Equivalent Fractions

Tenth Planet Explores Math

Number

Teaching Guide

Tenth Planet TM Explores Math

Number

© 1998 Tenth Planet Explorations, Inc. All rights reserved.

No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America.

Tenth Planet and the Tenth Planet logo are trademarks of Tenth Planet Explorations, Inc. All other trademarks are the property of their respective owners.

ISBN: 1-888618-82-5

1 2 3 4 5 6 7 8 9 10 L 02 01 00 99 98

2 Equivalent Fractions

Multimedia Activities

Interactive Multimedia 11

Children create equivalent fractions by dividingshapes, putting together totem pole pieces,comparing distances at a baseball game, makingorders of food, and cutting and labeling food for a Spacemarket.

OverviewNavigation 14Control Panel 15The Journal 16

IntoEquivalent Fractions 18

ThroughEquivalent Pieces 22

BeyondChoices for Equivalent Fractions 24Fraction Toss 26Spacemarket 28Diner 30Totem Creations 32Totem Creations: Problem Creation 34

Reflection TimeFraction Skits 36

Getting Started 5

Getting Started provides an introduction toEquivalent Fractions. You will find an overview ofthe mathematics explored in this Investigationand information that will help you begin usingthe program materials.

How to Begin 8How to Teach the Big Idea 9

Hands-on Activities

Totem Poles Theme 37

Students hear a story about a totem pole andexplore how the totem pole illustrates equivalentfractions. They use equivalent fractions to design their own totem poles, compare fractionsof a totem pole and prove whether they areequivalent, and make plans for a park bygenerating equivalent fractions. Using a deck of cards that shows equivalent fractions,students play games where they apply their understanding.

IntoTotem Tale 38

ThroughPuzzling Totem Poles 42

BeyondTotem Pole Controversy 44Totem Pole Park 46 Link to Home: Totem Pole Game Pack 48

Reflection TimeTell All About It 50

Additional Resources 51

Equivalent Fractions 3

Assessment Activity

Performance Assessment 53

In this performance assessment activity studentsdemonstrate their understanding of equivalentfractions by comparing two fractions to figureout whether Thunderbird and Frog have trickedRaven in a trading situation. This activityprovides an opportunity to assess what thestudents have learned over the course of theInvestigation. A scoring rubric is provided toguide your evaluation of student work.

The Trick 54Rubric 55

Math Content 57

This section provides a summary of the threeTenth Planet Number Investigations forfractions. You’ll also find charts identifying themathematics in each of the multimedia andhands-on activities, as well as the NCTMStandards addressed in Equivalent Fractions.

About Fractions 58Multimedia Activities Chart 59Hands-on Activities Chart 60

Supplies 61

This section includes blackline masters neededfor the hands-on activities. Spanish translationsof all stories are included, as well as a FamilyLetter in English and Spanish.

Blackline MastersThe Challenge (Spanish) 62Totem Parts 64Daily Divide News Flash (Spanish) 65Totem Pole Card Set 1 66Totem Pole Card Set 2 67The Trick (Spanish) 68

Family LettersLink to Home: Totem Pole Game Pack 69Link to Home: Totem Pole Game Pack (Spanish) 70

Internet Resources and ContactInformation 72


Getting Started

Getting Started describes the big mathematical idea and important understandings studentsencounter as they do the activities. Information on how to begin and a chart showing the organization of the activities follows.

Getting Started


The Big Idea:

Fractions are equivalent if they represen

t the same relative amount of a w

hole

The K–4 instruction should

help students understand

fractions . . . and build initial

concepts about order and

equivalence . . . (using) physical

materials, diagrams, and real-

world situations . . . to relate

their learning experiences to

oral language and symbols.

This K–4 emphasis on basic

ideas will reduce the amount of

time currently spent in the

upper grades in correcting

students’ misconceptions and

procedural difficulties.

—National Council of

Teachers of Mathematics


In this Investigation, students develop their understanding of equivalent fractions through experiences with concrete objects, geometric

regions, distance or length models, and sets of objects. Prior to this Investigation, students must understand basic fraction concepts so

they can create representations of fractions and interpret fraction symbols.

Through experiences with concrete objects and with diagrams, students compare amounts to determine equivalence. When

students have a solid understanding of equivalence, they can mentally create equivalent fractions without the use of objects or diagrams.

Students can eventually use their understanding of the meaning of fractions and equivalence to create ways to add, subtract, multiply, and

divide fractions in real-world situations.

Equivalent Fractions provides lessons that integrate experiences with real-world objects, manipulatives, cross-curricular activities, and

multimedia. Students make connections with math concepts in a variety of ways to accommodate a range of learning styles and abilities.

Throughout the Investigation students will have opportunities to:

� Identify equivalent fractions

� Explore equivalent fractions represented symbolically andvisually

� Create equivalent fractions by trading

� Use fractions to prove equivalence

� Find equivalent fractions by dividing an area into equal parts

� Use different equivalent fractions to name a fraction of a whole

� Compare distances in a fraction chart to determine equivalentfractions

� Explore fractions as a ratio

Important Understandings

� We can name every fraction in infinite equivalent ways.

� We can represent equivalent fractions with word names,drawings, objects, and symbols.

� We can create equivalent fractions by trading parts, groupingparts, repeating parts, and dividing parts and the whole.

� When we divide one part of a whole into smaller pieces, wecan divide all other parts of the whole into the samenumber of pieces to determine what fraction we have.

� We can find equivalent fractions for a set of two types ofobjects by changing the number of objects in the set whilekeeping the ratio of the two types the same. For example, a set of two pencils is half yellow if one of the pencils isyellow, and a set four pencils is half yellow if two are yellow.

Getting Started Investigate*


• Personalized student journals• Math connections to the world• Open-ended math challenges• Powerful tool set• Interactive problem-solving• Guidance and help

Hands-on Activities

• Theme-based curriculum, e.g., art, architecture, nature, and diverse cultures

• Incorporate common classroommaterials, such as math manipulatives

• Cross-curricular activities• Support materials: blackline masters,

Additional Resources• Link to Home

Internet Resourceshttp://www.tenthplanet.com/Teachers/

• Tips on teaching with technology• Links to related Web sites• Math challenges• Annotated resources

3


1 2Install

• Multimedia Investigation• Teacher Resource Tool

- create class list- review student journals- customize multimedia

activities(Refer to the Technical Guidefor more information.)

How to Begin

Integrate

• Complementary hands-onand multimedia activitiesbased on a consistentlearning model

• Assessment throughoutmultimedia and hands-onactivities

• Internet activities to extendteaching and learning

• Activities that align withNCTM Standards

* For a step-by-step preview of the program, see the At a Glance document contained in your package and on the Investigation CD.


How to Teach the Big Idea

IntoExplore andConnect

ThroughDo andUnderstand

BeyondCreate andApply

Assessment


• Equivalent Fractions

• Equivalent Pieces

• Fraction Toss• Spacemarket• Diner• Totem Creations• Totem Creations: Problem Creation

• Reflection Time:Fraction Skits

Hands-on Activities

• Totem Tale

• Puzzling Totem Poles

• Totem PoleControversy

• Totem Pole Park• Link to Home: Totem

Pole Game Pack

• Reflection Time:Tell All About It

• The TrickPerformance Assessment Activity

Integrating Activities

The multimedia activities aredesigned to be integratedwith the hands-on activities.Both types of activities followthe same learning model ofInto-Through-Beyondsupported by assessmentactivities. You can use boththe multimedia and hands-onactivities in this sequenceinterchangeably. You can alsosubstitute these hands-onactivities for those of yourregular math program.


Equivalent Fractions

Interactive Multimedia


Beyond

Students createequivalent fractionswhile solving problemsinvolving area, distance,and sets of objects.

Through

Students createequivalent fractions for a part of a givencircle or square,checking their solutionsby comparing areas.

Into

Students exploreequivalent fractionsrepresentedsymbolically and inreal-world contexts.

Home

Students select theirnames from a class listand sign in to theirpersonal journals.

The interactive multimedia activities in this Investigationinvolve children actively in developing and applying theirunderstanding of basic fraction concepts. The learningmodel—Into, Through, Beyond—serves as the framework.In Into, children explore short vignettes that illustrateequivalent fractions; in Through, they create equivalentfractions; and in Beyond, they apply and extend what theyhave learned about equivalent fractions to solve problems.

Auditory and visual learning feedback is provided within thecontext of a problem in all of the Through and Beyondactivities. The feedback helps students be successful in solvingproblems. For example, if the student creates an equivalentfraction, the result is recorded. If the fraction is not equivalent,the student can adjust or start over.

The diagram on the next page shows the choices that childrenhave at each phase of the multimedia activities. On the INTO

screen, for example, children have a choice of four objects theycan click to explore. The pages that follow describe themultimedia activities in detail and provide teaching strategies.

Other Tenth Planet Resources

A companion text, the Technical Guide, covers topics such as installing the software and setting up computers, using theTeacher Resource Tool, setting audio preferences, creating class lists, and using student journals.

Tenth Planet's Web site contains more teaching resources forextending the math concepts in this Investigation and providesa forum for communicating with other educators. Our Website includes a Customer Support area with answers tofrequently asked questions, special instructions, and the latestinformation on technical issues.

The address on the World Wide Web ishttp://www.tenthplanet.com/Teachers/

Activity Customization

You or your students can change the target fractions thatappear in each multimedia activity by pressing Command-N(Macintosh) or Ctrl+N (Windows). The activities are preset toproper fractions which appear each time you open theInvestigation CD. You can also choose improper fractions.

The choice of proper or improper fractions can be changed atany time when the Investigation CD is open. Changes remain in effect until you make a new choice or you close theInvestigation CD.

An Explore feature allows you or the children to set theparameters for a problem. Explore can be accessed by clickingon the telescope in the BEYOND activities.

Overview

InteractiveMultimedia


The Big Picture


This diagram shows the choice screensfor activities you will find at each phaseof the learning model. Clicking on anactivity illustration leads to that activity.

INTO BEYONDTHROUGH

To navigate in the multimedia activities, place the cursor overthe rocket at the bottom of the control panel. The symbol inthe rocket porthole shows the section you are working in—Into (key), Through (door), or Beyond (light bulb). Pressing the yellow buttons above the rocket takes you to theother sections of the program.

Return to the Home screen. The Home screen is thegateway to the multimedia activities.

Ask students to return to the Home screen at the end of eachsession. From here, the next student or team can choose theirname and open their journal, view the introductory movie, orstart the activities.

Go to INTO activities. Into activities introduce mathconcepts using video, animation, and stories.

Go to THROUGH activities. Through activities givestudents practice with new math concepts in an

interactive environment. Work created in Through activitiescan be recorded in the journal.

Go to BEYOND activities. Beyond activities givestudents opportunities to solve problems and apply

concepts explored in Into and Through activities. Workcreated in Beyond activities can be recorded in the journal.



THROUGH

BEYOND

Homescreen

INTO

You are herein BEYOND

Navigation

When the rocketdisplays a miniatureversion of your lastlocation you canclick on it to go backto that location.


Try Again

Click on the Sun to try the same

problem again.

New Problem

Click on the Moon to get a new

problem in the same activity.

Undo

Click on the Star to undo

your last action.

Explore

Click on the Telescope to explore

an activity in a new way.

Show Tools

Click on the Space Ship to show the

tools you can use in the activity.

Help

Click on the Radio Tower to get help.

In THROUGH activities you see a hint or

sample solution. In the BEYOND

activities instructions are repeated.

Journal

Click on the journal to record your work.

A control panel on the left-hand side of the

screen contains buttons that children can click

to use the various features of the activities or

to go to something new. Buttons that cannot

be activated are grayed out.

Control Panel

The journal is provided in the Through and Beyond activitiesas a valuable assessment tool for

� Self-assessment:

Students can keep a record of their work and assess theirown learning.

� Ongoing Assessment:

You may use the journal as a window into studentthinking and progress. Journals are also a useful tool forconferences with parents.

� Portfolio Assessment:

Journal entries accumulated over time can illustrate astudent’s growth and change.

� Performance Assessment:

Most performance assessment activities may be done withconcrete materials or the computer. The journal can serveas a student’s performance assessment product.

Students access their personal journal by clicking on the booklocated on the control panel. Inside the journal, they can take a picture of the work they just completed, and they canwrite about what they did or tell about it by talking into themicrophone. When they do so, visual feedback in the form of an animation unique to each activity rewards their recording efforts.

The journal includes an illustrated glossary of key words usedthroughout the Investigation. Students can view the words andpictures and hear them. If they wish, they can drag a word toplace it directly into their journals.

The journal always opens to a new page. If the journal is full,the work shown on previous pages can be deleted to makeroom for newer work. This gives your students an opportunityto reflect on what they have done and discard journal entries that no longer show their level of understanding. You determine the number of pages in each student journaland the maximum audio recording time with the TeacherResource Tool.

Students may print out their journal entries. Each printed pageautomatically includes the student’s name, the name of theInvestigation, and the date. Students may save printed pages touse for Reflection Time and to take home to share with theirfamilies. You may collect pages in portfolios forstudent/teacher conferences.



The Journal as an Assessment Tool Using the Journal

Journal


Copy workto journal

Open to a specific page

Scroll throughjournal

Scroll throughjournal

New page

Deletecurrentpage

Closejournal

Printcurrentpage

Hear comments

Recordcomments

Recordingtime

Scroll through glossary

Scroll through glossary

Open and closeglossary book

Gethelp

Glossary

Current page


Math connectionsFractionsGeometry and Spatial Sense

Curriculum connectionsSocial StudiesLanguage Arts

Related activitiesRefer to the hands-on Intoactivity on p. 38 for more onequivalent fractions.

Movie controlTo stop and restart movie,press Command-M(Macintosh) or Ctrl+M(Windows).

Into Equivalent Fractions

Equivalent Fractions� Explore equivalent fractions represented symbolically

and visually

Fractions of Sets� Explore a set of objects divided into various equivalent

fractions

Invite students to explore equivalent fractions through thesefive brief vignettes. Encourage students to view the vignettes asoften as they wish. You may use the questions that follow eachdescription to facilitate discussion.

A Gift for Raven This movie introduces theconcept of equivalent fractions through a storydepicted in the carvings on a totem pole. An

authentic totem pole carver cuts an image of a raven into a logas the narrator begins the story of how Raven and Froglearned the meaning of equivalence.

The story characters take over the screen, involving students inan important choice that a generous Chief gives Raven—oneexquisite cedar box or two boxes half its size. Frog, who willhave second choice, understands from the Chief ’s explanationthat the two gifts are equivalent. Raven chooses the two boxes,believing that two boxes make a bigger gift than one.

Ask students to explore other equivalent gifts the Chief couldhave offered to Raven and Frog that illustrate equivalentfractions other than one whole equals two halves.

Did Raven get a bigger gift than Frog?

Why was Frog satisfied with either gift?

You may wish to extend the totem poles theme by gatheringseveral of the books listed in Additional Resources on page 51.Show these books during a class discussion or have studentsrefer to them as they look for examples of equivalent fractionsin the world.

Your students may recall times when they have divided thingsto distribute among their siblings or friends. For example,when brownies have been divided into different-sized pieces,one child may get two half brownies and another may get onewhole. Some children might have felt cheated until theycompared the total amounts of food to see that they were thesame. Discuss the fact that the number of pieces a whole objectis divided into does not change the size of the whole. Havestudents draw representations of objects they have shared andillustrate how two identical objects can be divided intodifferent numbers of equal parts and still be shared equally.


The nativepeoples of thePacific Coastrecorded theirhistory andpreserved theirlegends, myths,and stories onspectacularlycarved totempoles.

Animals represent specific characteristics. The Raven, forexample, is a trickster who tendsto be restless, tough, curious,easily bored, and often greedy.

Stacking the boxesdemonstratesthat two halvesare equivalentto one whole.

Lasagna Party A girl learns about equivalentfractions as the number of guests coming to herdinner party keeps changing. After she has sliced

a lasagna into sixths, one of her friends calls to ask if he canbring six more people. She cuts each of the pieces in half tomake twelfths, only to be informed that the extra peoplecannot come after all. She resolves the problem by servingeach person two pieces of lasagna, or �� , which is the same asthe serving size she started with, �� .

What fraction of the whole lasagna will each guestget? How do you know?

What other fractions describe that same amount?

Sly Fox A sly fox sells carrot cakes. His sign saysthat half a cake costs $1.00. The fox tries to trickthe rabbit into taking him up on a special deal—�� of a cake for $1.00. Students see the pictures

the rabbit draws in her mind as she divides the cake intoeighths, reminding herself that � is the same as one whole and � is the same as �� . It’s clear that �� is less than � and thereforeless than �� . The rabbit knows too much about equivalentfractions to fall for the fox’s trick.

How did the rabbit figure out the trick?

Name a fraction that would give the rabbit more than �� of the cake, but less than one whole cake.

Almost There Two sisters traveling by car are hopingthey’re close to their destination. When the youngersister reads a sign that says they are �� of a mile away,

the older sister informs her that they are really �� , � � , or even ��

of a mile away. As the older sister recites equivalent fractionsfor �� and �� , students see them displayed: �� = �� = �� = � = �� .As the numerals in the equivalent fractions get larger, theyounger sister believes they are not at all close to theirdestination. Just before they arrive, her sister explains that the fractions are all equal.

How can the same distance be described with differentfractions?

How many equivalent fractions do you think there arefor each of those fractions? Why?

Can you draw a picture to prove that �� = �� ? �� = � ?�� = �� ?

Marching Ants A troop of eighteen red and blackmarching ants illustrates equivalent fractions by forming

three groups. In each group of ants, the ratio of �� red ismaintained, even though the total number in each group isdifferent: 3, 6, and 9. The ant “sergeant” drills each group on what fraction of them is red and how to say that fractionanother way. Use the movie controller to stop the movie sostudents can point to the group representing each fraction—�� , �� , and �� —and use the rows and columns of ants to explainhow each group represents �� .

What does it mean when we say that two fractions areequivalent?

At the end, when all 18 ants form one group, whatfraction of the group is red?

How many equivalent fractions do you think there arefor �� ?

Movie controlTo stop and restart thefollowing movies, click onthe Space Ship.




Sly Fox

Lasagna Party

Marching Ants

Almost There

The INTO choice screen

Home screen

THROUGH activitiesINTO activities

You are here in INTO.

Dividing an Area� Create equivalent fractions by dividing and shading a

part of a circle or square� Decide whether two fractions are equivalent by

comparing areas

In this activity, students shade in fractional parts of a shape toequal the colored fraction of an identical shape. Students aregiven a fraction, a circle or square with that fractional partcolored in, and a blank circle or square of the same size with afraction label they can adjust. As students change the denomi-nator in the fraction label, they divide their shape into thatnumber of parts. As they change the numerator, they shade inthat number of parts in the shape.

By dragging the given shape over their shape, students cancompare the shaded areas to see if they are equivalent fractions ofthe whole. This visual feedback helps students adjust theirfractions until they are equivalent and the two areas match exactly.

Students record their fractions by clicking on the camera. Theylearn from the number sentence that appears under theirpicture whether the fraction they created is equivalent to thegiven fraction. For example, they might see “ �� = � ” or “ �� ≠ �� .”Encourage students to move from guess-and-check strategiesto mental and visual reasoning. For example, suppose a studentis given �� . She might mentally cut each fourth of the shape inhalf, making a total of 8 pieces. Thus, the new shape can becut into eighths. Three of the original pieces were colored, so2 + 2 + 2, or 6 pieces, will be shaded in the new shape.Clicking on the camera will give the report, “ �� = �� .”

As students work, ask questions like the following:

Can you figure out how many pieces to cut the newshape into before you use the computer? Explain yourreasoning.

Suppose a circle is given that is �� colored. Try dividingyour circle in different ways—into thirds, fourths, fifths,eighths, or sixteenths, for example. Which ones canyou shade to show a fraction that is equivalent to �� ?Why?

How many equivalent fractions are there for �� ?

Tip from a teacher I challenge my students to see howmany different equivalent fractions they can make for certainfractions, such as �� , �� , �� , and �� . We make a poster for eachchosen fraction, then print out pictures of the equivalentfractions from the journal and attach them to each poster. The students can add drawings that show pairs of equivalentfractions and later add printouts from some of the Beyond activities.

What to look for Can students create several equivalent fractions for eachgiven fraction? Do students know when the solutions theyhave recorded are incorrect? Do students understand thatthere are an infinite number of equivalent fractions for anygiven fraction? Can students figure out how many parts todivide their shape into and how many parts to shade beforethey use the computer?


Through Equivalent Pieces


Curriculum connectionsLanguage Arts

JournalEncourage students to recordtheir work.

Related activitiesRefer to the hands-onThrough activities on pp. 42–43 for more onequivalent fractions.


Click on the doorand then the circlesor the squares toget to an activity.

This student’s journalreads, “One fraction canequal another fraction butit is realy the same thing.”

This student’s journal reads,“I made two equivalent fractions that is �� and � � .”

Trading� Create equivalent fractions by trading� Combine fractional parts to create equivalent fractions

for a target fraction

Demonstrating Equivalence Using Distance� Given a linear model of a fraction, enter equivalent

fractions using symbols� Compare distances in a fraction chart to determine

equivalent fractions

Dividing an Area� Create equivalent fractions by dividing each fraction of

a whole into smaller, equal parts

Demonstrating Equivalence Using Sets� Explore fractions as a ratio� Create equivalent fractions using sets of objects

Students apply their understanding of equivalent fractions tosolve challenging problems in various contexts. In all fouractivities, you have the option of having students work withproper fractions only or including improper fractions.

Fraction Toss Students play the role of a ballparkvendor, tossing hot dogs, baseballs, popcorn, andpennants to a customer. In order to reach the

customer who is sitting a given fraction of the way down therow, they enter an equivalent fraction. A fraction chart isprovided to help students find equivalent fractions.

Spacemarket It’s not easy running a supermarketin outer space, where creatures can only eat theirfood in certain fractional-sized chunks. Studentsare given a fraction of a pan of Neon Nutriglop

and challenged to divide it into many equivalent fractions sothat lots of creatures can buy it.

Diner A diner uses data from a customer survey toget the right types of food ready on the grill. Theinformation is presented to the cook as a fraction, for example, �� cheeseburgers, the rest hamburgers.

Student cooks decide how many of each to place on the grill.They ring the bell when the order is done to see if the fractionof cheeseburgers they cooked is equivalent to the target fraction.

Totem Creations Students build totem poles thatare equivalent to a target fraction of a decameterby stacking fractional pieces. They have a choiceof four different fractional units. As students build,

they can click on a tape measure to compare their totem pole’sheight to the target fraction. They build next to a scale showingthe same number of units as the denominator in the targetfraction. When they are done, they click on a pencil to see anumber sentence that compares their fraction with the target.

Explore Each problem has an Explore option whichallows students to set the parameters for a problem.

Students can use Explore to become familiar with theenvironment of a problem or to create a problem for anotherstudent to solve. A small “e” on the screen indicates that achild is in Explore.

Math connectionsFractionsGeometry and Spatial SenseNumber Sense and

NumerationProblem Solving

Curriculum connectionsLanguage ArtsSocial StudiesArt

Related activitiesRefer to the hands-on Beyond activities on pp. 44–49 for more onequivalent fractions.

Activity customization Proper fractions are presentedin all activities unless youinclude improper fractions by pressing Command-N(Macintosh) or Ctrl+N(Windows).

Beyond Choices for Equivalent Fractions




The choice screen for BEYOND activities

Home screen

THROUGH activities

You are here in BEYOND.

INTO activities

Diner

Fraction Toss

Spacemarket

Totem Creations

Demonstrating Equivalence Using Distance� Given a linear model of a fraction, enter equivalent

fractions using symbols� Compare distances in a fraction chart to determine

equivalent fractions

Students play the role of a ballpark vendor as they enter afraction to toss merchandise to their customer. The customer issitting a fraction of the distance from the vendor to the end ofthe row of seats. Whether it’s a hot dog, a pennant, popcorn,or a baseball, the item won’t reach the customer unless thestudent enters a fraction that is equivalent to the fractionrepresenting the customer’s location.

Students may use a fraction chart that is the same width as therow of seats to identify the target fraction and equivalentfractions. By dragging a vertical line along the row to thetarget fraction, students can clearly see the other fractions onthe chart that represent the same distance. They can keeptossing items to the customer to find different equivalentfractions. Encourage them to use the patterns they see toextend their list of equivalent fractions for each target fraction.

To help your students make their list of equivalent fractions,you could ask:

How can you use the fraction chart to indicate wherethe customer is located?

Do any of the fractions in other strips on the chartcover exactly the same distance?

Can you think of an equivalent fraction that isn’trepresented on the chart? How many are there?

Tip from a teacher As a warm-up, I pass out one row ofan egg carton (six sections) and a bag of small candies to eachpair of students. Each pair uses a marker to color a fraction ofthe row and writes their fraction symbol on paper— � , forexample. They put one candy in each of the six sections andwrite the fraction of candies that are in colored sections. Theythen add another candy to each of the six sections and writethe new fraction for the number of candies in colored sections.They keep going as long as they have enough candies to add acomplete row. In a follow-up discussion, the students noticethat they have each developed a list of equivalent fractions—� = �� = �� , for example.

In the Explore version of this activity, students canenter any fraction they choose and find equivalent

fractions on their own or have a partner solve their problem.Students can also explore the use of the fraction chart to locateequivalent fractions.

What to look for Can students represent the customer’s location on thefraction chart? Can they identify equivalent fractions on thefraction chart? Do students accurately enter the fractionsymbol they have represented on the chart? Can theyidentify equivalent fractions without using the fraction chart?

Math connectionsFractionsNumber Sense and




Related activitiesRefer to the hands-on Beyondactivities on pp.44–49 formore on equivalent fractions.

Activity customizationProper fractions are presentedin all activities unless youinclude improper fractions by pressing Command-N(Macintosh) or Ctrl+N(Windows).

Beyond Fraction Toss




Click on the light bulb andthen the baseball fans to getto this activity.

This student’s journal reads,“The customer got his flagand his baseball but he didn’tget his hotdog because thehotdog wasn’t equivalent.”

This student’s journalreads, “I looked at theline and I counted all theparts until I got to theline. Then I put thatnumber on the numeratorand looked at thenumbers on the side and I put it on thedenomenator.”

Dividing an Area� Create equivalent fractions by dividing each fraction of

a whole into smaller, equal parts

This activity takes students to a planet where the creatures’mouths are many different sizes. The Spacemarket sells NeonNutriglop in fractional parts of a pan. Some creatures can eat only halves, some only thirds, others only twelfths, and soon, one unit piece at a time. Therefore, each problem presentsa pan with a different fraction of Nutriglop in it— �� of a pan,for example. Only creatures who eat thirds will buy a pan cutand labeled as �� . Tell students to cut the Nutriglop into smallerpieces and label it so that creatures with smaller mouths willbuy it. Each time students cut and relabel the Nutriglopcorrectly, they create an equivalent fraction.

Students can move the arrows under the pan from one sectionto the next to cut the Nutriglop into equal parts. Whenstudents try to label the Nutriglop as a fraction of the wholepan, they realize the need to cut all sections into the samenumber of pieces. This realization is the foundation forunderstanding the multiplicative nature of equivalent fractions.

Encourage students to cut and label the pan of Nutriglop threedifferent ways and record their solutions by clicking on thelabel maker. The fraction they have entered is compared to thegiven fraction with a statement such as “ �� = �� ” or “ �� ≠ �� .”Advise students to divide the pan first to give them visualinformation that will help them name the fraction of the panthat is filled with Nutriglop.

Ask the following questions to evaluate children’s understanding:

Do you need to divide the Nutriglop as well as theempty sections of the pan to figure out what fraction ofthe whole pan the Nutriglop is? Why or why not?

If the given fraction and the solution are not equivalent, youcan ask:

After making all of your cuts, into how many parts isthe whole pan divided?

How many of the total number of parts in the pan haveNutriglop in them?

Tip from a teacher I bring in a pan of bar cookies cut intothirds, �� frosted in red, �� frosted in green. We cut the cookiesinto smaller and smaller equal pieces. Each time we cut thecookies, we label the fraction of the pan that contains redcookies and the fraction of the pan that contains green cookies.

In the Explore version of this activity, studentsdetermine the given fraction. This allows them to gain

practice making equivalent fractions for the fractions they aremost familiar with and then challenging themselves or apartner at their own pace.

What to look for Are students able to create equivalent fractions by cuttingthe pans of Nutriglop? Can students record severalequivalent fractions? Can students explain why everysection in the pan must be divided equally to createequivalent fractions?





Related activitiesRefer to the hands-on Beyondactivities on pp. 44–49 formore on equivalent fractions.


Beyond Spacemarket




This student’s journal reads,“First I thought it was ��because there were 2 coloredand 2 not. Then I realizedthere were 4 squares and 2 colored and I was right.Then I made 6 pieces so I had �� .”

This student’s journalreads, “I learned that �� , �� , and �� are equivalent.”

Click on the light bulb andthen the grocery bag to getto this activity.

Demonstrating Equivalence Using Sets� Explore fractions as a ratio� Create equivalent fractions using sets of objects

The setting for this activity is a restaurant that has gathered dataon customer preferences. In the case of cheeseburgers versushamburgers, the data might report that �� of the customersprefer cheeseburgers. Students play the role of the cook, placinghamburgers and cheeseburgers on the grill so that the ratio of ��cheeseburgers is maintained. Each problem presents the cookwith a different target fraction describing the ratio of one typeof food to another.

Encourage students to think about the total number of piecesof food they will put on the grill to make the target fraction.By representing the target fraction first, students build a visualfoundation for creating equivalent fractions. After they placethe food on the grill, remind them to click on the spatula to organize the order into equal parts. If the target order is �� cheeseburgers, the spatula will organize the grill into foursections. Find out if students understand why this is so.

After completing an order, students click on the bell to recordit. A number sentence tells whether their representation isequivalent to the target fraction, for example, “ �� = �� ” or “ �� ≠ �� .” Encourage students to create and record threedifferent equivalent fractions for each food order.

Students can duplicate items on the grill by highlighting eachitem and then clicking the red and purple button, the

Duplicate tool. After duplicating the first order, students might see “ �� = �� ” when they click on the bell. After anotherduplication, they would see “ �� = �� .” Students who use theDuplicate tool this way are displaying an understanding of themultiplicative nature of equivalent fractions.

As students work, or when they are finished, ask the followingquestions to evaluate their understanding:

How does the spatula help you?

Into how many parts will the spatula divide the grill?Why?

Tip from a teacher My students like to act out similarproblems by setting up a candy store. They divide into groupsand take turns running the store. Each group of customersgives the store an order— �� lemon suckers, the rest orange, for example. Each person in the store must fill the order usinga different amount of candy.

In the Explore version of this activity, studentsdetermine their food order. Note that students can

enter only fractions that are in their simplest form. You maywant to have them figure out which fractions the activity willaccept and which ones it won’t.

What to look for Can students create an order which is equivalent to thetarget fraction? Can they use visual and symbolic feedbackto adjust incorrect solutions? Can students create equivalentfractions quickly with the Duplicate tool?

Math connectionsFractionsNumber Sense and




Related activitiesRefer to the hands-on Beyondactivities on p. 44–49 formore on equivalent fractions.

Activity customizationSince the whole in thisactivity is one order of food,only proper fractions arepresented.

Beyond Diner




Click on the light bulband then the diner signto get to this activity.

This student’s journalreads, “ �� = �� I learnedthere is a pattern in eachfraction. Keep adding �� s.”

This student’s journal reads,“I saw �� before I decidedthat 12 + 12 = 24 and 1 more to the top it was �� . So I just doubled thebottom and added 1 to the top.”

Trading� Create equivalent fractions by trading� Combine fractional parts to create equivalent fractions


Students are challenged to build totem poles that are equivalentin height to a target height. The target height is given as afraction of a decameter. They have four different-sized sectionsto choose from. Students build their totem pole by stacking thefractional sections next to a scale showing the same number ofunits as the denominator in the target fraction.

As they build, students may click on the tape measure tocompare their totem pole to the target height. They will findthat several of the pieces offered can be used to make anequivalent totem pole, while at least one piece cannot. Forexample, if the target fraction is �� of a decameter, the studentcould choose the �� piece and stack six of them to make �� , butif the student chooses the �� piece, the target fraction cannot bemade. After students have constructed a totem pole, they canclick on the pencil to compare their totem pole with the targetfraction. For example, students might see “ �� = �� ” or “ �� ≠ � .”

Ask the following questions to evaluate children’s understanding:

How do you decide which totem piece to use?

How do you decide how many totem pieces to stack?

If the totem pole is not the target size, ask:

How can the white lines on the scale help you decideon a totem piece to use?

Can you find a totem piece that is exactly half theheight of the sections between the white lines? Howmany of those pieces would you stack to reach thetarget height?

Tip from a teacher My class uses the printouts of theirjournal pages from this activity to add to posters they createthroughout our study of equivalent fractions. Each child isassigned either �� , �� , �� , or �� , and illustrates various equivalentfractions with pairs of objects. The totem poles in theirjournals provide exact, colorful examples for their equivalentfraction posters.

In Explore, students can set their own pace bychoosing both the target fraction and the fractional

sizes of the totem pole pieces. You may want to introduceequivalent fractions to students by going into Explore and building different equivalent fractions for �� as a class.Students can also use Explore to create problems for a partner.See page 34 for suggestions on how to do this with your class.

What to look forDo students use their number sense or the marks on thetotem pole scale to estimate which pieces will work? Do they use trial and error? Can students tell you whichpieces will make an equivalent totem pole before stackingthem up? Can students construct more than one equivalenttotem pole for a given target fraction?



Curriculum connectionsLanguage ArtsArtSocial Studies




Beyond Totem Creations




Click on the light bulband then the totem poleto get to this activity.

This student’s journal reads,“Fractions can be made outof many different sizes andstill make the equal sizes.”

This student’s journal reads,“ �� = �� I learned thatequivalent fractions is two fractions that equal each other.”

Trading � Choose fractional parts that will form equivalent

fractions for a target fraction� Create equivalent fractions by trading� Combine fractional parts to create equivalent fractions


By clicking on the Explore button, students can createproblems for others to solve. Students choose both the targetheight of the totem pole and the fractional pieces given tosolve the problem. By creating a solvable problem, studentsdemonstrate their knowledge of equivalent fractions. It ispossible to choose a target height that cannot be matched withthe pieces provided. For example, a student could choose atarget height of �� of a decameter, and offer the followingtotem pole pieces: �� , �� , �� , and �� . An equivalent totem polecannot be created without, for example, the �� piece or the �� piece.

After students have created a problem, you may want to ask thefollowing questions:

Is your problem solvable?

How many different ways can the target fraction becreated with the pieces you provided?

How many of those pieces will your partner need tostack up to reach the target height?

Tip from a teacher My students love making problems foreach other. I give two students the same amount of time tocreate a problem on each other’s computer and then have themswitch back to solve the problems. When they record theirwork in their journals I ask them to include the name of theperson who created the problem.

What to look forDo students check to see whether the target fraction can bemade using the pieces they have chosen? Can studentsdetermine ahead of time which pieces will stack up tomake equivalent fractions for their target fraction? Can theyadjust the choices of totem pieces to make their problemssolvable? Can students solve their own problems?



Curriculum connectionsLanguage ArtsArtSocial Studies




Beyond Totem Creations: Problem Creation




This student’s journal reads, “We tried different fractiontotem poles and we foundthree ways to make four-eighths of a decameter.”

This student’s journal reads, “Michelle made this problemand I found two ways to makeequivalent fractions.”

Click on the light bulb, the totem pole, and thenthe telescope to get to this activity.


Acknowledging and celebrating what students have learnedwhile working with the multimedia activities is an importantaspect of Reflection Time. You may wish to display examplesof student work on equivalent fractions to serve as a reminderduring the discussion. Samples of work from the hands-onactivities can be included.

Tell students that they will work in groups of four to make upa skit that will help explain equivalent fractions. As a class, goover the hands-on and multimedia activities that helpedstudents understand and create equivalent fractions. Studentsmay want to begin their brainstorming by taking turnsexplaining the concept of equivalent fractions to each otherand choosing the key points to include in their scripts.

Ask the following questions to help students prepare their skits:

Which situations in the INTO movies on the computer, in books you looked at, or in the stories we read,helped you understand equivalent fractions?

Which activities in this Investigation particularly helpedyou understand equivalent fractions?

How would you explain equivalent fractions tosomeone? For example, is there a situation in whichyou traded something using equivalent fractions thatwould help others understand the concept?

Provide poster paper, construction paper, markers, crayons, andstring for students to make props for their skits. You may wantto have available some of the books suggested in AdditionalResources on page 51 to help spark ideas. Suggest thatstudents use printouts from their multimedia journals as propsfor their skits.

When students are ready, invite each group to present their skitto the class or to a visiting class or the principal.

While students are talking about equivalent fractions, look foropportunities to highlight the important understandings thatyou have been building throughout the Investigation.


� We can represent equivalent fractions with word names,diagrams, objects, and symbols.

� We can create equivalent fractions by trading parts, groupingparts, repeating parts, and dividing the parts and the whole.

� When we divide one part of a whole into smaller pieces, we can divide all other parts of the whole into the samenumber of pieces to determine what fraction we have.

� We can find equivalent fractions for a set of two types ofobjects by changing the number of objects in the set whilekeeping the ratio of the two types the same. For example, a set of two pencils is half yellow if one of the pencils is yellow, and a set of four pencils is half yellow if two are yellow.

The Reflection Time discussion may lead to new questions toexplore. You can help develop students’ mathematical powerby encouraging them to explore questions that they generate.



Reflection Time Fraction Skits


Equivalent Fractions

Into

Students hear a story that isrepresented by a totem pole.Students then explore howthe totem pole illustratesequivalent fractions.

Through

Students create totem polesby applying the concept ofequivalent fractions to tradepieces of paper representingdifferent fractions.

Beyond

Students use equivalent fractions to compare fractions of a totem pole, make plans for a park, and play games using a deck of cards.

Totem Poles Theme

Hands-on Activities


Math connectionsFractions

Curriculum connectionsLanguage ArtsScienceSocial Studies

What you needIllustration of Totem Pole

on p. 39“The Challenge” on p. 40,

p. 62Overhead transparencies

made from Totem Partsmaster, p. 64

Overhead projector and pensLinking cubes (optional)

MultimediaRefer to the multimedia Intoactivity on p.18 for more on equivalent fractions and totem poles.

Into Totem Tale

Equivalent Fractions� Identify equivalent fractions� Explore equivalent fractions represented symbolically

and visually

In this activity students will hear a story represented by a totempole and use fractions to describe the pole. Show students thepicture of the totem pole on the following page and read thestory on page 40.

Explain that the totem pole is divided into 12 animal carvingsor “parts,” and that all 12 parts make up one whole totem pole.Ask students what fraction of the pole is made of mammals ( �� , �� ). Project overheads that show the totem pole dividedinto twelfths and into halves. Show students these two poles are the same size.

Count and outline six of the one-twelfth sections that aretouching each other and color them. Compare this with theoverhead that shows the totem pole divided into halves.

You may want to give students 12 linking cubes to representthe totem pole. Have students divide their linking cube totempoles into two equal parts. Point out that �� is the sameamount of the pole as the area called �� . Explain that �� and ��are called equivalent fractions, fractions that represent the sameamount of a region or area.

Engage the class in a discussion about equivalent fractions.

What fraction of the totem pole has animals withbeaks? ( �� , �� )

What fraction of the totem pole has animals with theireyes open? ( �� , �� )

What fraction of the totem pole has animals withclaws? ( �� , � )

You may wish to extend this activity by asking students if theycan find any other equivalent fractions on the totem pole.

Tip from a teacher To introduce the theme, I share thebook Totem Poles. I have my students create a totem pole thatreminds them of a story that we have read together. (For book,see Additional Resources, page 51.)

What to look for Can students use fractions to describe characteristics of the totem pole? Do they understand that parts of the totem pole can be described with different, but equivalent, fractions?

Totem Poles

K i l l e r Whale

Th unde r b i rd

F rog

Sal mon

B ear Woman

B ear M an

G oat

B eave r

Wol f

L i zard

R ave n

E ag le



T h e C h a l l e n g e

Killer Whale wept great saltwater tears as he listened to the crashing thunder. He squeezed his

eyes tight against the mighty lightning flashes that leapt from Thunderbird’s eyes as Thunderbird

ate another killer whale.

He called his friends Salmon and Frog to a council. “How,” Killer Whale asked, “can we of

the water world stop Thunderbird from eating the whales?”

Salmon told Killer Whale to call upon his land cousins, the animals with warm blood and fur.

“Ask your land cousins to challenge Thunderbird to see who is the most powerful. If

Thunderbird loses, he must leave the whales in peace.”

“Frog,” said Killer Whale, “you can travel in the world of water and the world of land. You

must invite the land cousins to a council.” So Frog called Bear Man and Bear Woman, Goat,

Beaver, and Wolf to a council.

“Thunderbird thinks he is a great hunter,” said Wolf. “We can trick him, with Lizard’s help.”

The animals listened to Wolf ’s plan. They nodded because they knew it was good.

The next day, the land cousins started yelling to the sky, “Thunderbird cannot hunt as well as

the animals with fur! Thunderbird hunts only very large animals like the killer whale because he does not have the

skill to track and find smaller land animals.”

When Thunderbird heard these insults he became angry. “I can hunt better than any of you!” he boomed in

his mightiest voice. “Oh?” questioned Wolf, raising one eyebrow. “Can you, now?” The land animals burst into

laughter. Thunderbird became angrier. “I can meet any hunting challenge!” he boasted.

“Well, let us think,” said Bear Woman. The animals pretended to think and discuss, although they already

knew what the challenge would be. After a few minutes, Wolf turned to Thunderbird and said, “One of the animals

with fur will hide in the forest. You must find him.”

“Foolish animals,” Thunderbird thought. “Only this morning it rained. Any animal walking upon the earth

will leave tracks I can follow.” But Thunderbird pretended to worry. “You are so very clever,” he told the animals.


“This is a difficult challenge. May I ask my brothers with beaks, Eagle and Raven, to help me?”

Sure of their trick, the animals agreed to Thunderbird’s request. Thunderbird whispered to his brothers with

beaks. “The earth is still damp from this morning’s rain. Look for footprints. Goat’s prints are hooves. The other

animals have claws. The footprints will lead to the hidden animal.”

But the animals with fur had a very special plan in mind. Beaver was chosen to hide. Beaver covered his tracks

by smoothing the ground behind him with his large flat tail. Meanwhile, the animals asked Lizard to hide. Lizard’s

little clawed feet looked much like the clawed prints of the animals with fur.

Thunderbird, Raven, and Eagle quickly spotted the claw footprints and trailed Lizard to his hiding place.

“Come out, animal with fur!” yelled Thunderbird. “We followed your tracks and found you so very easily.”

At the sound of Thunderbird’s loud boasting, all the animals with fur emerged and laughed at Thunderbird.

“How can you have found one of us? We are all right here!” At that point Lizard came out of hiding.

Thunderbird and his brothers with beaks roared with anger. “Now, none of you are safe!” the brothers with

beaks bellowed. “We used to just hunt Killer Whales. Now we will hunt all animals for we have been tricked unfairly!”

The animals’ eyes went wide with fear. Now they were all in danger of being hunted. Thunderbird, Raven,

and Eagle returned to their mountaintops to sleep, relaxed in a feeling of power over the animals bound to land

and sea.

The End

Trading� Create equivalent fractions by trading� Combine fractional parts to create different

arrangements of the same whole

Tell students they are each going to get a strip of paper for atotem pole. However, a totem pole puzzle with only one coloris not very interesting. Thus, they are going to use equivalentfractions to make trades and create a colorful totem pole.

Have students work in groups of four. Each group selects fivedifferent colored construction paper strips. Each child selectsone of the colors. The remaining strip will serve as a guide forthe finished length of each totem pole.

Model how to fold the paper strip in half, lining up the topedge to the bottom. The folded strip will be 2 inches wide and6 inches long. Unfold the strip and explain that even thoughthe strip is broken into parts, the parts still represent onewhole.

Ask students to tear strips into different fractions. In eachgroup have one child divide a strip into halves, one intofourths, one into eighths and one into sixteenths. Then askstudents to label each of the pieces to identify what fraction ofthe whole totem pole it represents.

Ask students to trade their fractional pieces with one anotherto make totem pole puzzles, so that each puzzle has:

� different colors

� the correct length

� the fraction �� represented by an equivalent fraction, such as ��

Explain that each trade must be a “fair” or equivalent trade.Encourage students to discuss their trades using fractions todescribe the pieces, not colors. Have students record eachtrade. Bring the class together for a discussion.

Can you describe one of your trades using fractions?

What part of your totem pole puzzle shows �� ?

What are equivalent fractions for �� ? ( �� , � � )

How do you know when fractions are equivalent?

Can you describe other “fair” or equivalent trades?

Tip from a teacher I have my students decorate each of their totem pole pieces to look like a real totem pole. Then they glue their totem pole pieces onto a large sheet of construction paper along with their recording sheet. I have them create a key that shows what fraction each color represents.

What to look for Do students label their fractional pieces correctly? Do they make equivalent trades? Do students use fractions todescribe their paper pieces instead of color?



Curriculum connectionsArtLanguage ArtsSocial Studies

What you need12" × 2" construction paper

strips in five colorsPaper

MultimediaRefer to the multimediaThrough activity on p. 22 formore on equivalent fractions.

Puzzling Totem Poles

Totem Poles

Through

Demonstrating Equivalence� Use fractions to prove equivalence� Find equivalent fractions by dividing an area into

equal parts

Tell students they are going to use equivalent fractions to solvea mystery. The mystery involves a totem pole found in thevillage of Tenth Planet.

The local newspaper, The Daily Divide, reports the story. Giveeach child a copy of the Daily Divide News Flash to read. Askstudents to identify the mystery that needs to be solved. Is thetotem pole the lost pole of Chief Eckwivalenz or is it a fake?

Compare the Mayor’s description of the lost totem pole withthe picture of the pole the Sheriff found. List the informationon the board. Then ask students to think about the following:

Can the Mayor and the Sheriff both be right?

Who do you think is right? What makes you think this?

Have the students select a role to play: detective, reporter, ormember of the historical society. Ask them to work with oneor more students who have selected the same role.

Give each group a set of fraction manipulatives to study thefacts in the case and visually prove whether or not the recentlydiscovered totem pole belonged to Chief Eckwivalenz. Forexample, they can divide the 12 cubes into three sets of four to see that �� is equivalent to � .

Once students decide whether or not the totem pole belongedto Chief Eckwivalenz, ask them to prepare a report usingwords and pictures to prove their position. Detectives can calltheir report a police report, newspaper reporters can write astory for the newspaper, and historical society members canwrite a letter to the editor.

Tip from a teacher My class role-plays a city councilmeeting to discuss the authenticity of the totem pole. Themayor, sheriff, detectives, and historical society memberspresent their cases. The newspaper reporters ask questions.Members of the council hear the presentations and vote onwhether or not the totem pole is authentic. I make a videotapeof the discussion and we share it with other classes.

What to look for Do students demonstrate equivalent fractions withmanipulatives? Do their reports prove their position on the case by using words and pictures to show that twofractions are equivalent? Do students label their fractionalparts correctly?



Curriculum connectionsLanguage ArtsSocial Studies

What you needCopies of Daily Divide News

Flash, p. 45, 65Linking cubes or other

fraction manipulativesCrayons or colored markersPaper

MultimediaRefer to the multimediaBeyond activities on pp. 24–35 for more on equivalent fractions.

Beyond Totem Pole Controversy

Totem Poles


Tenth Planet—Last night a special ceremo-ny was held in front of city hall. The Mayorsputtered in anger when she unveiled whatthe local Sheriff claimed to be the long-losttotem pole of Chief Eckwivalenz. TheChief ’s prized totem pole disappeared myste-riously ten years ago, just before he died.

According to the Mayor, the totem poledoes not match the description the chief gaveher on his deathbed. She explained that thechief said one-third ( � ) of his totem pole wascarved with bears.

At that point the Sheriff stood up andshouted, “But one-third of this pole is carvedwith bears!”

The Mayor became even more upset atthis point in the meeting. “This is absurd!”she said. “First of all the pole is carved intotwelve equal parts, not three. Looking at thispole it is obvious that four-twelfths ( �� ) ofthe pole is carved with animals that havetheir tongues sticking out.

“We all know that our people alwayscarved bears with their tongues sticking out.So there are four bears on this pole. Thatalone proves that this totem pole is a fake.

“The Chief also told me that one-quarter( �� ) of the pole is carved with frogs. Anyonecan see that three-twelfths ( �� ) of this pole isdecorated with carvings of frogs.

“The Chief also said that three-quarters( �� ) of the totem pole is carved with animalsthat walk on land. How absurd! There arenine-twelfths ( � � )!

“Last, but not least, I was told by the Chiefthat two-thirds ( � ) of the totem pole is carvedwith animals that have their eyes open. Lookfor yourselves to see that eight-twelfths ( �� )have their eyes open.”

The Mayor and the Sheriff agreed to forma committee to study the totem pole and theChief ’s description in order to determinewhether or not the totem pole is authentic.

Killer Whale

Bear Man

Frog

Salmon

Bear Woman

Bear Man

Frog

Frog

Eagle

Salmon

Raven

Bear Woman

Dividing an Area� Create equivalent fractions by dividing an area into

equal parts� Use different equivalent fractions to name a fraction of

a whole

Tell students that they are going to make a plan for a TotemPole Park. They need to decide what fraction of the park willbe set aside for a viewing area for each of the totem poles. Theclass will create one plan together and then each student willcreate more plans.

Model a plan for the park with the class. Each plan must havean area to display the totem poles and an area for picnic andplay. Have a student fold a piece of paper into thirds. Have thestudent shade two-thirds of the whole. Point out that one-thirdof the park planning sheet is not shaded. This is the area whereone totem pole will be displayed.

When the park is divided into thirds, explain that � of the parkcan fit one totem pole and � of the park is for picnic and playareas. Point out that each totem pole needs an equal amount ofspace. Ask students how they can divide the park again inorder to fit more totem poles. You may want to fold the parkinto sixths before students proceed independently. Showstudents how to record the information.

Number of Fraction of park Fraction of parktotem poles for totem poles for play and picnic

1 � �

2 ��

Distribute pieces of paper. Ask students to create park plans bydividing the park first into thirds, then shading two of them.Next have them divide the park into more equal parts. Askstudents to record information about their park as they divideit. Tell students that they can fold their park into as many equalparts as they wish. You may ask:

How many poles can you fit into the totem pole area inyour park?

How many equal parts do you have in your park?

What fraction now describes the totem pole viewingarea in your park?

What do you notice about these fractions: � , �� , � � , �� ?

Emphasize that students must fold the entire park into equalparts, or imagine the park folded, to identify the fraction forthe totem pole viewing area. Some students may fold theirpark into as many parts as possible. Check that they are stillmaking equal parts.

As an extension, students may wish to create park plans basedon halves or fifths. Give each child additional sheets of paper to use as park plans. Ask students to record their results.

What to look for Do students divide the entire park plan into equal parts each time? Do they label the fractional parts of the totempole display area as a fraction of the whole park? Canstudents describe the totem pole viewing area usingequivalent fractions?




What you need8 �� " × 11" paperCrayons or colored markers


Beyond Totem Pole Park

Totem Poles

Matching Equivalent Fractions� Match fractions that are equivalent� Compare two fractions to determine which is more

Tell students they are going to play equivalent fraction cardgames. Then they will teach one or more of the games tosomeone at home.

Distribute the Family Letter with the game rules and the TotemPole Card Set handouts. There are four different games that can be played with this set of 36 cards, which contains 18 pairsof equivalent fractions. These games are Equivalent FractionConcentration, Fraction Fish, Greedy Raven, and Totem Solitaire.

Have the students cut out the playing cards to make their ownTotem Pole Game Pack. As students are doing this, have themdecide which cards show equivalent fractions.

In each game, players need to match a card with another that illustrates an equivalent fraction. Remind students thatequivalent fractions describe the same amount of a region or area.

To introduce the first three games, divide the class into groups.The maximum number of players for each game is five. Assign

46

23

each group the rules for one game. The group members readthe rules and play the game until each child understands how itis played. As students play, you may ask:

Do the shaded parts represent the same amount?

What fractions did you find that are equivalent?

Then rearrange the class into groups of three with one expertfor each game per group. Next the experts teach their game toothers in their new group.

Allow time for each child to play and understand the first threegames. This can be done over several days, so that students gaina good sense of which cards are equivalent pairs.

When students understand how the games are played, havethem take the card set and the Family Letter with the gamerules home so that they may teach the games to a familymember. You may also want to review the rules for Totem Solitaire, which children can play by themselves.

Tip from a teacher I have my students glue each of theircards to a piece of construction paper slightly larger than thecards they cut out. This makes the cards more durable andprevents students from seeing through the paper when playingthe games. I prepare stacks of colored rectangles in advance.

What to look for Can students compare two fractions to determine which is more? Are students able to match fractions that areequivalent? Are students able to explain game rules toanother child?



What you needCopies of Totem Pole Card Sets

1 and 2, pp. 66-67Copies of Family Letter with

game rules pp. 69, 70Envelopes for card sets


Beyond Link to Home: Totem Pole Game Pack

Totem Poles


Students make pairs of equivalent fractions in agame of Fraction Fish.

Totem Poles


Reflection Time provides an opportunity for students to shareand celebrate what they learned during this Investigation.Display student work, such as the colorful totem poles madefrom construction paper strips and the Totem Pole Controversyarticles and letters. If you have examples of work done usingthe multimedia activities, display them and include them inyour discussions.

Invite students to reflect on activities they did to findequivalent fractions through trading and dividing parts of thewhole. Discuss the definition for equivalent fractions. Remindthem that equivalent fractions represent the same amount of aregion or area. Help them to recall both multimedia andhands-on activities.

Ask students to invite the principal or another school staffmember to view their work. Use index cards, folded in half, for display captions. Have students write about theirlearning experiences inside the cards by reflecting on the following questions:

How can you explain equivalent fractions to our visitors?

What are some ways you can find equivalent fractions?

How can you prove two fractions are equivalent?

As the students discuss the questions and share what theylearned, look for opportunities to bring out the followingpoints, or highlight these ideas as students touch upon them.




� When we divide one part of a whole into smaller pieces, wecan divide all other parts of the whole into the samenumber of pieces to determine what fraction we have.

The Reflection Time discussion may lead to new questions toexplore. You can help students develop their mathematicalpower by encouraging them to explore questions that theygenerate.



Reflection Time Tell All About It


Additional ResourcesTotem Poles Theme

Fiction

Frog Girl, Paul Owen Lewis. Hillsborough, OR: BeyondWords, 1997.

Seal Oil Lamp, Dale de Armond. San Francisco: Sierra ClubBooks, 1988.

Very Last First Time, Jan Andrews. New York: McElderry,1986.

Non-Fiction

Art of the Totem Pole: Totem Poles of the Northwest CoastalIndians, Marius Barbeau. Blaine, WA: Hancock House, 1984.

Paper Animal Masks from Northwest Tribal Tales, Nancy LynRudolph. New York: Sterling, 1996.

Tlingit Totem Poles, Stephen Brown, editor. Santa Barbara, CA:Bellerophon Books, 1992.

Totem Poles, Diane Hoyt-Goldsmith. New York: HolidayHouse, 1990.

Totem Pole Indians of the Northwest, Don Beyer. New York:Franklin Watts, 1989.

Totem Poles: an Ancient Art, Carol Batdort. Blaine, WA:Hancock House, 1990.

Totems, Decoys and Covered Wagons: Cardboard Constructions fromEarly American Life, Jeremy Comins. New York: Lothrop, Lee& Shepard, 1976.

Where the People Gather: Carving a Totem Pole, Vickie Jensen.Seattle, WA: University of Washington Press, 1993.

Reference

Looking at Totem Poles, Hilary Stewart. Seattle, WA: Universityof Washington Press, 1993.

Totem Poles: an Illustrated Guide, Marjorie Halpin. Seattle, WA:University of Washington Press, 1981.

Fractions

Fraction Fun, David A. Alder. New York: Holiday House, 1996.

Fraction Skills, Helene Chiriman. Los Angeles, CA: Price, Stern,Sloan, 1992.

Fractions with Tangrams, Larry Ecklund. Cypress, CA: CreativeTeaching Press, 1994.

Hooray for Fraction Facts!, Becky Daniel. Carthage, IL: GoodApple, 1990.

Picture Pie, Ed Emberley. Boston: Little, Brown and Company,1984.

Tenth Planet Resources

Check Tenth Planet’s Web page for more resources to extendthe totem poles theme and fraction concepts.

http://www.tenthplanet.com/Teachers/

*Book referenced in the activities

Totem Poles

*


Students demonstrate their understanding of equivalent fractions by comparing two fractions tofigure out whether Thunderbird and Frog have tricked Raven in a trading situation. Thisactivity gives you an opportunity to assess what the students have learned over the course of theInvestigation. A rubric is provided to guide your evaluation of student work.

Performance Assessment

Assessment Activity


PerformanceAssessment



What you needCopies of The Trick

on p. 56, p. 68Fraction strips or other

manipulativesPaper

MultimediaThe journal can also be used for recording.

The Trick


� When we divide one part of a whole into smaller pieces, we can divide all other parts of the whole into the samenumber of pieces to determine what fraction we have.

� We can find equivalent fractions for a set of two types ofobjects by changing the number of objects in the set whilekeeping the ratio of the two types the same.

As an extension, pose additional challenges by asking studentswhich fractions of things they would rather have. For example,would they rather have �� of a bag of jelly beans or �� of a bagof jelly beans? Would they rather miss �� or � of a recess?

Tip from a teacher I like to invite another class to comefor a potlatch celebration of what we have learned. Each of my students is partnered with a child from the other class. Mystudents explain the equivalent fractions in the displays. For thepotlatch feast each child creates totem treats (fruit kabobs) andexplains to the visiting partner how to describe the treats usingequivalent fractions.

What to look for Are students able to determine when two fractions areequivalent? Do they accurately represent fractions withdiagrams or manipulatives? Do their explanations prove their position?

Tell students that they will use what they learned aboutequivalent fractions to help them figure out a trick thatThunderbird and Frog play on Raven.

Distribute copies of The Trick and ask students to read thestory. Then discuss the fact that the story ends with Raventhinking he got the better deal. Ask students:

Who ended up with the most food? Raven or Frog?

Or, did they both get the same amount?

Have the students review the information in the story usingfraction strips or other manipulatives. Then have them record their answer to the questions using diagrams, words, and symbols.

Invite students to share their explanations. Some students maychoose to demonstrate the fractions with cubes. They will find that � is physically less than � and conclude they are notequivalent. Point out that the food that Raven and Frog shared is the same size whole, so cubes are not a good model in thiscase since a set of 6 cubes is less than a set of 8 cubes.

During the class discussion, keep in mind the importantunderstandings that students develop as they do thisInvestigation. Look for opportunities to bring out thefollowing points, or highlight these ideas as students raise them.




Rubric

Level 3: Student identifies, represents, and compares fractions accurately and clearly

communicates using pictures, words, and symbols how two fractions are equivalent.

Level 2: Student identifies, represents, and compares fractions, but has some

difficulty explaining how two fractions are equivalent.

Level 1: Student identifies, represents, and compares fractions with difficulty

and cannot explain how the fractions are equivalent.


It was a time of celebration for the Killer Whale

Clan. Thunderbird was asked to come and dance at the

potlatch.* Thunderbird loved to dance. The Killer Whale

Clan hoped the dancing and food would please him and

he would stop snatching the Killer Whales from the sea

and taking them to the mountaintop to eat. When

Thunderbird arrived, Bear Woman passed out an equal

portion of food to each guest. As everyone began to eat,

two of the guests started to quarrel. Frog boasted

to Raven that he had more copper** than any other

creature. Raven knew Frog was right, but didn’t like to

listen to his boasting. He thought he would get even

by convincing Frog to give him some of his food. Each

guest had been given exactly the same amount of food.

“Frog, because you are so wealthy, you should share

some of your food with those less fortunate than you,” said

Raven, trying to look weak and forlorn. “There is

enough food here for everyone,” argued Frog. “I will

eat all of my share as I am very hungry from counting

my copper.” Raven was not quieted so easily. He

opened his mouth to demand that Frog give up some of

his food. “Stop your bickering at once!” Thunderbird

demanded. “I will solve your silly argument. Frog, I

want you to trade four-eighths ( �� ) of your dinner for

three-sixths ( �� ) of Raven’s dinner.” Greedy Raven

thought quickly that 4 and 8 are greater than 3 and 6.

“That trade seems good to me,” agreed Raven. Frog

looked at Thunderbird, shook his head and smiled. As

Raven and Frog began to divide their food, Thunderbird

began his dance and set the skies thundering with laughter

at Frog and Raven’s foolishness. Thunderbird also

laughed at the Killer Whale Clan. In spite of all the pot-

latch feasts, he would never stop feasting upon the mighty

Killer Whale.

* potlatch:

The name for

a great dinner

and celebration

among the native

Americans of

the Northwest.

** copper:

A valued

treasure among

the people of

the totem

pole world.

T h e T r i c k

© 1998 Tenth Planet Explorations, Inc. All rights reserved. This page may be reproduced for classroom use only.


In this section you will find a summary of the three Number Investigations for fractions,along with two charts providing an overview of the multimedia and hands-on activities,including the connections to NCTM Standards.

Math Content

MathContent

Fraction concepts involve complex relationships. For thisreason, students need time to explore the interrelated conceptsand relationships that make up the topic of fractions. TenthPlanet’s math series emphasizes fractions as a relationshipbetween parts and wholes. Further study of fractions to express division or ratios is best in later grades. The fractionInvestigations in Tenth Planet Explores Math are carefullysequenced to help students develop a strong conceptualunderstanding of fractions.

In the Representing Fractions Investigation, students learn thatfractions can be represented by words, symbols, objects, andpictures. They develop an understanding of the meaning offractions through these different representations. Students learnthat a whole must be divided into equal parts and develop skillin using fraction word names and symbols. They learn, forexample, that a fraction such as two-thirds consists of a wholedivided into three equal parts, with two of the one-third partsemphasized.

Throughout Representing Fractions and the other fractionInvestigations, students explore fractions represented by threemodels: the area model (for example, a circle or square), the setmodel (a group of objects), and the length or distance model.

The work in Representing Fractions is continued in the EquivalentFractions Investigation. Students learn that fractions areequivalent if they represent the same relative amount. Studentshave many opportunities to generate or simplify equivalentfractions by dividing a region into smaller or larger equal partsor by trading smaller parts for larger parts or larger parts forsmaller parts.

In the Fraction Operations Investigation, students build on theirexperience in Representing Fractions and Equivalent Fractions.Students use what they know about the meaning of fractionsand equivalence to perform operations through the use ofmodels. Using area models, students can visually see how five-sixths plus one-half is one and one-third. Students experiencereal-world problems where they take a fraction of a fractionand divide a fraction into smaller equal pieces. Theseexperiences build a deep understanding of fractions.

Problem solving is central to the field of mathematics andshould be integrated with all mathematical activity. In theseInvestigations, students solve meaningful problems and developproblem-solving skills as they learn about fractions. They alsohave opportunities to formulate problems for other students tosolve. Communication about problems and solutions is a keyingredient for developing power and ease with mathematics.


When children possess a sound

understanding of fraction and

decimal concepts, they can use

this knowledge to describe

real-world phenomena and

apply it to problems involving

measurement, probability, and

statistics. An understanding

of fractions and decimals

broadens students’ awareness

of the usefulness and power

of numbers and extends

their knowledge of the

number system.

—National Council of

Teachers of Mathematics

About Fractions


Activity

BeyondTotem Creations:Problem Creation• Create a problem foranother student to solve

• Trading

• Explore fractions as a ratio• Create equivalent fractions using sets of objects

IntoEquivalent Fractions• Explore brief vignettesof a lasagna party, a sly fox,two sisters, and a troop ofmarching ants

Math Focus

• Trading

• Create equivalent fractions by trading• Combine fractional parts to create equivalentfractions for a target fraction

ThroughEquivalent Pieces• Create equivalentfractions

• EquivalentFractions• Fractions of Sets

What Children Do

• Choose fractional parts that will form equivalentfractions for a target fraction• Create equivalent fractions by trading• Combine fractional parts to create equivalentfractions for a target fraction

BeyondFraction Toss• Find an equivalentfraction to toss items at a ballpark

• Dividing an Area

• Explore equivalent fractions representedsymbolically and visually• Explore a set of objects divided into variousequivalent fractions

Math Connections toNCTM Standards

BeyondSpacemarket• Cut and label food in a spacemarket usingequivalent fractions

BeyondDiner• Create orders of foodequivalent to a targetfraction

• DemonstratingEquivalence UsingDistance

• Dividing an Area

• Create equivalent fractions by dividing andshading a part of a circle or square• Decide whether two fractions are equivalent bycomparing areas

• Given a linear model of a fraction, identify thefraction using symbols• Compare distances in a fraction chart todetermine equivalent fractions

• Fractions• Geometry and Spatial Sense

• Fractions• Geometry and Spatial Sense

• DemonstratingEquivalence UsingSets

• Create equivalent fractions by dividing eachfraction of a whole into smaller, equal parts

• Fractions• Number Sense andNumeration• Problem Solving

BeyondTotem Creations• Create totem poles thatare equivalent in height to a target fraction

• Fractions• Geometry and Spatial Sense• Number Sense andNumeration• Problem Solving

• Fractions• Number Sense andNumeration• Problem Solving

• Fractions• Geometry and Spatial Sense• Number Sense andNumeration• Problem Solving

• Fractions• Geometry and Spatial Sense • Number Sense andNumeration• Problem Solving

CurriculumConnections

Social StudiesLanguage Arts

*Language Arts

*Language Arts

*Language Arts

*Language Arts

*Language ArtsArtSocial Studies

*Language ArtsArtSocial Studies

*The multimedia journal provides a language arts connection.

Multimedia Activities Chart: Equivalent Fractions


MathContent

Hands-on Activities Chart:Equivalent Fractions

Activity Math Focus What Children Do Math Connectionsto NCTM Standards

IntoTotem Tale• Use fractions to describea totem pole

• EquivalentFractions

• Identify equivalent fractions• Explore equivalent fractionsrepresented symbolically andvisually

• Fractions

ThroughPuzzling Totem Poles• Use equivalent fractionsto make equal trades andcreate a totem pole

• Trading • Create equivalent fractions bytrading• Combine fractional parts tocreate different arrangements ofthe same whole

• Fractions• Geometry andSpatial Sense

• DemonstratingEquivalence

• Use fractions to proveequivalence• Find equivalent fractions bydividing an area into equal parts


BeyondTotem Pole Park• Make a plan for a parkbased on fractions

• Dividing an Area • Create equivalent fractions bydividing an area into equal parts• Use different equivalent fractionsto name a fraction of a whole


• MatchingEquivalent Fractions

• Match fractions that areequivalent• Compare two fractions todetermine which is more


BeyondLink to Home: Totem Pole Game Pack• Play equivalent fractioncard games

Curriculum Connections

Language ArtsScienceSocial Studies

ArtLanguage ArtsSocial Studies

Language ArtsSocial Studies

Language ArtsSocial Studies

BeyondTotem Pole Controversy• Use equivalent fractionsto solve a mystery

Supplies

This section includes all blackline masters particular to the hands-on activities. Spanishtranslations of “The Challenge,” Daily Divide News Flash, and “The Trick” are included, aswell as a Family Letter in English and Spanish.


62 Equivalent Fractions© 1998 Tenth Planet Explorations, Inc. All rights reserved. This page may be reproduced for classroom use only.

Orca lloró grandes lágrimas de agua salada mientras oía retumbar los truenos. Había apretado

bien los ojos para protegerse de los rayos poderosos que saltaban de los ojos de Pájaro de Trueno,

mientras Pájaro de Trueno comía a otra orca.

Orca llamó a sus amigos Salmón y Rana para considerar este problema. “¿Qué podemos

hacer nosotros —preguntó Orca—, los habitantes del mundo de las aguas, para impedir que

Pájaro de Trueno siga comiendo ballenas?”

Salmón le dijo a Orca que llame a sus primos de la tierra, los animales de sangre caliente y

pelos. “Pídele a tus primos de la tierra que desafíen a Pájaro de Trueno para ver quién es más

poderoso. Si Pájaro de Trueno pierde, deberá dejar a las ballenas en paz.”

“Rana —dijo Orca—, tú puedes viajar por el mundo de las aguas y el mundo de la tierra. Tú

debes invitar a nuestros primos de la tierra para que participen en un consejo.” Entonces, Rana

llamó a Oso y Osa, Cabra, Castor y Lobo para que participaran en un consejo.

“Pájaro de Trueno cree que es un gran cazador —dijo Lobo—. Nosotros podemos engañarlo

con la ayuda de Lagarto.” Los animales prestaron atención al plan de Lobo asintieron, porque

sabían que era un buen plan.

Al día siguiente, los primos de la tierra comenzaron a gritar hacia el cielo: “¡Pájaro de Trueno no caza

tan bien como los animales con pelo! ¡Pájaro de Trueno caza solamente animales muy grandes, como la orca, porque

no sabe cómo seguir y encontrar animales más pequeños!”

Cuando Pájaro de Trueno escuchó estos insultos, se enojó. “¡Yo puedo cazar mejor que cualquiera de

ustedes!” —dijo con la más poderosa de sus voces—. “¡Oh!” —dijo Lobo, arqueando una de sus cejas—. “¿De verdad

que puedes?” Los animales de la tierra echaron a reír. Pájaro de Trueno se sintió más enojado. “¡Yo puedo desafiar a

cualquiera para ver quién caza mejor!”, alardeó.

“Bueno, veamos” —dijo Osa—. Los animales simularon pensar y discutir, aunque ya sabían cuál sería

el desafío. Después de unos minutos, Lobo se dirigió a Pájaro de Trueno y dijo, “Uno de los animales con pelo se

E L D E S A F I O

The Challenge (Spanish)

Equivalent Fractions 63© 1998 Tenth Planet Explorations, Inc. All rights reserved. This page may be reproduced for classroom use only.

ocultará en el bosque, y tú tienes que encontrarlo.”

“¡Qué animales tan tontos! —pensó Pájaro de Trueno—. Acaba de llover esta mañana, y cualquier animal que

camine sobre la tierra dejará huellas fáciles de seguir.” Pero Pájaro de Trueno simuló estar preocupado. “Ustedes son

tan listos —dijo a los animales—. Este es un desafío difícil. ¿Puedo pedirle a mis hermanos con picos, Águila y

Cuervo, que me ayuden?”

Confiados en el truco, los animales aceptaron el pedido de Pájaro de Trueno. Pájaro de Trueno susurró a sus

hermanos con picos: “La tierra todavía está húmeda por la lluvia de esta mañana. Busquen las huellas. Las huellas de

Cabra son pezuñas. Los otros animales tiene garras. Las huellas nos conducirán hacia el animal escondido.”

Pero los animales con pelo habían ideado un plan muy especial. Castor debía esconderse. Castor borró sus

huellas alisando el terreno detrás suyo con su larga cola chata. Entre tanto, los animales le pidieron a Lagarto que se

esconda. Las patas pequeñas con garras de Lagarto parecían mucho a las de los animales con pelo.

Pájaro de Trueno, Cuervo y Águila vieron rápidamente las huellas de garras y siguieron a Lagarto hasta su

escondite. “¡Sal de ahí, animal con pelo!” —gritó Pájaro de Trueno—. “Hemos seguido tus huellas y fue muy fácil

encontrarte.”

Mientras Pájaro de Trueno alardeaba a viva voz, todos los animales con pelo aparecieron y se rieron de Pájaro

de Trueno. “Cómo puedes haber encontrado a uno de nosotros si todos estamos acá!” En ese instante, Lagarto salió

del escondite.

Pájaro de Trueno y sus hermanos con picos gritaron de rabia. “¡Ahora, ninguno de ustedes estará seguro! —

bramaron los hermanos con picos—. Nosotros sólo cazábamos orcas, pero ahora cazaremos animales, ¡porque fuimos

engañados injustamente!.”

Los ojos de los animales se agrandaron de miedo. Ahora todos corrían peligro de ser cazados. Pájaro de

Trueno, Cuervo y Águila regresaron a las cimas de sus montañas para dormir, tranquilos y sintiendo que dominaban a

los animales de la tierra y el mar.

Fin

The Challenge continued (Spanish)


Totem Parts


Daily Divide News Flash (Spanish)

Alcaldesa IndignadaDenuncia Que el Tótem Es Falso

Flash de Noticias de Diario DivisionalFlash de Noticias de Diario DivisionalFlash de Noticias de Diario Divisional Orca

Oso

Rana

Salmõn

Osa

Oso

Rana

Rana

Ãgu i la

Salmõn

Cue rvo

Osa

Tenth Planet— Anoche se realizó unapequeña ceremonia frente al edificio del ayun-tamiento. La Alcaldesa se mostró indignada aldescubrir el tótem anunciado por el Alguacillocal como el tótem del Cacique Eckwivalenz,perdido hace mucho tiempo. El valioso tótemdel Cacique, desapareció misteriosamente diezaños atrás, poco antes de su muerte.

Según la Alcaldesa, el tótem no coincide conla descripción que el Cacique le había dado ensu lecho de muerte. Ella explicó que elCacique había indicado que un tercio ( � � ) del tótem tenía tallas de osos.

En ese instante, el alguacil se puso de pie ygritó: “¡Pero si un tercio del tótem tiene tallasde osos!”

La alcaldesa se mostró aún más contrariada.“¡Esto es absurdo! —dijo—. En primer lugar,el tótem está tallado en doce partes iguales, noen tres. Observando a este tótem es obvio quecuatro dozavos ( �� ) del tótem está tallado con animales que muestran la lengua.”

“Todos sabemos que nuestra gente siempretalló osos que muestran la lengua. Este tótemtiene cuatro osos. Eso sólo demuestra que eltótem es falso.”

“El Cacique también me contó que uncuarto del tótem ( � � ) tiene tallas de ranas. Cualquiera puede ver que tres dozavos ( �� ) del tótem está decorado con tallas de ranas.”

“El Cacique también dijo que tres cuartos( � ) del tótem está tallado con animales que cam-inan por tierra. ¡Qué absurdo! ¡Y en cambio haynueve dozavos ( �� )!”

“Por último, pero igualmente importante,tenemos que considerar que el mismo Caciqueme dijo que dos tercios ( �� ) del tótem tiene tallasde animales con los ojos abiertos. Miren ustedesy verán que ocho dozavos ( �� ) tienen los ojosabiertos.”

La Alcaldesa y el Alguacil convinieron en for-mar un comité para estudiar el tótem y ladescripción del Cacique para poder así determi-nar si el tótem es auténtico o no.


18

14

46

25

45

78

35

12

5 7

17

610

69

56

16

58

34

13

Totem Pole Card Set 1


Totem Pole Card Set 2

216

416

23

410

810

1416

610

612

10 14

214

35

23

1012

2424

212

1016

912

26


The Trick (Spanish)

Era época de festejos para el Clan de las Orcas. Pájaro

de Trueno había sido invitado para bailar durante el pot-

latch*. A Pájaro de Trueno le gustaba bailar. El Clan de las

Orcas esperaba que el baile y la comida fueran de su agrado

y que, entonces, dejara de atrapar Orcas en el mar para lle-

varlas a la cima de la montaña y comérselas. Cuando

Pájaro de Trueno llegó, Osa sirvió una porción de comida

del mismo tamaño a cada invitado. Ni bien todos empezaron

a comer, dos de los invitados comenzaron a discutir.

Rana alardeaba delante de Cuervo que ella tenía más piezas

de cobre** que ningún otro animal. Cuervo sabía que Rana

decía la verdad, pero le disgustaba oírla alardear, y pensó que

una manera de desquitarse sería convenciendo a Rana de

que le diera algo de su comida. Cada invitado había recibido

exactamente la misma cantidad de comida. “Rana,

como tú eres tan rica, deberías compartir algo de tu comida

con quienes son menos afortunados que tú” —dijo Cuervo,

simulando estar débil y triste—. “Hay suficiente

comida para todos —respondió Rana—. Y yo comeré

todo lo que me han servido, ya que tengo mucha

hambre por haber contado mis piezas de cobre.”

Pero Cuervo no era de contentarse fácilmente, y

abrió su boca exigiendo que Rana le diera algo de su

comida. “¡Dejen de reñir de una buena vez! —

ordenó Pájaro de Trueno—. Yo me encargaré de vuestra

ridícula discusión. Rana, quiero que cambies cuatro

octavos ( � ) de tu cena por tres sextos ( � ) de la cena

de Cuervo.” Cuervo, siendo muy avaro, pensó

rápidamente que 4 y 8 eran mayores que 3 y 6. "Eso me

parece bien" —dijo Cuervo—. Rana miró a Pájaro

de Trueno, hizo un gesto de fastidio y sonrió. Mientras

Cuervo y Rana dividían su comida, Pájaro de Trueno

comenzó a bailar, y el cielo estalló en truenos de risa

por las tonterías de Rana y de Cuervo. Pájaro de

Trueno también se rió del Clan de las Orcas. A pesar de

todas las fiestas de potlatch, nunca dejaría de

comer las poderosas Orcas.

* potlatch:

Nombre dado para

una gran cena y

festejos entre los

indios americanos

del noroeste.

** cobre:

Un tesoro muy

apreciado entre la

gente del mundo

de los tótems.

E L T R U C O


Dear Family,

During math time we are learning aboutequivalent fractions. Equivalent fractions

are fractions that describe the same amount of a whole. Forexample, �� is equivalent to �� .

In our lessons we used fractions to describe the differentways a totem pole is divided. Students have traded equivalentfractional parts to create colorful totem pole puzzles. Theirknowledge of equivalent fractions even helped them solve amystery.

Your child has prepared the Totem Pole Game Pack. This is aset of equivalent fraction playing cards that can be used toplay several entertaining games. Please play one or more ofthe games with your child. We have practiced the first threein class so your child can teach the games to you.

While playing the games ask your child the followingquestions:

� How do you say this fraction?

� How do you know these two fractions are equivalent?

Thank you for your help and interest.

Link to Home: Totem Pole Game Pack

E q u i va l e n t F r a c t i o n C o n c e n t r a t i o nLay all 36 cards face down in six rows of six cards. Players take turns flipping two cards at atime. If the two cards are equivalent fractions, the player keeps the match and then playsagain. If the cards are not equivalent fractions, the cards are turned back over and the nextplayer takes a turn. The game is played until all of the cards have been matched. The playerwith the most matches wins.

F r a c t i o n F i s hDeal 7 cards to each player. Place the remaining cards face down in a “fish pond.” Playersmatch equivalent fraction pairs in their hands and display all their matches face up. Playerstake turns asking each other for cards with equivalent fractions that match the cards in theirhands. If the person asked does not have the requested card, he/she tells the player “Gofish.” After the player takes a card from the fish pond, the turn moves to the next player. If a player runs out of cards, three more cards may be drawn from the fish pond. The game ends when all of the cards are matched. The player with the most matches wins.

G r e e d y R av e nDeal out all the cards, face down. Set a time limit for the game of at least five minutes. All players turn over the top card in their pile at the same time. The player with the largestfraction showing takes and keeps all of the cards played in the round. If two of the cardswith the largest fraction are equivalent, the players with the equivalent cards flip overanother card. The player with the largest fraction keeps all of the cards played in thisround. The winner is the player with the most cards at the end of the playing time.

Another game? YES!

T o t e m S o l i ta i r eA game to play by yourself!

Put the cards in order from the smallest to the largest. Which cards are equivalent?

T O T E M P O L E G A M E P A C K

© 1998 Tenth Planet Explorations, Inc. All rights reserved. This page may be reproduced for classroom use only.


Estimada Familia:

Durante las clases de matemáticas,estamos estudiando fracciones

equivalentes. Las fracciones equivalentes son las quedescriben la misma cantidad de un todo. Por ejemplo, ��es equivalente a �� .

Durante las lecciones utilizamos fracciones para describirlas diferentes maneras en que un tótem está dividido.Los estudiantes han intercambiado piezas fraccionales equivalentes para crear rompecabezas coloridos detótems. Su conocimiento de fracciones equivalentes lesfue útil hasta para resolver un misterio.

Su niño ha preparado el Paquete del Juego del Tótem.Es un conjunto de naipes con fracciones equivalentesque puede utilizarse para jugar varios juegosentretenidos. Por favor, juegue uno o más de estos jue-gos con su niño. Hemos practicado los tres primerosjuegos en clase, de modo que su niño podrá enseñarlesel juego.

Mientras jueguen, hagan las siguientes preguntas a suniño:

� ¿Cómo se dice esta fracción?

� ¿Cómo sabes que estas dos fracciones son equiva-lentes?

Muchas gracias por su interés y ayuda. Link to Home: Totem Pole Game Pack (Spanish)

P a q u e t e D e l j u e g o d e l t ó t e mC o n c e n t r a c i ó n d e F r a c c i o n e s E q u i va l e n t e s

Distribuya los 36 naipes cara abajo en seis líneas de seis naipes cada una. Los jugadores se turnanen dar vuelta dos naipes a la vez. Si los dos naipes son fracciones equivalentes, el jugador se losguarda y vuelve a jugar. Si los naipes no son fracciones equivalentes, el jugador los vuelve a colo-car cara abajo y pasa el turno al siguiente jugador. El juego termina una vez que todos los naipeshan sido ordenados en pares. El jugador que tenga la mayor cantidad de pares de fracciones gana.

P e s c a d e f r a c c i o n e sDé siete naipes a cada jugador. Coloque las naipes restantes hacia abajo en el "vivero". Losjugadores deben formar pares de fracciones equivalentes con los naipes en la mano y, cuandoestén formados los pares equivalentes, mostrarlos cara arriba. Los jugadores se turnan para pedira los otros jugadores los naipes con fracciones equivalentes que servirán de pareja para los naipesque tienen en la mano. Si el jugador al que se le solicita un naipe no tiene el naipe solicitado,deberá responder "Anda a pescar". Una vez que el jugador tome un naipe del vivero, le corre-sponde el turno al siguiente jugador. Si un jugador queda sin naipes, debe retirar tres naipes delvivero. El juego termina cuando todos los naipes están ordenados en pares. El jugador conmayor cantidad de pares, gana.

e l C u e r v o G u l o s oDistribuya todos los naipes cara abajo. Asigne un límite de tiempo para la duración del juego depor lo menos cinco minutos. Cada jugador da vuelta el naipe que está arriba de su pila al mismotiempo. El jugador con la fracción mayor se guarda todos los naipes de esa mano. Si dos de losnaipes con la fracción mayor son equivalentes, los jugadores que tienen los naipes equivalentesdan vuelta otro naipe de sus pilas. El jugador con la mayor de las fracciones guarda todos losnaipes jugados en esa mano. Gana el jugador con mayor cantidad de naipes al concluir el tiempode juego.

¿Otro juego? ¡Sí!T ó t e m S o l i ta r i o

¡Un juego para jugar a solas!Pon los naipes en orden, de menor a mayor. ¿Qué naipes son equivalentes?


We at Tenth Planet would like to thank the followingeducational advisors and contributors for their counsel,creativity, support, and hours of sharing their expertisethat helped Equivalent Fractions become what it is.

We also would like to thank the following educators andtheir students for their commitment, ideas, and enthusiasmfor testing and implementing Equivalent Fractions inclassrooms throughout the various phases of development.

AcknowledgmentsAnn Carlyle, Teacher, Goleta, CA

Anne Goodrow, Consultant and Author, Watertown, MA

Natalia Jacopetti, Teacher, Oakland, CA

Sheri Leafgren, Teacher, Akron, OH

Mary Lindquist, Professor, Columbus State University, Columbus, GA

Anne Linehan, Consultant and Author, Belmont, CA

Joanne Lobato, Assistant Professor, San Diego State University, San Diego, CA

Raven totem carver, Shane Eagleton/Protect All Lifeforms

Mosquito totem pole photograph, © 1997 Pat Kramer, author Totem Poles

Black and white historical photograph, Afternoon Drive, © 1994 FPG International

Cathy Bertonneau, Teacher, Foster City, CA

Jan Kaay, Teacher, Cupertino, CA

Image Acknowledgments


Internet Resources andContact InformationTenth Planet Internet Resources

We invite you to visit Tenth Planet on the World Wide Web.Find new teaching resources, ask questions, share ideas, andfind out what’s new at Tenth Planet. Our Internet address is:

http://www.tenthplanet.com

Teaching ResourcesWe’ve organized our Teaching Resources area of ourWeb site so that you can easily find ideas for integrating theInternet with the themes and math concepts in thisInvestigation. You'll find carefully selected content,information, and ideas for activities to integrate with yourcurriculum.

The Teaching Resources area is also a forum for sharingyour ideas and questions with other educators.

You can go directly to the Teaching Resources area of TenthPlanet’s Web site at http://www.tenthplanet.com/Teachers/

SupportOur Web site includes a Support area with answers tofrequently asked questions, special instructions, and the latestinformation on technical issues.

You can go directly to the Support area of Tenth Planet’sWeb site at http://www.tenthplanet.com/Support/

Tenth Planet Contact Information

If you have a question, need technical support, or just wantinformation about Tenth Planet, ask us directly. When youcontact us, be sure to let us know the best method and time to get back in touch with you.

E-mail Addresses

For general information about Tenth Planet:[email protected]

For sharing your tips for success with the program:[email protected]

For technical assistance:[email protected]

Telephone800-546-2317650-726-5891

Fax650-726-5995

U.S. MailTenth Planet Explorations, Inc.625 Miramontes StreetHalf Moon Bay, CA 94019

Equivalent Fractions

Documents

Transcript of Equivalent Fractions