AK STAR Math Grade 9 common mistakes
AK STAR Math Grade 9 · Understand the traps and how to avoid them.
The student rounds an irrational intermediate value, such as √2 or π, to two decimal places and then continues calculating, so the final answer is a rounded approximation even when an exact form is expected.
Calculator habits and a preference for decimal answers lead the student to treat rounded values as exact and to lose track of when approximation is acceptable.
What to do instead
Require the student to keep exact forms until the final step unless the problem explicitly asks for a decimal. Have them check by squaring or substituting back, and label any decimal answer as an approximation.
The student applies exponent rules across addition, for example writing 2^3 + 2^4 = 2^7 or x^2 + x^3 = x^5.
The student overgeneralizes the product rule for exponents and treats addition of powers as if it followed the same rule.
What to do instead
Have the student expand each power into repeated factors before simplifying, then compare with the product rule. Use side-by-side examples such as 2^3·2^4 versus 2^3+2^4 to show why the rules differ.
When distributing a negative sign, the student changes only the first term inside the parentheses, for example rewriting -(2x - 5) as -2x - 5.
The student visually attaches the negative sign to the first term only and does not treat the parentheses as a grouped quantity multiplied by -1.
What to do instead
Have the student rewrite the expression explicitly as -1(2x - 5), draw arrows to each term, and then check by substituting a simple value such as x = 1 into both the original and simplified expressions.
When solving an inequality, the student divides or multiplies both sides by a negative number and keeps the inequality symbol pointing in the same direction.
The student follows the same procedure used for equations and does not connect the reversal rule to the meaning of multiplying or dividing by a negative value.
What to do instead
After each operation, have the student test a value from the proposed solution set in the original inequality. Alternatively, rearrange terms so the variable coefficient stays positive, avoiding division by a negative.
The student interprets f(3) as f(x) multiplied by 3, or substitutes 3 for the wrong variable, instead of replacing x with 3 in the function rule.
Function notation is unfamiliar, and the parentheses look like multiplication rather than an instruction to evaluate the function at a given input.
What to do instead
Use a function-machine model: input 3 goes in, the rule is applied, and the output comes out. Have the student write f(3) = (rule with x replaced by 3) and compare with f(3) = 3·f(x) to show the difference.
The student computes slope as y divided by x from the origin rather than as the change in y divided by the change in x between two points, especially when the axes have different scales.
The student confuses slope with a ratio of coordinates and does not attend to the scale markings on each axis.
What to do instead
Have the student label two points, write Δy and Δx explicitly, and compute slope as Δy/Δx. Require them to state the vertical and horizontal scale before calculating and to check that the slope matches the graph’s steepness.
The student applies the Pythagorean theorem to any triangle without first checking for a right angle, or puts the longest side in as a leg rather than the hypotenuse.
The student recognizes the formula and pattern-matches it to triangle problems without verifying the conditions and identifying the hypotenuse.
What to do instead
Require the student to mark the right angle and label the hypotenuse before substituting. Use the converse to test whether a triangle is right, and include examples where the theorem does not apply so the condition is explicit.
The student scales area or volume by the same factor as the side lengths, for example multiplying area by 3 when the scale factor is 3 instead of by 9.
The student treats all measurements as changing linearly and does not distinguish between one-dimensional, two-dimensional, and three-dimensional scaling.
What to do instead
Have the student compare dimensions explicitly: length factor k, area factor k², volume factor k³. Use a rectangle or rectangular prism with numerical side lengths to verify the effect of doubling or tripling each dimension.
The student adds probabilities of overlapping events without subtracting the intersection, or multiplies probabilities for dependent events as if the events were independent.
The student memorizes the addition and multiplication rules without checking whether events overlap or whether one event changes the probability of the other.
What to do instead
Have the student represent the situation in a Venn diagram or two-way table before choosing an operation. Ask whether the events can occur together and whether the first event changes the sample space for the second.
When a two-way table asks for a conditional probability, the student divides by the grand total instead of by the total of the given row or column.
The student focuses on the numerator and does not identify the condition that restricts the relevant group in the denominator.
What to do instead
Have the student underline the phrase that states the condition, then shade the corresponding row or column and use only that subtotal as the denominator. Compare the conditional result with the marginal result to show why the denominators differ.