Problem Types • The structure of a problem determines its level of difficulty. • The structure of a problem determines the strategies that students may develop/use to solve it. • The context of the problem affects the difficulty of a problem. • Abstract concepts (time, distance, and volume) can make a problem more challenging to solve as opposed to problems where students can count discrete objects. • By including measurement (time, distance, and volume) you encourage students to explore new tool and develop their understanding of the geometry content standards. Join (add to) problems are action problems. The action words support students’ ability to act out the problem. These problems have three distinct quantities, any one of which can be unknown. Join Result Unknown (JRU) Sabrina has ___ dinosaurs. Lina gave Sabrina ___ more dinosaurs. How many dinosaurs does Sabrina have now? John has ____ Legos. Lisa gave John ____ more. How many Legos does John have now? Jerry has ____ toy cars. His sister gave him ___ more toy cars. How many toy cars does John have now? Trey put___ trucks in a bag. Then Trey put ____ more trucks in his bag. How many trucks are in the bag now? Nicole put ____ cherries in a bowl. Her mom put ____ more cherries in the bowl. How many cherries are in the bowl now? Chad put ___ baseball cards in a notebook. His brother put ___ more baseball cards in the notebook. How many baseball cards are in the notebook now? ____ birds are sitting on a wire. ___ more birds land on the wire. How many birds are on the wire now? ____ kids are playing on the playground. ___ more kids run onto the playground. How many kids are on the playground now? ____ ants are crawling up a wall. _____more ants start crawling up the wall. How many ants are crawling up the wall now? Join Change Unknown (JCU) may encourage kids to count up from any given number. Mary has ___ beads on her shirt. How many more beads does she need to have ___ beads on her shirt? Eric cleaned ___ desks. How many more desks does he need to clean to reach a total of ___ desks? Sabrina had ___ crayons. Lina gave her some more crayons. Now Sabrina has __ crayons. How many Crayons did Lina give her? Meghan read ___ pages Monday morning. In the afternoon she read some more pages. By Monday evening she had read a total of ___ pages. How many pages did Meghan read in the afternoon? Lionel drank ___ grams/kilograms of water. How much more water does he need to drink to reach ___ grams/kilograms? Ally woke up at ___ o’clock. She fell asleep at __ o’clock. How long was Ally awake? Candace fell asleep at ____ o’clock. She woke up at ___ o’clock. How long was Candace asleep? Bobby threw the ball ___ feet/yards/meters. How many more yards does he need to throw to reach ___ feet/yards/meters? Melissa jumped ___ feet. How many more feet does she need to jump to reach ___ feet? Daniel drew a line that was ___ inches long. Then he drew the line a little longer. Now the line is ___ inches/feet long. How many inches/feet longer did he draw the line? Join Start Unknown (JSU) Problem Types -­‐ JSU problems have verbs in them but may be more challenging to solve than JRU and JCU problem types because they do not follow a typical story pattern. In addition, students have little exposure to this type of problem structure with the unknown being located at the beginning. Students are confounded as to where to begin. Students typically learn to solve this problem type through trial and error and student shares. 1. Mary had some books on her bookshelf. Her mom put ___ more books on her bookshelf. Now Mary has __ books on her shelf. How many books did Mary have on her shelf? 2. Andy drew some pictures in a book. John drew ___ more pictures in the book. Now there are ___ pictures in the book. How many pictures did Andy draw in the book? 3. Cary caught some insects. Her brother caught __ more. Together they caught a total of ___ insects. How many insects did Cary catch? 4. Michael brought some comic books to park. Andrew brought ___ comic books to the park. Together they have ___ comic books at the park. How many comic books did Michael bring to the park? 5. Janis had some money. Her dad gave her ___ more dollars/cents. Now she has ___dollars/cents. How much money did Janis have before her dad gave her money? *What coins or bills could he have used? 6. Jordan ran in the relay race. Michelle ran ___ feet/yards/meters in the relay race. Together they ran a total of ___ feet/yards/meters. How many feet/yards/meters did Jordan run? 7. Miesha walked some miles/kilometers during the walkathon. Janet walked ___ miles during the walkathon. Together they walked ___ miles/kilometers during the walkathon. How many miles/kilometers did Miesha walk? 8. Linda drank some cups/liters of water afterschool. Roman drank __ cups/liters of water after school. Together they drank a total of ___ cups/liters. How much water did Linda drink after school? 9. Mary caught a fish. Damon caught a fish that weighs ___ grams/pounds/kilograms. Mary’s fish and Damon’s fish have a combined weight of ___ pounds. How many pounds/ grams/kilograms does Mary’s fish weigh? 10. Mark gained some weight during the summer. He gained ___more grams/pounds/kilograms in the fall. He gained a total of ___ grams/pounds/kilograms. How much weight did Mark gain during the summer? Separate problems (take from) are action problems. The action words support students’ ability to act out the problem. These problems have three distinct quantities, any one of which can be unknown. Separate Result Unknown (SRU) Problem Types 1. Erica had ___ dresses. She donated ___ dresses to charity. How many dresses does Erica have now? 2. Melissa made ___ cupcakes. Her friends ate ___ cupcakes. How many cupcakes does Melissa have now? 3. Brian had ___ books. He donated ___ books to the library. How many books does Brian have now? 4. Brian is reading a book that is ___ pages long. He already read ___ pages. How many pages does he have left to read? 5. Stacey bought ___ dozen eggs. She dyed ____ eggs. How many eggs does she have left? 6. Gary earned $___. He spent $__ on groceries. How much money does he have left? 7. The trip to Grandma’s house is ___ miles/kilometers long. The family already drove ___ miles. How much farther do they have to drive to get to Grandma’s house? 8. The wedding cake needs ___ cups/grams of flour. The baker already added ___ cups/grams. How many more cups/grams of flour does the cake need? 9. The street is___ feet/yards/meters long. The workers have already paved ___ feet/yards/meters. How many more feet do they have to pave? 10. John weighs ____ pounds/kilograms. He lost ___ pounds/kilograms. How much does John weigh now? 11. Susan has ___minutes/hours to get ready for a party. She spent ___minutes/hours doing her hair. How much longer does she have to get ready? Separate Change Unknown (SCU) problem types-­‐ may encourage students to count up or back from any given number. 1. Erica had ___ dresses. She donated some to charity. Now she has __ dresses. How many dresses did Erica donate to charity? 1. Melissa made ___ cupcakes. Her friends ate some cupcakes. Now Melissa has ___ cupcakes. How many cupcakes did Melissa’s friends eat? 2. Brian had ___ books. He donated some books to the library. Now he has ___ books left. How many books did Brian donate to the library? 3. Brian is reading a book that is ___ pages long. He already read some pages. Now he has ___ pages left to read. How many pages did Brian read already? 4. Stacey bought ___ dozen eggs. She cracked some eggs. Now she has ___ eggs left. How many eggs did Stacey crack? 5. Gary earned $___. He spent some money on video games. Now he has $____. How much money did he spend on video games? 6. The trip to Grandma’s house is ___ miles/kilometers long. The family already drove some miles. Now they have ___ miles/kilometers to drive. How far did they already drive? 7. The wedding cake needs ___ cups/grams of flour. The baker already added some cups/grams to the batter. Now the cake needs ___ grams/cups of flour. How many more cups/grams of flour did the baker add? 8. The street is___ feet/yards/meters long. The workers have already paved some feet/yards/meters of the street. Now there are ___ feet/meter/yards left to pave. How many feet did the workers pave already? 9. John weighs ____ pounds/kilograms. He lost some pounds/kilograms. Now John weighs ___ pounds/kilograms. How much weight did John lose so far? 10. Susan has ___minutes/hours to get ready for a party. She spent some minutes/hours doing her hair. Now she has ___ minutes/hours left to get ready. How much time did Susan spend doing her hair? Separate Start Unknown (SSU) problem types-­‐ have an action stated in them but can be challenging for students to solve because they do not follow a typical story flow and the unknown that they are solving for is not in the usual place. Students typically learn to solve this problem through trial and error and student share. Erica had some dresses in her closet. She donated ___ to charity. Now she has __ dresses. How many dresses did Erica have in the beginning? 1. Melissa made some cupcakes. Her friends ate ___ cupcakes. Now Melissa has ___ cupcakes. How many cupcakes did Melissa make? 2. Brian bought some books. He donated ___ books to the library. Now he has ___ books left. How many books did Brian buy? 3. Brian bought a book that has many pages. He already read ____ pages. Now he has ___ pages left to read. How many pages does Brian’s book have? 4. Stacey bought some eggs. She cracked ____ eggs. Now she has ___ eggs left. How many eggs did Stacey buy? 5. Gary earned some money. He spent $___ on video games. Now he has $____. How much money did Gary earn? 6. The trip to Grandma’s house is very long. The family already drove ___ miles/kilometers. Now they have ___ miles/kilometers to drive. How long is the drive to Granma’s house? 7. The street is many feet/yards/meters long. The workers have already paved ___feet/yards/meters of the street. Now there are ___ feet/meter/yards left to pave. How long is the street? 8. John weighs many pounds/kilograms. He lost ___ pounds/kilograms. Now John weighs ___ pounds/kilograms. How much did John weigh before he lost weight? 9. Susan has some minutes/hours to get ready for a party. She spent ___minutes/hours doing her hair. Now she has ___ minutes/hours left to get ready. How much time did Susan have to get ready all together? Part-­‐Part-­‐Whole problem types describe parts of a set. Part-­‐part whole problems do not have an action or describe a change over time, they are static. This can make them more challenging for students to solve, since students do not have an action to imitate. Part-­‐Part Whole (Whole Unknown) Sabrina has ___ sparkly stickers and ___ plain stickers. How many stickers does Sabrina have? Damon has ___ fiction books and ___ informational books. How many books does he have? Erica has ___ dollars in the bank and ___ dollars in her purse. How much money does Erica have? John had basketball sprinting practice for ___ minutes and basketball shooting practice for ___ minutes. How long was John at basketball practice. Stacey has ___ clear marbles and ___ blue marbles. How many marbles does Stacey have? There are __ boys in 3rd grade and __ girls in 3rd grade. How many kids are in third grade? There are ___ boys in the elementary school and ___ girls in the elementary school. How many kids are in the elementary school? There are ___ cups/grams of brown sugar in the recipe and __ cups/grams of white sugar in the recipe. How much sugar is in the recipe? The parking lot has ___ parking spaces on the first level and ___ parking spaces on the second level. How many parking spaces does the parking lot have? There are ___ red insects, ___ green insects, and ___ black insects in the collection. How many insects are in the collection? There are ___ miles/kilometers of bike paths in the city and ___ miles/kilometers of walking paths in the city. How many miles/kilometers of walking and biking paths are in the city? The city park is has ___ square feet for pets and ___square feet for humans. How many square feet are in the park total? Part-­‐Part Whole (Both Addends Unknown) Sabrina wants to buy ___ stickers. Some can be sparkly stickers and some can be plain stickers. How many sparkly stickers could Sabrina buy, how many plain stickers could she buy? Damon wants to buy ___ books. Some of the books can be fiction and some of the books can be informational books. How many of his books could be fiction, how many books could be informational? Erica has ___ dollars. She can spend some money on shoes and some money on a purse. How much money could Erica spend on shoes, how much money could she spend on a purse? John has ___minutes of basketball practice. How much time could he spend on dribbling practice and how long could he spend on basketball shooting practice? Stacey wants to buy ___ marbles. Some of the marbles could be blue marbles and some of the marbles could be clear. How many marbles could be blue and how many marbles could be clear? There are __ kids in third grade. How many could be boys and how many could be girls? There are ___ kids in the elementary school. How many could be boys and how many could be girls? There are ___ cups/grams of sugar in the recipe. How many cups/grams could be white sugar and how many cups/grams could be brown sugar? The parking lot has ___ parking spaces on the first and second levels. How many parking spaces could be on the first level and how many spaces could be on the second level? There are ___ insects in the collection. Some are red, green, and black. How many of the insects could be red, how many could be green, and how many could be black? There are ___ miles/kilometers of bike paths and walking paths in the city. How many miles could be walking paths and how many could be biking paths? The city park has ___ square feet for pets and humans to share. How many square feet could be for pets and how many square feet could be for humans? Part-­‐Part-­‐Whole (Part Unknown) Sabrina has ___ sparkly stickers and ___ plain stickers. How many stickers does Sabrina have? Damon has ___ fiction books and ___ informational books. How many books does he have? Erica has ___ dollars in the bank and ___ dollars in her purse. How much money does Erica have? John had basketball dribbling practice for ___ minutes and basketball shooting practice for ___ minutes. How long was John at basketball practice. Stacey has ___ clear marbles and ___ blue marbles. How many marbles does Stacey have? There are __ boys in 3rd grade and __ girls in 3rd grade. How many kids are in third grade? There are ___ boys in the elementary school and ___ girls in the elementary school. How many kids are in the elementary school? There are ___ cups/grams of brown sugar in the recipe and __ cups/grams of white sugar in the recipe. How much sugar is in the recipe? The parking lot has ___ parking spaces on the first level and ___ parking spaces on the second level. How many parking spaces does the parking lot have? There are ___ red insects, ___ green insects, and ___ black insects in the collection. How many insects are in the collection? There are ___ miles/kilometers of bike paths in the city and ___ miles/kilometers of walking paths in the city. How many miles of walking and biking paths are in the city? The city park is has ___ square feet for pets and ___square feet for humans. How many square feet are in the park total? Compare Problem Types -­‐ These problems typically involve comparing two sets. The third quantity is the difference between the two sets. These problems typically do not involve a physical action and can be challenging for students to solve. Students may learn through trial and error or through student shares. Compare (Difference Unknown) Sabrina baked __ cupcakes. Ryan baked ___ cupcakes. How many more cupcakes did Sabrina bake than Ryan? Luis read ____ pages of his book. John read ____ pages of his book. How many more pages did Luis read than John? Joey got ___ likes on his page. Gina got __ likes on her page. How many more likes did Joey get than Gina? Melissa got ___ views of her video. Michael got ___ views of his video. How many more views did Melissa get than Michael? Mary has ___ friends on her account. Bill has ___ friends on his account. How many more friends does Mary have than Bill? The red team has ___ points in the basketball game. The blue team has ___ points. How many more points does the red team have than the blue team? A new car cost ___ dollars. An old car costs ___ dollars. How much more is a new car than an old car? Jane weighs ___ pounds/kilograms. Erica weighs ___ pounds/kilograms. How much more does Jane weigh than Erica? Chad ran __ miles/kilometers. Ryan ran ___ miles/kilograms. How much farther did Chad run than Ryan? The school is ___ miles/kilometers away from home. The store is ___ miles/kilometers away from home. How much farther from home is the store than the school? Compare Quantity Unknown Sabrina baked __ cupcakes. Ryan baked ___ more cupcakes than Sabrina. How many cupcakes did Ryan bake? Louis read ____ pages of his book. John read ____ more pages of the same book than Louis. How many pages did John read? Joey got ___ likes on his page. Gina got __ more likes on her page than Joey. How many likes did Gina get? Melissa got ___ views of her video. John received ___ more views of his video than Melissa. How many views did John get? Mary has ___ friends on her account. Bill has ___ more friends on his account than Mary. How many friends does Bill have? The red team has ___ points in the basketball game. The blue team has ___ more points than the red team. How many points does the blue team have? An old car cost ___ dollars. A new car costs ___ more dollars than an old car. How much does a new car cost? Jane weighs ___ pounds/kilograms. Erica weighs ___ more pounds/kilograms than Jane. How much does Erica weigh? Chad ran __ miles/kilometers. Ryan ran ___ more miles/kilograms than Chad. How far did Ryan run? The school is ___ miles/kilometers away from home. The store is ___ more miles/kilometers farther away from home than the school. How far away is the store from home? Compare (Referent Unknown) Sabrina baked __ cupcakes. She baked ___ more cupcakes than Ryan. How many cupcakes did Ryan bake? Louis read ____ pages of his book. Louis read ____ more pages of the his book than Richard. How many pages did Richard read? Joey got ___ likes on his page. Joey got __ more likes on his page than Gina. How many likes did Gina get? Melissa got ___ views of her video. Melissa got ___ more views of her video than Michael did. How many views did Michael get? Mary has ___ friends on her account. Mary has ___ more friends on her account than Melissa. How many friends does Melissa have? The red team has ___ points in the basketball game. The red team has ___ more points than the blue team. How many points does the blue team have? A new car cost ___ dollars. A new car costs ___ more dollars than an old car. How much does an old car cost? Jane weighs ___ pounds/kilograms. Jane weighs ___ more pounds/kilograms than Erica. How much does Erica weigh? Chad ran __ miles/kilometers. Chad ran ___ more miles/kilograms than Ryan. How far did Ryan run? The school is ___ miles/kilometers away from home. The school is ___ more miles/kilometers farther away from home than the store. How far away is the store from home?