Advantages Of Using Manipulatives - UC Davis School Of .

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udiesindicate: eputtogooduse Manipulativesimprovechildren’slong- ‐termandshort- ‐termretentionofmathematics oms oreducesmathanxiety Childrenbetterunderstandmath ryandeducationalresearchintheclassroom ovefromconcreteexperiencestoabstractreasoning.” s rstandingofmathematicalconcepts.” ofamanipulatives“makessense”forthattopic. oblemsolvingtests rsknowledgeableabouttheiruse

Aunt Alice and the Silver CoinsVersion 1Aunt Alice gave each of her three nieces a number of silverdollars equal to their ages. The youngest felt that this was unfair.They agreed to redistribute the money. The oldest would split halfof her silver coins evenly with the other two sisters. The middlesister would then give each of the others 3 silver coins. Finally,the youngest would give each of the others 1 silver coin. Afterexchanging money, each girl had 14 silver dollars. How old arethe sisters?

Aunt Alice and the Silver CoinsVersion 2Aunt Alice gave each of her three nieces a number of silverdollars equal to their ages. The youngest felt that this was unfair.They agreed to redistribute the money. The youngest would splithalf of her silver coins evenly with the other two sisters. Themiddle sister would then give each of the others 4 silver coins.Finally, the oldest was to split half of her silver dollars equallybetween the two younger sisters. After exchanging money, eachgirl had 16 silver dollars. How old are the sisters?

Basic Model Drawing1.Tyrone had 17 in his piggy bank. He added 8 more. What was his total savings?2.Ray has 465 tractors and his brother Ben has 289. How many tractors do they havealtogether?3.Jennifer went shopping with 42. She came home with 9. How much money didshe spend?4.Hansel read 235 pages of his book over the weekend. Gretel read 198 pages of herbook over the weekend. How many more pages did Hansel read than Gretel?

5.A total of 100 raffle tickets were sold over a 3-day period. If 21 raffle tickets weresold on Monday, and 67 tickets were sold on Tuesday, how many raffle tickets weresold on Wednesday?6.There are 5 plates of cookies on the shelf. If there are 4 cookies on each plate, howmany cookies are there in all?7.There are 20 chairs. Kayla wants to put the chairs into 4 rows. How many chairswill be in each row?8.Twelve students need rides to an after school event. If only 4 students can ride ineach car, how many cars are needed to transport the students?

Fractions and Model Drawing1. Harry cuts a pizza into 8 equal parts. If Harry eats 3 pieces of pizza, whatfraction of the pizza does he eat? What fraction of the pizza is left?2. Jonas divides a batch of cookies evenly into 9 groups. If he gives away 4groups of cookies, what fraction of the cookies does he give away? Whatfraction of the cookies is remaining?3. If23of a number is 4, what is the number?4. In the school chorus, 3 5 of the students are girls. If there are 10 boys in thechorus, how many girls are there?5. David spent25of his money on a storybook. The storybook cost 20. Howmuch money did he have at first?6. At the zoo, 3 7 of the visitors saw the panda exhibit. If 1,245 people saw thepanda exhibit, how many people visited the zoo?7. On Friday, 3,684 people visited the Statue of Liberty. 5 6 of the visitors climbedto the top of the Statue. How many visitors climbed to the top?8. One number is one fifth of another number. If the difference between the twonumbers is 20, find the two numbers.9. Handy Mann bought some boards at the lumber store. He used 1 2 of them tomake bookshelves, and 1 4 of the remaining boards to make a workbench. Hehad 6 boards left. How many boards did Handy buy at the lumber store?10. 2 3 of Brett’s savings is equal to 2 5 of Andrew’s savings. If they have 160 insavings altogether, how much more money does Andrew have in savings thanBrett?

HORSE g,thehorsetraderboughtahorsefor 60.Laterthatday,thehorsetradersoldthehorsefor ecidedtobuythehorsebackfor 80.Butwhensomeoneofferedhim shorse,ordidhemanagetojustbreakeven?

ModelDrawingLevel11. goalsdidtheykickinall?2. dthepuppychew?3. heyreadinall?4. 5. heartsdidtheyeatinall?

ModelDrawingLevel16. owmanydinnersdidthecookserve?7. school.HowmanycoloredpencilsdoesJanahaveinall?8. sdidhehaveleft?9. Tylerhad 10.Hespent 2attheschoolstoreand 0. Dennishas7yo- ‐yos.Hissisterhas2feweryo- ‐yosthanDennis.Howmanyyo- ‐yosdotheyhaveinall?

ModelDrawingLevel111. gsisthatinall?12. chshelf.Howmanyicecreamtubsareonthe3shelves?13. .Howmanypineappleswereinthe6baskets?14. inall?

ModelDrawingProblemsLevel21. pencilsdoesThaliahaveinall?2. Ateacherspent 4.48onglueand 3. ?4. away,howmanystudentswilltherebeintheschool?5. Mysistergot 100forherbirthday.Shespentsomeofthemoneyandhad 36left.Howmuchmoneydidshespend?

ModelDrawingProblemsLevel26. venow?7. e?8. ocket.HowmuchmoremoneydoesMarthahavethanKris?9. manymorepackagesdoesAaronneedtowrap?10. ydishesdoesshestillhavetowash?

ModelDrawingProblemsLevel211. ellafor19ofthosedays.HowmanydayswasBellaalone?12. Sevenchildrenwenttothemovies.Theirticketscost 3each.Howmuchdiditcostallofthemtogotothemovies?13. Twopeoplecanfitinaroller- ontherollercoaster?14. ckageswouldyouneedtobuytohave24lollipops?15.

ModelDrawingLevel31. .Howmanytotalfishdidthealligatorsinmydreameat?2. anywordproblemsdoesNikohavelefttosolve?3. dMr.Alvarezdrive?4. Mysisterwon 2,000.Shespentsomeofthemoneyandhad 1,136left.Howmuchmoneydidshespend?5. ,howmanyfishwillbeleft?6. applesdidLisetteuse?7. yfewerblueblocksthanredblocksareinthecabinet?

ModelDrawingLevel38. gcontest?9. Bao- 10. manybagswillthezookeeperneed?11. ,howtallwouldtheybe?12. Asteakcosts 18.27whilealobstercosts . Nikomakes eywillhemake?14. jellybeans,howmanyjellybeanswillbeleft?

Level 4 (4th Grade)1.Donald has 595.95. He has 148.35 more than Daffy. How muchmoney do they have altogether?2.The distance around a track is 1 kilometer 600 meters. If Fabian ranaround the track 5 times, how far did he run?3.Mrs. Alvarez brought 5 kilograms 450 grams of Skittles into her class.She packed them equally into 5 bags. How much did each bagweigh?4.Jonathan spends 45 minutes on his homework every night. Vigneshspends 30 minutes on his homework every night. How much time dothe 2 boys spend altogether on homework each night?5.Grace had 3 times as many strawberries as Tyler. After Grace eats50 strawberries, she has half as many strawberries as Tyler. Howmany strawberries does Grace have left?6.A piece of ribbon 1,624 inches long was cut into 2 parts. The longerpiece was 312 inches longer than the shorter piece. What was thelength of each piece of ribbon?7.On the airplane, there are 258 adults and 65 children. How manymore adults than children are on the airplane?8.Frank has 1,578 in the bank. Marco has 269 more in the bank thanFrank. How much money do they have in the bank altogether?

9.Holly had 5,000. She bought a table for 299. She also bought 4chairs for 89 each. Now Holly wants to buy a cabinet for 1,500.After purchasing the other 5 items, does she have enough money?10.Jonas divides a batch of cookies evenly into 9 groups. If he givesaway 4 groups of the cookies, what fraction of the cookies does hegive away? What fraction of the cookies is remaining?11.Thomas counted 487 leaves on the tree in his backyard. Brittanycounted 365 leaves on her tree. How many leaves did the childrencount on their trees in all?12.At the flower shop, there are 5 times as many roses as sunflowers. Ifthere are 252 roses and sunflowers altogether, how many roses arethere? How many sunflowers are there?13.Sheila was making place-value disks. She colored 2/6 of the disksred. She colored 1/4 of the disks yellow. If she colored 40 red placevalue disks, how many disks did she color yellow? How many disksdoes she still have left to color?14.Harry cuts a pizza into 8 equal parts. If Harry eats 3 pieces of pizza,what fraction of the pizza does he eat? What fraction of the pizza isleft?

Level 5 (Grade 5)1.Angle DEF is a straight angle. Angle CEF is 37 degrees. What is themeasure of angle CED?2.The hot dog vendors at the state fair sell an average of 78 hot dogsevery 30 minutes. At this rate, how many hot dogs can they expect tosell in 9 1/4 hours?3.David’s birthday party was held at the bowling alley from 3:30 to 5:30.His family rented 2 lanes for that time. Lane rental for 1 lane is 10 anhour for 10:00 AM to 5:00 PM and 14 an hour from 5:00 PM tomidnight. What was the total lane rental for 2 lanes for those 2 hours?4.If 7/8 of a number is 63, what is 3 times that number?5.A dozen watermelons weigh an average of 8.5 kilograms. If 8 of thewatermelons weigh an average of 9.1 kilograms, what is the averageweight of the other 4 watermelons?6.At the farmer’s market there were 75 apples in each of a farmer’s 5baskets. There were 200 apples still on his truck. How many apples inall did the farmer bring to the market?7.Astronaut Alex has been in space 375 days. Astronaut Debi has beenin space 190 fewer days than Alex. Astronaut Sue and Astronaut Luhave been in space an equal number of days; their total is 422 days.How many days has Debi been in space? How many days has Lubeen in space? How many total days have these 4 astronauts been inspace?

8.The folks at the concession stand sold 480 cans of soda during thetournament. They sold 150 more bottles of water than cans of soda.How many drinks did they sell in all?9.The party store had many balloons. Of all the balloons, 10% were red,25% were orange, 15% were green, and 20% were yellow. The other30 were blue. How many balloons did the store have in all?10. Tickets for animated to non-animated films at the Movie Star Theatersold last month at a ratio of 4:7. There were 14,700 movie-goers whosaw non-animated films. How many saw animated movies? How manypeople total went to the movies at the Movie Star Theater last month?11. Rocco’s Rock Shop has 725 igneous rocks on display. Rocco hasdivided the rocks up evenly among 5 shelves. The rocks on the bottom2 shelves are 3 each. If he were to sell all the rocks on those 2shelves, what would Rocco’s income be? The rocks on the other 3shelves are 2 each. What would the total sales be if Rocco sold all ofhis rocks?12. Of the people at the water park, 3/7 are children ages 3 through 12and 2/7 are teens. Another 1/7 are children under the age of 3. Theother people are adults. There are 87 adults at the park. How manychildren ages 3 through 12 are at the park? How many people in allare at the park?13. Nidhi bought 3 baseball caps for 12.45 each and 4 jerseys for 27.75each. What was her total bill?14. The ratio of flutists to trumpeters to drummers in the school band is3:4:1. If there are 5 drummers, how many flutists are there? How manytrumpeters are there?

Problem- ‐SolvingWebsitesSingaporeMath–ModelDrawing http://www.thesingaporemaths.com/- ‐ answers)fromgrades3- .k12.nm.us/instruction/singaporemathbook.cfm- ‐ 20minutesaday)- 2).http://www.thinkingblocks.com/- ‐ terials/Documents/Singapore%20Math%20- ‐%20Forsten.pdf- ‐ del- thebook–Step- ‐By- /EdServices/Math/MultiStepWordProblemsGr23.pdf- ‐ awing.Itincludesteachertalkandexamples.

Problem- ‐SolvingStrategies http://library.thinkquest.org/4471/learn.htm- ‐ s,whowanttosharpentheirproblem- ntproblem- mathlearnnc.sharpschool.com/cms/One.aspx?portalld 4507283&pa- ‐ adesK- opics.http://www.mathstories.com/- ‐ TMcompliantmathwordproblemsforstudentsgrades1- sh.Pricerangesfrom 24- ‐ 48forayearmembership.VirtualManipulatives http://nlvm.usu.edu/en/nav/vlibrary.html- ythingonthesiteitcosts 39.95foraone- picsthatarecoveredonthissite. http://www.glencoe.com/sites/common assets/mathematics/ebook assets/vmf/VMF- ‐Interface.html- ‐ Onthissite,youpickagradelevel(K- tudentscanuseitindividually. http://www.mathplayground.com/games.html- ‐ dentstouse.Itiscolorfulandhascartoon- ‐likecharactersthatstudentswouldreallyenjoy. http://www.ct4me.net/math manipulatives.htm- ‐ lativesontheweb.

Learning Model Drawing1.Abdul put 3 apples into a bowl. He added 2 oranges. How many total pieces offruit did Abdul put in the bowl?2.Tyrone had 17 in his piggy bank. He added 8 more. What was his total savings?3.Jennifer went shopping with 42. She came home with 9. How much money didshe spend?4.Carla has 19. She wants to buy a computer game that costs 36. How much moremoney will Carla need to buy the computer game?5.Hansel read 235 pages of his book over the weekend. Gretel read 198 pages of herbook over the weekend. How many more pages did Hansel read than Gretel?6.A total of 100 raffle tickets were sold over a 3-day period. If 21 raffle tickets weresold on Monday, and 67 tickets were sold on Tuesday, how many raffle tickets weresold on Wednesday?7.Mutt and Jeff both have money. Mutt has 15 more than Jeff. If Jeff has 30, howmuch money do they have altogether?8.There are 5 plates of cookies on the shelf. If there are 4 cookies on each plate, howmany cookies are there in all?9.There are 20 chairs. Kayla wants to put the chairs into 4 rows. How many chairswill be in each row?10.Twelve students need rides to an after school event. If only 4 students can ride ineach car, how many cars are needed to transport the students?11.James is half as old as Henry. Edward is twice as old as Henry. If Edward is 20years old, how old is James?12.If13.In the school chorus, 3 5 of the students are girls. If there are 10 boys in the chorus,how many girls are there?14.One number is one fifth of another number. If the difference between the twonumbers is 20, find the two numbers.23of a number is 4, what is the number?

15.There are 240 students in the fifth grade. If 60% of them are girls, how many boysare in the fifth grade?16.The ratio of boys to girls at a party was 2:3. If there were 9 girls, how many boyswere at the party?17.Together, Sally and Susie collected 32 seashells at the seashore. If Sally collected 6more seashells than Susie, how many seashells did Susie collect?18.Handy Mann bought some boards at the lumber store. He used12 ofthem to makebookshelves, and 1 4 of the remaining boards to make a workbench. He had 6 boardsleft. How many boards did Handy buy at the lumber store?19.Debbie, Eddie, and Fred have earned a total of 51 AR points. Eddie has twice asmany points as Fred. Debbie has 15 more points than Fred. How many AR pointsdid each person earn?20.Amy, Betty, and Carla have a total of 67 marbles. Amy has 4 more than Betty.Betty has three times as many as Carla. How many marbles does each person have?21.Tom has four times as many baseball cards as Greg. Together, they have a total of40 cards. How many cards did Greg have?22.Bill has 12 more than three times the number of baseball cards Chris has. Bill has42 more cards than Chris. How many baseball cards does Chris have? How manybaseball cards does Bill have?23.The difference between two numbers is 24. If one number is three times the othernumber, find the sum of the two numbers.24.David spent 2/5 of his money on a storybook. The storybook cost 20. How muchmoney did he have at first?

- s.” ingofothers.” Studentscananalyzecomplex,real- modelstointerpretandsolveproblems.” ate,extend,andapply

Using Model DrawingModel Drawing provides a systematic approach to solving word problems.1.Read the entire problem, first ignoring the actual numbers so we can“visualize” the problem conceptually (reflective reading) and then readingit again with all the numbers included2.Rewrite the question in sentence form leaving a space for the answer.3.Decide and write down (label) who and/or what the problem is about4.Draw unit bars of equal length that we’ll eventually adjust as we constructthe visual image of the problem5.Chunk the problem and adjust the unit bars to reflect the information inthe problem6.Determine exactly “what” we’re being asked to find and place aquestion mark in the place on the model drawing that reflects the “what”7.Compute the problem to come up with an answer (show all work!)8.Write the answer in a complete sentence that clearly states the solution

Illustrative Mathematics Websitehttp://illustrativemathematics.org A work in progress Designed to provide examples that help to clarify what each CCSS meansTools for the Common Corehttp://commoncoretools.me Bill McCallum’s blog Contains articles, up to date information and links to viable sites related to theCCSSInside Mathematics websitewww.insidemathematics.orgNew resources on Common Core Standards for Mathematical Practice, including: Classroom video examples illustrating the math practice standards, includingcommentary Videos of exemplary lessons integrating multiple math practicesNew resources on Common Core Standards for Mathematical Content, including: Common Core-aligned tasks, searchable either by grade level or by CommonCore content area, such as “Operations and Algebraic Thinking,” or“Geometry - Congruence.”Additional classroom videos of Number Talks Four new classroom videos of “Number Talks” showing students engaged inmental math exercises and conversations about math, including one from abilingual Spanish-English classroom.Mathematics Assessment ProjectUC Berkeley & Shell Centre for Mathematical ex.php Primarily middle school and high school materials Tasks Lessons aligned to CCSSSmarter Balance Consortium Websitehttp://www.smarterbalanced.orgSmarter Balanced Assessments Tab – Sample Items and Performance Tasks Sample Items on Computer that model what students would see and do onthe computer based assessmentsSmarter Balanced Assessments Tab Scroll down on page until you see Item/Task Specifications Under that heading, keep scrolling down until you see Mathematics The zip files contain sample items for each of the 4 claims

California Mathematics New Resources and Webinars on Standards and se-tools/professional-development-modulesASCD Common Core Resources Websitehttp://educore.ascd.org

5. A total of 100 raffle tickets were sold over a 3-day period. If 21 raffle tickets were sold on Monday, and 67 tickets were sold on Tuesday, how many raffle tickets were sold on Wednesday? 6. There are 5 plates of cookies on the shelf. If there are 4 cookies on each plate, h

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