FAST Grade 7 Math common mistakes
FAST Grade 7 Math · Understand the traps and how to avoid them.
When subtracting integers, the student rewrites 5 − (−3) as 5 − 3 and answers 2, or computes −7 + 4 as −11 by adding magnitudes and keeping the sign of the larger number regardless of direction.
The two-minus pattern looks like a double negative in English, so students 'cancel' it. Subtraction of integers is usually taught as a rule to memorize ('keep-change-flip') without a number-line or chip model, so the rule collapses when signs vary.
What to do instead
Have the student re-solve every subtraction on a number line and say the move aloud: 'start at 5, remove three negative units, so move right 3.' Then connect to the equivalent addition 5 + 3. Drill mixed-sign sums in short sets of 8 with immediate checking, and require a written sign check ('same signs add, different signs subtract') before each answer.
The student evaluates −3² as 9 and writes (−3)² = −9, or computes 2³ as 6 by multiplying base times exponent.
Students see the exponent as 'the thing that makes it negative' or as a multiplier of the base. The distinction between the negative sign being inside or outside the parentheses is a notational convention that is stated once and rarely revisited.
What to do instead
Use the read-aloud convention: −3² means 'the opposite of three squared'; (−3)² means 'negative three, squared.' Build a two-column table of paired expressions with values and have the student explain the difference in words. For base-times-exponent confusion, expand 2³ = 2·2·2 and compare to 2·3 side by side.
When setting up a percent problem, the student computes the part but answers with the percent, or solves 'what percent of 80 is 12' by writing 80/12. The percent equation is set up with the unknown in the wrong position.
Percent problems come in three interchangeable forms (find part, find whole, find percent), and students pattern-match on the numbers rather than identifying which quantity is missing. The word 'of' is treated as a cue for multiplication without deciding what the whole is.
What to do instead
Require a labeled setup before any arithmetic: part = percent × whole, with the unknown marked. Have the student write 'the whole is ___ because it follows of' in every problem. Practice the same numbers in all three forms (12 is what percent of 80; 12 is 15% of what; what is 15% of 80) so the structure, not the numbers, drives the setup.
The student treats a non-proportional relationship as proportional, e.g., claims a table with points (1, 3) and (2, 5) is proportional by checking only that it increases, or assumes any line on a graph is proportional.
Students equate 'goes up steadily' with 'proportional' and skip the two tests that define it: a constant ratio y/x and passing through the origin. Additive patterns (add 2 each time) and multiplicative patterns (multiply by 2 each time) look identical in a short table.
What to do instead
Enforce a two-check routine for every table and graph: (1) compute y ÷ x for at least three rows and confirm the quotient is identical; (2) confirm the graph passes through (0, 0). Give mixed tables where some add and some multiply, and have the student sort them into proportional and non-proportional with the ratio written next to each row.
When solving 3(x + 4) = 27, the student distributes to only the first term (3x + 4 = 27), or when solving 2x + 5 = 3x − 1, moves a term without changing its sign.
The distributive property is learned as 'multiply the thing in front' and the second term inside the parentheses is visually less salient. For equation solving, students perform operations on one side only or forget that subtracting a term from both sides flips its sign.
What to do instead
Have the student draw an arrow from the outside factor to each term inside the parentheses before distributing, and check by substituting the solution back into the original equation. For moving terms, require writing each step on its own line with the operation labeled ('subtract 2x from both sides'), and verify with substitution every time until it becomes automatic.
The student solves an inequality such as −2x > 8 and reports x > −4, failing to reverse the inequality symbol when dividing by a negative number.
The reversal rule has no intuitive anchor for students who treat inequalities as equations with a different symbol. Because the algebra steps look identical to equation solving, the symbol is carried down unchanged.
What to do instead
Test the rule numerically: pick x = −5 and x = −3, substitute into −2x > 8, and let the student see which values actually work. Then state the rule as 'dividing or multiplying by a negative flips the sign' and require the student to circle every negative divisor before writing the final answer. Graph the solution on a number line so the direction is visible.
The student confuses area and circumference formulas for a circle, using 2πr for area or πr² for circumference, and uses the diameter in place of the radius.
The two formulas share the same symbols and are introduced close together; πr² and 2πr differ only by a squared symbol and a factor of 2. Problems that give the diameter invite the substitution error because the number is used directly.
What to do instead
Anchor the formulas in meaning: circumference is the distance around (one-dimensional, so no square), area covers the inside (two-dimensional, so squared units). Have the student label every answer with units (cm vs cm²) as a self-check, and require a first step that writes 'r = d ÷ 2 = ___' whenever the diameter is given.
When finding the surface area of a prism or the volume of a composite solid, the student counts only the visible faces, omits the top and bottom, or adds volumes of overlapping regions twice.
Students work from a picture rather than a net, and in a drawing the back and bottom faces are hidden. Composite figures require decomposing the shape, and students often cannot decide where one solid ends and another begins.
What to do instead
Require drawing the net and labeling each face with its dimensions before computing, then summing the labeled areas. For composite solids, have the student sketch the decomposition and mark the shared boundary, computing each piece separately and recording which faces are internal (and therefore not part of the surface). Check units: surface area in square units, volume in cubic units.
The student assumes two angles are equal because they 'look the same' - treating any pair of angles formed by a transversal as congruent, or calling supplementary angles complementary.
Angle relationships (vertical, adjacent, alternate interior, corresponding, supplementary, complementary) are visually similar in textbook diagrams, and students rely on appearance rather than on the defining position of the angles relative to the parallel lines and transversal.
What to do instead
Have the student name the relationship and its reason before writing an equation: 'these are vertical angles, so they are equal' or 'these are a linear pair, so they sum to 180°.' Practice on diagrams deliberately drawn not to scale so appearance cannot be trusted, and keep a reference sheet that defines each relationship by position, not by size.
The student generalizes from a biased or small sample, or reports that a sample is representative without checking how it was selected - for example, concluding about all students from a survey of the basketball team.
Students conflate 'a sample' with 'a random sample' and have no framework for judging bias. The idea that sample size and selection method both affect validity is usually stated but not practiced.
What to do instead
Use a two-question check on every sampling scenario: who was asked, and how were they chosen? Have the student identify the population, the sample, and one specific reason the sample might not represent the population. Compare predictions from samples of size 5, 25, and 100 drawn from the same population to make variability visible.
The student describes a data distribution using only the mean, ignoring spread and shape, or compares two data sets by mean alone when one has much greater variability.
Mean is the first measure taught and the easiest to compute, so it becomes the default summary. Students rarely connect a larger mean absolute deviation or interquartile range to a statement about consistency or overlap between distributions.
What to do instead
Require a three-part description of every distribution: center (mean or median), spread (range, IQR, or MAD), and shape (symmetric or skewed, any outliers). For comparison tasks, have the student write two sentences, one about center and one about spread, and then judge whether the difference in centers is meaningful given the spread.
The student computes the probability of a compound or complementary event incorrectly - adding probabilities for independent events, or failing to use 1 − P(event) for 'at least one' and complement questions.
The addition rule for mutually exclusive events is over-applied to independent events, and the complement is not recognized as a shortcut because students try to enumerate every favorable outcome instead.
What to do instead
Have the student classify the event first: single event, complement, or compound. For complements, require writing 'P(not A) = 1 − P(A)' before computing. For compound events, use a sample space (list, table, or tree diagram) so the count of favorable outcomes is visible, and compare the result to the multiplication rule for independent events.