2/23/2017 3 Comments KEY SKILL: Teaching Students to Test Cases And Use Examples & Counterexamples in Math Class
The easiest and most concrete way I've found to do this is through "always, sometimes, never" questioning.
Kids cannot think through these types of questions without mentally testing cases. And if you go a step beyond to have them support their answers, they will be forced to justify by using both examples and counterexamples. BONUS: This type of question almost ALWAYS inspires some solid "math talk," so make sure to let your classes work in pairs or showcase their reasoning at the board to get that conversation going. Here are some examples to try to incorporate: Testing Cases in Algebra:
Often, students may think they know the answer and then in testing, they'll come across a different situation that violates the "rules" they were imagining originally.
Make sure to have them try every case they can think of that fits the "premise" criteria and then see if they can reach a conclusion about whether each statement is always true, sometimes true, or never true.
Samples with Quadratics: Always true, sometimes true, or never true??
PreAlgebra Examples:
Have kids try these samples, and be sure that they support their claims by providing both examples and counterexamples.
See if they can develop some systems to organize their work and be sure that they have tested every possible case.
Samples with Integers: Always, sometimes, or never true??
Students will need to test in different orders and justify their answers by showing each case and whether it turned out to be a true or false statement.
Geometry Challenges:
These ones work well at the board, because students can watch one another draw the figures and see which cases work out.
If you have access to geometry software like GeoGebra, you can use that as well. Have students draw each triangle and see if they can drag a vertex to meet the criteria.
Samples with Triangles: Always, sometimes, or never true??
Best Ways to Try It:
All the sets I develop with this type of questioning include 36 statements to classify.
My favorite 2 ways for students to work through these types of questions are in partners in a worksheet, or as teams with a sorting activity with a notebook on the side. PARTNER WORKSHEET: I lay the statements out in a grid so that when they color each statement that is NEVER true RED, each statement that is ALWAYS true BLUE, and each statement that is SOMETIMES true PURPLE, it creates a pattern that I can check really easily in just 2 seconds. This makes it really easy to check it over as I walk around and just point to ones that I can see are incorrect. They go back to the drawing board on those statements and continue their (sometimes heated) discussions!
SORTING ACTIVITY:
I cut the statements (larger print version) into cards and have small teams or pairs of students sort them into categories based on whether they are always, sometimes, or never true. Be sure that they have notebooks available, because at any moment they may be required to provide examples to support their answers. You can also hand out the cards and have students come up to the front in a wholeclass setting to tackle challenging ones. These are pretty versatile. It's nice to hear a student model an explanation and show the class how they tested different cases to reach a conclusion. Get PreMade Sets
Each time I develop a set, I pull together my worksheet version and my sorting version so both are included.
Browse the sets that will work well for your classroom and get your kids thinking critically! PreAlgebra: Algebra: Geometry: Get Inspired:To Read Next:
3 Comments
Lise Riehle
8/26/2018 01:49:22 pm
6  (4) is 10!!!
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8/26/2018 02:14:44 pm
Hi Lise,
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11/7/2021 03:30:02 am
showing how the number sentence proves that the statement above is false. The task is to see whether it is always true, so students try to find examples supporting it and counterexamples that prove the original statement false
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