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The errors mentioned in this post are mistakes that I have seen from my Precalculus and Calculus students.
Some resources you may be interested in:
- My Favorite Mistakes Posters to hang in your classroom.
- Get a FREE SAMPLE here.
- Read my blog post High School Math Classroom Decorating Ideas.
- Algebra Error Analysis / Common Mistakes
1) Incorrectly reducing rational expressions
Examples:
- Students may not understand the definition of a factor.
- Students may be confusing “terms” with “factors.”
- Teachers should avoid using phrases like “cancel” instead say “reduce to one” or “sum to zero.”
In expressions like this
x² + 3x – x²
Instead of saying, “x² and negative x² cancel,”
say, “x² and negative x² sum to zero.”
In expressions like
instead of saying, “x² and x² cancel,”
say, “x² over x² reduces to 1.”
This should help students to avoid thinking they can “cancel” the x² in an expression like
2) Not understanding what “like terms” are
Examples
x² + x ≠ x³
sinθ + sin²Î¸ ≠ sin³Î¸
Diagnosis
- This is an example of students not knowing what are “like terms.”
- Remind your students that if they are unsure if x² + x is x³, they should substitute a number for x. For example, let x = 3.
On the left: 3² + 3, which is 12.
On the right: 3³ is 27
Since 27 ≠ 12, then you know that x² + x ≠ x³.
3) Incorrectly expanding binomials
Example
Diagnosis
- This error stems from students learning this property of exponents: (ab)² = a²b², then thinking that one can “distribute” the exponent to every term in the parentheses.
- Avoid using the word “distribute” when teaching properties of exponents.
- Remind students that exponents are a fast way to multiply, so (x + 2)² is the same thing as (x + 2)(x + 2)
4) Incorrectly taking roots
Example
Diagnosis
- This error happens after students learn this property of radicals:
then thinking that one can always put the radical on each inside term.
Treatment
- Remind students of the definition of a square root. We know that
because if we square 3, we get 9.
Students can use this logic to check their answer by squaring their answer (e.g. x+2) to see if they get the expression under the radical (x² + 4).
5) Not using the Distributive Property correctly
Examples
- Students either do not understand the Distributive Property or have forgotten to use it.
- In the second example, students often get confused by the negative because nothing is written between the negative sign and the open parenthesis.
- Students need to be reminded of the Distributive Property.
- In the second example, show students how a 1 can be written just before the open parenthesis, which should help them see that they will distribute a negative one in the first step.
6) Not following the Order of Operations
Examples
Diagnosis
- Students want to do the first operation they see on the left and ignore the Order of Operations.
- Remind students to think deeply while they are working and not attempt to work too fast (check out my blog post: Why You Should Not Reward Speed in Math Class)
- If students forget to follow the Order of Operations, they are likely thinking too fast.
- It would also be a good idea to give your students a list of common errors and ask them to correct them, something like this: Algebra Error Analysis / Common Mistakes.
7) Cross multiplying every time two fractions are next to each other
Example
does NOT lead to
6x = 35
Diagnosis
- Students who think they should cross multiply when multiplying fractions have forgotten about the Properties of Proportions.
- My on-level calculus students always said “cross multiply” when I would ask them how to multiply fractions.
- I made a poster of the Properties of Proportions (available for FREE) to remind them about when (and why) we can cross multiply.
8) Not using the Zero Product Property
Example
x(x – 1) = 2
does NOT lead to
x = 2 or x – 1 = 2
Diagnosis
- Students have translated the Zero Product Property (ZPP) to mean “set each factor equal to whatever is on the other side of the equal sign.”
- This is a frustrating error for me because I saw many of my on-level calculus students do this even though they had learned the ZPP years before.
- One way to overcome this error with students is to say the name of the property every time it is used – emphasizing the word “zero.”
- You could even remind students why the property works (if the product of two quantities is zero, of course, one of them must be zero) so they can then reason that it doesn’t work with other numbers (e.g., if the product of two quantities is 12, it does not mean that one of them must be 12.)
9) Dividing by a variable or a function
Examples
x = 4x – x²
does NOT lead to
1 = 4 – x
sin x cos x = sin x
does NOT lead to
cos x = 1
Diagnosis - Students think that since they can divide an entire equation by a constant, they can also divide an entire equation by a variable (or a function).
- I tell students that this is illegal because they are dividing by an unknown quantity and could be dividing by zero.
- It can be difficult for students to realize they made a mistake in this process since they will likely be correct when checking their answers, but they will not have all the answers.
- As a teacher, you can train students to factor out common factors instead of dividing by them.
For example, 2x + 4 = 0
Instead of dividing by 2, you could factor out the 2 to get:
2(x + 2) = 0, then use the ZPP to get
2 = 0 or x + 2 = 0
Obviously, 2 ≠ 0, so that can be disregarded.
10) Not understanding properties of logarithms
Example
log x = log x + log(x – 4)
does NOT lead to
x = x + (x – 4)
Diagnosis - Students who make this mistake are not familiar enough with the Properties of Logarithms. They forget to apply the Product Property before applying the One-to-One Property.
- Remind students of the Properties of Logarithms and ensure they get plenty of opportunities to practice before an assessment.
- Formative assessments (whose grades don’t go in the gradebook) are an excellent way to stop these errors before they cost students points. Read my blog post How to Give a Formative Assessment in Secondary Math
11) Not understanding function notation correctly
Examples
- Students are not understanding function notation and think that sin x represents multiplication.
- Always say the word "of" when talking about a function so that students can make the connection between f(x) and sin(x) or log(x).
- Usually, students work so much with f(x) that they learn early that f(2) ≠ 2f. Therefore, use that to help them understand that sin(2) does not mean multiplying sine times 2 (which is nonsense anyway). This should help them avoid a mistake like (x)(sin(x) ≠ sin(x²)
- Always write the parentheses around the x when using a trig function or log function to emphasize that the x is attached to the “sin” or “log.”
12) Not understanding how to undo operations with rational expressions:
Example
Diagnosis
- Students have “split a fraction” in the past, such as
or
- But they don’t realize that one can only “split a fraction” with addition or subtraction when the denominator is a monomial.
- What makes it even more confusing is you CAN do this:
and this:
Treatment:
- Reinforce with the students that when “splitting a fraction,” they are just undoing either addition/subtraction or multiplication of fractions.
- Students should check their answers to determine whether they would get the given expression if they “operated” with their answer.
General suggestions for fixing errors
- Make an activity where students are given expressions and asked how to simplify each expression - they don't have to actually simplify them, just say *how* to simplify or if it can be simplified.
- If I am limited on time, then I will do it as a whole class activity. Below are some examples I have used in my calculus class:
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| After doing u-substitution examples for integration, I went through this slide with my students. |
Some resources you may be interested in:
- My Favorite Mistakes Posters to hang in your classroom.
- Get a FREE SAMPLE here.
- Read my blog post High School Math Classroom Decorating Ideas.
- Algebra Error Analysis / Common Mistakes
In summary
To correct most of these procedural errors, students must become more familiar with the rules, properties, and theorems in mathematics. In other words, you've got to know the rules if you want to play the game!Think of the first time you learned to play Monopoly or Uno. Probably someone was there to teach you. Perhaps that person read the rules and How to Play that came in the box. Or maybe someone just said, "the best way to learn to play is to play," and then taught you the rules as you went along.
Well, learning algebra is a bit like that. You have to know what the rules, properties, theorems, and constraints are before you can "play" algebra.
