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.
"Simplify the following expressions. Use the Distributive Property if needed."
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.
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>
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?
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?
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
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.