Advanced Math 5

Singapore Dimensions Math 5A / 5B

Exam: NC Grade 5 EOG. 15 units.

Units

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   Whole numbers to one billion

   place-value-billion

   Same place-value machine, three more zeros. Multiplying by a power of ten is a shift, not a superstition about adding zeros. Students read and round through one billion and multiply and divide by 10, 100, 1,000, and 10^k.

   Watch for

   $4.2 \times 100 = 4.200$; $56{,}000 \div 100 = 56{,}0$.

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   Writing and evaluating expressions

   order-of-operations

   The first algebra that is not a missing box. Order of operations is a convention so everyone gets the same number. Students evaluate expressions with parentheses, brackets, and nested grouping, and translate a sentence into an expression with a variable.

   Watch for

   $8-2\times 3 = 18$; left-to-right through mixed × and ÷ ignored.

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   Multiples and factors (prime factorization)

   prime-factorization-gcf-lcm

   Prime factorization is the canonical name of a number. GCF is the shared primes at their lowest powers; LCM is all primes at their highest powers. Students use factor trees and continuous division, write index notation, and solve tiling (GCF) and repeating-cycle (LCM) stories.

   Watch for

   LCM of 8 and 12 as 24 vs 96 (missing vs extra factors); GCF of 8 and 12 as 2.

   +−2 formulas

   GCF from primes

   $$\text{GCF}(2^a 3^b,\ 2^c 3^d) = 2^{\min(a,c)}3^{\min(b,d)}$$

   LCM from primes

   $$\text{LCM}(2^a 3^b,\ 2^c 3^d) = 2^{\max(a,c)}3^{\max(b,d)}$$
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   Multiplication (multi-digit)

   multi-digit-multiplication-5

   Fluency chapter. The interesting part is decomposition (48 × 25 = 12 × 100). Students multiply 3- and 4-digit numbers by 2- and 3-digit numbers, estimate, and catch an impossible product.

   Watch for

   a 3-digit × 3-digit product with four digits and no comment.

   Qty 3Practice 3 items →
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   Division (2-digit divisors)

   two-digit-division

   Two-digit divisors. Remainders become mixed numbers and decimals, not only R. Students divide 4-digit by 2-digit, estimate quotients, write the remainder as a fraction or decimal, and choose the interpretation from the story.

   Watch for

   compatible-number estimate that is off by a factor of 10; always rounding the quotient up.

   Qty 3Practice 3 items →
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   Fractions as division

   fraction-as-division

   a/b equals a ÷ b. This is the sentence that makes decimals, percents, and slopes possible. Students rewrite a division as a fraction, place fractions on the line, and compare unlike denominators with a common denominator or by cross-multiplying.

   Watch for

   $3 \div 4 = 1\frac{1}{3}$ (inverting out of habit).

   +−1 formula

   Fraction as division

   $$\frac{a}{b} = a \div b$$
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   Adding and subtracting fractions (stories)

   fraction-add-sub-5

   Unlike mixed numbers with renaming, now inside stories that need two operations. Students handle unlike denominators and mixed-number regrouping, and draw part–whole, comparison, and remainder bar models — the remainder model drawn correctly before any arithmetic.

   Watch for

   remainder-model (“after $\frac{2}{5}$ left, $\frac{1}{3}$ of the remainder…”) drawn as $\frac{1}{3}$ of the original.

   Qty 3Practice 3 items →
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   Multiplying fractions

   fraction-mul-div

   Area model: a/b × c/d is a rectangle inside a unit square. Students multiply fractions and mixed numbers, cross-cancel first, divide with unit fractions in both directions, and solve sequential of-the-remainder stories.

   Watch for

   $\frac{2}{3}\times\frac{4}{5} = \frac{6}{8}$ (adding); $3\div\frac{1}{2} = \frac{1}{6}$.

   +−1 formula

   Fraction product

   $$\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$$
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   Line graphs, line plots, and the mean

   line-plots-mean

   Mean as fair share: the total, split equally. Students build fractional line plots, read single and double line graphs, and compute a mean from a plot — total over count, not marks over distinct values.

   Watch for

   mean of a line plot computed as (number of Xs) / (number of distinct values).

   +−1 formula

   Mean

   $$\text{mean} = \frac{\text{sum}}{n}$$
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    Four operations of decimals

    decimal-four-operations

    The point moves when you multiply or divide by a power of ten; when you multiply two decimals the decimal places add. Students run all four operations, including decimal × decimal and decimal ÷ decimal by making the divisor whole.

    Watch for

    $0.3 \times 0.3 = 0.9$; $1.2 \div 0.3 = 0.4$.

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    Geometry and angle theorems

    angle-theorems-triangles

    You may not measure; you may use facts. Angles on a line, at a point, vertically opposite, the triangle sum, isosceles base angles, and the exterior-angle theorem. Students chase angles in a diagram with one or two given measures, name the special quadrilaterals, and construct a triangle given SAS or ASA.

    Watch for

    exterior angle treated as $180^\circ$ minus one interior; “vertically opposite” used on adjacent angles.

    +−2 formulas

    Triangle sum

    $$\angle A + \angle B + \angle C = 180^\circ$$

    Exterior angle

    $$\text{exterior} = \text{sum of the two remote interior angles}$$
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    Data analysis and coordinate graphs

    mean-quadrant-i

    Mean, then the first coordinate plane — Quadrant I only. Students use mean = sum ÷ n and sum = mean × n, plot (x, y) in Quadrant I, and graph a simple linear pattern from a table.

    Watch for

    origin as $(1,1)$; swapping coordinates when the axis labels are “time” and “distance.”

    +−1 formula

    Sum from mean

    $$\text{sum} = \text{mean} \times n$$
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    Ratio

    ratio-tape-diagrams

    Ratio is multiplicative comparison. Bars make before–after and constant-difference problems thinkable. Students write a:b and a:b:c in lowest terms, distinguish part-to-part from part-to-whole, and solve before–after problems with a changing quantity, a constant total, or a constant difference.

    Watch for

    $2:3$ treated as $\frac{2}{3}$ of the whole rather than 2 of 5 parts; before–after drawn as one bar.

    Qty 3Practice 3 items →
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    Rate

    unit-rate-work

    A rate compares different units; a unit rate has 1 in the denominator. Students compute and compare unit rates and solve combined-work problems by adding rates, never times.

    Watch for

    3/4 of a tank in 2 hours → rate as 3/4 per hour; work problems adding times instead of rates.

    +−1 formula

    Combined work

    $$\frac{1}{t_A} + \frac{1}{t_B} = \frac{1}{t_{\text{together}}}$$
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    Percentage

    percent-financial-intro

    Percent is a ratio with denominator 100. Students convert among fraction, decimal, and percent, find p% of a quantity, and work discount, tax, markup, and simple-interest stories — knowing which amount the tax applies to.

    Watch for

    5% of 80 as 5; “increase by 20%” as +20; tax computed on the discounted price vs original, mixed up.

    +−1 formula

    Percent

    $$p\% = \frac{p}{100}$$
    Qty 3Practice 3 items →