Unit 2: Multiplication and Division: Models within 100elementarymath.dmschools.org/uploads/1/3/2/2/1322452… · Web viewUnit 2: Multiplication and Division: ... Use multiplication

3rd Grade MathematicsUnit 2: Multiplication and Division: Models within 100

Teacher Resource Guide

2012-2013

In Grade 3, instructional time should focus on four critical areas:

1. Developing understanding of multiplication and division and strategies for multiplication and division within 100;Students develop an understanding of the meanings of multiplication and division of whole numbers through activities and problems involving equal-sized groups, arrays, and area models; multiplication is finding an unknown product, and division is finding an unknown factor in these situations.

2. Developing understanding of fractions, especially unit fractions (fractions with a numerator of 1); Students are able to use fractions to represent numbers equal to, less than, and greater than one. They solve problems that involve comparing fractions by using visual fraction models and strategies based on noticing equal numerators or denominators.

3. Developing understanding of the structure of rectangular arrays and of area;Students understand that rectangular arrays can be decomposed into identical rows or into identical columns. By decomposing rectangles into rectangular arrays of squares, students connect area to multiplication, and justify using multiplication to determine the area of a rectangle.

4. Describing and analyzing two-dimensional shapes;Students compare and classify shapes by their sides and angles, and connect these with definitions of shapes. They also relate their fraction work to geometry by expressing the area of part of a shape as a unit fraction of the whole.

3rd Grade 2012-2013 Page 1

Unit 2: Multiplication and Division: Models within 100 October 15- November 16 (5 weeks)

Unit Time Frame Test ByTR

IMES

TER

1 1: Addition and Subtraction (Within 1,000)

7 weeks 8/27 – 10/12 October 12

2: Multiplication and Division: Models within 100 5 weeks 10/15 – 11/16 November 16

TRIM

ESTE

R 2 3: Geometry/Measurement 4 weeks 11/19-12/21 December 21

4: Multiplication and Division: Properties within 100 5 weeks 1/2 – 2/8 February 8

5: Fractions 8 weeks 2/11 – 4/12 April 12

TRIM

ESTE

R 3

6: Multiplication and Division: Application & Fluency within 100 7 weeks 4/15 – 5/30 May 30

3rd Grade Mathematics 2012 - 2013



Big Ideas Essential QuestionsMultiplication is repeated addition. How are addition and multiplication related?Multiplication and division are inverse operations. How are multiplication and division related?



Identifier Standards Mathematical PracticesST

ANDA

RDS

3.OA.3

3.OA.1

3.MD.3

3.OA.4

3.OA.5

3.OA.6

3.OA.9

3.NBT.3

3.MD.1

Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown to represent the problem.

Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each.

Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each.

Determine the unknown whole number in a multiplication or division equation relating three whole numbers.

Apply properties of operations as strategies to multiply and divide.1 Examples: (Commutative property of multiplication.) (Associative property of multiplication.) (Distributive property.)

Understand division as an unknown-factor problem.

Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.

Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations.

Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram.

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 and express regularity in repeated reasoning.

Identifier Standards Mathematical Practices

1



Identifier Standards Mathematical PracticesST

ANDA

RDS

(con

tinue

d)3.OA.8

3.MD.2

Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.3

Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l).2 Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem.4



Identifier Standards Bloom’s Skills ConceptsST

ANDA

RDS

3.OA.3

3.OA.1

3.MD.3

3.OA.4

3.OA.5

3.OA.6

3.OA.9

3.NBT.3

3.MD.1

Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown to represent the problem.

Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each.

Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each.

Determine the unknown whole number in a multiplication or division equation relating three whole numbers.

Apply properties of operations as strategies to multiply and divide.2 Examples: (Commutative property of multiplication.) (Associative property of multiplication.) (Distributive property.)

Understand division as an unknown-factor problem.

Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.

Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations.

Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram.

Apply (3) Solve (mult. & div. word problems)

multiplicationdivision

3.OA.8

3.MD.2

Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.3

Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l).2 Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem.4

Apply (3) Solve (two-step word problems with the four operations)

equationsunknown quantity

2



Instructional Strategies for ALL STUDENTS



Critical Reading Prior to Instruction –Wiki: Student-Centered Mathematics Grades K-3, Van de Walle & Lovin, Pearson, 2006, p. 77-79, 111-119 (Building Resource SpEd)Children’s Mathematics, Carpenter, Heinneman, 1999 (CGI Year One text)

Use of models to build conceptual understanding of multiplication – Drawings, counters, unifix cubes, and number lines are typically used to represent multiplication concepts. It is essential for students to understand the relationship between addition and multiplication. (See examples of models below.) To make clear the connection to addition, early multiplication work should include writing an addition sentence and a multiplication sentence. It is not necessary to write the products, but rather write one sentence that expresses both concepts at once, for example,3 + 3 + 3 + 3 = 3 x 4.

Real-world context – For students to reach the level of rigor intended for the operations of addition and subtraction in the new Iowa Core, they must develop understanding of the operations within real-world contexts. A lesson built around word problems focuses on how students solve the problem. They may use words, pictures, and numbers to explain how they solved the problem and why they think they are correct. Allow students to use physical materials or drawings. Someone else should be able to understand how they solved the problem when looking at their paper.

Multiplication and division problem types – There are four structures for multiplication and division problems: Equal Groups, Comparison, Partitive (How many in each group?), and Measurement (How many groups?). See page 9 of this guide for further explanation of the problem types. Students need regular opportunities to solve all of the different types of problems in order to reach the level of rigor described in the Iowa Core.

Teaching multiplication and division simultaneously – Multiplication and division are taught separately in most traditional programs, with multiplication preceding division. It is important, however, to combine multiplication and division soon after multiplication has been introduced in order to help students see how they are related (Van de Walle, 2006). Studying multiplication and division together makes the process faster because each division is just finding an unknown factor.



Routines/Meaningful Distributed PracticeDistributed Practice that is Meaningful and PurposefulPractice is essential to learn mathematics. However, to be effective in improving student achievement, practice must be meaningful, purposeful, and distributed.

Meaningful: Builds on and extends understanding Purposeful: Links to curriculum goals and targets an identified need based on multiple data sources Distributed: Consists of short periods of systematic practice distributed over a long period of time

Routines are an excellent way to achieve the mandate of Meaningful Distributed Practice outlined in the Iowa Core Curriculum. The skills presented during routines do not necessarily reinforce the lesson concept for that day. Routines may be used to address a need for small increments of exposure to a skill or review of skills already taught. Routine activities may be repeated several days in a row, allowing for a build-up of conceptual understanding, or can be visited and re-visited over a period of time. Routines can be inserted as the schedule allows; in short intervals throughout the day or as a lesson opener or closer. Selection of the routine should be made based on informal teacher observation and formative assessments.

Skill StandardMultiplication and division facts 3.OA.7Tell and write time 3.MD.1Graphing 3.MD.3



Lesson Bank

ResourceTeacher’s

EditionPages

Lesson Standards Addressed

Math Wiki - Things That Come in Groups 3.OA.3, 3.OA.1Math Wiki - Circles and Stars 3.OA.3, 3.OA.1Math Expressions 468 Unit 7, Lesson 2, Activities 1-3 3.OA.3, 3.OA.7, 3.OA.1Math Wiki - Candy Boxes 3.OA.3, 3.OA.1Math Expressions 478 Unit 7, Lesson 3, Activities 2-4 3.OA.3, 3.OA.7, 3.OA.1, 3.OA.5Math Expressions 490 Unit 7, Lesson 4, Activities 2-4 3.OA.2, 3.OA.4, 3.OA.6Math Wiki - Patterns in Multiples (Focus on twos only) 3.OA.3, 3.OA.1, 3.OA.9Math Expressions 501 Unit 7, Lesson 5, Activities 2-3 3.OA.3, 3.OA.7, 3.OA.1, 3.OA.9Problem Bank (p. 10-13 of this guide)

- Pose multiplication and division problems focusing on twos.Students solve and share strategies.

3.OA.3,3.OA.7, 3.OA.1, 3.OA.5, 3.OA.6, 3.OA.9

Math Wiki - Patterns in Multiples (Focus on fives and tens only) 3.OA.3, 3.OA.1, 3.OA.9Math Expressions 458 Unit 7, Lesson 1, Activities 1 and 3 (add division as in Lesson 5

above)3.OA.3, 3.OA.7, 3.OA.1

Math Expressions 510 Unit 7, Lesson 6, Activities 3-5 3.OA.3, 3.OA.7, 3.OA.1, 3.OA.6, 3.OA.9, 3.NBT.3Problem Bank (p. 10-13 of this guide)

- Pose multiplication and division problems focusing on twos, fives, and tens. Students solve and share strategies.

3.OA.3,3.OA.7, 3.OA.1, 3.OA.5, 3.OA.6, 3.OA.9

Math Expressions 528 Unit 7, Lesson 8, Activities 1-4 (Twos, fives, and tens only) 3.OA.3, 3.OA.7, 3.OA.1, 3.OA.6, 3.OA.9Math Wiki - Patterns in Multiples (Threes) 3.OA.3, 3.OA.1, 3.OA.9Math Expressions 537 Unit 7, Lesson 9, Activities 3-4 3.OA.3, 3.OA.7, 3.OA.1, 3.OA.6, 3.OA.9Problem Bank (p. 10-13 of this guide)

- Pose multiplication and division problems focusing on threes. Students solve and share strategies.

3.OA.3,3.OA.7, 3.OA.1, 3.OA.5, 3.OA.6, 3.OA.9

Math Wiki - Patterns in Multiples (Fours) 3.OA.3, 3.OA.1, 3.OA.9Math Expressions 560 Unit 7, Lesson 12, Activities 2 and Alternate Approach

p. 562 (this is a key strategy for fours!)3.OA.3, 3.OA.7, 3.OA.1, 3.OA.6, 3.OA.9

Problem Bank (p. 10-13 of this guide)

- Pose multiplication and division problems focusing on twos and fours. Students solve and share strategies.

3.OA.3,3.OA.7, 3.OA.1, 3.OA.5, 3.OA.6, 3.OA.9

Math Expressions 576 Unit 7, Lesson 14, Activities 2-4 3.OA.3,3.OA.7, 3.OA.1, 3.OA.5, 3.OA.6, 3.OA.9



Math Expressions Activities(use as centers, re-teaching/extension support, etc.)

Activity Standards Activity StandardsActivity Card 7-1 Intervention, On-Level 3.OA.7 2, 5, 9, 10 Mult. Spin p. 533 (Practice multi. of 1- and 2-digit #s) 3.OA.7Activity Card 7-2 Intervention, On-Level 3.OA.7, 3.OA.1 Unit 7 Lesson 15, Activity 3 p. 589 (Three-in-a-Row) 3.OA.7Activity Card 7-3 Intervention 3.OA.7 Connecting Equations p. 597 (Solving equations) 3.OA.7Activity Card 7-4 Intervention, On-Level 3.OA.7, 3.OA.1, 3.OA.2 Count by Cross Out p. 465 (Practice multiplication using count-bys) 3.OA.7Activity Card 7-5 On-Level, Challenge 3.OA.3, 3.OA.7 Divide by 2s and 5s p. 507 (Practice division) 3.OA.7Activity Card 7-6 Intervention, On-Level, Challenge 3.OA.3, 3.OA.7 Division Tic-Tac-Toe p. 933 (Practice solving division problems) 3.OA.7Activity Card 7-7 On-Level 3.OA.7 Guess My Number p. 743

(Using inverse operations to solve division problems)3.OA.6

Activity Card 7-8 On-Level, Challenge 3.OA.7 What’s My Rule p. 742 (Identifying rules using a function table) 3.OA.7Activity Card 7-9 Intervention, On-Level 3.OA.7 Hex a Product p. 573 (Number and operations) 3.OA.7Activity Card 7-11 On-Level 3.OA.3, 3.OA.7 Mult. and Div. Scramble p. 713 (Practice multiplication and division) 3.OA.7Activity Card 7-12 Intervention, On-Level, Challenge 3.OA.7, 3.OA.1 Take the Tens p. 681 (Practice multiplication) 3.OA.7Activity Card 7-15 Intervention, On-Level 3.OA.7 Square # Cube Toss p. 707

(Solving multiplication equations with the correct product)3.OA.7

Activity Card 7-16 Intervention, On-Level 3.OA.7

Partner Games District Created Activitites (from 2010-2011)A file of these activities is on the Wiki

Activity Standards ActivityGroups Galore p. 28 3.OA.7 Representing MultiplicationFishy Multiplication p. 32 3.OA.7 Representing DivisionArray Game p. 36 3.OA.7 Slide ArraysProduct Comparing p. 38 3.OA.7 Strategies for Learning Multiplication and Division FactsSide by Side p. 40 3.OA.7 Using True/False Number SentencesKeep the Leftovers p. 42 3.OA.7 Using Open Number Sentences

Dice MultiplicationKryptoMultiplication Tic-Tac-ToeMultiplication ChallengeConnect FourEquation Creation



Table 2. Common multiplication and division situations.7

Iowa Core Mathematics, p. 93; www.iowacorecurriculum.iowa.gov

Unknown Product Group Size Unknown("How many in each group?"

Division)

Number of Groups Unknown("How many groups?" Division)

3 × 6 = ? 3 × ? = 18, and 18 ÷ 3 = ? ? × 6 = 18, and 18 ÷ 6 = ?

EqualGroups

There are 3 bags with 6 plums in each bag. How many plums are there in all?

Measurement example. You need 3 lengths of string, each 6 inches long. How much string will you need altogether?

If 18 plums are shared equally into 3 bags, then how many plums will be in each bag?

Measurement example. You have 18 inches of string, which you will cut into 3 equal pieces. How long will each piece of string be?

If 18 plums are to be packed 6 to a bag, then how many bags are needed?

Measurement example. You have 18 inches of string, which you will cut into pieces that are 6 inches long. How many pieces of string will you have?

Arrays,4

Area5

There are 3 rows of apples with 6 apples in each row. How many apples are there?

Area example. What is the area of a 3 cm by 6 cm rectangle?

If 18 apples are arranged into 3 equal rows, how many apples will be in each row?

Area example. A rectangle has area 18 square centimeters. If one side is 3 cm long, how longis a side next to it?

If 18 apples are arranged into equal rows of 6 apples, how many rows will there be?

Area example. A rectangle has area 18 square centimeters. If one side is 6 cm long, how longis a side next to it?

Compare

A blue hat costs $6. A red hat blue hat. How much does the red hat cost?

Measurement example. A rubber band is 6 cm long. How long will the rubber band be when it is stretched to be 3 times as long?

A red hat costs $18 and that is 3 times as much as a blue hat costs. How much does a blue hat cost?

Measurement example. A rubber band is stretched to be 18 cm long and that is 3 times as long as it was at first. How long was the rubber band at first?

A red hat costs $18 and a blue hat costs $6. How many times as much does the red hat cost as the blue hat?

Measurement example. A rubber band was 6 cm long at first. Now it is stretched to be 18 cm long. How many times as long is the rubber band now as it was at first?

General a × b = ? a × ? = p, and p ÷ a = ? ? × b = p, and p ÷ b = ?

7The first examples in each cell are examples of discrete things. These are easier for students and should be given before the measurement examples.4The language in the array examples shows the easiest form of array problems. A harder form is to use the terms rows and columns: The apples in the grocery window are in 3 rows and 6 columns. How many apples are in there? Both forms are valuable.5Area involves arrays of squares that have been pushed together so that there are no gaps or overlaps, so array problems include these especially important measurement situations.



Multiplication/Division Problem Bank

Multiplication

Joel has ____ boxes of chocolates. There are ____ pieces in each box. How many pieces does Joel have?

Each 4th grader has ____ binders. Each binder holds ____ folders. How many folders does each 4 th grader have?

Kate has ____ boxes of toys. There are ____ toys in each box. How many toys does Kate have?

A shark can eat ____ fish for lunch. How many fish will ____ sharks eat for lunch?

Tom has ____ packages of whiteboard markers. There are ____ markers in each package. How many whiteboard markers doe Tom have?

Each child’s box has ____ cubes. How many cubes altogether are in ____ boxes?

Zoe got ____ boxes of crayons. There were ____ crayons in each box. How many crayons does Zoe have?

Mom made ____ cupcakes. There are ____ sprinkles on each cupcake. How many sprinkles does she need?

There are ____ octopi in the cove. How many tentacles in all?

Ms. Blair had ____ plates of cookies. There were ____ cookies on each plate. How many cookies did she have altogether?

There are ____ people in Mario’s family. If each member of his family has ___ pairs of shoes, how many shoes does Mario’s family own?



Division, Measurement (How many groups?)

Megan is making cupcakes for her birthday. ____ cupcakes fit on one plate. How many plates will Megan need for ____ cupcakes?

I am making a necklace with ____ blue beads. How many necklaces can I make with ____ blue beads?

The teacher is sorting cubes into bags. ____ cubes can fit in each bag. How many bags are needed for ____ cubes?

Maggie is organizing her CD collection. ____ CDs will fit on one shelf. How many shelves will she need for ____ CDs if she wants to put all her CDs on one shelf?

Bryan had ____ treats for his classmates. ____ treats fit on each plate. How many plates did he need?

Jacob is making apple pies. He can fit ____ apples in each pie crust. How many pie crusts will Jacob need for ____ apples?

Mohamed is sorting his stamp collection. ____ stamps can fit on each sheet. How many sheets will Mohamed need for ____ his stamps?

Syd is getting new fish. ____ fish can fit in one tank. How many tanks will Syd need for ____ fish?

Keontre is organizing his pictures into photo albums. ____ pictures can fit on one page. How many pages will he need for ____ pictures?

Stephanie has ___ rocks in her collection. She wants to put them in display boxes. Each display box holds ____ rocks. How many full boxes will she have?

Sylvia wants to buy a tackle box with compartments. If she puts ___ fishing lures in each compartment, how many compartments will she need for ____ lures?



Division, Partitive (How many in each group?)

Jackson is having a sleepover and ordered pizza. There are ____ pieces of pizza. How many pieces will each of his ____ friends get?

The team has ___ players. We need to make ___ different teams. How many players will be on each team?

Anne has ____ gummy bears. She put them into ____ bags so there were the same amount in each bag. How many did she put in each bag?

Salem had ____ golf balls. He placed the same number of balls in ____ baskets to take to the driving range. How many baskets did he fill?

I have ____ beads. I want to make ____ necklaces. How many beads can go on each necklace?

Quintin has ____ cookies. He wants to put them in ____ boxes. How many cookies can go in each box?

There are ____ students in our class. If I divide them into ____ groups, how many will be in each group?

Ryan has ____ sheep. He put them into ____ pens so there are the same number in each pen. How many sheep can Susie put in each pen?

Mom has ____ cupcakes. She put them into ____ boxes so there are the same number of cupcakes in each box. How many cupcakes can mom put in each box?

We have ____ books for ____ students to share equally. How many books will each student get?

Sean has ____ baseball cards. If he has ____ page protectors how many cards will be on each page?



Multiplication, Comparison

Jill picked ____ apples. Mark picked ____ times as many apples as Jill. How many apples did Mark pick?

This month Mark saved ____ times as much money as last month. Last month he saved $____ . How much money did Mark save this month?

Valerie found ____ seashells. Roberto found ____ times as many seashells as Valerie. How many seashells did Roberto find?

This week I read ____ times as many books as last week. Last week I read ____ books. How many books did I read this week?

Division, Comparison

Mark picked ____ apples. He picked ____ times as many apples as Jill. How many apples did Jill pick?

This month Mark saved ____ as much money as he did last month. If he saved ____ dollars this month, how much money did he save last month?

Mark picked ____ apples, and Jill only picked ____. How many times as many apples did Mark pick as Jill?

This month Mark saved $____ . Last month he saved ____ dollars. How many times as much money did he save this month as last?

Susan’s age is ____ times Victor’s age. Susan is ____ years old. How old is Victor?

Jose walked ____ miles. He walked ____ times as far as Tim. How far did Tim walk?


Documents

Unit 2: Multiplication and Division: Models within 100elementarymath.dmschools.org/uploads/1/3/2/2/1322452… · Web viewUnit 2: Multiplication and Division: ... Use multiplication