of 4. 4. Tell your partner how to solve using the area model.

(A rectangle is partitioned into hundreds, tens, and ones. I’ll multiply 4 times 3 hundreds, 4 times 1 ten, and 4 times 6 ones and add the three products together for the answer.)

5. Work with a partner to multiply.6. Circulate, providing assistance, while students work.7. 4 times 3 hundreds is..? (12 hundreds)8. 4 times 1 ten is? (4 tens)9. 4 times 6 ones is? (24 ones)10. Solve 316 times 4 using the standard algorithm and

compare your answer to the area model. (I got the same answer using both ways. The areal model doesn’t let me show how to regroup 24 ones

UDL – Multiple Means of Action and Expression: One advantage to the area model is its flexibility for learners. Students can represent their partial products as arrays of number disks, in unit form, or standard form. Though it is not as efficient as the standard algorithm, it may be an effective scaffold for students working below grade level.

Source: https://www.engageny.org/resource/grade-4-mathematicsGrade 4 Unit 3: Block 11

Ongoing Learning & Practice

for 2 tens and 4 ones, but the algorithm does. I can regroup in the area model. I can draw an arrow to regroup 20 ones as 2 tens. Now my area model looks like a place value chart because I regrouped to show 6 tens. The area model aligns better to the partial product method, but the algorithm is still the quickest way to solve.)

11. Repeat with 5,263 x 5 drawing the area model and comparing it to the algorithm or the partial products method.

Problem 3: Solve a word problem using the standard algorithm, area model, or partial products strategy.

1. Display and read problem aloud: A cafeteria makes 4,408 lunches each day. How many lunches are made Monday through Friday?

2. Discuss with your partner how you would solve this problem.

3. What are some methods that can help you solve this problem? Try to get students to give the three options we have been working with today, area model, algorithm, and partial product.

4. You could also use the distributive property to help you solve by breaking apart your answer. Choose your method and solve.

5. When debriefing the possible solution types, make a note of how to draw an area model without a digit in the tens column.

Problem Set: Allow students to finish Problem Set 3.11. Students should do their personal best to complete the problem set in groups, with partners, or individually. For some classes, it may be appropriate to modify the assignment by specifying which problems they work on first. Some problems do not

Before circulating, consider reviewing the reflection questions that are relevant to today’s problem set.

Source: https://www.engageny.org/resource/grade-4-mathematicsGrade 4 Unit 3: Block 11

Share Summarize

specify a method for solving. Students solve these problems using the RDW approach used for Application Problem.

REFLECTION TEACHER NOTES1. Invite students to review their solutions for

Problem Set 3.11. They should check their work by comparing answers with a partner before going over answers as a class.

2. Guide students in a conversation to debrief the Problem Set and process the block. You may choose to use any combination of the questions below to lead the discussion.

Can you solve any of the problems in problem 1 using a different method or strategy?

In Problem 1, how does the area model connect with the second number sentences? How could the distributive property be used to solve problems without drawing the area model?

For problems 4-6 which method did you sue to solve? Why?

How is finding the area of a rectangle similar to finding the product using the area model?

3. Allow students to complete Exit Ticket 3.11 independently.

Look for misconceptions or misunderstandings that can be addressed in the reflection.

Source: https://www.engageny.org/resource/grade-4-mathematicsGrade 4 Unit 3: Block 11

Name: ________________________ Date: ________________Problem Set 3.11 – page 1Solve the following problems using the standard algorithm, the partial products method, and the area model.

1. 425 x 4

4 (400 + 20 + 5)

(4 x _____) + (4 x _____) + (4 x ____)

2. 534 x 8

7 (_____ + ______ + ______)

(7 x _____) + (7 x _____) + (7 x ____)

3. 209 x 8

____ (_____ + ______ + ______)

(__ x _____) + (__ x _____) + (__ x ____)

Problem Set 3.11 – page 2Solve using the partial products method.

Cayla’s school has 258 students. Janet’s school has 3 times as many students as Cayla’s. How many students are in Janet’s school?

Model with a tape diagram and solve.

4 times as much as 467.

Solve using the standard algorithm, the area model, the distributive property, or the partial products method.

1. 5,131 x 7

2. 3 times as many as 2,805

3. A restaurant sells 1,725 pounds of spaghetti and 925 pounds of linguini every month. After 9 months, how many pounds of pasta does the restaurant sell? Write your answer as a statement.

Name: _____________________________ Date: ___________Exit Ticket 3.11

1. Solve using the standard algorithm, the area model, the distributive property, or the partial products method.

2,809 x 4

2. The monthly school newspaper is 9 pages long. Mrs. Smith needs to print 675 copies. How many sheets of paper will she use?

Name: _____________________________ Date: ___________Exit Ticket 3.11

1. Solve using the standard algorithm, the area model, the distributive property, or the partial products method.

2,809 x 4

2. The monthly school newspaper is 9 pages long. Mrs. Smith needs to print 675 copies. How many sheets of paper will she use?

Name:__________________________________ Date:________

Additional Practice 3.11 – page 11. Solve the following expressions using the standard algorithm, the partial products

method, and the area model.

Additional Practice 3.11 – page 22. Solve using the partial products method.

a. 3 0 2 × 8

b. 2 1 6 × 5

c. 5 9 3 × 9

5 ( ____ + ____ + ____ )

( __ × _____ ) + ( __ × _____) + ( __ × ____ )

On Monday 475 people visited the museum. On Saturday there were 4 times as many visitors as there were on Monday. How many people visited the museum on Saturday?

3. Model with a tape diagram and solve.

6 times as much as 384.

Solve using the standard algorithm, the area model, the distributive property, or the partial products method.

4. 6,253 × 3

5. 7 times as many as 3,073.

6. A cafeteria makes 2,516 pounds of white rice and 608 pounds of brown rice every month. After 6 months, how many pounds of rice does the cafeteria make? Write your answer as a statement.