FAST Math Grade 8 common mistakes
FAST Math Grade 8 · Understand the traps and how to avoid them.
When simplifying expressions with exponents, the student adds the exponents in a product such as 2^3 · 2^4 and writes 2^7 correctly, but then applies the same 'add' rule to a power of a power and writes (2^3)^4 = 2^7 instead of 2^12; similarly they multiply bases in 3^2 · 4^2 and write 12^2 instead of recognizing the bases differ.
The three exponent laws look alike on the page, so students memorize one operation ('add') and apply it to every exponent expression. The underlying meaning - repeated multiplication - is never used to check the result, and textbooks often present the laws as a list to memorize rather than as consequences of counting factors.
What to do instead
Have the student expand each expression once in full: (2^3)^4 = (2·2·2)(2·2·2)(2·2·2)(2·2·2), count twelve factors, and only then state the shortcut. Build a two-column table of 'expanded form' versus 'exponent form' for at least eight mixed examples, including one product with unlike bases to show no law applies. On every practice item, require a one-line justification naming the law used.
The student converts between scientific notation and standard form by counting zeros incorrectly, e.g. writing 3.2 × 10^4 as 320,000 or 0.00032, or writing 45,000 as 4.5 × 10^3. When multiplying numbers in scientific notation they multiply the coefficients but forget to combine the powers of ten, or leave a coefficient such as 24 × 10^5 without rewriting it as 2.4 × 10^6.
Place-value counting is done by eye rather than by reasoning about the exponent's meaning. Students also treat scientific notation as a formatting trick rather than as a product, so they do not see that the powers of ten must be multiplied and that the final coefficient must lie between 1 and 10.
What to do instead
Anchor the exponent to movement of the decimal point: positive exponent moves right, negative moves left, and the exponent equals the number of places moved. Practice with a place-value chart where the student writes the digit string and physically marks each jump. For operations, insist on the two-step check: multiply coefficients, add exponents, then normalize the coefficient and adjust the exponent. Include items where the answer must be renormalized.
Solving multi-step linear equations, the student distributes only to the first term inside parentheses, as in 3(x + 4) = 3x + 4, or drops a negative sign when distributing a negative, as in −2(x − 5) = −2x − 10. When variables appear on both sides, they move terms but forget to change the sign of the term they moved.
Distribution is learned as a quick notation rule rather than as repeated addition, and negative signs are treated as labels attached to numbers rather than as operations. Under time pressure, students do the visually salient step and skip the second term.
What to do instead
Require rewriting every distribution as repeated addition before simplifying: 3(x + 4) = (x + 4) + (x + 4) + (x + 4). For negatives, have the student write the subtraction as adding the opposite first. For moving terms, mandate showing the same operation on both sides on separate lines rather than 'moving' anything. Give a five-item sign audit at the start of each practice set.
When solving a system of two linear equations, the student finds a value for x and stops, reporting x alone as the solution, or substitutes back into the equation they just manipulated and gets an identity, concluding the system has infinitely many solutions. They also accept a graphical intersection read off a hand-drawn sketch without checking it algebraically.
The habit of 'solve for x' from single-equation work carries over, and the ordered-pair nature of a system's solution is not reinforced. Substituting into the same equation used for elimination always yields a true statement, which mimics the infinitely-many-solutions case.
What to do instead
Adopt a fixed solution protocol: solve, substitute into the other equation, write the answer as an ordered pair, then verify in both original equations. Have students classify a system (one solution, none, infinitely many) before solving, using slope and intercept comparison. Include at least one no-solution and one infinite-solution system in every practice set so the distinction is practiced, not just described.
The student treats a function as any rule or graph, accepting a relation such as {(1, 2), (1, 5), (3, 4)} or a circle as a function, and confuses the vertical line test with the horizontal line test. When comparing two functions given in different forms (table versus equation versus graph), they compare only the y-intercepts or only the slopes and declare one 'greater'.
The definition of function is memorized as a phrase without being tied to the input-output mapping, so repeated inputs go unnoticed. Comparison questions require translating between representations, a skill that is assumed rather than taught.
What to do instead
Drill the definition with sorted input lists: for each relation, list the inputs and circle any repeats with different outputs. Practice the vertical line test on graphs of a circle, a parabola opening sideways, and a parabola opening upward, stating what each result means. For comparison items, require a table with columns for slope, y-intercept, and a stated point of comparison before any conclusion is written.
Applying the Pythagorean theorem, the student always computes a^2 + b^2 = c^2 even when the unknown is a leg, and adds the squares instead of subtracting. They also apply the theorem to triangles that are not right triangles, or use the hypotenuse as a leg because it is drawn at the bottom of the figure.
The formula is recalled as a string rather than as a relationship about the side opposite the right angle. Diagrams in textbooks usually place the hypotenuse on a slant, so a rotated triangle breaks recognition.
What to do instead
Before any calculation, have the student mark the right angle and label the side opposite it as c. State the equation in words first ('leg squared plus leg squared equals hypotenuse squared'), then substitute. For leg-unknown problems, require the rearrangement c^2 − a^2 = b^2 to be written before numbers appear. Include rotated triangles and one non-right triangle as a discrimination item.
Performing transformations, the student applies a translation in the wrong direction (moving left when the rule says x + 3), reflects across the wrong axis (negating y when reflecting over the x-axis), or rotates 90° clockwise when the item specifies counterclockwise. When describing a sequence that maps one figure to another, they give only one transformation when two are needed.
Coordinate rules are memorized as sign patterns without a spatial image, and clockwise versus counterclockwise is not tied to a physical turn. Sequence problems require composing transformations, which students rarely practice because single-transformation items dominate.
What to do instead
Have students perform every transformation on graph paper with a labeled pre-image and image, then record coordinates and derive the rule themselves. Use a physical quarter-turn of the paper to fix rotation direction. For sequences, require naming the intermediate image explicitly (A to A′ to A″) and checking congruence of side lengths and angles at each stage.
Reading a scatter plot, the student describes association using only the number of points in each quadrant, calls any downward trend 'no correlation', or treats a strong correlation as proof that one variable causes the other. When asked to interpret a line of best fit, they report the slope as a prediction rather than as a rate of change per unit.
Scatter plots are often taught as vocabulary (positive, negative, none) without practice in describing strength, form, and outliers together. Causation is a familiar phrase but is not connected to the specific context of the data.
What to do instead
Use a three-part description frame for every scatter plot: direction, strength, and any outliers or clusters, each supported by a specific observation. For the line of best fit, require a sentence of the form 'for each additional [x-unit], the predicted [y] changes by [slope]' and a separate sentence for an actual prediction. Include a data set with a strong association but a plausible confounding variable to discuss causation explicitly.
Working with a two-way table, the student computes a joint relative frequency when the question asks for a conditional relative frequency, or divides by the grand total when the question restricts to a row or column. They also read the wrong cell for 'given that' questions and report a percentage without identifying the denominator.
Joint, marginal, and conditional frequencies share the same table and differ only in the denominator, which is invisible in the arithmetic. Students focus on locating the cell and skip identifying the group being conditioned on.
What to do instead
Require writing the denominator before dividing: for 'of the students who play a sport, what fraction also ...', underline the conditioning phrase and write 'denominator = row total for sport players'. Practice labeling all three frequency types on the same table with color-coded denominators. Include items where the answer is a count rather than a relative frequency so the distinction stays visible.
In multi-step modeling problems, the student extracts numbers and operates on them in the order they appear in the text, ignoring units and the question actually asked. They give a bare number with no unit, round too early in intermediate steps, or answer 'how many' with a decimal such as 4.5 buses.
Word problems are read for keywords rather than for the situation, and there is no habit of planning before computing. Early rounding and unit neglect are invisible when the final answer is checked only against the answer choices.
What to do instead
Enforce a written plan before computation: define the variable, list given quantities with units, state the target quantity and its unit. Require carrying full precision through intermediate steps and rounding only at the end, with the rounding decision justified by context. Finish every item with a reasonableness check ('does a fractional number of buses make sense?') written in one sentence.