Advanced Math 6

Singapore Dimensions Math 6A / 6B

Exam: NC Grade 6 EOG. 13 units.

Units

1. not started01

   Whole numbers: primes, GCF, and LCM

   gcf-lcm-grade6

   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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2. not started02

   Fractions: division and multi-step stories

   fraction-operations-6

   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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3. not started03

   Decimals: terminating and repeating

   decimals-terminating-repeating

   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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4. not started04

   Negative numbers and absolute value

   integers-absolute-value

   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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5. not started05

   Ratios

   ratios-grade6

   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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6. not started06

   Rate and speed

   rate-speed

   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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7. not started07

   Percent and percent change

   percent-change

   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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8. not started08

   Algebraic expressions

   algebraic-expressions-6

   A variable is a number we have not picked yet. Students write expressions from sentences, evaluate them at given values, and combine like terms — terms with the same variable part. The distributive property runs both ways: a(b + c) = ab + ac expands, and ab + ac = a(b + c) factors out a common monomial, which is why GCF comes first in the year. The three standard mistakes: reading 3x at x = 4 as 34, adding 3x + 2x to get 5x², and distributing 2(x + 3) as 2x + 3. Evaluating at negative inputs waits for Math 7.

   Watch for

   $3x+2$ at $x=4$ as 32; $3x+2x=5x^2$; $2(x+3)=2x+3$.

   +−3 formulas

   Distributive property

   $$a(b+c) = ab + ac$$

   Factoring out a monomial

   $$ab + ac = a(b+c)$$

   Like terms

   $$3x + 2x = 5x$$
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9. not started09

   Equations and inequalities (one step)

   one-step-equations-inequalities

   An equation is a balance: whatever you do to one side you do to the other. Students solve x + a = b and ax = b, with decimal and fraction coefficients, by undoing the one operation — subtract 7 to solve x + 7 = 10, not add it. An inequality is a region on the number line. Students graph x < a and x ≥ b with an open circle for strict and a closed circle for inclusive, and write a one-step inequality from a constraint (“riders must be more than 48 inches tall”). Multi-step equations are Math 7.

   Watch for

   $x+7=10$ solved by adding 7; closed circle on a strict inequality.

   +−2 formulas

   Undo addition

   $$x + a = b \implies x = b - a$$

   Undo multiplication

   $$ax = b \implies x = \frac{b}{a}$$
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10. not started10

    Coordinates and graphs (four quadrants)

    four-quadrant-plane

    The coordinate plane now has four quadrants, numbered counter-clockwise from the upper right. The signs of x and y are the quadrant: (−3, 4) has negative x and positive y, so it sits in Quadrant II. Horizontal and vertical distances between points come from absolute value — the distance from (2, 5) to (2, −1) is |5 − (−1)| = 6, not 4. Students plot polygons on the plane and find their perimeters, and represent a linear rule three ways: as a table, an equation, and a graph. Slope, as a number, waits for Math 7.

    Watch for

    $(-3,4)$ in Quadrant III; distance between $(2,5)$ and $(2,-1)$ as 4.

    +−2 formulas

    Vertical distance

    $$d = \lvert y_2 - y_1 \rvert \quad (x_1 = x_2)$$

    Horizontal distance

    $$d = \lvert x_2 - x_1 \rvert \quad (y_1 = y_2)$$
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11. not started11

    Area of plane figures

    area-triangles-trapezoids

    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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12. not started12

    Volume and surface area

    volume-surface-nets

    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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13. not started13

    Displaying and comparing data

    data-distributions-6

    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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