Monthly Archives: December 2013

While lesson planning for calculus class, I was thinking this morning about common mistakes students make during simplification. Teaching students how to combine like terms can be a challenge. I'd like to share how I approach teaching combine like terms and factoring. Consider the Algebra 1 exercise below.

Textbook instructions:
"Simplify the following expressions. Use the Distributive Property if needed."

Problem:

How often do we see struggling students claim the above binomial is equivalent to 13x? We can quickly diagnose the misconception the student has... the student saw the addition symbol and combined the integers 7 and 6 as they did in their youth. The not-so-helpful hint "use the Distributive Property if needed" may distract the student from what they are being asked to do. I would pose the exercise in a different way.

Modified instructions:
"Simplify."

Problem:

Note: Some may argue writing "simplify if possible" would be better since we may encounter some problems like this one where there really isn't any 'work' to be done. Instead of writing "not possible," I would rather see my students recognize the quantities 7x and 6 are relatively prime (with GCF of 1) because the value of x is unknown. To facilitate the teaching of factoring later in algebra, the student must also recognize when to "stop." Students will often ask, "How do I know when I can't factor it any further?" I like to revisit the definitions of prime and composite to address this notion of knowing when to stop.

I have my students write the following:

Physical models like algebra tiles are one way to approach modeling the reason why we cannot combine 7x and 6. Let's look at another approach as to why we should not combine 7x and 6. Here's a contrived exchange between a student and a teacher that looks like how I approach this in my classroom verbally.

Teacher: What is the exponent on the x? <points to the x in 7x>

Student: Zero!

Teacher: If the exponent were a zero, we know any nonzero number to the zero power is 1. Then we would be multiplying the 7 by a 1, and we know multiplying any number by 1 does not affect the number's value. Also, we use zero to represent 'none.' So, if the exponent were a zero, then there would be 'no' x.

Student: Then the power must be a 1.

Teacher: That makes sense. What is the power of x on the 6? <points to 6>

Student: There's no x there!

Teacher: Would it be legal to draw a  ghost x0 to mean there is no x there?

Student: Okay.

Teacher: On page 89 of our algebra text, the author states "3x and 5x are like terms because they contain the same form of the variable x." The author then says 3x + 5x = 8x. What do you think the author means by 'same form'?

Student: The powers of x match in both.

Teacher: If that's true, then is it possible to combine 7x and 6 using addition or subtraction?

Student: No.

Then the curtains fall. The teacher and students move on to another problem. But let's take a second look. We still need a convincing argument to demonstrate the expressions 7x and 6 cannot be combined. Let's look at some specific cases.

Suppose x = 1. Then the student is correct, since 7(1) + 6 = 13. But this substitution does not allow x to vary freely. Another way we can convince the student we cannot combine 7x and 6 would be to use a graph.

The fact that the red line y = 13 and the blue line y = 7x + 6 is pretty convincing evidence the two quantities 13 and 7x + 6 are not the same for any x value other than 1, the intersection point on the graph. We can counsel the student that y = 13 is constant while y = 7x + 6, the line with slope 7 and y-intercept 6, depends on x. Let's consider another special case.

Suppose for some integer . We can look at some specific cases to help our thinking about the consequences of this selection for x.

We can now discuss the earlier statement, "Use the Distributive Property if needed." If we consider the case where x is an integer multiple of 6, then we have

7x

 

 

 

7k and 1 are not like terms, just as 7x and 6 are not like terms. 7k + 1 only equals 8 if k = 1. Why even think this way? Because we can tie the original problem back to factoring.I tell my students factoring is like playing tennis. Metaphorically speaking, we know a volley is over with in tennis based on what the ball does. We know we are 'done' factoring when we have an expression made of prime factors. I tell my students factoring is essentially a two step process:

Step 1. Greatest Common Factor (GCF)

Step 2. Depends on terms that remain. Applying a possible strategy may involve the whole collection of terms or a subset of the terms in the problem.

We volley back and forth - do both steps - until we obtain expressions that are prime or constants that can be written as the product of primes. For instance, in the problem , we could rewrite 6 as the product of 2 and 3... but the binomial can be rewritten as the product of prime linear factors x + 4 and x - 4.

Why do we consider x + 4 prime? Since the value of x varies, if we only consider integer values of x, then x + 4 may give a composite number or a prime number. Since we do not know the value of x, we think of x + 4 more conservatively as prime.

It does not take much to convince students that all numbers are divisible by 1. I insist my students look for a GCF every time, even in trivial problems, and identity the GCF is 1, or more specifically, . I try to emphasize that when we factor expressions with multiple terms, we look to factor out the "lowest" power the variable.

If we try and factor the original problem using this approach, we have

since GCF(7, 6) = 1 and GCF(x, x^0) = x^0 = 1. This approach is particularly useful for later work in calculus, specifically when taking derivatives of expressions. My calculus students sometimes struggle with reconciling their solutions with the answer the text provides because they may not totally understand the factoring necessary for simplification. Students stumble when asked to find the GCF of the expression . If students are accustomed to identifying the GCF, then problems of this type aren't as troublesome.

My self-imposed holiday blogging break is over. <cracks knuckles>

The holiday break affords teachers the opportunity to reflect. One dimension I think many overlook in teacher reflection is to sit and think deeply about one's content area. I would even be bold enough to say the best math teachers are mathematicians at heart. When I sit down to think about mathematics, as a high school math teacher, I like to approach it like a person that can solve a Rubik's Cube. When you watch this feature below on a speed solver, Chester Liam makes the following claim about speed solving a Rubik's Cube:

As a speed solver? No, there is no math involved, no thinking involved. It's just finger dexterity and pattern recognition. There is nothing, no thinking involved in the entire solving process. [1:02 - 1:14]

We can unpackage Chester's thoughts about the Rubik's Cube and how he solves it so quickly by applying mathematical structure to the cube. However, we should ask: what is the objective? What are we trying to do? There's gobs and gobs of mathematics wrapped up in speed solving a Rubik's Cube, but if Chester were to pay attention to the procedures he is applying, it would inhibit his ability to solve the cube quickly.

But consider this: what if Chester attempts to teach how to solve the cube to another person? What would he have to do? What examples, explanations, and demonstrations would he utilize to teach his pupil speed solving? What implications does this thinking have on teaching mathematics? There would be many features of speed solving the learner may not perceive until Chester brings it to the learner's attention. And if Chester's choices are calculated, deliberate, purposeful... the learner may be helped or hindered dependent upon Chester's ability to communicate his thinking which leads to the automaticity of the procedures he applies to solve the cube. Just like learning how to read or learning how to drive a car, we want to teach learners how to do these tasks so well they become 'automated' at some level.  To understand mathematics deeply, I believe it is often necessary to unpackage some of these automated tasks. I will share an example of such a mental exercise below.

Today I've been thinking about procedures we accept as true while doing math at the high school level, algebra in particular. As an example, suppose we want to determine the location of the x-intercept for the line 5x - 3y = 7. We might approach this 'task' in the following way:

And we might even graph the line to confirm our solution...

Yep. There it is. The x-intercept at (1.4, 0). As a student, we might simply yawn and move on to the next exercise. The student must recognize the y-coordinate of the line will be zero when the line crosses the x-axis. Yes, we have a solution, but I'm not so sure as a math teacher I'm satisfied to stop there. What if we take a different approach? Let's turn the Rubik's Cube and look at the problem another way. A 'typical' algebra student might subscribe to the church of y = mx + b and do the following...

Just another equally valid path to the value of the x-intercept. I'd like to focus on something in one of the above lines.

This statement says five-thirds of some mystery number is seven-thirds. So, what's the mystery number? I think if we asked a room full of high school math teachers to draw a diagram explaining why we multiply each side of this equation by the multiplicative inverse of 5/3, namely 3/5, we would get some really surprising results. It's not an indictment of teacher education. Rather, it's to say some teachers may not have considered the how's and why's of this procedure before for the same reason someone learning to speed solve a Rubik's Cube may miss a key structure. They may not have the experience of needing to know why it works. Rather, it was more important that they can find the x-intercept of the line; the multiplication by the multiplicative inverse was viewed as "below" the task or level at hand. Perhaps a student never asked "why?" at the critical moment to give the teacher pause.

I struggled to produce the corresponding fraction diagram without trying to reverse engineer the solution. When I write this blog, I often worry I might make a mistake that will be indelibly written into the electronic space of the Internet. But this worry violates the spirit of my blog's theme: that we need to make mistakes in a good direction to evolve our mathematical understanding. Here's the images of my failed by-hand attempts to generate this fraction diagram:

ChickenScratchesPage_1Chicken scratches, page 1

ChickenScratchesPage_2

 

 

 

 

 

 

 

 

 

 

 

 

 

Chicken scratches, page 2

I had trouble thinking about making the diagram because 5/3 and 7/3 have the same denominator, so I wrote out some equivalent fractions. Then, I wanted to use "ninths" because that made sense to me in terms of the grid on the graph paper I had cut apart. But, I then realized I would need to cut fifths to find the mystery number, and I was REALLY struggling with trying to free-hand cut fifths with the grid in the background. That led me to coordinatize the points of the polygon. Then I had a problem with relationships between the linear units on the horizontal axis and the area (the fact the polygon is not "one" unit vertically... which is basically the notion of a unit fraction we see emphasized in CCSSM). So I abandoned the paper approach in favor of Geogebra because I could generate better precision and be more efficient with respect to time. <Sorry for the sloppiness of this paragraph, but it does describe my thinking and the mistakes I made.>

Below is an image of the fraction diagram I constructed using Geogebra.

When stating the equation , we should think of it as a verbal statement: "Seven-thirds is five-thirds of what mystery number?" Well, if the green polygon represents a whole, then the orange polygon is one-and-two-thirds of that whole. Then the green area of 1.4, which equals 7/5, corresponds to the solution. Mentally, how in the world did I end up with fifths, then? How did I know to cut the horizontal into fifths using vectors and vertical lines in the coordinate plane?

We can think of a fraction in the most basic way. Consider 5/3. If the denominator indicates the number of pieces we partition from a whole, and the numerator indicates how many pieces we possess, then I knew we needed to cut the orange rectangle in a way that would make five pieces. It gets to the root cause of WHY we invert and multiply. The numerator 5 becomes the desired number of pieces, considered in a denominator.

For the sake of time, I will stop my mental exercise there. Between working the problem, generating the fraction diagram by hand and on Geogebra, and typing this up, I've spent about two hours roughly on this article. This professional development is incredibly powerful for me as a teacher, and it's absolutely free (well, not quite free, I do pay for the website hosting, but you get the idea).

Let's end by stirring the discussion pot. Consider our understanding of how to find the x-intercept of the given line. Does our understanding, or lack of understanding, of the fraction diagram and how to construct the fraction diagram (essentially "invert and multiply" in many high school classrooms) inhibit our ability to solve the original problem? Is it still possible to understand the original solution without knowing all the nuts and bolts of the fraction procedure? Stephen Wolfram argues in favor of using computers to automate trivial computation procedures to help us access problems in the world outside school.  How will our teaching of mathematics change as computers continue to become faster and more powerful?

Professional_Pursuits_Venn_DiagramEach year, like many teachers, I approach the winter break with high aspirations. Retool and revamp all my lesson plans for a particular class to help save me time in the spring. Start and finish books that have spent a long time in the queue on my reading list. Inevitably, there is not enough time to get to all of it. The down time is necessary to refresh, to spend time with family, to remember the reasons why I teach in the first place. This year, I am approaching break a little differently. I am collecting resources to help make my reflection on my teaching more efficient because this time issue will always be there. I want to grow as a math teacher in an optimal way with what little time I have.

The break offers time for reflection on teaching. What should we be after as a society? To prepare students for a world with future problems we can't currently conceive, we need our students to think critically, to persevere, to be problem solvers. For example, when mankind gains the ability to clone organs, what are the ethical implications? This dilemma and many more do not have answers for look-up in the back of a textbook to confirm a 'correct' approach. It is truly in our American society's best interest, all competition issues aside, for all students to do well in math class because high performance in math class will improve life outcomes for all. Here are a few things I have encountered the past few days I plan to spend some time with during my break.

Sahlberg - 3 Fallacies of Teacher Effectiveness in Under-

Performing Nations

Leinwand - Accessible Mathematics: 10 Instructional Shifts that Raise Student Achievement

Hanushek - How Much is a Good Teacher Worth?

Fuller - Teaching Isn't Rocket Science. It's Harder.

Baliga - Math is Not an Innate Skill, Has to be Practiced

Lakoff & Nunez - Where Mathematics Comes From

 

 

 

 

 

 

 

1 Comment

Each semester, I stand in awe of how many students do not understand how to calculate the impact a semester final exam has on their semester grade. Our math department grading policy roughly breaks down in each class to the following:

90% Formative/Summative measures from the semester
10% Cumulative Final Exam

Kids carry many misconceptions about the final exam. A common one is that the final can somehow miraculously overwrite a semester's effort (or lack of effort). Here's a visual representation, to scale, of the 90%/10% model.

Exhibit A: The orange team blows out the red team, 90 to 10.

I have often instructed students to make the following program to help them forecast the impact of the exam on their semester standing.

Exhibit B: Program on the TI-84 for computing a 90%/10% weighted grade.

Here's a numeric example for how this sometimes surprises a student.

Exhibit C: Not much movement of the overall grade, despite a solid final exam score.

This is where the discussion gets interesting. Students will often use trial and error, over and over and over again, with the above program to compute the final exam score they need in order to get an A (which is 90% in our grading scale).

Exhibit D: An A doesn't appear to be in the cards, especially since extra credit does not exist in my class. (I'll share my views on EC on a different day). But I digress.