Problem Solving-Working Backwards
Introduce the lesson by asking the following questions to students and discuss how students solve the problems:
- Have you ever forgotten your lunch? What did you do?
- Have you ever been unable to do your homework because you didn't understand it? What did you do?
- Have you ever lost something that is very important to you? What did you do?
After a discussion, show students the following video. Tell students to watch carefully to be able to describe how the animal solved it's problem. Pick one way to view the video:
- www.educanon.com/public/8885/37937 (shortened version)
- http://www.youtube.com/watch?v=Eh4x7fDilpI (full version on YouTube)
- (link to webpage with App Flow materials)
Discuss how the rat solved it's problem. Relate how problems can be solved in different ways and from different viewpoints. Be sure to stress that fact--different ways and viewpoints--as it will tie into the wrap-up section of this App Flow. Then be sure to state the learning goal and learning activity.
2 Direct Instruction
Introduce students to the problem solving video. Students will watch a video where a character named Andy is trying to raise some money by selling cupcakes. Play the full length video once for students to watch carefully.
After the first viewing, inform students that everyone will watch it one more time together. However, this time the video will have some questions that pop up. So be sure to pay close attention! It will be helpful for students to have a piece of scratch paper and a pencil to answer the questions using the EduCanon video. Pick one way to view the video:
- http://www.educanon.com/public/8885/20570 (EduCanon version)
Once students have viewed the video, discuss the problem presented in the video as a class. Remind students that they will have to work backwards in order to solve Andy's problem. Then have students use Edmodo to write a prediction to whether Andy solved his problem correctly or if he did not. Students can do this individually, with a partner, or as a small group. Be sure students explain why they think the way they do in their Edmodo posting.
Afterwards, present students with another short video that describes the problem solving method students will use to assist in working backwards: UPSC (Understand, Plan, Solve, and Check).
Then pass out the graphic organizer and complete the Understand and Plan sections as a class.
- Answer for understand: Did Andy have the correct amount of money?
- Answers for information we know:
- Each cupcake cost $3
- Andy sold a total of $123
- Robin bought half of the cupcakes baked
- Uncle Leo bought the other half of the cupcakes baked
- Uncle Leo bought 3 cupcakes for himself
- Nikki, Sam, and Mrs. Green each bought 1/3 of the remaining cupcakes from Uncle Leo
- Mrs. Green ended up with 5 cupcakes
- Answers for plan: Working backwards
3 Guided Practice
Next, have students work individually, with partners, or in small groups to solve Andy's problem by working backwards. Be sure to inform students to use the map outline on the back side of the graphic organizer. Students are able to use Edmodo to access the video and graphic organizer. They may replay the video as many times as needed to help them solve the problem. While students are using Edmodo to view the video, guide them through working backwards using the outline on the back side of the graphic organizer.
If students are struggling with the concept of working backwards, use the app MathLands (free on itunes: https://itunes.apple.com/us/app/mathlands-kids-logic-game/id489905188?mt=8) and have student(s) try the math comics activities within the app. In the math comics, students will read a comic strip that is based off of working backwards to solve problems. Try the level 1 comic and using the manipulatives on the app to assist in understanding how to work backwards. Sample pictures below:
Continue to guide students through outline of the graphic organizer. I have found it helpful to use two different color writing utensils (or highlighter) to solve this problem. With one color, start from the beginning of the video and write down all of the information that is known. This information is the same as the information written in the Understand section of the graphic organizer. Afterwards, use the other color utensil and work backwards from Mrs. Green back to the beginning of the outline to figure out how many cupcakes each person had. Then you will be able to calculate how many cupcakes Andy had originally and how much money he should have raised.
If you find that students would like to use money manipulatives to solve the problem, encourage students to try this website that offers money manipulatives online. Once the website opens, click on the "manipulatives" tab and then scroll down to the "currency." After, money and coins will appear for students to use. You may also want to have paper money manipulatives for groups to use if they do not like the online version.
4 Independent Practice
Continue to facilitate and guide students through solving Andy's problem using the graphic organizer. As students finish the problem, ask the student, partners, or group to explain how they know that their answer is correct. Does that answer make sense? How do you know? After a brief discussion with you, have student, partners, or group post their explanation on how they solved the problem on Edmodo. You may choose to have students post a reply under their initial posting for their prediction or have students reply to a new thread you have previously created on Edmodo. In addition, encourage students to collaborate and discuss their thoughts to the problem by having them reply to other student postings on Edmodo.
Did Andy receive the right amount of money? Do you agree or disagree with Inspector Blue that the amount of money does not seem correct? Show or explain how you know if the money is correct and how you solved the problem.
Just so you are aware, Andy did not receive the correct amount of money in the video from selling cupcakes. The video states that Andy raised $123. After working backwards, you will find that Andy should have raised $108 from selling cupcakes.
As an extension to the students' explanation, you can prompt students to discuss and/or post on Edmodo how Andy could have earned the extra $15 ($123-$108).
Also, to check students for their understanding of working backwards to solve problems, you should create a quick quiz on Edmodo for students to solve problems that require them to work backwards. Some examples are listed below:
- John is four years younger than Carmel but Jane is 24 years older than Carmel. If Jane is 35, how old is John?
- Answer: John is seven years old.
- Four students in the class weighed themselves. Carter was 15 kilograms lighter than Adrian. Gary was twice as heavy as Carter and Jeremy was seven kilograms heavier than Gary. If Jeremy weighed 71 kilograms what was Adrian’s weight?
- Adrian weighs 47 kilograms.
- When three dogs jumped on a weighing scale together at the veterinarian, the scale read 164 kilograms. One dog walked off and the scale moved down to 104 kilograms. One more dog jumped off and the scale showed 55 kilograms. What was each dog’s weight?
- 1st dog= 55 kg; 2nd dog= 60 kg; 3rd dog= 49 kg
- Daniel has lots of pets. He has four more goldfish than he has turtles. He has one less canary than goldfish. Six of his pets are birds (canaries and parrots). He has two parrots. How many pets does Daniel have in all?
- 12 pets in total
After all students or groups have finished the problem, discuss everyone's reaction to the problem. Were you or your group correct? Ask students to share and view the postings on Edmodo.
To extend student learning, have students use the app MathLands (as previously mentioned in the Guided Practice section of this App Flow). Within that app, students can play the math comics game and try to solve each comic (there are seven levels).
Also, students can explore the app Where's My Water?, in which student have to help Swampy by guiding water to his broken shower. Each level is a challenging physics-based puzzle where students cut through dirt to guide fresh water, dirty water, toxic water, steam, and ooze through increasingly challenging scenarios. (https://itunes.apple.com/us/app/wheres-my-water/id449735650?mt=8)
In addition, posted on the following webpage are examples of working backwards problems students or teachers can print out for extra practice.
Furthermore, to wrap-up the lesson, share the following poem with students. The poem is written by Jordan Nichols who was in 8th grade when the poem was written. The poem is called "Our Generation." Before you read the poem with your students, encourage students to think differently, just as they had to do in order to solve Andy's problem.
After reading the poem from top to bottom, read the poem again. But this time read from the bottom line (left to right) to the top. The poem takes on a totally different viewpoint...Enjoy!