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
||x + 4
|Multiply by 2
||2x + 4
||2x + 4 – 3
||2x + 4 – 3 – x
||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.
I teach a low-level Algebra class (Algebra Essentials). Half of the thirty students are Special Ed or English Language Learners. I was teaching Order of Operations, and rather than offer notes on a subject that they had seen several times before (and still failed), I taught the topic with the 4-Digit Problem instead. (created by College Prepetory Mathematic program, UC Davis)
The 4-Digit Problem goes something like this: You must use four of the same digit (I use 8) to form expressions that have designated values. You may use any of the standard operations represented in “PEMDAS;” you may combine eights to make 88; and you may use any number other than 8 only as an exponent (which then makes roots legal). I did offer some hints:
8 ÷ 8 = 1 80 = 1 √(8 + 8) = 4 3√8 = 2.
I had the students practice by first attempting to express the value 19 with four 8’s. I then had them independently attempt to express the values 1-5. Here is what this supposedly low-level group came up. (multiple solutions shown)
1 = 8 ÷ 8 · 8 ÷ 8 or 8 – 8 + 8 ÷ 8
2 = 880 + 880 or 8 ÷ 8 + 8 – 8
3 = 8 ÷ 8 + 80 + 80 or 80 + 8 ÷ 8 + 80
4 = 80 + 80 + 80 + 80 or 8 ÷ 8 · √(8 + 8)
5 = 3√8 + 3√8 + 3√8 – 80
Their class/homework that day was to then generate the values 6-10, and submit for a grade.
Great day for learning Algebra!
I teach in an inner city high school with heterogeneous classes (in which there are all skill levels), poor attendance, a transient population (students enter and leave all year long), and sadly, many students who, due to different social problems, are unmotivated to learn. The projects which I have seen in your journal assume basic math skills. Unfortunately many of my students have poor skills, and yet are expected to pass Algebra I (which is our lowest level math offered). Needless to say, we have a high failure rate in our school. I would love to do some type of projects with these students, so if you have any other ideas, I would be very happy to hear them.
Marie Feragne (North Providence, Rhode Island)
I have yet to find a teacher anywhere that feels that students come to the course with the proper fundamental math skills. Albiet, your situation is tougher than most. Attack the problem of basic skills; but instead of stopping to reteach them, review the skills briefly. In our Algebra classes at Great Oak High School, we commit Mondays to practicing the arthimetic skills needed for the topics that week. (i.e. integers when solving equations, fractions when simplifying rational expressions). Don’t be afraid to use a project or activity. By constantly presenting complex problems, the students will acquire the necessary skills quicker than you think.
Work hard at getting kids to understand the concepts behind the mathematics. It has been our experience the students have an easier time understanding concepts than memorizing algorithms.
Do more activities in the classroom, and assign less homework. Dynamic lessons are themselves a motivating factor in the classroom. Think of all the faculty meetings and professional development sessions you’ve been through. What could have been done to engage your mind more. The same answer will work for your students.
As far as motivating students, nobody has found the complete answer to that one. We have laid out in detail our grading policies in an article “Nuts & Bolts.” Here is a brief glimpse of two key components, both the carrot and the stick, of that grading system:
- Cumulative Tests & Quizzes: This requires both teacher and students to revisit material continually through the course, instead of just “testing & forgetting.” It also offers the slower student an opportunity to acquire necessary skills. The incentive that we offer is that the student test score (50% of the overall grade) is assigned the test average or the most recent test, whichever is higher. This encourages students to not quit, and offers a light at the end of the tunnel, which is rare in traditional grading systems.
- Incomplete Policy: The student must submit an appropriate product for each major project assigned, or the student will receive an incomplete which equates to an ‘F’ on the report card. In other words, the students may not skip major assignments.
Chris Shore, Editor