Using Models and Algorithms to Solve Multiplication Problems

Objectives

Students will learn how to apply models for multiplication and division. They will:

appropriately choose a strategy to solve a variety of multiplication and division problems.
demonstrate an understanding of at least two strategies for computing that use properties of operations and estimation.

Essential Questions

What makes a tool and/or strategy appropriate for a given task?
How can data be organized and represented to provide insight into the relationship between quantities?

Related Unit and Lesson Plans

Related Materials & Resources

The possible inclusion of commercial websites below is not an implied endorsement of their products, which are not free, and are not required for this lesson plan.

NCTM Curriculum Focal Points

http://www.nctm.org/standards/content.aspx?id=270

Formative Assessment

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Multiple-Choice Items:

Which number sentence does this model represent?

A	2 × 14
B	14 × 4
C	(10 + 4) × (10 + 4)
D	(2 + 7) × 4

Which is the same as (30 + 2) × (10 + 6)?

A	32 × 16
B	16 × 16
C	32 × 60
D	300 + 12

Darcy had seven packages of straws. Each package had 48 straws. How many straws did Darcy have?

A	364
B	280
C	316
D	336

Use the array to answer the question.

Which number sentence shows the number of hearts?

A	8 × (10 + 7) = 136
B	8 × (17 + 7) = 192
C	7 × 8 = 56
D	10 × (7 + 8) = 150

What is the solution to 18 × 23?

A	390
B	414
C	314
D	369

What is the solution to 9 × 34?

A	274
B	363
C	306
D	236

Jose used the partial-products method to solve a multiplication problem. His work is shown below but it is partially hidden. What problem was Jose trying to solve?

A	40 × 57
B	58 × 47
C	28 × 74
D	48 × 57

Courtney is multiplying 82 × 3 as shown. What is the next step?

A	Add 240 and 6
B	Multiply 240 and 6
C	Multiply 20 by 3
D	Add 2 and 3

Greg made four batches of taffy. There were 36 pieces of taffy in each batch. How many pieces of taffy did Greg make?

A	126
B	252
C	144
D	420

Multiple-Choice Answer Key:

1. B

2. A

3. D

4. A

5. B

6. C

7. D

8. A

9. C

Short-Answer Items:

Priscilla has eight boxes of mints. There are 22 mints in each box. Priscilla must solve 8 × 22 to find the total number of mints. Solve 8 × 22 in two different ways and show your work for each.

Solve the multiplication problem 5 × 11 on the rectangular grid below. Shade in the spaces to show the multiplication problem. Label the parts. Then explain how the drawing represents
5 × 11.

Solve 45 × 36 using the partial-products method. Show your work. Explain what you did and why the steps are needed.

Short-Answer Key and Scoring Rubrics:

Priscilla has eight boxes of mints. There are 22 mints in each box. Priscilla must solve 8 × 22 to find the total number of mints. Solve 8 × 22 in two different ways and show your work for each.

Answers will vary.

Sample responses:

Points	Description
2	Examples are complete and detailed. The student response demonstrates thorough understanding of multiplication of multidigit numbers. The student response meets the requirements of the problem (includes two strategies).
1	Examples are correct but brief or simplistic. The student response demonstrates partial understanding of multiplication of multidigit numbers. The student response partially meets the requirements of the problem (includes at least one strategy).
0	Examples are incorrect or missing. The student response demonstrates no understanding of multiplication of multidigit numbers. The student response does not meet the requirements of the problem (includes less than one full strategy).

Solve the multiplication problem 5 × 11 on the rectangular grid below. Shade in the spaces to show the multiplication problem. Label the parts. Then explain how the drawing represents 5 × 11.

Answers will vary.

Possible solution for 5 × 11:

I know this drawing represents 5 × 11, because it is 5 rows of 11. In other words, I have 5 groups of 11. I took 11 and broke it apart into 10 and 1. So I have 5 groups of 10 and 5 groups of 1. My answer would be 50 + 5 = 55; therefore, 5 × 11 = 55.

Points	Description
2	The drawing accurately and clearly represents the product of 5 × 11. The written explanation is complete and detailed. The student response demonstrates thorough understanding of using a rectangular grid to model a multiplication problem. The student response meets all the requirements of the problem.
1	The drawing is present and shows the correct product of 5 × 11. The written explanation is correct but brief or simplistic. The student response demonstrates partial understanding of using a rectangular grid to model a multiplication problem. The student response partially meets the requirements of the problem.
0	The drawing may not accurately represent the product of 5 × 11. The written explanation is incorrect or missing. The student response demonstrates minimal to no understanding of using a rectangular grid to model a multiplication problem. The student response does not meet the requirements of the problem.

Solve 45 × 36 using the partial-products method. Show your work. Explain what you did and why the steps are needed.

Answers may vary depending on where a student decides to start.

Sample response:

I know that 45 can break apart into 40 + 5. I also know that 36 can break apart into 30 + 6. Using this information can help me when using the partial-products method.

To begin with, I did all the multiplication. I multiplied 5 × 6 = 30 because I had to multiply the ones value for both factors together. Then I multiplied the ones by the tens. I did 6 × 40 (240); then I did 30 × 5 (150). Finally, I had to multiply the tens from both factors, so I multiplied 30 × 40 and the product was 1,200. Then I added all the partial products together in order to get the product for the original multiplication problem. After adding 30 + 240 + 150 + 1200, I found the sum to be 1,620. The product of 45 × 36 is 1,620.

Points	Description
2	The student response demonstrates thorough understanding of multiplication of multidigit numbers using the partial-products method. All the steps needed to solve the problem are neatly and accurately shown in a logical manner. The explanation is specific and shows understanding of why the steps are needed. The student response meets the requirements of the problem.
1	The student response demonstrates partial understanding of multiplication of multidigit numbers using the partial-products method. The steps needed to solve the problem are shown with no major errors. The explanation is correct but may be brief or simplistic. The student response partially meets the requirements of the problem.
0	The student response demonstrates minimal understanding of multiplication of multidigit numbers using the partial-products method. The steps needed to solve the problem may be missing or have major errors. The explanation is unfinished or incorrect. The student response does not meet the requirements of the problem.

Performance Assessment:

Using Models and Algorithms to Solve Multiplication Problems

You are responsible for setting up trays of bakery goods at the local bakery. You have loaves of bread, cupcakes, cookies, bagels, and donuts to arrange. There are different strategies you can use to solve multiplication problems: repeated addition, break-apart, arrays, and partial products. Calculate how many of each type of bakery item will be on the display trays. You can use any multiplication strategies you like, as long as you use at least two different strategies throughout the assessment. The following table gives information helpful in setting up the trays. After solving the multiplication problems on the following pages, record the answers in the final column.

Item	Number of Trays Needed	Number of Items on Each Tray	Total Number of Items
Loaves of Bread	12	8
Cupcakes	6	18
Cookies	5	36
Bagels	15	18
Donuts	15	12

Loaves of Bread