High School Math 1

NC Math 1 (18 chapters)

Exam: Math 1 EOC · Grade 6 EOG in the Scholars HSM1 section. 28 units.

Units

1. not started01

   Exponents

   laws-of-exponents

   Exponent laws are names for counting factors; negative exponents are reciprocals. Students simplify monomial quotients, convert between rational exponents and radicals, add and multiply simple radicals, and operate in scientific notation.

   Watch for

   $2^3\cdot 2^4=2^{12}$; $3^{-2}=-9$; $\sqrt{a+b}=\sqrt a+\sqrt b$.

   +−3 formulas

   Product and quotient

   $$a^m a^n = a^{m+n},\quad \frac{a^m}{a^n} = a^{m-n}$$

   Power of a power

   $$(a^m)^n = a^{mn}$$

   Negative and rational

   $$a^{-n} = \frac{1}{a^n},\quad a^{m/n} = \sqrt[n]{a^m}$$
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   Algebraic expressions (Math 1)

   algebraic-expressions-m1

   Term, factor, coefficient, degree. Translate, then simplify, then evaluate — including at negative and fractional inputs, and with grouping phrases like “twice the sum of.”

   Watch for

   “twice the sum of $x$ and 3” as $2x+3$.

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3. not started03

   Polynomials

   polynomial-operations

   Standard form is decreasing degree; multiplication is the Math 4 area model with letters. Students classify, add, subtract, and multiply polynomials, and use the special products (a ± b)² and a² − b².

   Watch for

   $(x+5)^2=x^2+25$; subtracting $-(x^2-3)$ as $-x^2-3$.

   +−2 formulas

   Square of a binomial

   $$(a \pm b)^2 = a^2 \pm 2ab + b^2$$

   Difference of squares

   $$(a-b)(a+b) = a^2 - b^2$$
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   Equations and inequalities (Math 1)

   multi-step-equations-literals

   Identity, contradiction, or one number. Students solve with variables on both sides, recognize 0 = 0 and 0 = k, rearrange literal equations, and solve compound AND/OR inequalities.

   Watch for

   dividing by a coefficient that might be zero; AND graphed as two rays.

   +−1 formula

   Literal equation

   $$d = rt \implies t = \frac{d}{r}$$
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5. not started05

   Linear equations and inequalities (two variables)

   linear-forms-half-planes

   Three forms, one line. Students convert among slope-intercept, point-slope, and standard form, use parallel and perpendicular slope relations, and graph a linear inequality with the correct boundary style and half-plane.

   Watch for

   perpendicular of $m=2$ as $m=-2$; shading the wrong half-plane after a correct dashed line.

   +−3 formulas

   Point-slope

   $$y - y_1 = m(x - x_1)$$

   Standard form

   $$Ax + By = C$$

   Perpendicular slopes

   $$m_1 m_2 = -1$$
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   Systems of equations and inequalities

   linear-systems

   One solution, none, or infinitely many. Students solve 2×2 systems by graphing, substitution, and elimination, classify them, model mixture and break-even stories, and shade the feasible region of a system of inequalities.

   Watch for

   elimination that subtracts but forgets to subtract the constant; “no solution” vs “infinitely many” swapped.

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   Relations and functions

   function-foundations

   A function assigns each input exactly one output, and f(x) is a number. Students find domain and range, apply the vertical-line test, evaluate f(a) and solve f(x) = k, distinguish discrete from continuous, and use interval notation.

   Watch for

   $f(x)=x+2$ at $x=3$ written $f(3)=3x+2$; domain of $\sqrt{x-1}$ as all reals.

   +−1 formula

   Function notation

   $$f(a) = \text{the output of } f \text{ at } a$$
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   Factoring polynomials

   factoring-polynomials

   Factoring is multiplying run backward; always GCF first. Students factor a difference of squares, perfect-square trinomials, x² + bx + c, and ax² + bx + c by grouping or the ac-method.

   Watch for

   $x^2-9=(x-3)^2$; $2x^2+10x+12$ factored without pulling 2.

   +−2 formulas

   Difference of squares

   $$a^2 - b^2 = (a-b)(a+b)$$

   Perfect square

   $$a^2 \pm 2ab + b^2 = (a \pm b)^2$$
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9. not started09

   Quadratic equations and functions

   quadratics-parabolas

   Three tools (factoring, square roots, the formula) and one discriminant. Students solve quadratics, read the number of real roots from D, find vertex, axis, and intercepts, sketch the parabola, and model projectile and area stories.

   Watch for

   forgetting $\pm$; $D$ as “the answer”; vertex formula with $a$ dropped.

   +−3 formulas

   Quadratic formula

   $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

   Discriminant

   $$D = b^2 - 4ac$$

   Vertex

   $$x = -\frac{b}{2a}$$
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    Exponential functions

    exponential-growth-decay

    f(x) = a·bˣ with growth for b > 1 and decay for 0 < b < 1, asymptote y = 0. Students evaluate and graph, read a as the start and b as the factor in a story, and write compound-interest and half-life models.

    Watch for

    $2^{x+1}$ graphed as a shift of $2^x$ to the right; linear “add 2 each time” called exponential.

    +−2 formulas

    Exponential model

    $$f(x) = a \cdot b^x$$

    Compound interest

    $$A = P\left(1 + \frac{r}{n}\right)^{nt}$$
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    Graphing functions

    function-features

    A graph is a set of answers; read it instead of recomputing. Students identify domain, range, intercepts, intervals of increase and decrease, extrema, end behavior, and average rate of change, for piecewise and step functions too.

    Watch for

    “increasing” as “positive”; average rate of change as $f(b)-f(a)$.

    +−1 formula

    Average rate of change

    $$\frac{f(b) - f(a)}{b - a}$$
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12. not started12

    Comparing functions

    comparing-function-families

    Linear has constant first differences, quadratic constant second differences, exponential constant ratios. Students classify tables, graphs, and stories, compare growth over an interval, and explain why exponential growth eventually wins.

    Watch for

    a table with almost constant first differences called linear; $r$ of a scatter plot used here (that is ch. 18).

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13. not started13

    Patterns, sequences, and series

    arithmetic-geometric-sequences

    Sequences are functions on the positive integers. Students write explicit and recursive forms for arithmetic and geometric sequences, find any term from two given terms, and sum a short arithmetic series.

    Watch for

    $a_n=a_1+nd$; geometric with $r<0$ skipped.

    +−3 formulas

    Arithmetic

    $$a_n = a_1 + (n-1)d$$

    Geometric

    $$g_n = g_1 r^{n-1}$$

    Sum of first n integers

    $$\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$$
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14. not started14

    Polygons

    polygons-angle-theorems

    Convex versus concave; regular means all sides and angles equal. Students find interior and exterior angles of regular n-gons, recover n from an angle, and place quadrilaterals in the hierarchy (a square is both a rectangle and a rhombus).

    Watch for

    “a square is not a rectangle”; exterior angle of regular octagon as $360/6$.

    +−2 formulas

    Interior sum

    $$S = (n-2) \cdot 180^\circ$$

    Regular interior angle

    $$\theta = \frac{(n-2) \cdot 180^\circ}{n}$$
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15. not started15

    Coordinate geometry

    coordinate-geometry

    Pythagoras on the plane. Students compute distance, midpoint, and a partition point, classify figures from their vertices, find polygon areas on the grid, and prove a quadrilateral is a parallelogram by slope and by distance.

    Watch for

    midpoint as a sum not a mean; distance missing the square root.

    +−2 formulas

    Distance

    $$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$$

    Midpoint

    $$M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$$
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16. not started16

    Euclidean geometry: transformations and congruence

    transformations-congruence

    Rigid motions preserve distance and angle; dilations preserve angle only. Students apply and compose translation, reflection, rotation, and dilation rules, name the motion between figures, and justify congruence (SSS, SAS, ASA, AAS, HL) and similarity (AA, SAS, SSS).

    Watch for

    $90^\circ$ rotation as $(x,y)\mapsto(-x,y)$; SAS~ using two sides and a non-included angle.

    +−3 formulas

    Rotate 90° CCW

    $$(x,y) \mapsto (-y, x)$$

    Reflect over y = x

    $$(x,y) \mapsto (y, x)$$

    Dilate by k

    $$(x,y) \mapsto (kx, ky)$$
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17. not started17

    Statistics: center and spread

    descriptive-stats-spread

    Center and spread, now with standard deviation. Students compute the five-number summary, mean, and s or σ by hand for a short list, apply the 1.5 × IQR outlier fence, and predict how shifting or scaling the data changes each statistic.

    Watch for

    $\sigma$ computed without the square root; outlier rule as $IQR$ beyond the min/max.

    +−2 formulas

    Population SD

    $$\sigma = \sqrt{\frac{\sum (x_i - \bar x)^2}{n}}$$

    Outlier fence

    $$[\,Q_1 - 1.5\,IQR,\ Q_3 + 1.5\,IQR\,]$$
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18. not started18

    Data analysis: two variables

    bivariate-regression

    Two variables: the line is a model and the residuals say whether it was honest. Students read two-way tables, describe scatter plots, fit and interpret a line, interpret r, read a residual plot, and refuse to confuse correlation with causation.

    Watch for

    $r=0.2$ called “strong”; residual plot ignored; “ice cream causes drowning” from a summer scatter.

    +−2 formulas

    Residual

    $$e = y - \hat y$$

    Correlation range

    $$-1 \le r \le 1$$
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    Whole numbers: primes, GCF, and LCM

    gcf-lcm-grade6

    From Math 6 — required for the Grade 6 EOG in the dual-exam section

    This is the last dedicated whole-number unit. Prime factorization should be automatic by now, written in index notation like 84 = 2² · 3 · 7. The greatest common factor is built from the primes two numbers share, each at its lowest power; the least common multiple uses every prime that appears, each at its highest power. Lists of factors and multiples still work for small numbers, but they miss things (GCF by listing and forgetting 12) and do not scale. The same unit revisits order of operations with nested grouping, because every later unit will evaluate expressions. On the AMC 8, GCF/LCM word problems — cycles that realign, ribbons cut into equal pieces — are regulars.

    Watch for

    GCF by listing and missing 12; LCM of 8 and 12 as 96.

    +−3 formulas

    GCF from primes

    $$\text{GCF} = \prod p^{\min}$$

    Shared primes, lowest power each.

    LCM from primes

    $$\text{LCM} = \prod p^{\max}$$

    Every prime that appears, highest power each.

    Product identity

    $$\text{GCF}(a,b) \times \text{LCM}(a,b) = ab$$
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    Fractions: division and multi-step stories

    fraction-operations-6

    From Math 6 — required for the Grade 6 EOG in the dual-exam section

    Division of fractions is the last arithmetic operation to learn, and the one most often done by rote. Keep the sentence “how many of these fit in that”: 3½ ÷ ¾ asks how many three-quarters are in three and a half, which is 4⅔. The rule — multiply by the reciprocal of the divisor — follows from that picture, and inverting the wrong fraction is the classic mistake. Mixed-number arithmetic should be fluent in all four operations. Multi-step bar-model stories (a fraction of the remainder, then what is left) are where the year's fraction work pays off; draw the bar before touching the arithmetic.

    Watch for

    invert the first fraction; $\frac{1}{2}\div\frac{1}{8} = \frac{1}{16}$.

    +−2 formulas

    Division of fractions

    $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$$

    Invert the divisor (the second fraction), never the first.

    Fraction product

    $$\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$$
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    Decimals: terminating and repeating

    decimals-terminating-repeating

    From Math 6 — required for the Grade 6 EOG in the dual-exam section

    All four decimal operations should be automatic, including mental multiplication and division by powers of ten. The new idea is that some fractions terminate as decimals and some repeat forever. After simplifying, a fraction terminates exactly when its denominator has no prime factors other than 2 and 5 — so ⅛ terminates (0.125) and ⅙ does not (0.1666…). Converting among fractions, decimals, and mixed numbers in both directions is the fluency goal. Two misconceptions to watch: writing 1 ÷ 3 as 0.3, and dismissing 0.999… as “not a real number.”

    Watch for

    $1\div 3 = 0.3$; $0.\overline{9}$ dismissed as “not a number.”

    +−2 formulas

    Terminating test

    $$\frac{a}{b} \text{ terminates} \iff b = 2^m 5^n \text{ (after simplifying)}$$

    Repeating notation

    $$\frac{1}{3} = 0.\overline{3},\quad \frac{1}{4} = 0.25$$
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22. not started04

    Negative numbers and absolute value

    integers-absolute-value

    From Math 6 — required for the Grade 6 EOG in the dual-exam section

    Negative numbers enter through context: temperature, elevation, debt. Minus three is not “less than nothing”; it is three units to the left of zero. Absolute value is distance from zero, so |−4| is 4 and never −4. Students order integers, negative fractions, and negative decimals on one number line, where −7 is less than −3 because it is farther left. Operations on negative numbers (adding, multiplying) are Math 7; this unit is about position, comparison, and opposites. Do not skip it to get to algebra sooner — negative numbers leak into every later unit.

    Watch for

    $-7 > -3$; $\lvert -4\rvert = -4$; “two minuses make a plus” used before operations are taught (operations on negatives are Math 7).

    +−2 formulas

    Absolute value

    $$\lvert x \rvert = \text{distance from } 0$$

    Always non-negative.

    Opposites

    $$-(-a) = a$$
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    Ratios

    ratios-grade6

    From Math 6 — required for the Grade 6 EOG in the dual-exam section

    A ratio compares quantities multiplicatively. The tools, in order: bar (tape) diagrams, then ratio tables, then “multiply both parts by k.” Students write a:b and a:b:c in lowest terms, build equivalent ratios, and share a quantity in a given ratio — which means adding the parts first to find the size of one part. Three-term ratios and sharing problems are the Grade 6 upgrades. The classic error is reading 2:3 of 20 as “2 and 3” instead of 8 and 12. Ratios, rates, and percents (the next two units) are one language.

    Watch for

    2:3 of 20 as 2 and 3 rather than 8 and 12.

    +−2 formulas

    Sharing in a ratio

    $$\text{one part} = \frac{\text{total}}{a+b+c}$$

    Equivalent ratios

    $$a:b = ka:kb$$
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    Rate and speed

    rate-speed

    From Math 6 — required for the Grade 6 EOG in the dual-exam section

    A rate compares two different units; a unit rate has one in the denominator (miles per hour, dollars per pound). Speed is the most important unit rate: distance over time. Students convert between m/s and km/h, solve unit-price comparisons, and read speed from a story. The trap is average speed. It is always total distance divided by total time — never the mean of two speeds unless the times happen to be equal. Going 60 km/h for one hour and 40 km/h for two hours averages 46⅔ km/h, not 50.

    Watch for

    60 km/h for 1 h then 40 km/h for 1 h averaged as 50 km/h (this one happens to be right because times are equal) vs 60 km/h for 1 h then 40 km/h for 2 h still called 50.

    +−3 formulas

    Speed

    $$\text{speed} = \frac{\text{distance}}{\text{time}}$$

    Average speed

    $$\bar v = \frac{\text{total distance}}{\text{total time}}$$

    Not the mean of the speeds.

    m/s to km/h

    $$1\ \text{m/s} = 3.6\ \text{km/h}$$
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25. not started07

    Percent and percent change

    percent-change

    From Math 6 — required for the Grade 6 EOG in the dual-exam section

    Percent is a ratio with denominator 100, so p% of a quantity is p/100 times it. Students find a percent of a number, find the whole from a part and a percent, and handle increase, decrease, tax, discount, and markup. Percent change is measured against the original amount unless the problem says otherwise. Two things to show with a bar: increasing 80 by 25% is not 80 + 25, and decreasing by 20% then increasing by 20% does not return to the start (it lands at 96%). Reverse-percent problems (the sale price is known, find the original) appear at a basic level here and in full in Math 7.

    Watch for

    increase 80 by 25% as 80 + 25; decrease 80 by 25% then increase by 25% and expect 80 back.

    +−3 formulas

    Percent of

    $$p\% \text{ of } Q = \frac{p}{100} \cdot Q$$

    Percent change

    $$\%\ \text{change} = \frac{\text{new} - \text{original}}{\text{original}} \times 100\%$$

    After a change

    $$\text{new} = \text{original} \times \left(1 \pm \tfrac{p}{100}\right)$$
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26. not started11

    Area of plane figures

    area-triangles-trapezoids

    From Math 6 — required for the Grade 6 EOG in the dual-exam section

    Three formulas — triangle, parallelogram, trapezoid — and one rule that governs all of them: the height is perpendicular to the base you chose. A slanted side is not a height. For a trapezoid, a and b are the two parallel sides, not the legs. Students decompose composite figures into these pieces and compute shaded regions as whole minus hole. Fractional dimensions are expected. The hardest items mark the height on a different side from the one students expect.

    Watch for

    slanted side used as height; trapezoid $a$ and $b$ as the legs.

    +−3 formulas

    Triangle

    $$A = \tfrac{1}{2} b h$$

    Parallelogram

    $$A = b h$$

    Trapezoid

    $$A = \tfrac{1}{2}(a + b) h$$

    a and b are the parallel sides.

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    Volume and surface area

    volume-surface-nets

    From Math 6 — required for the Grade 6 EOG in the dual-exam section

    Volume is base area times height, and that stays true with fractional edges: a 2½ × 1⅓ × 3 prism has volume 10. A net is the surface unfolded, so surface area is the sum of the face areas on the net. Students match nets to cubes, rectangular and triangular prisms, and square pyramids, and compute surface area from a net or from dimensions. Surface area of a non-cube box is not 6ℓw; it is 2(ℓw + ℓh + wh). A net whose faces would overlap when folded is not a valid net.

    Watch for

    surface area as $6\ell w$ for a non-cube; a net that folds into overlapping faces accepted as valid.

    +−3 formulas

    Prism volume

    $$V = \ell w h = B h$$

    Box surface area

    $$S = 2(\ell w + \ell h + w h)$$

    Cube surface area

    $$S = 6 s^2$$
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    Displaying and comparing data

    data-distributions-6

    From Math 6 — required for the Grade 6 EOG in the dual-exam section

    Center and spread are two different questions. A statistical question expects variation in the answers (“how many hours do sixth graders sleep?”), unlike “how old am I?”. Students compute mean, median, and mode for center, and range, interquartile range (IQR = Q₃ − Q₁), and mean absolute deviation for spread. They build dot plots, histograms, and box plots from the five-number summary, and describe shape, skew, and outliers. The mean follows every point; the median does not — so when one value is far from the rest, the median is usually the better typical value, and students should be able to say why.

    Watch for

    mean always “better”; IQR as $Q_3+Q_1$; histogram bars with gaps for continuous data.

    +−3 formulas

    Mean

    $$\bar x = \frac{\sum x_i}{n}$$

    Interquartile range

    $$IQR = Q_3 - Q_1$$

    Subtract, do not add.

    Mean absolute deviation

    $$MAD = \frac{\sum \lvert x_i - \bar x \rvert}{n}$$
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