Advanced Math 7

Singapore Dimensions Math 7A / 7B

Exam: NC Grade 7 EOG. 16 units.

Units

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   Factors and multiples: Euclid's algorithm

   euclid-gcd

   Euclid's algorithm — gcd(a, b) = gcd(b, a mod b) — finds a GCF faster than prime trees for large pairs. Students run it to completion (not one remainder too early), then get the LCM from the product identity, and solve cycle and partition stories without swapping HCF and LCM.

   Watch for

   stopping Euclid one remainder too early; HCF and LCM swapped in a story.

   +−2 formulas

   Euclid

   $$\gcd(a,b) = \gcd(b,\ a \bmod b)$$

   LCM from GCD

   $$\operatorname{lcm}(a,b) = \frac{ab}{\gcd(a,b)}$$
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2. not started02

   Real numbers

   reals-radicals-sci-notation

   Rationals versus irrationals: √2 is a point on the line, not a broken number. Students classify numbers, estimate square and cube roots between integers, operate on signed numbers, and write scientific notation with 1 ≤ a < 10.

   Watch for

   $\sqrt{16/25}$ called irrational; $\pi=\frac{22}{7}$; $3.2\times 10^3$ written $32\times 10^2$ as “scientific.”

   +−1 formula

   Scientific notation

   $$a \times 10^k,\quad 1 \le a < 10$$
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   Introduction to algebra

   algebra-foundations-7

   Expression, equation, formula, identity — four different kinds of sentence, and what you may do with each. Students name term, coefficient, constant, and variable, translate phrases, and evaluate expressions including at negative inputs.

   Watch for

   $x=5$ called an expression; $2(x+3)=2x+6$ called an equation to solve.

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   Algebraic manipulation

   expand-factor-7

   The skill Math 1 factoring sits on. Students combine like terms with integer and fractional coefficients, distribute with negative factors, expand (ax + b)(cx + d), factor out a common factor, and factor four terms by grouping.

   Watch for

   $(x+3)(x+4)=x^2+12$; $-2(x-5)=-2x-10$; grouping that never rearranges.

   +−2 formulas

   Binomial product

   $$(ax+b)(cx+d) = acx^2 + (ad+bc)x + bd$$

   Negative distribution

   $$-a(b - c) = -ab + ac$$
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   Simple equations in one variable

   linear-equations-7

   Variables on both sides, and fractions. Students solve ax + b = cx + d and fractional equations by multiplying every term by the common denominator, check by substitution, and model consecutive-integer, age, and dimension stories.

   Watch for

   clearing fractions by subtracting denominators; forgetting to multiply every term.

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   Ratio, rate, and speed (motion graphs)

   speed-motion-graphs

   On a distance–time graph the slope is speed; on a speed–time graph the area is distance. Students handle changing ratios and compound rates, read both graph types, and compute average speed as total distance over total time — 48 km/h for an equal-distance out-and-back at 40 and 60, not 50.

   Watch for

   averaging 40 km/h and 60 km/h for an out-and-back of equal distance (the answer is not 50).

   +−1 formula

   Average speed, equal distances

   $$\bar v = \frac{2 v_1 v_2}{v_1 + v_2}$$
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   Percentage: reverse percent and simple interest

   percent-simple-interest

   Reverse percent: the 80% is the new amount, so the original is new ÷ 0.8. Students find an original from a changed amount, work profit/loss, discount, and tax, and use simple interest I = Prt and A = P(1 + rt).

   Watch for

   “price after 20% off is $80, original?” answered $96$ (they took 20% of 80).

   +−2 formulas

   Simple interest

   $$I = Prt$$

   Amount

   $$A = P(1 + rt)$$
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   Angles, triangles, and quadrilaterals

   euclidean-angles-polygons

   Parallel lines with a transversal are a kit: corresponding angles equal, alternate interior equal, co-interior supplementary. Students chase angles, use the triangle inequality, classify triangles and special quadrilaterals, and apply the interior sum (n − 2)·180° and exterior sum 360°.

   Watch for

   co-interior called equal; exterior sum of a pentagon as $540^\circ$.

   +−2 formulas

   Interior angle sum

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

   Exterior angle sum

   $$360^\circ \text{ (convex)}$$
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   Number patterns

   sequences-patterns-7

   A sequence is a function whose inputs are 1, 2, 3, …. Students write the nth term of an arithmetic sequence, recognize a common ratio, use triangular and square numbers, and match a growing diagram to a formula.

   Watch for

   $T_n$ written as $n+d$; triangular numbers as $n^2$.

   +−2 formulas

   Arithmetic term

   $$T_n = a + (n-1)d$$

   Triangular numbers

   $$T_n = \frac{n(n+1)}{2}$$
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    Coordinates and linear graphs

    slope-linear-graphs

    Slope is a rate. Students compute m from two points, graph y = mx + c from slope and intercept, handle horizontal (slope 0) and vertical (undefined slope) lines, and read slope and intercept in a story such as water draining.

    Watch for

    $m=\frac{x_2-x_1}{y_2-y_1}$; vertical line called slope 0.

    +−2 formulas

    Slope

    $$m = \frac{y_2 - y_1}{x_2 - x_1}$$

    Slope-intercept

    $$y = mx + c$$
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    Inequalities (multi-step)

    linear-inequalities-7

    Multiply or divide by a negative: reverse the inequality, always. Students solve multi-step inequalities, graph on a number line, and write compound inequalities a < x ≤ b.

    Watch for

    $-2x>6 \Rightarrow x> -3$ (did not reverse).

    +−1 formula

    Reversal rule

    $$a < b,\ k < 0 \implies ka > kb$$
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    Perimeters and areas of plane figures (circles)

    circles-arcs-sectors

    π is a number. An arc and a sector are fractions of the circle, so their length and area scale by θ/360°. Students use circumference, area, arc length, and sector area, and compose them with triangles and rectangles.

    Watch for

    area of a sector using $\frac{\theta}{360}2\pi r$; mixing diameter and radius.

    +−4 formulas

    Circumference

    $$C = 2\pi r = \pi d$$

    Circle area

    $$A = \pi r^2$$

    Arc length

    $$s = \frac{\theta}{360^\circ} \cdot 2\pi r$$

    Sector area

    $$A = \frac{\theta}{360^\circ} \cdot \pi r^2$$
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    Volume and surface area of solids

    volume-surface-3d

    Prisms, cylinders, pyramids, cones, spheres — keep a table rather than memorizing in isolation. Students find volume and surface area, derive the slant height of a cone from r and h, and decide surface versus volume from the story.

    Watch for

    pyramid $\frac12 Bh$; cone lateral area with $h$ instead of $\ell$.

    +−4 formulas

    Cylinder

    $$V = \pi r^2 h,\quad S = 2\pi r^2 + 2\pi r h$$

    Pyramid / cone

    $$V = \tfrac{1}{3} B h$$

    Cone surface

    $$S = \pi r^2 + \pi r \ell,\quad \ell = \sqrt{r^2 + h^2}$$

    Sphere

    $$V = \tfrac{4}{3}\pi r^3,\quad S = 4\pi r^2$$
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    Proportions

    direct-inverse-proportion

    Direct proportion is a line through the origin; inverse proportion is a hyperbola with xy constant. Students recognize each from a table (constant ratio versus constant product), find k, and fill a missing value.

    Watch for

    inverse graphed as a line; “twice $x$, twice $y$” said of an inverse situation.

    +−2 formulas

    Direct

    $$y = kx$$

    Inverse

    $$y = \frac{k}{x} \iff xy = k$$
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    Data handling

    data-handling-7

    Outliers pull the mean and barely move the median. Students distinguish discrete from continuous data, build stem-and-leaf plots (including back-to-back), compute center and range for grouped and ungrouped data using midpoints, and spot a misleading graph.

    Watch for

    mean of grouped data using interval width instead of midpoint.

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    Probability

    probability-trees

    Probability is a number in [0, 1]. A tree multiplies along a branch and adds across complete branches. Students compute P(E) and P(not E), list outcomes in tables and trees, and handle two-stage events with and without replacement.

    Watch for

    $P(\text{H then T})=\frac12$ for two coins; adding along a branch.

    +−2 formulas

    Probability

    $$P(E) = \frac{\text{favorable}}{\text{possible}}$$

    Complement

    $$P(E') = 1 - P(E)$$
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