Nix the Tricks

Fix: there is no need for students to memorize a rule here. They should be able to reason about adding integers (and extrapolate to the reals) without difficulty. Students are comfortable adding integers on the number line, all they need to add to their previous understanding is that a negative number is the opposite of a positive number.

2 + 5 ⇒ start at 2, move to the right 5 spaces

2 + (–5) ⇒ start at 2, move to the right –5 spaces ⇒ start at 2, move to the left (opposite of right) 5 spaces

2 – 5 ⇒ start at 2, move to the left 5 spaces

2 – (–5) ⇒ start at 2, move to the left –5 spaces ⇒ start at 2, move to the right (opposite of left) 5 spaces

Nix Two negatives make a positive (Integer Addition)

Because it’s meaningless

2 – (–5) ⇒ start at 2, move to the left –5 spaces ⇒ start at 2, move to the right (opposite of left) 5 spaces

start at 2, move to the right 5 spaces ⇒ 2 + 5

Thus, 2 – (–5)=2 + 5

Nix Two negatives make a positive (Integer Multiplication)

– · – = + etc.

Because many students apply this to addition as well as multiplication / division

negative + negative = positive (right?)

Fix: there is no need for students to memorize a rule here. They should be able to reason about multiplying integers (and extrapolate to the reals) without difficulty. One option is to use opposites:

(+2) ∙ (+3) = +6

so what should (–2) ∙ (+3) equal?

the opposite of +6 of course!

(–2) ∙ (+3) = –6

so what should (–2) ∙ (–3) equal?

the opposite of –6 of course!

Another option is to use patterning since students are already familiar with the number line and can continue to skip count their way right past zero into the negative integers.

5 ∙ 2

5 ∙ 1

5 ∙ 0

5 ∙ –1

5 ∙ –2

Nix PEMDAS “Please Excuse My Dear Aunt Sally” (aka ‘BEDMAS’)

Because students read it in order as presented, meaning they multiply before dividing regardless of order

6 ÷ 2 · 5 = 0.6 even though · 5 = 3 · 5 = 15

students also don’t know how to “do parentheses first” when the expression is 4(3)

Fix: Replace with GEMA. G is for grouping, which is better than parentheses because it includes brackets, absolute value and expressions under square roots, in the numerator of a fraction, etc. Grouping also implies more than one item, which eliminates the confusion presented by 4(3). Since only one letter appears for both operations, it is essential to emphasize the important relationship between multiplication/division as well as addition/subtraction. As soon as fractions are introduced talk about the equivalence of dividing and multiplying by the reciprocal. As soon as integers are introduced talk about the equivalence of subtraction and adding a negative.

Nix The log and the exponent cancel

Because cancel is a vague term, it invokes the image of terms/values/variables magically disappearing

Fix: Refer to inverse functions or rewrite the log in exponent form, then root each side. This is another reason to teach the definition of logarithms; then there is no confusion about the relationship between logs and exponents.

log2(x) = 4

[students think this looks scary and it is rather ridiculous to write this if you know the definition]

The log asks the question ‘base’ to what ‘power’ gives ‘value’

logbase(value) = power

basepower = value

24 = x

[invoke the definition, it’s much simpler!]

Nix The square root and the square cancel

Because cancel is a vague term, it invokes the image of terms/values/variables magically disappearing

Plus- the square root is only a function when you restrict the domain!

Fix: Insist that students show each step instead of cancelling operations.

Example 1: because = = 5

Example 2: x2 = 25

if you “cancel” the square with a square root you miss a solution.

x2 = 25 x =

Example 3: : Have partners graph and 2 in their graphing utilities and compare results.

Ratios and Proportional Relationships

Nix Butterfly Method aka Jesus Fish

Because students have no idea why it works and there is no mathematical reasoning behind the butterfly, no matter how pretty it is.

Fix: If students start with visuals they will see that they need to have like terms before they can add. Then the concept of getting common denominators will make sense. Don’t insist on least common denominators, many visual/manipulative methods won’t give least common denominators (instead using the product of the denominators) and that’s just fine!

Nix Least Common Denominator

Because fractions can be compared and added/subtracted with any common denominator, there’s no mathematical reason to limit students to the least common denominator

Fix: Accept any denominator that is computationally accurate. Students will eventually gravitate towards the least common denominator as they look for the easiest numbers to work with. Encourage students to compare different methods — do different common denominators give different answers? Are they really different or equivalent? How did that happen?

Fractions can even be compared with common numerators — a fascinating discussion to have with students of any age!

Nix Cross Multiply (fraction division)

Because division and multiplication are different (albeit related) operations, one cannot magically switch the operation in an expression. Plus, students confuse “cross” (diagonal) with “across” (horizontal).

Fix: Use the phrase “multiply by the reciprocal” but only after students understand where this algorithm comes from. The reciprocal is a precise term that should also remind students why we are switching the operation.

[easy!]

[makes sense]

[not as obvious, but following the above example we are dividing the 4 by 3]

[no idea! but if it looked like the above examples, it would be easier -

get common denominators]

[makes sense]

So what did we do?

Alternate approach: compound fractions

[this method uses the reciprocal as the fraction that will turn the denominator into one]

Nix Flip-And-Multiply aka Same-Change-Flip

Because division and multiplication are different (albeit related) operations, one cannot magically switch the operation in an expression. Plus, students get confused as to what to “flip”.

[easy!]

[makes sense]

[not as obvious, but following the above example we are dividing the 4 by 3]

[no idea! but if it looked like the above examples, it would be easier - get common denominators]

[makes sense]

So what did we do?

Alternate approach: compound fractions

[this method uses the reciprocal as the fraction that will turn the denominator into one]

Nix Cross Multiply (solving proportions)

Because students confuse “cross” (diagonal) with “across” (horizontal) multiplication, and/or believe it can be used everywhere (such as in multiplication of fractions). More importantly, you’re not magically allowed to multiply two sides of an equation by different numbers but still get an equivalent equation.

Correct multiplication of fractions:

Common error:

Fix: Instruct solving all equations (including proportions, they aren’t special!) by opposite operations. Encourage students to look for shortcuts such as common denominator, common numerator or scale factor. Let students extrapolate. Once students know when and why a shortcut works, skipping a few steps is okay, but students must know why their shortcut is “legal algebra.”

Shortcuts:

x=3 [no work required, think about what it means to be equal]

x=6 [multiply a fraction by 1 and it’s equivalent]

x=5 [students are quick to recognize ½ but any multiplicative relationship between

numerator and denominator will have to be the same in both fractions]

[take the reciprocal of both sides of the equation to get

the variable into the numerator]

Arithmetic With Polynomials

Nix FOIL (Binomial Multiplication)

Because it implies an order (some Honours PreCalc students were shocked to learn that OLIF also worked) and is less transferable to later work such as polynomials larger than binomials, and factoring by grouping.

Fix: Replace with “distributive property”, it’s even teachable at the same time as straight distribution.

Example:

(2x + 3)(x – 4)

(2x + 3)(x) + (2x + 3)(–4) [distribute first binomial to each term in the second binomial]

2x2 + 3x – 8x – 12 [distribute each monomial]

2x2 – 5x – 12 [combine like terms]

The box method is also good; remember students learn an array model for multiplying numbers in elementary. Build on their knowledge of partial products.

Example: With variables:

23 · 45 = (20 + 3) · (40 + 5)

= 20 · (40 + 5) + 3 · (40 + 5)

= 20 · 40 + 20 · 5 + 3 · 40 + 3 · 5 (2x + 3)(x – 4)

= 800 + 100 + 120 + 15 2x2 + 3x – 8x – 12

2x2 – 5x – 12

Reasoning with Equations and Inequalities

Nix: ‘Hungry’ inequality symbols

Because: students get confused with the alligator/pacman analogy. Is the bigger eating the smaller, how is the inequality the PacMan, would it eat the smaller or bigger, is it the one it already ate or the one it’s about to eat?, etc.

Fix: Instead of equal (same distance on both sides), the bars have tilted to make a smaller side and a larger side. The greater number is on the wider (or open) side.” Note that “bigger” language can be especially confusing when integers are involved. 2 is bigger than –5?

Nix: Follow the arrow (Graphing Inequalities)

Because: students should understand what the inequality symbol means. Plus, x > 2 and 2 < x are equally valid but would point in opposite directions.

Fix: Ask students “Are the solutions greater or lesser than the endpoint?” This is a great time to introduce test points. The symbolic representation is more abstract than the number lines, so having students practice going from context or visual representations to symbolic ones will support student understanding of the symbols. While it is true that x > 2 is the more natural way to say the sentence - “the solutions are greater than two” - students need the versatility for compound inequalities. For example, 0 < x < 2 requires students to consider both 0 < x and x < 2.

Nix: Cancel

Because: cancel is a vague term that hides the actual mathematical operations being used, so students do not know when or why to use it. To many students, cancel is digested as “cross-out stuff”. Magic!

Fix: Instead, require a mathematical operation or description along with those lines drawn through things that “cancel”. In fractions, we’re not canceling out a factor in top and bottom, we’re dividing to get 1. On opposite sides of an equation, we’re subtracting the same quantity from both sides or adding the opposite to both sides. Simplifying is also good language to use here, such as “ simplifies to 1” or “3 + –3 simplifies to 0.”

Nix: Take/Move over to the other side

Because: taking or moving is another vague description. When is it allowed? Why does it work?

Fix: Be more precise! How is it getting there? Using mathematical operations and properties to describe what we’re doing will help students develop more precise language. “We add the opposite to both sides. That gives us zero on the left and leaves the on the right.” Or “If I divide both sides by 3 here, that will give , so 1, on the right and a 3 in the denominator on the left.”

Nix: Switch the side and switch the sign

Because: This hides the actual operation being used. Students might have no idea what they are doing, so will misapply the idea and be unable to generalize appropriately.

Fix: Talk about inverse operations or getting to zero (addition)/one (multiplication) instead. Preserve the idea of maintaining the equality by doing the same operation to both quantities.

Nix: Equals means answer or next step

Because: The equal symbol does not mean “find the result of the calculation given to the left of this symbol.” It indicates that both sides of the symbol are the same.

Non-Examples:

12 + a = 25 – 12 = 13. [Does a = 13 or 12 + a = 13?]

12 + 3 = 15 ∙ 2 = 30. [12 + 3 is not 30]

Fix: Use arrows or write a new equation, usually on a new line. “Run on math sentences” should be discouraged, always with an explanation of what the equals sign means.

Example:

12 + a = 25 a = 25 – 12 a = 13

12 + 3 = 15

15 ∙ 2 = 30

Nix: The “New” equation

Because: new does not indicate any connection or operation. Common student response, “where did that come from?”

Fix: Stop describing equivalent equations as new, instead describe the connection or mathematical process that occurred.

Note: sometimes we are making a new equation, such as in linear systems when we combine two different equations to make a new (non-equivalent) equation.

Example: Completing the square

2x2 + 12x + 5 = 19

2(x2 + 6x) + 5 = 19 [by factoring 2 from the variable terms]

2(x2 + 6x + 9) + 5 – 18 = 19 [since we added 2 · 9 we also have to subtract 18]

2(x + 3)2 – 13 = 19 [now the trinomial is a perfect square of a binomial but the solutions are the same as

those of the original equation, it’s an equivalent equation in a new form]

Interpreting Functions

Nix: What is b?

Because: b is a letter, it doesn’t have any inherent meaning, ask students exactly what you’re looking for.

Fix: y = mx + b is an equation of a line, but it isn’t the only one! Feel free to ask what the intercept is, but it’s much easier to ask students to pick a value for x (eventually recommend 0 since it’s easy!) and then solve for y. This method works for any equation and it shows why we put b on the y-axis. Remind students that, while b is a number, we are concerned with (0, b) which is a point.