5th Grade Mathematics - Investigations

5th Grade Mathematics - InvestigationsUnit 1: Multi-Digit Multiplication and DivisionTeacher Resource Guide2012 - 2013

In Grade 5, instructional time should focus on three critical areas:Developing fluency with addition and subtraction of fractions, and developing understanding of the multiplication of fractions and of division of fractions (limited to unit fractions divided by whole numbers and whole numbers divided by unit fractions);Students apply their understanding of fractions and fraction models (set model, area model, linear model) to represent addition and subtraction of fractions with unlike denominators as equivalent calculations with like denominators ( + 2/8 = + ). They develop fluency in calculating sums and differences of fractions, and make reasonable estimates of them. Students also use the meaning of fractions and the relationship between multiplication and division to explain why the procedures for multiplying and dividing fractions make sense. Extending division to 2-digit divisors, integrating decimal fractions into the place value system and developing understanding of operations with decimals to hundredths, and developing fluency with whole number and decimal operations;Students develop understanding of why division procedures work based on place value and properties of operations. They are fluent with multi-digit addition, subtraction, multiplication, and division. They apply their understandings of models for decimals, decimal notation, and properties of operations to add and subtract decimals to hundredths. Students use the relationship between decimals and fractions, and the relationship between decimals and whole numbers (i.e., a decimal multiplied by an power of 10 is a whole number) to understand and explain why procedures for multiplying and dividing decimals make sense. They compute products and quotients of decimals to hundredths efficiently and accurately.

Developing understanding of volume.Students understand that volume is an attribute of three-dimensional space and can be measured by finding the total number of same-size units of volume required to fill the space without gaps or overlaps. They understand that a 1-unit by 1-unit by 1-unit cube is the standard unit for measuring volume. They decompose three-dimensional shapes and find volumes of right rectangular prisms by viewing them as decomposed into layers of arrays of cubes.5th Grade Mathematics 2012-2013 UnitTime FrameTesting WindowTRIMESTER 11: Multi-Digit Multiplication and Division7 weeks8/22 10/12October 122: Measurement/Geometry4 weeks10/15 11/9November 9TRIMESTER 23: Addition and Subtraction of Fractions 8 weeks11/12 1/18January 184: Decimals 8 weeks1/22 3/14March 14TRIMESTER 35: Multiplication and Division of Fractions9 weeks3/25 5/30May 30

DMPS Wiki: http://dmps-mathematics.wikispaces.com/

Big IdeasEssential QuestionsMultiplication and division are inverse operations.How are multiplication and division related?

IdentifierStandardsMathematical PracticesSTANDARDS5.NBT.54.OA.25.NBT.1Fluently multiply multi-digit whole numbers using the standard algorithm.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.Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left. Note: This unit will only focus on multiplying by 10 and 100 of whole numbers.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.5.NBT.6Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.5.OA.25.OA.1Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.*4.OA.3Solve multi-step word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. 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.

IdentifierStandardsBloomsSkillsConceptsSTANDARDS5.NBT.54.OA.25.NBT.1Fluently multiply multi-digit whole numbers using the standard algorithm.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.Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left. Note: This unit will only focus on multiplying by 10 and 100 of whole numbers.Apply (3)Multiply (multi-digit whole numbers fluently)multiplymulti-digitwhole numbers5.NBT.6Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.Apply (3)Understand (2)Divide (whole numbers w/ up to four-digit dividends andtwo-digit divisors)Explain (using equations, arrays, or area models)dividerectangular arraysarea modelsequations5.OA.25.OA.1Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.Understand (2)Write (simple expressions)Interpret (numerical expressions w/out evaluating)expressionsnumerical expressionsevaluate *4.OA.3Solve multi-step word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. 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.

Instructional Strategies for ALL STUDENTSCritical Reading Prior to Instruction Math Expressions, Teacher Edition Volume I, Houghton Mifflin, 2009, p. 639L-MTeaching Student-Centered Mathematics Grades 3-5, Van de Walle & Lovin, Pearson, 2006, p. 113-128 (Building Resource Sp Ed)Real-world context For students to reach the level of rigor intended for the operations of multiplication and division in the new Iowa Core, they must develop understanding of the operations within real-world contexts. The tendency in the United States is to have students solve a lot of problems in a single class period. The focus of these lessons seems to be on how to get answers. In Japan, however, a complete lesson will often revolve around one or two problems and the related discussion (Reys & Reys, 1995). A lesson built around word problems focusing on how students solve the problem is ideal. When the teacher focuses the discussion of the word problems on students strategies, attention is turned toward the reasoning and knowledge of the number system necessary to solve the problems. Multiplication and division problem types There are two types of multiplicative structures involving groups of equal size: equal groups and multiplicative comparison. Typically the equal groups problem types are used for multiplication and division in earlier grades. Students will need support in realizing that the equal groups concept is not the only situation for multiplication. Comparative situations in previous grades were additive (How many more? How many less?), so students will need many opportunities to work with multiplicative comparison problems in order to recognize the problem type and become proficient. (See p. 8 in this guide for further explanation of the problem types.)There are two types of division problems: measurement and partitive. In measurement problems, the group size is known and the number of groups is unknown. For example: If 18 plums are to be packed 6 to a bag (group size known), then how many bags are needed (number of groups unknown)? In partitive problems, the number of groups is known and the group size is unknown. For example: If 18 plums are shared equally into 6 bags (number of groups known), then how many plums will be in each bag (group size unknown)? Students need to recognize both types of problems as division situations, should be able to model and solve each type of problem, and should know the units of the result: Is it 3 bags or 3 plums per bag? (See p. 8 in this guide for further explanation of the problem types.)Use of models to solve multiplication and division problems It is essential for students to learn methods for computation that are meaningful. The primary model for multi-digit multiplication used in 4th grade is the area model. In this unit, students will continue to work with the area model and work toward shortcuts (traditional algorithms) that do not require drawing the area model. The traditional multiplication algorithm is much easier for students when they have had plenty of opportunities to work with the area model.Students began their work with multi-digit division in 4th grade. The focus was one-digit divisors. The rectangular sections model and the expanded notation model were the emphasis. Work with these models will continue in this unit as students work to become more proficient with division up to two-digit divisors. Keep in mind that the rectangular sections model and the expanded notation model are based upon place value much like the area model for multi-digit multiplication. The digit-by-digit model (traditional algorithm) will be harder for students because it is based on digits instead of place value. The expanded notation and digit-by-digit model are efficient ways to divide multi-digit numbers. With the new Iowa Core, 6th graders are expected to be fluent in whole number operations. Therefore, multi-digit multiplication and division are critical priorities for 5th grade.

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 understandingPurposeful: Links to curriculum goals and targets an identified need based on multiple data sourcesDistributed: Consists of short periods of systematic practice distributed over a long period of timeRoutines 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. Concepts taught through Meaningful Distributed Practice during Unit 1:SkillStandardResourceRequired: These concepts align to the supporting standards in this unitMultiplicative comparison4.OA.2Resource Guide p. 12Assessing reasonableness of answers (mental computation & rounding)4.OA.3Interpreting remainders4.OA.3Expressions: Unit 7; Resource Guide p. 13-14Order of operations 5.OA.1Expressions: Unit 8Powers of ten (whole numbers only)5.NBT.1Expressions: Unit 7Additional: These concepts are optional, based on student needMultiplication and Division factsOther skills students need to develop based on teacher observations and formative assessments.

Investigation Resources for Unit 1

Instructional Plan

Resource

Standards

Grade 5, Unit 1

Investigation 1 (sessions 1 4)

Investigation 2 (sessions 1 4A)

Investigation 3 (session 1 4, 6 7)

Investigations

CGI Resource Guide

Investigations and the Common Core State Standards (pages: CC4)

5.NBT.5

5.NBT.6

5.OA.2

5.NBT.1 imbedded, not explicit

5.OA.1

Grade 5, Unit 7 Investigation 2 (session 3)

Investigation 3 (sessions 1 6)

Investigation 4 (sessions 1 4)

Investigations

CGI Resource Guide

Investigations and the Common Core State Standards (pages: C103)

Grade 5, Unit 8

Investigation 2 (1 8)

During this Investigation teachers will want to have a conversation about Order of Operations. Refer to Order of Operations materials including Math Conversation for Order of Operations

Investigations

CGI Resource Guide

Additional Focus Needed On:

Solving multi-step word problems using all operations, including interpretation of remainders.

CGI Resource Guide

4.OA.3

Unit 1: Multi-Digit Multiplication and Division August 22-October 12 (7 weeks)

Table 2. Common multiplication and division situations.[footnoteRef:1] [1: ]

Iowa Core Mathematics, 2010, p. 93; www.corecurriculum.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 = ?

Equal

Groups

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,2

Area3

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 long

is 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 long

is 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 = ?

The first examples in each cell are examples of discrete things. These are easier for students and should be given before the measurement examples.

2The 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.

3Area 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.

5th Grade 2012-2013Page 8

Multiplication Problem Bank

There are ____ pairs of shoes is Mrs. Bakers closet. How many individual shoes are there all together?

Everyday each child in Mrs. Bakers class reads _____ pages in their book. How many pages will they have they read after ____ days?

There are _____ snowmen on the playground. Each snowman is made with three big snowballs. How many snowballs were used to make all the snowmen?

Jonah has a lot of money in his piggy bank. All of the money is in five dollar bills. Jonah has _____ bills in his piggy bank. How much money does he have?

There are five days in a school week. We have _____ weeks until our next test. How many days until the next test?

Mrs. Baker is making cookies for the cookie exchange. She is making one dozen (12) of____ different types of cookies. How many cookies is Mrs. Baker making?

Mr. Lingwall was getting ready to decorate his Christmas tree. He carried up all of the ornament boxes from his basement. There were ____ boxes of ornaments. Each box had ____ ornaments in it. How many ornaments did Mr. Lingwall have?

There were ______ children sitting at ______ tables. How many children were there sitting in the cafeteria?

There are ____ children in the room sitting in rows of _______. How many rows are there?

Mrs. Baker was baking cookies. Only _____ cookies would fit on each pan. She made ____ pans. How many cookies did she bake?

There were _____ towels in the closet. If mom puts them in piles of _____ how many piles will there be?

Michael went shopping for his friends birthday. He was buying baseball cards that cost 9 cents each. If he bought ____ cards, how much would he spend?

Jonah has a lot of money in his piggy bank. All of the money is in five dollar bills. Jonah has _____ bills in his piggy bank. How much money does he have?

There were _____groups of turkeys out in the farm field. There were ___ turkeys in each group. How many turkeys were in the field?

At the pet store there were _____cages of mice with _____mice in each cage. How many mice were in the pet store?

There are ____ pieces of gum in a pack. Mrs. Baker has ____ packs of gum. How many pieces of gum does Mrs. Baker have?

Morgan has ____ jars with ____ gumballs in each jar. How many gumballs does Morgan have all together?

Division Problem Bank

Partitive Division / Group Size Unknown

There are ______ caterpillars that need to be put into groups for second grade science. We need ____ groups. How many caterpillars will be in each group?

There are _____teams in the soccer clinic. ______children signed up to play. How many kids will be on each team?

There were _____ mats at the wrestling tournament. ____boys are assigned to each mat for the day. How many wrestlers are at the tournament?

Patricia has _____ erasers. If she shares them among _____ friends, how many erasers does each friend get?

Michelle is inviting _____ friends to a party. She has _____ cookies. How many cookies will each friend get?

There are _____ students in the class and _____ candies. If the candies are divided equally among the students, how many does each student get?

John has _____ marbles stored in boxes. If there are _____ boxes, how many marbles must go in each box?

A class has _____. If they are divided into _____ sections, how many students are there in each section?

A class picnic is estimated to cost _____. If there are _____ children in the class, how many dollars should each child contribute?

If _____ children in a class are to be assigned _____ science projects, how many children should work on each project?

For a school day parade, _____ students are arranged in _____ rows. How many students are there in each row?

The parking garage can hold _____cars when its full. There are _____levels in the garage. How many spaces are on each level?

_____ buses brought a total of _____passengers to the city. About how many passengers were on each bus?

Division Problem Bank

Measurement Division / Number of Groups Unknown

Theresa wants to split a collection of candies into groups of _____. Theresa has _____ candies. How many groups will be created?

Earl wants to split some Skittles into groups of _____. Earl has _____ Skittles. How many groups will be created?

The math teacher wishes to solve _____ math problems. If she solves _____ math problems each day, how many days will she need to solve

the problems?

Joshua is hanging up _____ towels. If each clothesline holds _____ towels, how many clotheslines are needed?

There were _____ brushes hanging from display racks. There were _____ brushes hanging from each rack. How many racks were there?

Mom baked _____ cookies. She put _____ cookies in each bag. How many bags were there?

There are _____ coconuts hanging from the trees. There are _____ coconuts on each tree. How many trees are there?7

Mrs. Russell planted _____ flowers in her garden. There are _____ flowers in each row. How many rows are there?

There are _____ pickles. There are _____ pickles in each jar. How many jars are there?4

The parks sprinkler system sprayed _____ gallons of water on the grass in _____ minutes. How many gallons does it spray in one minute?

Travis built a tower of blocks _____centimeters high. Each block in the tower is _____ centimeters tall. How many blocks were used to build the tower?

7

The laser printer can print _____ pages each minute. Johns term paper is _____ pages long. How many minutes will it take to print?

Multiplication / Division Comparison Problem Bank

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

Susans age is ____ times Victors age. Susan is ____ years old. How old is Victor?

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

Interpreting the Remainder Problem Bank

Examples for rounding up the remainder:

You are organizing a school trip to a museum for 122 students, teachers, and parents. You need to order enough buses to seat

25 people to a bus. How many buses do you need?

122 divided by 25 = 4 remainder 22. You have to decide what to do with the 22 people left over. To ignore the 22 people would mean they wouldnt get to go. You have to round up the number of buses needed so they can go. You would need to order 5 buses.

1) 95 people need to ride the elevator to the top of a skyscraper. The elevator can hold 4 people at a time. How many times will the elevator need to go to the top, so everyone can go to the top?

2) You are organizing the tables in the cafeteria to seat 12 students at a table. How many tables would you need to seat 211 students?

3) A total of 58 awards will be given away to the soccer team. Each box holds 8 awards. How many boxes are needed to hold all the awards?

4) If 13 parents want to go on the field trip and each car holds 6 people. How many cars will they need?

5) Carol makes 26 cupcakes. Each cupcake tray holds 4 cupcakes. How many trays are needed to hold all the cupcakes?

6) There are 47 books in the box. Mr. Red wants to put the books in baskets. 7 books will fit in each basket. How many full baskets will Mr. Red have?

Interpreting the Remainder Problem Bank

Examples for ignoring the remainder:

You have 65 baseball cards that you want to share equally with your 3 friends. How many cards would each of your 3 friends receive?

65 divided by 3 = 20 remainder 5. Since you want each of your friends to receive the same number of cards, you will ignore the 5 remaining cards and give each one 20 cards.

1) There are 62 students in the Football League. How many teams of 6 can they make?

2) There are 76 people seated at tables. Each table holds 10 people. How many full tables are there?

3) Mrs. Freebern has 52 seeds to plant in her garden. She wants to plant the seeds in rows of 5. How many rows will she make?

Examples for the remainder as the answer:

Sams Soda has 43 liters of soda to put into bottles. Each bottle holds 8 liters. After filling as many bottles as possible, how many liters of cola will be left over?

43 divided by 8 = 5 remainder 3. Since you want to know how many liters will be left over, the answer is the remainder, 3.

1) At an amusement park, a group of 659 people wants to ride the roller coaster. If each car on the roller coaster holds 6 people, how many people will be in the partially full car?

2) At the fair, Drew has 604 ride tickets. Each ride on the Ferris wheel costs 3 tickets. After riding the Ferris wheel as many times as possible, how many tickets will Drew have left?