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Algebraic Sense I

Early Years/Benchmark 1

Contents

Introduction

Purpose of the Mathematics Classroom-Based Evidence Model

Link to Essential Academic Learning Requirements in Mathematics

Overview of the Problem-Solving Process

How to Use This Classroom-Based Evidence Model

Post-Problem Solving Questions

. Materials Needed

Evaluating Students' Work

Annotated Student Masters (for Teachers) Introduction to Problem Solving SM 1 Introduction to Problem Solving SM 2 Five Steps for Solving Mathematics Problems SM 3 Step 1 Study and define the problem. SM 4 Step 2 Choose one or more strategies. SM 5 Step 3 Apply your strategy or strategies. SM'6 Step 4 Show your steps. SM 7 Step 5 Check to see if you answered the question. SM 8 A Closer Look a t Different Problem-Solving

Strategies SM 9 Make a Chart/Look for a Pattern SM 10 Working Backward SM 11 Solve a Simpler Problem SM 12 Use Estimation SM 13 Reviewing the Five Steps for Problem Solving SM 14 Review Checklist

1

3

4

5

6

7

7.

8

Group Activities SM 15 Build a Pen SM 16 How Many Beans in the Cup? SM 17 Are AU the Parts the Same Size? SM 18 How Many Tricycles and Bicycles? SM 19 Five Square SM 20 Who Took the Last Bean? SM 21 How Many Necklaces Can You Make? SM 22 How Many Cookies Are There?

Math Fair SM 23 Number Cube Toss SM 24 ‘20 Circles SM 25 Odd Wins SM 26 Which Jar is Heavier? SM 27 Guess the Number SM 28 Help the Farmer SM 29 The Great Cover-up SM 30 Mathematics Re-Problem Solving Questions SM 31 Mathematics Post-Problem Solving Questions SM 32 Student Worksheet SM 33 Sample Student Checklist SM 34 Student Scor in~ Criteria (1 page) SM 35 Grids or Graphpaper for the Activity “Build a Pen” (5 pages)

Teacher Master: Observation Checklist 89

Thanks to the following people who assisted in reviewing this document.

Washington Student Assessment System Classroom-Based Assessment Review Committee

Veronica Bernabie

Barbara Chamberlain

Judy Dagnon

Deborah Gonzales

Emilie Hard

Cheryl Hewitt

Phyllis Keiley-Tyler

Linda Lee

. Mary Niemeyer

Beverly Neitzel

Mary Jo Ormsby

SuzAnne F’ayseur

Pat Reistroffer

Stephen Rushing

Nancy Scott

Nancy Skemtt

Holly Stein

Johnnie Tucker

Peggy Vater

Kate Venneri

Preparing t o Solve Math Problems

Introduction This Wahingmn Modelj6r c[arnwm-Baed Evidence of Sludenf Laming is an important part of the Washington state assessment system As a part of the Cksroom-Based Assessment Tool Kif+ it provides 8 model for understanding the Essential Academic Learning Rquimnents and rccognizing thecharacteristics of quality work that define the standads in Mathematics

General Definition of the Models for Classroom-Based Evidence of Student Learning Washington models for dasroom-based evidence of student leaning provide teachers with:

Tools to organize, collect and evaluate student work by applying scoring criteria (rubrics) to gather evidence of what students know and can do; - Examples of classroom activitiesltasks that are tied to the Essential Academic Learning Requirements, especially those that are diffcult to test on the state-level -mens or are more appropriately addressed in the classroom

Content of the Models Classroom-based evidence models provide information from oral

exhibitions of student work cdlected over a week. a month, or the interviews, presentations work products, experinmu and projects, or

entire school year. The models include paper and pencil tasks, generic cheddisu of skills and haits, observation assessment strategies, simple rating scales and generic ptomls for oral communications and personal interviews Teachers can also use the models as effective, high quality "tests" that reflect student progress toward the Essential Academic Learning &quirements and the instructional programs offered to students.

Instructional strategies While these models for classoom-based evidence of student learning offer suggestions for instruction. they do not prescribe specific teaching methods Instead, they enmurage teacher choice and creativity. Using these models in the classmom encourages leachers to evaluate student learning in both independent and group settings. It is recommmded that teachers anend professional development sessions designed to help them understand how to use the materials effectively.

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Differences Between the State-Level Assessment and this Model for the Classroom-Based Evidence of Student Learning

Keyed to Washington's Essential Academic Learning Requirements, the activities contained in this model for the classroom-based evidence of student learning are meant to target benchmarks that are not adequately or thoroughly assessed in the state-level azsessment. Those benchmarks would include the following:

b search for patterns in simple situations

$. use a e e t y of strategies,and approaches

& recognize when an attempted approach is unproductive and try a new approach

identify questions to be answered in familiar situations

0 define problems in familiar situations

b identify the unknowns in familiar situations

sele i t and use appropriate mathematical tools

Teachcrs will be able to observe and evaluate students' ability to s o l v e problems in individual and collaborative settings using dassroom objects, manipulatives, measuring tools, and calculams

Link to Essential Academic Learning Requirements in Mathematics

Washington's Essential Academic Learning Requirements are listed below.

Essential Academic Learning Requirement 1.The student understands and applies the concepts and procedures of mathematics To meet this standani, the student will:

11 understand and apply concepts and procedures from number

I 2 understand and apply concepts and procedures from

13 understand and apply concepts and procedures from geomehic

1.4 understand and apply conceptvand procedures from probability

I5 understand and apply concepts and procedures from algebraic

imse

measurement

sense

and statistics

sense

Eyential Academic Learning Requirement 2The student wes mathematics to define and solve problems.

Essmtial Academic Learning Requirement 3. The student uses mathematical reasoning.

Essential Academic Learning Requirement 4 The student conununicates knovAedge and understanding in.both everyday and mathimatical language.

E s s e n t i a l Academic Learning Requirement I The student understands how mathematical ideas connect within mathematics, to other subject BMS. and to real-life situations

The activities provided in this classroom-based evidence model encourage Students' active involvement in solving problems. These activities will help teachers assess their students' ability to investigate situations, formulate questions, and construct solutions as students apply the mathematics they have l e a r n e d to a variety of contexts

Essential Academic Learning Requirement 2 The student uses mathematics to define and soh problem.

To meet this standard the student will:

21 iwestigate situations

p search for pat~ems in simple situations

bl use a variety of strategies and approaches

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@ recognize’when information is missing or extraneous

!$ recognize when an approach is unproductive and try a new approach

2.2 formulate questions and define the problem

9 identify questions to be answered in familiar situations

p define problems in familiar situations

B1. identify the unknowns in familiar Situations

2.3 construct solutions

’@ organize relevant information

B select and u,se appropriate mathematical tools

$. use a variety of strategies

@- apply appropriate methods, opeptions, and processes to consmct a solution

Overview of the ProblernSolving Process

According to The National Council of Supervisors of Mathematics, problem solving is “a process of applying previously acquired knowledge to new and unfamiliar situations” (NCSM, 1989). To solve a problem, you must k t understand the task and determine what information is needed to solve the problem. Next, it is helpful to decide on a game plan or strategy for how you will usi the information to find a solution. Once you arrive at a solution. it is important to look back at your answer to see if it is reasonable.

In this model for classmm-based evidence, students will be introduced to the following five steps for solving problems:

I . Study and define the problem. 2. Choose one or more strategies. 3. Apply your strategy or scrategies. 4. Show your steps. 5 . Check to see if you answered the question.

Students will be encouraged to use and internalize these five steps as they complete the activities included in this model. Student Master 32 could be used to guide students in applying these steps.

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Students need to developstrategic ways of thinking about problems (step 2 in the five-step process).Thii model for classroom-based evidence includes an introduction or review of common problem- solving strategies that students can use as tools to help them solve problem Since people need to solve a variety of problems using a variety of methods, it is imperative that teachers supplement the activities contained in this model with other nonroutine problems that foster higher-order mathematical thinking skills.

How to Use This Classroom-Based Evidence Model

The materials for this mathematics classroom-based evidence model are designed to be used with either small groups of students or the class as a whole Once students have direction, they can work on the student activities in small groups pairs, or individually.

Because these materials are offered as black line masters, you can pick and, choose what you want to use The first portion of the Student Masters presents instructional (modeling) text on the five steps for solving mathematics problems and an overview of different problem-solving strategies The second portion of the Student Masters contains student activity pages. Please feel free. to use any or all of these materials. to tailor them to your students' needs, or to supplement them with your own materials The order of use and the time line will be determined by you based on classroom population and curriculum.

Prior to distributing the student masters, have students share the strategies or approaches they have used to solve mathematics problem Have them complete Student Master 30 Mathematics Pre-Problem Solving Questions.

The student masters are divided into 3 sections:

Student Masters 1-14 will lead students through five basic steps for problem solving and will describe different problem- solving strategies they can use for solving problems.

Student Masters 15-22 consist of problems students can work on individually, in pairs or small groups, or as a home activity. Suggstions for manipdatives are given for each activity. However, you may provide other manipulative that you feel are appropriate for these tasks An Observation Checklist for Group Activities found on page 89. can be used to record students' progress and group interaction during these. activities.

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Student Masters 23-29 consist of problems that center around a mathematics fair scenaria You can have students do these activities on their own, or you may choose to have them work in pairs or small groups

Feel free to clarify any material covered in the student masters, provide additional examples to illushate the fwe steps for problem s o I v i i or any of the problemsolving strategies, and reword or rephrase the directions for the activities.

PostProblern Solving Ou'estions

Ask students to complete the Mahematics Post-Problem Solving Questions on Student Master 31 after the problem-solving process has k e n taught and applied many times. This information can be shared with students during student-teacher conferences 6r with parents during parent-teacher conferences Also, consider making this information part of the students' folders or portfolios

Materials Needed

The following materials will be needed by each student to complete the suggested activities

I? Pencil

p Crayons

:e Red. blue, and green crayons or markers

p Glue stick, glue, or tape

p Scissors

@ 16 paper clips or toothpicks

28 33 chips, counters, or other small, round objects

5% 21 beans or pennies

k? Sning

!,* I-inch tiles

. .

Evaluating Students' Work

A sample student checklist on Student Master 33, modeled after the generic scoring criteria. is provided for students' self-evaluation and reflection. Students' work may be evaluated or scored'using any of the generic scoring criteria shown on the following pages. To help students better understand the scoring criteria, have them create a simplified version modeled after the generic rubric and then apply the smring criteria to their own papers or those of their classmates. AII example of student scoring aiteria is found on Student Master 34 Also, samples of task-specific scoring criteria for two of the activities are found on pages 12 and 13.

Mathematics Scoring Criteria for Solving Problems

A &point response * shows' clear and accurate.understanding of the task - effectively explores the problem, evaluates strategies, and clarifies patterns and problems * organizes information, selects appropriate tools and s&tegies, and uses appropriate mathematical concepts and procedurcs - develops M effective solution

A fpoint response shows understanding of the task and how to explore problems shows understanding of how to d u a t e strategies, clarify patterns and problem'organize information. or select appropriate and strategies uses appropriate mathematical concepts and procedures

develops a viable solution, but minor gaps or errors may limit solutions

A 2-point response * shows some understanding of the task and how to explore problems * shows some understanding of how to evaluate strategies, clarify patterns and problems, organize information, or select appropriate tools and strategies - uses mostly appropriate mathematical concepts and procedures dwelops a solution, but gaps or errors limit viability of solutions

A 1-point response shows little understanding of the task or how to explore problems, evaluate strategies, clarify patterns and problems. organize information, or select appropriate tools and strategies develops a solution, but major gaps, ermrs, or poor conceptual understanding prevent a viable solution

A 0-point or Not Seonble (NS) response shows little or no understanding of the tark provida no evidence of problem-solving

"I don't know" or a question mark (?). ' skills the prompt may simply be recopied. or the response may indicate

General Scoring Criteria for Concepts & Procedures (Accuracy and Appropriateness)

Accuracy and Appropriateness Performance Criteria

Number Sense: computation (accurately applies operations through mental arithmetic, papertpencil. andlor technology)

Measurement systems and took (uses appropriate to&, units of measurement)

Geomenic Sense: relationships and transformations (uses appropriate tools; makes accurate constructions)

Robability and Statistics: data analysis (uses appropriate displays for data:

clearly and accuratelylabek displays)

Algebraic Sense: operations (accurately solves equations, inequalities. and expressions)

4 points Meets all relevant criteria solutions, construaions andlor results are complete and accurate choices of procedures are appropria~ IO the requirements of

displays if required, are clear, accurate+ and appropriate uses appropiiaIe units, opmtions andlor tools skillfully and

the task

accurately

3 points Meets most &levant criteria solutions constructions. andlor results are complete and mostly

choices of procedures are mostly appropriate to the

displays, if required, are appropriate, clear, and mostly accurate 9 uses appropriate units, operations, and/or tools accurately

accurate

requirements of the task

2 points Meets some relevant criteria solutions, constructions, and/or results are partially complete

choices of procedures may not be appropriate to the

. displays, if required, are appropriate, but may be unclear or

uses units, operations, andor tools that are partially appropriate

and/or partially acciuate

requirements of the tark

inaccurate

or partially accurate

1 point Meetsfov reletant criteria solutiong constructions, and/or results are incomplete andlor

choim of procedures may not be appropriate to the

displays, if required, are somewhat appropriate, but miy be

sometimes uses appropriate units operations andlor tools

0 points or Not Scorable (NS) completely inaccurate, unreadable, or off target

mostly inaccurate

rquiriments of the task

unclear or inaccurate

General Scoring Criteria for Concepts and Procedures

4 points Meets all relevant criteria shows understanding of concepts and procedures both within

consistently and purposefully applies appropriate concepts and and beyond the task

procedures

3 points Meets most relevant criteria . .

shows thorough understanding of concepts and procedures

consistently applies appropriate concepts and procedures

2 points Meets some relevant criteria shows general understanding of concepts and procedures

generally applies appropriate concepts and procedures

1 point Meetsfew relevant criteria shows rote or partial understanding of concepts and procedures

occasionally applies appropriate concepts and procedures

0 point or Not Scorable (NS) attempted with no understanding of the concepts and

required by the task



procedures, no attempt, off topic, can't be read

TaskSpecific Scoring Criteria for “Build a Pen” (Concepts and Procedures]

A 4-point response shows a thorough understanding of area and perimeter by showing and labeling all 4 possibilities of pens with Idfoot perimeters and by indicating that a 4 x 4 pen has the pces! am (I6 sq ft).

A $point response sh0ws.a thorough understanding of area and perimeter by showing and labeling more than one rectangular pen with a perimeter of 16 and by identifying the one with the g r e a t e s t areaorbyindicatingthata4~4penhasthegreacestam(16sqA).

A >point response response ah& generai understanding of : paimcccr by showing one or more rectangubx pens with a perimeter of 16. There is little indication of understanding of area.

A 1-point rrsponse shows partial understanding of perimeter by showing rectangles that may or may not have a perimeter of 16 Then is little indication of understanding of area.

A &point or Not Scorable @IS) response shows no understanding of the task or the concepts of perimerer and area.

Task-Specific Scoring Criteria for "Odd Wins" (Solving Problems)

A &point response does the following: - Thoroughly investigates the situation by showing all the ways a person can get an odd and even sum: o d d 1 + 2 2 + 3 even: 1 + 1.1 + 3,2 + 2 3 + 3

Applies hisher understanding of probability to explain thii choice better chance of losing the &ne

using words, numbers, or pictures.

* Correctly uses this information to determine that a person has a

A >point response meets most of the criteria, but may do one of the following (or similar):

Does not thoroughly investigate the situation by showing all the ways a person can get an odd or even s u m One or two of the 6 number pairs is incorrect or missihg. May show a minor flaw in using the information gathered to determine whether a person has a better chance of winning or losing the game, cg. the student may haw assumed that an even sum results in a win.

A 2-point response meets some of the criteria, but may do one of the following (or similar): - Less than 4 of the number pairs are correct, but the concept of

probability is correctly applied to the pairs given. * 4, I or 6 of the number pairs are correcf but the application of the

concept of probability demonstrates some uncenainty or confusion as to how this information can be used to determine whether a puson has a better chance of winning or losing.

* Does not thoroughly investigate the situation by showing all the wys

pairs is inaxrect or missing, AND shows a minor flaw in using the a person can get an odd or even sum One or two of the 6'number

chance of winning or losing the game, cg. the student may have information gathered to determine whether a person has a better

assumedthatanevensumdtsinawin.

A 1-point mponse does one or more of the following: Less than 4 of the number pairs are correct some understanding of the concept of probability is demonstrated, but there is uncertainty or confusion as to how this information can be used to determine

4 5. or 6 of the number pairs are correct, but no understanding of the whether a p e k n has a better chance of winning or losing.

concept of probability is shown.

A &point or Not Scorable (NS) response shows no ewidence of problem-solving skills and very little or no understanding of the mk, eg., student may l i t 2 or 3 of the number pairs but demonstrate no understanding of Probability.

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Solving problems is very much a part of your life. Every day you are faced with different kinds of problems-How can .I fmd a faster way to bike.to

to watch my favorite TV show? How can I earn school? How can I get my homework done in time

enough money to buy the toy I want?

Today you will begin reviewing five important steps for solving mathematics problems. You will see how different problem-solving strategies can be usedto solve problems.

Remember that solving a mathematics problem is like working on a puzzle or trying to solve a riddle. It's the challenge of making the pieces fit together that makes the task fun and rewarding.

Let's begin by looking at some steps you can use to solve mathematics problems.

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check to see if YOU answered the q&tion

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Step .i:7i Study and define the problem.

Before you can begin to solve a mathematics

understand it. As you read a problem, think problem, you have to make suriyou meally

about the following questions:

B What is the problem really asking? * Can you picture what is happening in the

problem? * How could you put the question in your own words or ask it in a different way?

* Is the problem asking for an answer, an estimate, a design, a picture, or a diagram?

What fads are given?

?& What facts must you know to solve the problem?

What fads are & needed to solve the problem?

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,them d k : g the:foUo%ng $Uptions:, Vfhich of.,these stiategiejhave youiusedto solve .:pioble,&?..What ,Other.strat&iei ha%,j.ou.used? How:didiyou'ke the strategy? Do you ,.. often.use.the samestrategy:b.iolve , ., . a:certain.kind of.problem?.JSit possible to use more

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Step '2 Choose one or more strategies.

Many wmmon strategies are listed below. Some of these strategies may be new to you. Others may have names M e r e n t from what you have called them.

You will read more about these strategies later in the booklet.

1

I

step d . . ~ p p ~ y your strategy or strategies.

Once you pick one or more strategies you want to use, the next step is to apply them.

Lefs aee how Detective Sam solves the following problem:

Kathy has just Gnished eating her lunch. Now she is thinking about what she can have for a snack. These are the things she can choose from:

bananas cookies apples raisins

She 'wants to eat two things. What are all the combinations she could have for a snack?

First, Detective Sam decides to draw lines to make pairs of words.

ARer he looks at his paper, he decides to change his strategy.

This time, he uses a strategy called Make an Organized List to help him solve the problem. He begins to write a list of all the possible combinations of snacks Kathy can eat.

Step '4. Show your steps.

When you solve a problem, it is impartant to clearly explain your thinkhg to others. Most of the time, you will show your work by using numbers or words. Other times, you will find it easier to use pictures, diagrams, charts, or graphs to show how you solved the problem.

When you work out a problem, remember to do the following

@ Use any combination of words, numbers, or pictures to explain your answer or show your work

%+ Show all the steps you took to solve the problem.

@ State your answer as c l k l y as possible. To communicate clearly, you must label your picture, diagram, c h a r t , or graph.

This is how Detective Sam’s friend wrote his answer to the problem. Does Sam’s Eend show the steps he took to solve the problem? Is his answer complete and clearly stated?

h

Look at Detective Sam’s answer below. How does Sam’s answer compare to his friend’s answer? Does Sam show all his work? Is his answer complete? Is it clearly stated?

step : .5;.. Check to see if you answered the question.

Read the problem again. Compare your answer to the facts given in the problem. Ask yourself these questions. (Not all of the questions will apply ti every problem.)

Did I find or do what the problem asked?

Does my answer make sense? Is it reasonable?

v

B+ Is there more than one answer to the problem?

Do'I need to go back and change my strategy? a Is my arithmetic correct? -

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. . .. . . . . . . . . . . . . i , . . . . . . , . . . . . . . . . . . . . . Ha&students +d StudMt h t e r . 8 ,m:r+d i! aloud.while stuaentsfollow.alon~.Have stu&nts, m e t additional:snategies.they hav&Sedin c l s s S . a n d ad&them t&e,list given. Point out that. :hey w i l l often n d . m use m o r e . t h a n one':problem:sohing sbategy.to'+h.e a problem . .

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Goover,the examples ~ f f ~ ~ n t . p r o t G n - ~ l ~ g iuatcgies on'StudentMasters9-12 with : the Clars so.students.fidly , . ~dektand how *&:of the suategies can be used Io solve problems

Use your judgmentas to how much class time should be spent on each of these strategies.

Supplement'the . . . examples &en on.theifollowing pages with additional problems that Crequiie or. en.~uragejthe.~e'of;~ese,stratepies: :; ..: . .; ..: '1, . . . . .

. . . . .;.. . . . .:~. . . . . . . . . . . . . . .... . . . . . . , .~ . . . . . . . . . . . . . . . .,, .. , . , . , . , . , . , . . . . , .

loser Look a t Different blem-Solving Strategies

How do you know which strategy to use to solve a problem? Choose whatever plan works best for you. Sometimes you may even have to use more than one strategy to solve a problem. As you do diEerent kinds of problems, you will find that some strategies work better with certain types of problems. ARer a while, you will notice .that you can use the same strategy to solve similar types of problems.

Detedive Sam made an organized list to help him solve his problem. Let's look again at some of the problem-solving strategies you can use.

!$. Guess, check, and revise.

i:. Draw a picture.

!? Act out the problem.

& Use objects.

: Solve a simpler or similar problem.

? Make a table or chart.

I. Look for a pattern.

i s Make an organized list.

j.:. Write a number sentence.

:. Use logical reasoning.

:.> Work backward.

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B. On the following pages, you will lo6k at how some of these problem strategies c ~ n be used to solve problems.

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Make a Chart/Look for a Pattern

Making a chart and looking for a pattern can be helpful when you are trying to work out a problem. Read the problem below.

Example:

Chris buys 3 cans of cat food every 5 days for her cats. How many cans of cat food will she buy in 30 days?

A student is working out this problem. He makes

Chris will have to buy every 5 days. a chart by writing down how many cans of food

9 -

e The student looks a t the chart and sees a D a t t e r n

in the number of cans.

What pattern do you see? How can you use this pattern to find the number of cans Chris will buy in 30 days? What other strategy could you use to solve this problem?

Working Backward

When you use this strategy, start with what you know and work back to the beginning. Look at the example below.

Example:

Hana has twice as many baseball cards as Mark. John has

Amy has 35 cards. How many cards d&s Hana have? 10 more cards than Mark Amy has 5 fewer cards than J o b .

Start with the facts given in the problem.

You know Amy has 35 cards. You also b o w Amy has 5 fewer cards than John. (Or you cansaythatJohnhas5morecardsthan Amy.) What can you do to find how many cards John has? That's right! Add 5 to 35.

35+5=40&' John has 40 cards.

@ You know now that John has 40 cards.

cards than Mark. What can you do to You also know that John has 10 more

find how many cards Mark has? The answer is, subtract 10 from 40.

40-10x30 Mark has 30 cards.

Now you know that Mark has SO cards. The problem tells you that Ham has mice as many baseball cards as Mark. How can you find out how many cards Hana has? Simply double the number of cards Mark. has.

30 + 30 = 60 So, Hana must have 60 cards.

Solve a Simpler Problem

Sometimes you can solve a problem more easily when you change it into a simpler problem and use smaller numbers. Read the problem below.

,Middleton School has 74 more students than Prairie School. Middleton has 388 students in all. How many students does Rairie s c h o o l have?

To solve this problem, first change it into a simpler problem. Look at the example below:

Middleton School has 5 more students than Prairie School. Middleton has 25 students in all. How many students does Prairie School have?

To solve this problem, you would do the following:

25-5=20 Therefore, Prairie School has 20 students.

Now use similar steps to solve the o w problem.

388-74=314 Prairie School has 314 students.

Always be sure to check your answer. Prairie School has 314 students. Middleton School has 74 more students. 314 + 74 = 388 Middleton has 388 students.

Use Estimation

Sometimes you can use simpler numbers to solve a problem because the problem is asking for an estimate. Read the problem below.

Your Wends are Coming to your house after school. You have $10.00 to buy snacks for you and your friends. At the store, you see the following items:

potato chips $1.99 a bag

Popcorn $1.59 a bag peanuts $3.39 a can

fruit bars . ' $2.29 a box cheese crackers $2.99 a box corn chips $1.89 a bag

Use estimation to decide which items you could buy for $10.00 or less.

potato chips $1.99 could be rounded

POPWrn $1.59 could be rounded to $2.00

to $1.60 cheese crackers $2.99 could be rounded

fruit bars $2.29 could be rounded

The total estimate is $9.00.

to $3.00

to $2.30

The example above shows one way you can estimate. There me other ways you can estimate as well.

You can use estimation to help you find other combinations of things that add up to $10.00 or less.

Use the Review Checklist below to check if you have uscd all five steps for problem solving. Not all the questions below will apply to ewery problem you do in mathematics.

: Step 1 : Study and define the problem. $2 Did I state the problem in my own words? $X Did I figure out what the question is?

'. Ei Did I identify the fads given in the problem? ,: 82 Did I decide which information is needed to ; solve the problem? 5 .E Did I decide if an exact answer or estimate i is needed? : ; Step 2: Choose one or more strategies. ; E Did I choose one or more strategies to solve ; the problem? 2.9 Which strategies did I choose?

i: Step 3: Apply your strategy or strategies. p; $:

:z ?% Did I carry out the strategy or strategies?

; Step 4: Show your steps. ?Z Did I use words, numbers, or pictures-to ... I I /

i~ explain my answer? !,

6 % Did I show all the steps I took tu solve the :! ; problem? p

.: F$ Did I state my answer as clearly as possible? F 5 3; Did I label my drawing, picture, or chart?

5 Step 5: Check to see if you answered $;

{. the question. j:

'; 2: Did I find or do what the problem asked for? f ;;

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Does my answer make sense? ......

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Group Ackvities

The following activities may be done individually, in pairs or small groups, or as a home activity. Use your judgment in selecting which group activities your students will do. based on your students' strengths and weaknesses in solving problems You may wish to have students complete the Student Worksheet on Student Master 32 as they do each of the activities. The purpose of the worksheet is to reinforce the 5 steps for problem solving Sample worksheets wmplehd by students are (or will be) provided as examples of student work.

After students have completed an activity, allow time for them to share their anstvex$ and methods or strategies for solving the problem. (In the Teacher Discussion Boxes, possible strategies are suggested. However. these are not the only strategies chat can be used by students.) Remind students to use the Review Checklist as they work on each activity.

Group Activities

Measurement

Build a Pen

You are asked to build a rectangular pen for your rabbit. Your parents want you to use al l 16 feet of fencing. Use materials to help you design different rabbit pens. Draw pictures of all the rabbit pens you can make and tell the size of each pen.

Describe the sue and shape of the rabbit pen with the greatest area.

Explain or show all the steps you used to find the pen with the greatest area.

There is a total of 9 beans in the green and blue cups. There is a total of 8 beans in the green and red cups.

How many beans are in the blue cup? ~

How many beans are in the green cup? How many beans are there altogether in the three cups?

Use the space below to solve the problem

How many beans are in the blue cup?

How many beans are in the green cup?

How many beans are there altogether?

!

When you are finished, play a game with your partner. Put a difFerent number of beans in each of the cups. Give your partner three clues like the ones given on the other page:Have your partner figure out how many beans you put in each of the cups.

Write the 3 clues you gave your partner.

How many beans did you place in each cup?

Geometric Sense

Are All the Parts the Same Size?

Look at the geoboard on the following page. The geoboard has been divided into 8 parts. Do all these parts have the same area?

Explain or show all the steps you used to solve the problem.

e YOU may cut out or draw on this page to help you complete the activity.

e

Number Sense or Algebraic Sense

How Many Tricycles and Bicycles?

Lisa sees some tricycles and bicycles a t the park. She notices 'that there are 4 more bicycles than tricycles. Lisa counts a total of 33 wheels altogether.

How many tricycles did she see? How many bicycles?

e

tricycle bicycle

Use the space below to solve the problem. Use words, numbers, or pictures to show how you found your answer.

How many tricycles did she see?'

How many bicycles did she see?

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Geometric Sense

Five 'Square

A pentomino is a figure made up of 5 squares. Each square must completely touch one side of another square.

These figures are pentominoes.

These figures are not pentominoes.

Use the grids on the next three pages to draw all the possible pentominoes.

When you have finished, cut out all the pentominoes you made. See which pentominoes can be folded into a box without a lid, like the one shown below.

-r

t I c

On this page, glue a l l the pentominoes that can be folded into a box without a lid.

h

On this page, glue all the pentominoes that cannot be folded into a box without a lid.

.. . . . . .. : , . . j . . . , , , . . .:,. . . .> : . . : . . . . . . . . . . . . . 'Essential Academic Learnihg~Require.ments~l.5 Algebraic:Sense, 2J.Solvjng Problems 'Possible Strategy or Strategies: Use~object$hok.for'.a,pattern; uSe:logical reasoning

Have students read the.directions for"Who Tookithe:LasZBean?".Foi.lhis acti~ty, students may use beans, pennies, or any other small obJects:flemind,students to use.the chart 10 record how many. beans each player takes during his or.her turn. Clarify he directions if needed. When students have played the game several &.have them answer andthen discuss the questions ontheznext page (Note: The an- &given on the following page after each question.) Ask students if they used a particular strateg'for playing'this game or,,discuss with them what strategy'they could have used to help them win. ' . '

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. . . , .. . .. . , . . . . . . . . . , . . . .: . .

Play this game with your partner. Count out 13 beans and put them into a pile. Each of you takes turns getting 1,2, or 3 beans from the pile. The person to take the last bean or beans from the pile wins the game. For each 'game, record

Can you be sure you will always be the winner? Why or why not?

jThe answer is no. I f there are 4 left, the opponent can win the

game.)

What happens if you change the total number of beans to 9 at the beginning of the game? Does it make a difference? If so, how?

Fhe answer is no. Again, if there are 4 left, the opponent can win.)

Suppose you have 13 beans. This time, each of you takes turns getting 2,3, or 4 beans from the pile. Does this change the game? Why or why not?

. . ;,.. ... .,.,... . ' . ,:.. :Ti::, ;.,:.,, , :.... . . .. ., .. . . &ential.Acade$c Learning Requirements 11 Number Sense; ul Solving'Problems Posible.Strategy . . . . or,SbaQies: Use:objects, draw a.picture, make a table or chart

Mdi manipulatives.for'this activity. If different '&iipul&es:aicused; such as paper clips, beans. and &ins. &:sure studenls.undenmd whichirpanipulatiGe represents each kind, of beid'sh& on thii!pag&'(You rnay.w&h to m a k e a c M . o n the:boiid to help.students make this.qmnection.) Ask students to shark the strategies they used to solve.this problem and disc& other sbategis that could,have been used to work this problem. (Note: The answm

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Number Sense

How Many Necklaces Can You Make?

For the art fair, you will be making necklaces exactly like the one below.

. .

The chart shows how many beads you have of each shape.

I l9 I

Use the space below to figure out how many necklaces you can make with the number of beads in the chart.

How many necklaces can you make?

46 necklaces)

Look at the number of beads you have left over. How many more of each kind of bead do you need to make 1 more necklace?

11 square bead, 2 round beads, and 0 oblong beads)

Number Sense

How Many Cookies Are There?

The school is having a bake sale. You bring fewer than 25 cookies to school for the sale. You need to put the cookies into bags. When you put 3 cookies into each bag, you have 1 cookie left over. When you put 4 cookies into each bag, you have 2 cookies left over.

Use the space below to find the number of cookies you could have altogether.

Tell or show another way you can group these cookies. Make sure that

there is more than 1 bag of cookies,

there is more than 1 cookie in each bag, and

there is the same number of cookies in each bag.

Jlf 22 cookies, you could have 5 bags with 4 cookies in each bag,

but you would have 2 lefl over. Or you could have 2 bags with 11

cookies in each bag, and none lefl over. I f there are a total of

10 cookies, you could have 2 bags with 5 cookies in each bag.There

are other possibilities as well.) . .

e

Math Fair

. .

Math Fair

"he school is having a Math Fair! Today you are going to try out some of the games that will be at the fair. You will use your understanding of mathematics and problem-solving strategies as you try out different games.

1 Color the number cube red. "hen cut on solid lines. Fold on dotted lines. Glue tabs inside or out.

. . . . . . . . , , . .

1 ~ Color the number cube blue.

Then cut on solid lines. Fold on dotted lines. Glue tabs inside or out.

Number Sense

Number Cube Toss

A number cube will be tossed 6 times. Write each of the numbers. Now do the following:

-~ - - - -

0 Arrange the numbers into two 3-digit numbers 0 Add or subtract the numbers you made.,, 0 ~'TY different ways to get the largest answer and the

smallest answer possible.

a

. .

Before s1udentsbegin the.acrivity, have them readthe.problem and d . e . t h e problem by .restating itin.iheir.own.words. . .

For Student Master 23% repeat the &e procedure as on Stydent.Master 23d. Each time you toss the number.cube, have students record that number in one of the blanks given. (As an option; each student can toss his or her cube six-times and:record each number in one of the blanks) Diiect.studenu.to arrange these 6 numbers in anyionier to make 2 three-digit numbers and then add or subtract the 2 numbers to by to get the.largest answer possible.

wen.stud.en& have finistied;have:them.write an explanation;showing the:steps they used.to solve the-pmblem or lellingw~:they.placed:ihe.numbers ini&e:boxes the,way they did. Students.may.$so . .. cell-what.they'could.have.done differentlyind explain why.

. . . .

A number cube will be tossed 6 times. Write each of the numbers. _ _ _ _ _ _ Now write each number in one of the boxes below to make two 3-digit numbers. Add or subtract the numbers you made t o get the largest number possible.

Explain the thinking you used to find the largest number possible.

. ..

.. ... . . . . . , . . . . . , . . ,. . . , . . . ..:Have . . . students~wmplete the.acti&y on.Student Master.23f. Repeat the same procedure. but

.,:_ 'thistim~students~shoula . . . . . . . . . .~ . . h y . t o . . . get.the':smallest answer possible. .. . . . . ~.

, .

A number cube will be tossed 6 times. Write each of the numbers. ------

Now write each number in one of the boxes below to make two 3-digit numbers. Add or subtract the numbers you made to get the smallest number possible. .

Explain the thinking you used to find the smallest number possible.

Introductory Activity for the Game "20 Circles"

When you toss two number cubes, you can get any s u m from 2 through 12. Show all the different ways you can get each sum with your two number cubes.

You are going to play a game called “20 Circles.” On Student Master 24c, you will be asked to draw 20 circles in any of the boxes on the game board. To begin the game, the teacher will toss two number cubes. You will determine the sum of the two numbers on the number cubes and find that sum on the game board. You will then fill in one of the circles next to that number. The first person to fill in all of the circles on his or her board wins the game.

Now let’s play the game! On your game board, draw 20 circles 0 in any of the boxes. Do not draw more than one circle in a box.

5 + 3 = ?

Game Board

Think about the activities you've done so far If you want to have the best chance of winning the game, where would you draw your circles on the game board?

phe answer is "next to the number 7.3

Explain your thinking using words, numbers, or pictures.

phere'are more ways to get the sum of 7 than any other number.)

Probability

Odd Wins

One of the games a t the Math Fair is called “Odd Wins.” A person gets to throw two darts at the dart board shown below. If the sum of the two numbers is odd, that person wins. If the sum of the two numbers is even, that person loses.

Show all the ways a person can get an odd or even sum.

9 I

. . . . . . . . . . I. . . . : . . . . . . . . :.._ . . , '

Hawstudents romplete'StudentMaster2jb.Remind.themto use'the work.they did on . . , . . . .

. . . . . '' the other.page to support their answers: .:.

. . . , ,*.. , . . . . . . . ::. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Look at your work. Remember that a person wins if he or she gets two numbers that have an odd sum.

Does a person have a better chance of winning or losing in this game?

Explain your thinking using words, numbers, or pictures.

. .

,&entia1 Academic Learning Requirements 1.4 Probability and Statistics, 20 Solving Problems Possible Strategy or Strategies: Draw.a picture; guess, check, and revise

I Have students read Student Master 25c and write the problem in their own words Review all the rules of the game ‘Ddd Wnr”Emphasize to students that they should draw the two dart boards such that a person has a better chance of winning the game with Dan board A than with Dart bard R (Note: One possible solution is to divideDan bard A into 4 equal parfs with numbers 4,I6 and 7; and divide Dan bard B into 3 equal parts with numbers 1,2 and 3.)

. .

Remember the d e s of the game: The person who gets two numbers that have an odd s u m wins the game.

This time, you will get a chance to make 2 dart boards for this game - Dart board A and Dart board B. Use a t least 3 different numbers on each dart board below. Draw the dart boards so that a person has a better chance of winning the game with Dart board A than with Dart board B.

Dan board A Dan board B

Write the problem in your own words.

TO @mplete Student Master 25d, students may wish to make a.chart, a picture, or an organized list to help them explain why their dart'board will give,someone a better chance of winning.

Explain why a person will have a better chance of winning with Dart board A than with Dart board B.

. . .

. , : . l .:. :: . . . . . . . .. ,,:,i.,;:: . ' . . . , . , , I ., , . . , . , . . . . . .

. : ~ t i ~ ~ ~ d e m i c ' L e ~ i n g R e q U i n m e n u . t Z . . . . . Measuitimenf . . . Sol;ing,Pmblems ..... :':.Po&ible.Strategy ..:: .... ..l.. . orSh$te~~:~Draw'a.pictui<~&lvea . . . . . . siniilar,pruble~u.ie:logid reasoning,

. . . . . . . . . . . . . . . . . . . . . . . . . . &:Objects . . . . . . . . .:,, .. : . . . . . . : . , . . "..:. . . . . :.;,, i . .

~ . : . ; ,. . . . . . . . . .

. . . . . . . . . . .:., . . . . . ,?;€&&&&nu ,>.;'a ... .< read Studentlvlaster .:I. . > . ..*, . . . . 26a;Gd writLthe~problem.in.their ,\, . . . . . . . . . . om'lwodiHave them ..:explain.how.they.would.goaboui d&d&gwhchjai.is h&i&'&igl& least..numk.of ' ,

&ighin~They may.ch&e'tb draw pictures or.write theiiexplanations (Note: One.possible solution is to put jars l.and 2 on theilefhide of scale andjars 3 and 4 on the right:side of &lc,If . . . both sidesare.b@&ced.therjar 5 is heavierAf l eh ide is heavier, put.jar 1 on left

:: ..... ,ijde:ofscale.and i i ~ . . . . . jar 2:on . . . . right *de of;scaieIf,right s'idelis he+&, putjar,3,on left side of

. . . . . . . . . . . .

.?, scaleahdjar,40n:right'si~~,ofscale)i,:.I 'c.:.:! :::: ': ,,, '.;! ',:,'':;\';.. '. ' ',. ' '

. . . . .

. . . . . . . . . . . . . . . . . ^ . . . . . . . . . . . . . . . : . :

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. . . .

Measurement

Which Jar i s Heavier?

In the game "Which Jar is Heavier?" there are five jars. Four of the jars weigh the same because they have the same number of jelly beans. Only one jar is heavier because it has more jelly beans than the others. e

1 2 3 4 5

You have to find which jar is heavier. To win a prize, you must weigh the jars on the scale the least number of times.

f


Tell the steps you could use to solve the problem.

Algebraic Sense

Guess the Number

One of the games at the Math Fair is called “Guess the Number.“ The following shapes are on a board:

Behind each shape is a number. AU the squares have the same number, all the triangles have the same number, and all the circles have the same number. You have to figure out what number is bidden behind each shape.


Show all the steps you would use to find the 3 numbers.

Now use A, 0, and 0 to make the number 15.


Explain or show how the farmer can get her animals across the river.

Describe another way the farmer can get her animals across the river. -

. .

Geometric Sense

The Great Cover-Up

For th is game, you will use cutouts that have t he following shapes.

Find as many ways as you Can to cover all of each rectangle below with the cutouts. (The cutouts should not overlap.)

Rectangle A Rectangle B

When you have finished the activity, answer the questions below.

How many different ways can you cover Rectangle A?

How many different ways can you cover Rectangle B? B

4

Color these shapes red.

L

Color these shapes blue.

L,

Color these shapes green.

a

.. Show all the ways you can cover each of the rectangles.

0

B

Mathematics Pre-Problem Solving Questions

Before you get started reading about the five steps for problem solving, think about how,you solve math problems. Read the questions below and write a short answer for each.

What is.problem solving in math?

Explain the steps you might take to solve a math problem.

f

What strategies are useful in problem solving? Name 3 and explain each.

What are some things you do well in solving math problems?

What are some things you need to work on when you solve math problems?

Mathematics PostProblem Solving Questions

You have learned the five steps.for problem solving and have read about different strategies you can use to solve math problems. Think about how you would now solve math problems. Read the questions below and write a short answer for each.

What is problem solving in math?

Explain the steps you might take to solve a math problem.

. .

What strategies are useful in problem solving? Name 3 and explain each.

What are some things you do well in solving math problems?

What are some things you need to work on when you solve math problems?

a

Student Worksheet

1.' Study and define the problem.

2. Choose a strategy.

3. Apply the strategy.

4. Show your steps or all your work.

Date

Sample Student Checklist

I read and study the problem carefully.

I find all the facts and information I need.

.I make a good plan or choose one or more strategies to help me solve the problem.

I use the mathematics I know to help me find the answer.

I use tools such as pictures, charts, or number sentences to help me explain my answer or show my work.

Student Scoring Criteria

A 4-point response: Explores the problem carefully Uses all of the necessaxy information Applies a useful strategy that best fits this problem Gives a solution that is complete, accurate, and clearly written

A 3-point response: Explores the problem Uses al l or most of the information I needed Applies a strategy that seemed to work Gives a solution that is accurate, but may not be complete

A 2-point response: Explores only part of the problem Uses some of the information Applies a strategy Finds a solution that may not completely solve the problem

A 1-point response: Does not really explore the problem carefully Uses little of the information Tries to use a strategy Tries to find an answer

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