Esabior Rivera
Strategy Overview
Eliminating possibilities means to get rid of options that you know for a fact can not be the correct answer to make finding the solution just a bit easier.
For what types of problems is this strategy helpful?
This problem-solving strategy is very useful when you have multiple solutions presented to you for a problem. You can start by eliminating the solutions that are obviously not correct, then move on to solutions that may be less obvious. The more you work on the problem, the more clear certain solutions appear to be incorrect. Use this strategy until the correct solution becomes obvious through the process of elimination.
When is it NOT very useful?
Eliminating possibilities is not very useful when the problem you are solving doesn’t have clear choices or possibilities as solutions. For example, a problem asking the student to define the term “irregular shapes” wouldn’t have potential possibilities to eliminate, so the strategy wouldn’t be very useful here.
Common mistakes or misconceptions to watch out for:
One common mistake seen when eliminating strategies is being too quick to eliminate certain options, when working through the problem with this strategy, be sure to double check your work so you aren’t accidentally eliminating the correct answer. Another common mistake to look out for is forgetting which possibilities have already been eliminated. Find a way to keep track of which solutions are still possible and which solutions you’ve already eliminated. If you aren’t carefully keeping track, you may forget what has already been eliminated, which would waste time and may lead to confusion.
How does this strategy help students become better mathematical thinkers? Give a sales pitch.
This strategy helps students become better mathematical thinkers by improving a student’s critical thinking skills as well as their information organization skills. Both of these lead to students that can solve problems quicker as they can find the solution through the process of elimination more efficiently that simply solving the problem. Information organization also keeps students work in check, allowing them to reduce mistakes made due to messy work or confusing problems.
Example Problems
Problem 1:
Liar Liar Pants on Fire
Jim tells lies on Fridays, Saturday, and Sundays. He tells the truth on all other days. Freda tells lies on Tuesdays, Wednesdays, and Thursdays. She tells the truth on all other days. If they both say, “Yesterday I lied,” then what day is it today?” (Johnson, Kerr, & Kysh, 2018, p. 56)
Problem 2:
Key Takeaways
Emil and Olive live on a farm with their father, Gordon. One day Emil asked his father, “Dad, what happened to that cat we used to have?” Olive over heard this and said, “Yeah, and we
used to have a horse too.” Gordon replied, “That tomcat and that old nag were no use. I traded them for our new goat.” Emil said, “Hey, that sounds like a good cryptarithmetic problem. Come on, Olive, let’s see if we can solve it.” They wrote down TOM + NAG = GOAT. Each letter stands for a different digit, 0 through 9. No tow letters stand for the same digit. Determine which digit each letter represents. (Johnson, Kerr, & Kysh, 2018, p. 57)
Worked Examples + Think-Alouds
Problem 1:
Watch the problem one think-aloud:
Problem 2:
Watch the problem two think-aloud:
Student Self-Talk Questions
Self talk question 1: How can I efficiently eliminate the most possibilities?
This question will help you get to the correct answer the quickest! For example, if you are playing twenty questions attempting to guess a number 1-100, asking ,”is the number 37″, will only eliminate one possibility if incorrect, wasting time. Instead, asking, “is the number odd?” is guaranteed to eliminate 50% of the possible solutions, getting you much closer to the correct answer!
Self talk question 2: How can I keep track of eliminated possibilities?
This question will help you keep track of solutions you’ve already eliminated. Keeping organized like this is the best way to stay on course to finding the correct answer, since, if you don’t, you are more likely to get confused and waste time. For example, keeping a list off eliminated possibilities or crossing them out may be very useful!
Teacher Notes and Instructional Tips
Lesson Launch Ideas
Start the lesson by providing students with an equation with multiple variables. Ask the students to ask yes or no questions in order to begin finding what digit each variable represents. As students begin uncovering more and more context to the equation, they should be able to use logic and general reasoning to find each variable by eliminating possibilities. They should be able to understand which variables are and aren’t plausible through each question asked.
Teacher Questions & Anticipated Student Responses
- Teacher: What can you ask to figure out the most possibilities?
- Student: Is the answer 5?
- Teacher: That question is pretty specific and won’t eliminate many possibilities. What is a broader question you can ask?
- Student: Is the answer odd?
- Teacher: Great question! What can you ask yourself to make sure you don’t get confused?
- Student: What have I already eliminated?
Differentiation Suggestions
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Support:
- Create graphs or charts for students to keep track of eliminated possibilities, provide an example question for the students as a scaffold
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Extend:
- Allow students to create their own questions for their partners to try and solve, ask students to attempt to find multiple solutions
Formative Assessment Tools
Provide students with example questions and ask if they think it is an efficient question to eliminate possibilities, allow students to ask eachother questions to eliminate possibilities and watch for efficiency and understanding.
Student Reflection Prompts
- What can I ask to eliminate the most possibilities?
- What do I know about the problem that makes certain solutions impossible?
- How can I keep track of the answers I’ve already eliminated?
What prior knowledge do students need?
Students need the ability to think outside of the box and use logic to consider all possible solutions before figuring out which solutions can be efficiently eliminated. They also must be able to organize/keep track of information so they don’t find themselves confused when eliminating possibilities.
What classroom challenges commonly arise?
Students struggling with learned helplessness may find themselves instinctually asking questions that aren’t efficient in order to avoid further questions from teacher, or may even explicitly avoid trying for fear of asking an inefficient question. Students may also harass others that ask question they deem illogical or foolish.
