Using Models and Algorithms to Solve Division Problems
Using Models and Algorithms to Solve Division Problems
Objectives
Students will learn how to apply models for division to solve a variety of multiplication and division problems. Students will:
- demonstrate an understanding of a minimum of two strategies for computing that use properties of operations and estimation.
- apply understanding of models of division, place value, properties, and the relationship of division to multiplication.
- employ inverse operations to solve multiplication and division equations.
- develop, discuss, and use efficient, accurate, and generalizable procedures to find quotients involving multidigit dividends.
- select and apply appropriate methods accurately to estimate, calculate mentally, and problem solve.
Essential Questions
How are relationships represented mathematically?
How can expressions, equations, and inequalities be used to quantify, solve, model, and/or analyze mathematical situations?
How can mathematics support effective communication?
How can patterns be used to describe relationships in mathematical situations?
How is mathematics used to quantify, compare, represent, and model numbers?
What does it mean to estimate or analyze numerical quantities?
What makes a tool and/or strategy appropriate for a given task?
When is it is appropriate to estimate versus calculate?
- How is mathematics used to quantify, compare, represent, and model numbers?
- How can mathematics support effective communication?
- How are relationships represented mathematically?
- What does it mean to estimate or analyze numerical quantities?
- When is it appropriate to estimate versus calculate?
- What makes a tool and/or strategy appropriate for a given task?
- How can patterns be used to describe relationships in mathematical situations?
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
- Interactive activity modeling different types of division problems
http://www.thinkingblocks.com/ThinkingBlocks_MD/TB_MD_Main.html
- Interactive activity designed to practice a variety of division problems
http://www.numbernut.com/basic/activities/div_4card_2x1word.shtml
- Worksheets with different versions that combine multiplication and division word problems using money at various levels
http://www.dadsworksheets.com/v1/Worksheets/Money%20Word%20Problems.html#MultiplicationandDivision
Formative Assessment
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View
Multiple-Choice Items:
1. A baker had 72 rolls and some bags. She put 8 rolls in each bag. How many bags of rolls did the baker have? Which equation represents this problem?
A
72 ÷ 8 = __
B
72 ÷ 9 = __
C
8 × __ = 72
D
9 × __ = 72
2. Stewart is decorating gingerbread houses. He knows he has 48 gumdrops. Choose the statement below that would make this a grouping division problem.
A
If there were 6 gingerbread houses and each house had the same number of gumdrops, how many gumdrops would be on each gingerbread house?
B
If each house had the same number of gumdrops, how many gumdrops would Stewart have to put on each house?
C
Stewart wanted each gingerbread house to have 8 gumdrops. How many gingerbread houses could he decorate this way?
D
Stewart wanted to make each gingerbread house the same. How many gingerbread houses would he need to make in order to use up all the gumdrops?
3. Jace is stocking coolers at the ice cream parlor with 30 gallons of ice cream. Each cooler can hold 6 gallons of ice cream. Which question below could be answered using this data?
A
How many more gallons of ice cream are needed to completely fill the coolers?
B
How many gallons of ice cream will the parlor use?
C
How many more coolers are there than gallons of ice cream?
D
How many coolers are needed if each cooler is completely filled?
4. In a certain division problem, the dividend is 48, the divisor is 8, and the quotient is 6. Which part of the division equation would be the product of the related multiplication sentence?
A
divisor
B
dividend
C
quotient
D
none of the above
5. A baker had a strip of dough that was 18 inches long from which she wanted to make pretzels. Each pretzel required 6 inches of dough, so she was able to make 3 pretzels. Which equation represents the number of inches of dough required for each pretzel?
A
6 × 3 = 18
B
18 ÷ 6 = 3
C
18 ÷ 3 = 6
D
6 + 3 + 9 = 18
6. There are 6 students playing marbles. Each student has 8 marbles. Figure out how many marbles they have in all. Which statement is a related division equation?
A
48 ÷ 6 = 8
B
48 ÷ ___ = 6
C
48 ÷ 8 = 6
D
all of the above
7. Finish the following problem. What is the quotient?
A
14
B
19
C
114
D
119
8. Jordan needs to read 375 pages of a book in 5 nights. How many pages should he read each night to be sure he finishes?
A
55 pages
B
65 pages
C
75 pages
D
77 pages
9. A balloon company was making balloon bouquets for a party. Each bouquet had 4 balloons. How many balloon bouquets could be made with 108 balloons?
A
22 balloon bouquets
B
27 balloon bouquets
C
28 balloon bouquets
D
32 balloon bouquets
Multiple-Choice Answer key:
1. A
2. C
3. D
4. B
5. C
6. D
7. C
8. C
9. B
Short-Answer Items:
- Take the following division problem of the sharing type, and rewrite it as a grouping problem. Then show the equation you would use to solve the problem.
At the carnival, Sophie decided to play the Lucky Ducky game. She saw that there were 54 duckies in the game. The duckies were floating in 6 inflatable pools. Each pool had the same number of duckies. How many duckies were in each inflatable pool?
- After reading the problem below, write a division equation and its related multiplication equation. Solve them.
A seamstress had 56 feet of ribbon. Each banner she made required 8 feet of ribbon. How many banners could she make?
- Use the partial-quotients method to solve the problem below.
Jenna made 192 brownies for a bake sale at her school. She put 6 brownies in each box. How many boxes of brownies could Jenna bring to the sale?
Short-Answer Key and Scoring Rubrics:
- Take the following division problem of the sharing type, and rewrite it as a grouping problem. Then show the equation you would use to solve the problem.
At the carnival, Sophie decided to play the Lucky Ducky game. She saw that there were 54 duckies in the game. The duckies were floating in 6 inflatable pools. Each pool had the same number of duckies. How many duckies were in each inflatable pool?
Possible answer: At the carnival, Sophie decided to play the Lucky Ducky game. She saw that there were a total of 54 duckies in the game. The duckies were floating in inflatable pools. Sophie counted 9 duckies in each inflatable pool. How many inflatable pools were there?
54 ÷ 9 = 6 pools
Points
Description
2
- A rewritten word problem is present and accurately reflects a grouping problem.
- The response shows complete understanding of the mathematics.
- The answer is correct.
- The response meets the requirements of the problem.
1
- A rewritten problem is present but does not reflect grouping, or appears to be grouping but is only partially written.
- The response shows partial understanding of the mathematics.
- The answer is correct, but a method other than grouping is used or no method is shown.
- The response partially meets the requirements of the problem.
0
- The rewritten problem is not present.
- The response shows no understanding of the mathematics.
- The answer is incorrect due to major mathematical error/misunderstanding or is missing.
- The response does not meet the requirements of the problem.
- After reading the problem below, write a division equation and its related multiplication equation. Solve them.
A seamstress had 56 feet of ribbon. Each banner she made required 8 feet of ribbon. How many banners could she make?
Answer: 56 ÷ 8 = ___
8 × ___ = 56 (7 banners)
Points
Description
2
- Division and multiplication equations are both shown and accurately represent the problem.
- The answer is correct.
- The response shows complete understanding of the mathematics.
- The response meets the requirements of the problem.
1
- A division or multiplication equation is shown. It accurately represents the problem, but the other equation is missing. Or both are present but one is illogical.
- The answer is correct, but a method other than grouping is used or no method is shown.
- The response shows partial understanding of the mathematics.
- The response partially meets the requirements of the problem.
0
- Both the division and multiplication equations are missing.
- The answer is incorrect due to major mathematical errors/misunderstandings or is missing.
- The response shows no understanding of the mathematics.
- The response does not meet the requirements of the problem.
- Use the partial-quotients method to solve the problem below.
Jenna made 192 brownies for a bake sale at her school. She put 6 brownies in each box. How many boxes of brownies could Jenna bring to the sale?
Estimates may vary.
Answer:
32 boxes of browniesPoints
Description
2
- The work or written explanation is thorough and clear.
- The response shows complete understanding of the mathematics.
- The answer is correct.
- The response meets the requirements of the problem.
1
- The work is shown but is somewhat difficult to follow, or is partially shown.
- The response shows partial understanding of the mathematics.
- The answer is incorrect due to minor mathematical error(s).
- The response partially meets the requirements of the problem.
0
- The work is not shown or is illogical.
- The response shows no understanding of the mathematics.
- The answer is incorrect due to major mathematical errors or is missing.
- The response does not meet the requirements of the problem.
Performance Assessment:
You have been invited to the school sports banquet next week. You offered to help your basketball coach and the sports director prepare. After discussing what still needs to be done, they asked you to help them solve a few problems.
Part I:
The first problem is that the coach has three numbers written down that have something to do with the number of parents attending the banquet and their tables. He was interrupted and cannot remember what he was about to calculate. Write down a grouping division problem that is likely to be a useful calculation he may have been working on earlier. Explain what each number represents in your problem and solve the problem.
28, 4, 7
Grouping division word problem:
________________________________________________________________________
________________________________________________________________________
Answer: _________________________________
The 28 represents _________________________________________________________.
The 4 represents __________________________________________________________.
The 7 represents __________________________________________________________.
Explain why this is an example of a division problem that would be classified as a grouping division problem.
________________________________________________________________________
________________________________________________________________________
Part II:
During the sports banquet each athlete is going to receive a participation certificate. The program director needs a total of 136 certificates and they come in packages of 8. How many packages of certificates should he buy?
Explain how the problem can be solved using division by sharing or division by grouping.
________________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
Write a division equation that could be used to solve this problem.
________________________________________________________________________
Now write a related multiplication sentence with an unknown factor.
________________________________________________________________________
Solve the division problem using the partial-quotients method. Be sure to show all the steps of your work.
Answer: ______________
Part III:
The coach has collected $492 from parents and athletes to cover the cost of the banquet meal. Write and solve a division question that may help provide some useful information regarding the banquet. (You may use information provided in Parts I and II as necessary.)
________________________________________________________________________
________________________________________________________________________
What did this help you find?
________________________________________________________________________
Explain when it is important to know how to divide.
________________________________________________________________________
________________________________________________________________________
Performance Assessment Key and Scoring Rubric:
You have been invited to the school sports banquet next week. You offered to help your basketball coach and the sports director prepare. After discussing what still needs to be done, they asked you to help them solve a few problems.
Part I: Solution
The first problem is that the coach has three numbers written down that have something to do with the number of parents attending the banquet and their tables. He was interrupted and cannot remember what he was about to calculate. Write down a grouping division problem that is likely to be a useful calculation he may have been working on earlier. Explain what each number represents in your problem and solve the problem.
28, 4, 7
Grouping problems will vary but may include:
The number of parents attending will be 28. If he wants to put 7 parents at each table, how many tables will he need?
Answer: Answers will vary. 4 tables
The 28 represents the total number of parents or dividend.
The 4 represents how many tables are needed for parents or quotient.
The 7 represents how many parents will sit at each table or divisor.
Explain why this is an example of a division problem that would be classified as a grouping division problem.
This is an example of a grouping division problem because you are given the total number of an item, and you are trying to find out how many groups (tables) you can make. You already know how many items will be in each group.
Part II: Solution
During the sports banquet each athlete is going to receive a participation certificate. The program director needs a total of 136 certificates and they come in packages of 8. How many packages of certificates should he buy?
Explain how the problem can be solved using division by sharing or division by grouping.
Answers will vary. When using division by sharing, one certificate is placed in each of the packages at a time until all the certificates have been shared and equally divided between the 8 packages. You are sharing 136 certificates equally between 8 packages.
Write a division equation that could be used to solve this problem.
136 ÷ 8 = ___
Now write a related multiplication sentence with an unknown factor.
8 × ___ = 136
Solve the division problem using the partial-quotients method. Be sure to show all the steps of your work.
Answer: 17 packages
Part III: Solution
The coach has collected $492 from parents and athletes to cover the cost of the banquet meal. Write and solve a division question that may help provide some useful information regarding the banquet. (You may use information provided in Parts I and II as necessary.)
What did this help you find?
Explain when it is important to know how to divide.
Answers will vary. Possible answer: There are 136 participants and 28 parents coming based on the other two parts of problem, for a total of 164 people. $492 ÷ 164 = 3. So each person paid $3.00 to attend the banquet.
Sample answer: It is important to know how to divide if you have to find out how many groups you need or how many sets of items to put in a group. Many occupations use division. For example, a baker may need to know how many packages she will make with a certain number of cookies. A fashion designer may need to know how many dresses he can make with a given amount of fabric. A party planner may need to know how many packages of plates, hot dogs, and buns he needs to purchase for a party.
Points
Description
4
- Mathematics is correct with complete and organized work shown.
- The response has accurate and thorough explanations that distinguish understanding of grouping and sharing type division; it includes examples or other clarifying information.
- The response displays advanced understanding of the concepts, processes, and purpose of division.
- The student performs beyond the problem requirements.
3
- Mathematics is correct with work shown.
- The response accurately explains grouping and sharing division problems.
- The response displays a good understanding of the concepts, processes, and purpose of division.
- The response meets all of the problem requirements.
2
- Mathematics is correct with little or no work shown, or partially incorrect due to minor error(s).
- The explanation of grouping and sharing is present and accurate but lacking detail.
- The response displays partial understanding of the concepts, processes, and purpose of division.
- The response meets most of the problem requirements.
1
- Mathematics is all present but incorrect due to several errors, or some parts are correct while other parts are missing. Little work is shown.
- Only one type of division is explained correctly; the other is incorrect or missing.
- The response displays little understanding of the concepts, processes, and purpose of division.
- The response does not meet a majority of the problem requirements.
0
- Mathematics is incorrect or missing. Little or no work is shown.
- The explanations of grouping and sharing division are both either incorrect or missing.
- The response displays no understanding of the concepts, processes, or purpose of division.
- The response does not meet the problem requirements.
Final 05/17/2013
