.png)
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” (expressions being added or subtracted) with “factors” (expressions being multiplied).
- Teachers should avoid using the word “cancel.” Instead, say “reduce to one” or “sum to zero.”
In an expression like
x² + 3x – x²
instead of saying, “x² and –x² cancel,”
say, “x² and –x² sum to zero.”
In an expression like
instead of saying, “x² and x² cancel,”
say, “x² over x² reduces to 1.”
This explicit language helps prevent students from thinking they can “cancel” the x² out of an expression like
2) Not understanding what “like terms” are
Examples
x² + x ≠ x³
sinθ + sin²θ ≠ sin³θ
Diagnosis 🔍
- Students treat exponents or functions as items that can be casually combined during addition, failing to understand that different powers represent entirely different quantities.
- Remind your students that if they are unsure if x² + x = x³, they should substitute a number for x.
For example, let x = 3.
Does 3² + 3 = 3³ ?
9 + 3 ≠ 27
This confirms that x² + x ≠ x³.
3) Misapplying Exponent Properties to Polynomials
Example
Diagnosis 🔍
- Students are incorrectly applying the Power of a Product Rule, (ab)ⁿ = aⁿbⁿ, to a binomial addition or subtraction problem (a + b)ⁿ.
- Avoid using the word “distribute” when teaching properties of exponents so students do not associate exponents around parentheses with distribution.
- Remind students that exponents are a shortcut for repeated multiplication. Therefore, (x + 2)² is the same thing as (x + 2)(x + 2). Have students write out the binomials fully and multiply using the distributive property (sometimes called FOIL) until it becomes second nature.
4) Incorrectly taking roots
Example
Diagnosis🔍
- This error is similar to the previous error and 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) Forgetting to Distribute Fully
Examples
- Students either do not fundamentally understand the Distributive Property or lose focus mid-problem and forget to multiply the outer term by all the inner terms.
- In the second example, students often get confused by the standalone negative sign because nothing is written between it and the open parenthesis.
- Students need a visual or structural reminder of the Distributive Property (such as drawing arrows from the outside term to all inside terms).
- In the second example, show students how a 1 can be written just before the open parenthesis: –(x – 5). This helps them visualize that they are distributing a negative 1 in the first step, leading correctly to –x + 5.
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) The "Cross-Multiply" Overuse
Example
does NOT lead to
Diagnosis 🔍
- Students learn "cross-multiplication" as a gimmick and try to use it as a universal fix for any problem that contains two fractions, regardless of the operation linking them.
- Ensure students understand that cross-multiplication is a property reserved exclusively for proportions (an equation where a fraction equals another fraction, like a b = c d
- You can use this poster of the Properties of Proportions (available for FREE) to remind students when and why cross-multiplication is mathematically valid.
8) Misunderstanding the Zero Product Property
Example
x(x – 1) = 2
does NOT lead to
x = 2 or x – 1 = 2
Diagnosis 🔍
- Students memorize the mechanical step of splitting factors into separate linear equations, but ignore the fundamental requirement: the equation must equal zero for the logic to work.
- Reinforce the definition of the property by pointing out its name: the Zero Product Property. Emphasize that if two things multiply to equal 4, those numbers could be 2 and 2, 1 and 4, 8 and 0.5, etc. There isn't a unique restriction.
- However, if two things multiply to equal 0, one of them absolutely must be zero. The equation must always be set to 0 before factoring and splitting.
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 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) Misapplying Logarithmic Properties
Example
log x = log x + log(x – 4)
does NOT lead to
x = x + (x – 4)
Also,
log (x + y) ≠ log x + log y
- Students are not familiar enough with the Properties of Logarithms. They forget to apply the Product Property before applying the One-to-One Property.
- In the second example, students confuse the internal arguments of logarithms with the external operations, incorrectly applying a distributive-style property to log(x + y).
- Explicitly contrast the properties side-by-side on an anchor chart:
- Product Property: log(a𑛀b) = log(a) + log(b)
- Quotient Property: log(a/b) = log(a) – log(b)
- Have them evaluate numerical expressions like log(2+8) vs. log(2) + log(8) on a calculator to see visually that the values are not equivalent.
- 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 do not understand function notation. Because they spent years learning that parentheses indicate multiplication, they read sin(x) or f(x) as "sin times x " or "f times x."
- Always say the word "of" when reading functions aloud (sin of x, log of x) to reinforce the input/output concept.
- Students usually learn early on that f(2) ≠ 2f. Use that structure familiarity to explain that sin(2) does not mean multiplying sine times 2. This should help them avoid a mistake like (x)(sin(x)) ≠ sin(x²)
- Insist that students write the parentheses around arguments, i.e., sin(x) instead of sin x, to visually clarify that the variable is safely locked inside an input/output relationship.
12) Incorrectly Splitting Fractions with Polynomial Denominators
Example
Diagnosis 🔍
- Students recall that fractions can sometimes be split into two pieces, but they fail to realize that you can only decompose a fraction across addition/subtraction if those operations are located in the numerator, over a monomial denominator:
or
- What makes it even more confusing is you CAN do this:
and this:
Treatment 🩹
- Show students what they are actually doing by looking at fraction arithmetic backward:
- Valid: a/x + b/x = (a+b)/x (This works because they share a common denominator).
- Invalid: Try adding x/a + x/b using actual fraction operations. The common denominator is ab, yielding (xb +xa)/(ab), which looks absolutely nothing like x/(a+b).
- Reinforce that splitting a fraction with addition is simply the inverse of adding fractions with a common denominator.
- Students should check their answers to determine whether they would obtain the given expression if they “operated” on their answers.
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, I will do this as a whole-class activity. Below are some examples I have used in my calculus class:
|
 |
| 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 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.
.png)
.png)
.png)
.png)