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Grade _4_ Lesson Title: Problem Solving and Multiplicative ComparisonUnit 1: Multiplication and Division Concepts Time Frame: 5-7 Days Essential Question: What real-life situations require the use of multiplication or division and how are they modeled?Targeted Content Standard(s): Student Friendly Learning Targets4.OA.1 Interpret a multiplication equation as a comparison, e.g. interpret 35=5×7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.

4.OA.2 Multiply or divide to solve word problems involving multiplicative comparison, e.g. by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.

I can… Solve word problems that involve

comparing multiplicative situations. Solve multiplication equations. Explain why a model accurately

represents a multiplicative situation.

Targeted Mathematical Practice(s): 1 Make sense of problems and persevere in solving them 2 Reason abstractly and quantitatively 3 Construct viable arguments and critique the reasoning of others 4 Model with mathematics 5 Use appropriate tools strategically 6 Attend to precision 7 Look for and make use of structure. 8 Look for an express regularity in repeated reasoning

Supporting Content Standard(s): (optional)3.OA.3

Explanation of Rigor: (Fill in those that are appropriate.)Conceptual:Students develop the concept of multiplicative comparison.

Procedural:Students represent multiplicative comparison using diagrams and equations.

Application:Students solve multi-step word problems, including those that involve multiplicative comparison.

Vocabulary:Multiplicative comparison

primearrayStandard formcomposite

area modelWritten formdivisorequation

Expanded formdividendproductFactor

remainderquotientMultiplebar model

Evidence of Learning (Assessment):Pre-Assessment: Additive Comparison Exit SlipFormative Assessment(s): Multiplication Situation Card Sort Activity Multiplicative Comparison Card ActivitySummative Assessment: Grade 4 Unit 1 Multiplicative Comparison AssessmentSelf-Assessment: Journal Prompt: What have I learned about Multiplicative Comparisons?

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Grade _4_ Lesson Title: Problem Solving and Multiplicative ComparisonUnit 1: Multiplication and Division Concepts Time Frame: 5-7 Days Essential Question: What real-life situations require the use of multiplication or division and how are they modeled?Lesson Segments:

1. Addition and Subtraction Problem Solving2. Mental Multiplication and Division using Reasoning3. Multiplication and Division Problem Solving4. Multiplicative Comparison

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Grade _4_ Lesson Title: Problem Solving and Multiplicative ComparisonUnit 1: Multiplication and Division Concepts Time Frame: 5-7 Days Essential Question: What real-life situations require the use of multiplication or division and how are they modeled?Lesson Procedures:Segment 1

Approximate Time Frame:

90 – 120 mins

Lesson Format: Whole Group Small Group Independent

Modeled Guided Collaborative Assessment

Resources:Problem Solving Structures (Common Core Glossary pg 88 table 1: Common addition and subtraction situations)

Handout 1: Addition and subtraction bar model blank sheetHandout 2: Exit Slip

Focus:

Students will review the common addition and subtraction situations found on pg. 88 of the Common Core Glossary, Table 1

Modalities Represented: Concrete/Manipulative Picture/Graph Table/Chart Symbolic Oral/Written Language Real-Life Situation

Math Practice Look For(s):

Make sense of problems and persevere: watch that students fill in the bar model correctly and can explain their thinking

Model with Mathematics: students transfer information found on the bar model into equations

Use Tools Strategically: students use the bar model graphic as a way to organize the information

Differentiation for Remediation:

Have manipulatives available to solve the problems as needed

Differentiation for English Language Learners:

Ensure students know the difference between hole and whole

Differentiation for Enrichment:

Potential Pitfall(s):

Set up their bar model incorrectly.

Addition errors.

Independent Practice (Homework):

Handout 2: Exit Slip

Steps:

Students will work in pairs as they engage in this review of addition and subtraction. They should have handout 1 with the bar model graphic or white boards and markers. Explain that this is an opportunity for them to re-engage with content from the past few years to get ready to extend that thinking into multiplication situations this year.

Put a bar model graphic up on display. Share the word problem examples below one at a time. Have the students work in pairs to come to agreement on how to fill in their bar model. After working in pairs, have the students share where to put the numbers into the graphic.

Teacher Notes/Reflections:

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Grade _4_ Lesson Title: Problem Solving and Multiplicative ComparisonUnit 1: Multiplication and Division Concepts Time Frame: 5-7 Days Essential Question: What real-life situations require the use of multiplication or division and how are they modeled?

Note that you will add the labels as you add them to the graphic. Have the students justify their thinking. Record the equation after the graphic is filled in – discuss the use of a letter (n) for the unknown. Make sure the students understand different situations require different parts of the graphic to be filled in.

Repeat this procedure for each example. This is an opportunity to support an environment rich in hearing one another’s thinking and justification, so be sure to prompt students to ask clarifying questions and re-state students’ explanations. Agreement should come from the students rather than the teacher telling them their solutions are correct.

ExamplesResult Unknown (Putting Together)124 cars and 84 trucks are in a parking lot. How many vehicles are there altogether? 208 vehicles124 + 84 = n

124 cars 84 trucks

n total vehicles 124 + 84 = ?

Discussion: Encourage the students to create their bar models with precision. The parts must match the total.

Change Unknown (Taking Apart)208 vehicles are in a parking lot. 124 are cars and the rest are trucks. How many vehicles are trucks? 84 trucks124 + n = 208

124 cars n trucks 124+?

208 total vehicles =208

Start UnknownSome cars were in a parking lot in the morning. 84 trucks drove into the lot in the afternoon. Now there are 208 total vehicles parked. How many cars were in the parking lot to start with? 124 cars n + 84 = 208

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n Cars 84 Trucks

208 Total Vehicles ?+84=208

Continue the lesson with the remaining structures.

Adding ToResult Unknown48 horses were in the pasture. 27 more horses came to join them. How many horses are in the pasture now? 75 horses48 + 27 = n

Change Unknown48 horses were in the pasture. Some more horses came to join them. There are now 75 horses. How many horses joined them? 27 horses48 + n = 75

Start UnknownSome horses were in the pasture. 27 more horses came to join them. Now there are 75 horses in the pasture. How many horses were there before? 48 horsesn + 27 = 75

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Taking FromResult Unknown48 horses were in the pasture. 27 of the horses left. How many horses are in the pasture now? 21 horses48 – 27 = n

Change Unknown48 horses were in the pasture. Some horses left. There are now 21 horses. How many horses left the pasture? 27 horses48 - n = 21

Start UnknownSome horses were in the pasture. 27 horses galloped away. Now there are 21 left. How many horses were in the pasture before? 48 horsesn – 27 = 21

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CompareDifference Unknown(“How many more?” version)Manny has 63 baseball cards. Jeff has 47 baseball cards. How many more baseball cards does Manny have than Jeff? 16 cards

63 – 47 = n 47 + n = 63 63 – n = 47(any of these equations are acceptable)

Manny’s Cards 63

Jeff’s Cards47

(“How many fewer?” version)Manny has 63 baseball cards. Jeff has 47 baseball cards. How many fewer baseball cards does Jeff have than Manny? 16 cards47 + n = 63 63 – n = 47 63 – 47 = n(any of these equations are acceptable)

Bigger Unknown(Version with “more”)Manny has 16 more baseball cards than Jeff. Jeff has 47 baseball cards. How many baseball cards does Manny have? 63 cards47 + 16 = n n – 16 = 47 (any of these equations are acceptable)

Manny’s Cards n

Jeff’s Cards 47

(Version with “fewer”)Jeff has 16 fewer baseball cards than Manny. Jeff has 47 baseball cards. How many baseball cards does Manny have? 63 cards47 + 16 = n n – 16 = 47 (any of these equations are acceptable)

Smaller Unknown7

Grade _4_ Lesson Title: Problem Solving and Multiplicative ComparisonUnit 1: Multiplication and Division Concepts Time Frame: 5-7 Days Essential Question: What real-life situations require the use of multiplication or division and how are they modeled?(Version with “more”)Manny has 16 more baseball cards than Jeff. Manny has 63 baseball cards. How many baseball cards does Jeff have? 47 cardsn + 16 = 63 63 – n = 16 63 – 16 = n(any of these equations are acceptable)

Manny’s Cards 63

Jeff’s Cards n

(Version with “fewer”)Jeff has 16 fewer baseball cards than Manny. Manny has 63 baseball cards. How many baseball cards does Jeff have? 47 cardsn + 16 = 63 63 – n = 16 63 – 16 = n(any of these equations are acceptable)

Exit Slip: Solve two different additive comparison problems. Fill in a bar model and write an equation for each problem.

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Grade _4_ Lesson Title: Problem Solving and Multiplicative ComparisonUnit 1: Multiplication and Division Concepts Time Frame: 5-7 Days Essential Question: What real-life situations require the use of multiplication or division and how are they modeled?Segment 2


10 minute opener (this should be an on-going opener for this unit to provide quick mental fluency opportunities)



Resources:

Handout 3: Multiplication ReasoningNumber gridNumber lineCounters

Resources for teachers on math talk www.mathsolutions.com

Eyes on Math: a visual approach to teaching math concepts by Marian Small

Focus:

Students select and accurately apply strategies to mentally calculate products. This develops fluency with efficient procedures.


Math Practice Look For(s): Differentiation for Remediation:

Differentiation for English Language Learners:

Differentiation for Enrichment:


Re-adjusting or compensating incorrectly.


Steps:

Visual Math:

Another opportunity for teachers to support students’ fluency with computation strategies is to provide short visual math number talks.

Show a pictorial array of a multiplication fact (Handout 3: Multiplication Reasoning)

XXXXXX XX XXXXXX XX XXXXXX XX


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http://www.mathsolutions.com/

Grade _4_ Lesson Title: Problem Solving and Multiplicative ComparisonUnit 1: Multiplication and Division Concepts Time Frame: 5-7 Days Essential Question: What real-life situations require the use of multiplication or division and how are they modeled? XXXXXX XX

Ask the following Questions: What multiplication does the picture show? We want students

to notice that the array can be seen as both 4 x 8 but also 8 x 4. This will support the understanding of commutative property of multiplication introduced in grade 3 (3.0A.5).

How can you rearrange the rows or columns to show the other ways to figure out 4 x 8? (We want students to know that any combination of rows or columns could be used such as (2 x 8) + (2 x 8) is the same as 4 x 8. Try to find students who have divided the array in different vertical or horizontal ways. This will support their understanding of the distributive property.XXXXXX XX

XXXXXX XX

XXXXXX XX XXXXXX XX

What does the picture tell you about how you can figure out multiplication facts you don’t already know? (We want students to come to realize that knowing smaller facts can help them with bigger facts.) Have them justify their thinking.

Teachers should use this time to do some informal observation of how comfortably students can use the strategy of breaking multiplication arrays into easier chunks. Students struggling with this should receive some explicit instruction in small group at a later time.

This activity will carry forward to later when they work with two-digit by two-digit multiplication. (This is a great connection to subsequent learning.)

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60 min



Resources:

Problem Solving Structures (Common Core Glossary pg 89 table 2: Common Multiplication and Division Situations)Color chips (1 bag for 2 students) Overhead (optional, but will be helpful) Overhead color chips (optional)Centimeter grid paper, to be used for arrays (several sheets per student,plus a transparency for the teacher)

Spiral notebooks (for students to write/draw examples)Handout 4: Multiplication Models

Formative Assessment: Handout 5: Multiplication Story Situation Sorting Cards

Focus:

Students will review the common multiplication and division situations found on pg. 89 of the Common Core Glossary, Table 2



Make sense of problems and persevere: watch that students fill in the bar model and area model correctly and can explain their thinking

Model with Mathematics: students transfer information found on the models into equations

Use Tools Strategically: students use the bar models, counters, and area models as ways to organize the information

Differentiation for Remediation: Give students a copy of the Problem Solving Structures handout to help them sort the Problem Solving Cards.

Differentiation for English Language Learners: Make Vocabulary Cards showing the different models for multiplication and division problem situations.

Differentiation for Enrichment: Have students create their own Problem Solving Cards using the different types of multiplication and division structures.

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Set up their models incorrectly.

Multiplication fact errors.


Handout 4: Multiplication Models

Steps:

Students will work in pairs as they engage in this activity. Explain that they are going to build on what they learned about addition and subtraction situations to model multiplication situations.

Using the examples below, build/draw the graphic support (bar model or array model) with the group for each problem. It will help to have the students write out the problems in their spiral notebooks (or notes) as you work through the examples.

For the EQUAL GROUPS examples, the students will draw a bar model. Share the word problem examples below one at a time. Have the students work in pairs to come to agreement on how to fill in their bar model. After working in pairs, have the students share where to put the numbers into the graphic. Note that you will add the labels as you add them to the graphic. Have the students justify their thinking. Record the equation after the graphic is filled in. Make sure the students understand different situations require different parts of the graphic to be filled in.

Repeat this procedure for the EQUAL GROUPS examples. This is an opportunity to support an environment rich in hearing one another’s thinking and justification, so be sure to prompt students to ask clarifying questions and re-state students’ explanations. Agreement should come from the students rather than the teacher telling them their solutions are correct.

For the ARRAYS, AREA examples, the students will use an array model to support their thinking about the multiplication situations. For each word problem, have the students write the problem, model the problem with color chips, use centimeter grid paper to draw the arrays, write the equation using a variable (e.g., 3×6=p) and figure out the answer. Again, ensure they know how to label their model. Pairs will again fill in the diagrams together and be prepared to explain their reasoning.

EQUAL GROUPS: EXAMPLESUnknown Product (Equal Groups)There are 6 boxes with 8 muffins in each box. How many muffins are there in all? 48 muffins

6 × 8 = n


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Group Size Unknown (Equal Groups)If 48 muffins are shared equally into 6 boxes, then how many muffins will be in each box? 8 muffins6 × n = 48 and 48 ÷ 6 = n

Number of Groups Unknown (Equal Groups)If 48 muffins are to be packed 8 to a box, then how many boxes are needed? 6 boxesn × 8 = 48 and 48 ÷ 8 = n

ARRAYS, AREA: EXAMPLESUnknown Product (Arrays, Area)There are 5 rows of chairs with 7 chairs in each row. How many chairs are there in all? 35 chairs5 × 7 = n

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Grade _4_ Lesson Title: Problem Solving and Multiplicative ComparisonUnit 1: Multiplication and Division Concepts Time Frame: 5-7 Days Essential Question: What real-life situations require the use of multiplication or division and how are they modeled?Group Size Unknown (Arrays, Area)If 35 chairs are arranged into 5 equal rows, how many chairs will be in each row? 7 chairs5 × n = 35 and 35 ÷ 5 = n

Number of Groups Unknown (Arrays, Area)If 35 chairs are arranged into equal rows of 7 chairs, how many rows will there be? 5 rowsn × 7 = 35 and 35 ÷ 7 = n

Observation Assessment: Students work in small groups. Each group gets a deck of the Multiplication Situation Sorting Cards. (Cut apart handout) As a group, they sort the cards under the shaded situation type cards. The teacher can informally observe and assess the students on their ability to recognize the various multiplication situations.

Note: There are examples of the three types of Multiplicative Comparison Problems in the Sorting Cards. You may wish to take those out and repeat this activity at the end of the lesson using all of the cards, or you may wish to keep the cards in and have a whole class discussion as an introduction to Segment 4.

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120 - 180 mins



Resources:Problem Solving Structures (Common Core Glossary pg 89 table 2: Common Multiplication and Division Situations)

Handout 6: Multiplicative Comparison Cards (1 set for 2 students)Handout 7: Multiplicative Comparison Workspace worksheets (1 per student)Handout 8: Multiplicative Comparisons Worksheet

Summative Assessment: Grade 4 Unit 1 Multiplicative Comparison

Focus:

Students will identify, solve and explain how to solve multiplicative comparison problems by using drawings and equations with a letter for the unknown number.



Make sense of problems and persevere: watch that students fill in the bar model correctly and can explain their thinking

Model with Mathematics: students transfer information found on the bar model into equations

Use Tools Strategically: students use the bar model graphic as a way to organize the information

Differentiation for Remediation: Make posters showing the models for multiplicative comparison situations for students to refer to when solving problems.

Differentiation for English Language Learners: Draw simple illustrations to show what “times as much” means. For example: If you have 3 apples, 4 times as much would be 12 apples. An illustration can help visualize that 4 times as much would be 12, and would not look like 4 more (7).

Differentiation for Enrichment: Give students multiplication equations and have them create multiplicative comparison word problems for each equation.


Set up their models incorrectly.

Multiplication fact errors.


Handout 8: Multiplicative Comparisons Worksheet

Steps: Teacher Notes/Reflections:15


1) Number Talk Openers

At the start of lessons during this unit students will be engaged in mental math activities to support fluency in multiplication strategies.

Day 1: Have students gather for a Math Talk. Show the multiplication fact below and ask them to solve it mentally. Students should share their strategies to deepen their understanding about why the procedures work based on place value and properties of operations. They will later use this skill to solve problems.

3 × 2 × ___ = 30

Day 2: Have students gather for a Math Talk. Show the multiplication facts below and ask them to express the equations as multiplicative comparisons. Discuss.

5 × 8 = 40(40 is 5 times as much as 8)

18 = 6 × 3(18 is 6 times as much as 3)

Show the multiplication statements below and ask them to write (on the board) the multiplicative comparisons as equations. Discuss.

12 is 3 times as much as 4(3 × 4 = 12)

28 is 4 times as much as 7(28 = 4 × 7)

2) Multiplicative Comparison – Bar Model and Equations

Technology Resource: http://www.thinkingblocks.com/index.html(online models for multiplicative comparisons)

A multiplicative comparison is a situation in which one quantity is multiplied by a specified number to get another quantity (e.g., “a times as much as b”). Students should be able to identify and verbalize which quantity is being multiplied and which number tells how many times.

Students should be given opportunities to practice the concept. The

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http://www.thinkingblocks.com/index.html

Grade _4_ Lesson Title: Problem Solving and Multiplicative ComparisonUnit 1: Multiplication and Division Concepts Time Frame: 5-7 Days Essential Question: What real-life situations require the use of multiplication or division and how are they modeled?Problem Solving Structures table can be used to provide students with more examples of the different types of multiplicative comparisons. Explain that “unknowns” in math problems can come in different forms. It can be the product (answer), group size, or the number of groups that need to be solved for in a multiplicative comparison. Variables should be used to hold the place of the “unknown” in any type of multiplicative comparison problem.

Using the examples below, draw the bar model with the group for each problem. (Following each example are more problems that can be used for guided practice/independent practice during instruction.) It will help to have the students write out the problems in their spiral notebooks (or notes) as you work through the examples. Share the word problem examples below one at a time. Have the students work in pairs to come to agreement on how to fill in their bar model. After working in pairs, have the students share where to put the numbers into the graphic. Note that you will add the labels as you add them to the graphic. Have the students justify their thinking. Record the equation after the graphic is filled in.

Unknown Product (Compare)A tennis ball costs $2. A soccer ball costs 6 times as much as the tennis ball. How much does the soccer ball cost? $126 × 2 = n

A. Raygen is 5 years old. Her mother is 8 times as old as she is. How old is Raygen’s mother? 40 years oldB. Garrett’s stack of coins is 4 times as high as Cade’s stack of coins. If Cade’s stack is 3 inches tall, how high is Garrett’s stack? 12 inchesC. Jaden has 7 baseball cards. Juan has 6 times as many baseball cards as Jaden. How many baseball cards does Juan have? 42 baseball cardsD. During the summer reading program at the library, children who read 10 books won a new book. To win the Grand Prize of a book bag, children had to read 5 times that amount. How many books did a participant need to read to win the Grand Prize? 50 books

Group Size Unknown (Compare)17

6

SOCCER BALL n

TENNIS BALL $2

Grade _4_ Lesson Title: Problem Solving and Multiplicative ComparisonUnit 1: Multiplication and Division Concepts Time Frame: 5-7 Days Essential Question: What real-life situations require the use of multiplication or division and how are they modeled?A soccer ball costs $12 and that is 6 times as much as a tennis ball costs. How much does a tennis ball cost? $26 × n = 12 and 12 ÷ 6 = n

A. Manny has lived in Illinois for 21 years and that is 7 times as many years as Lucas. How many years has Lucas lived in Illinois? 3 yearsB. Jackie has saved 4 times as much money as Heather. If Jackie has saved $32, how much money has Heather saved? $8C. Amy has 35 marbles. She has 7 times as many marbles as Scott. How many marbles does Scott have? 5 marblesD. Jake’s grandmother lives 48 miles away from him. She lives 8 times as far away from him as his aunt. How many miles away from Jake does his aunt live? 6 miles

Number of Groups Unknown (Compare)A soccer ball costs $12 and a tennis ball costs $2. How many times as much does the soccer ball cost as the tennis ball? 6 times as muchn × 2 = 12 and 12 ÷ 2 = n

A. A tree in Ashton’s yard is 15 feet tall. Ashton is 3 feet tall. The tree is how many times as tall as Ashton? 5 times as tallB. A marshmallow was 4 cm long to start. Now it has been stretched to be 16 cm long. How many times as long is the marshmallow now than it was at first? 4 times as longC. The fourth grade class has two class pets, a turtle and a hamster. The turtle weighs 12 ounces and the hamster weighs 4 ounces. The turtle weighs how many times as much as the hamster? 3 times as muchD. 40 is __________ times as much as 4. 10Note: Provide more examples of each type of problem if the students seem to be having trouble understanding the concepts taught in this lesson. Use the examples as the guide to making up new problems.

3) Multiplicative Comparison Cards

18

6

n

SOCCER BALL $12

TENNIS BALL n

SOCCER BALL $2 $12

TENNIS BALL $2


Distribute the Multiplicative Comparison Cards and Multiplicative Comparison Workspace worksheets when students are ready for additional practice. Students work in groups of 2 to answer the questions. Students should draw a bar model for each problem and write an equation using a variable for the unknown number in the problem (see answer key). The teacher may want to check student answers before allowing students to go on to the next card OR the teacher can give each group of students an answer key, so students can check their own work when they have completed each problem. It would not be beneficial to have students do all of the problems without checking.

Note: There are many ways that problem-solving cards can be used. Listed are a few resources/strategies:

*Partner work - http://www.kaganonline.com/index.php

*Math Scavenger Hunt - http://www.mathnspire.net/2012/01/scavenger-hunt-revisited-review-game.html

*Group Work – Assign a problem for each group to work on. Each group member completes a step to solve the problem and then passes the work to the next member, repeat until the problem is complete.

Formative Assessment: Assign one problem from the Multiplication Comparison Cards for students to complete independently. Evaluate their ability to draw and label a bar model, write an equation, and solve the problem.

Self-Assessment: Journal Prompt - What have I learned about Multiplicative Comparisons?

Summative Assessment: Grade 4 Unit 1 Multiplicative Comparison Assessment

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http://www.mathnspire.net/2012/01/scavenger-hunt-revisited-review-game.html

http://www.mathnspire.net/2012/01/scavenger-hunt-revisited-review-game.html

http://www.kaganonline.com/index.php

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