Number Tricks is a lesson that involves writing and simplifying expressions. It demands the higher order thinking skills called for in the Common Core in several ways. 1) The students are to write a mathematical model for a trick given to them. 2) They are to create their own trick and offer the algebraic expression that represents it. 3) It presses the students to understand the concept of a variable; in this case, the variable represents the number originally chosen. 4) The students are asked to compare their simplified expression to the pattern generated by the various numbers tested. The lesson offers a great opportunity for a high level of critical thinking with a rather low level piece of content.
Here is an erroneous submission from my Algebra class. I want to analyze the mistake and discuss why this lesson was so very good for this student even though the “answer was wrong.”
This was Dewey’s response to creating his own Number Trick, including 3 numbers to generate the pattern, and the algebraic expression it represents:
|Pick a number||3||10||-7||x|
|Add 4||7||14||-3||x + 4|
|Multiply by 2||14||28||-6||2x + 4|
|Subtract 3||11||25||-9||2x + 4 – 3|
|8||15||-2||2x + 4 – 3 – x|
|Simplified:||x + 1|
|Common Result: one more than the number picked|
Now of course we can see that the student should have included the parenthesis when multiplying by 2. The final expression should have been:
2(x + 4) – 3 – x, simplified: x + 5
So the positives? The student is showing that he is solid in his operations with negative integers, that he can simplify correctly and that he is interpreting the final expression properly (x + 1 means 1 more than the original number). The Big Negative? The pattern of numbers does not support the students simplified expression. The resultant numbers are NOT one more than the original number; they are 5 more.
My judgement call here was to ask Dewey if his expression matches the pattern. He couldn’t answer right away. There was disconnect between generating the expression and actually knowing what the expression represented. Once I pointed out that the last numbers in each column where not 1 more than the first, I asked him to find his own mistake, which he did. Dewey was then able to correctly simplify and without assistance verify that his new expressions supported the pattern of numbers.
Dewey did a great deal of complex thinking with a topic as simple as simplifying expressions.