Mathematics 4

The first digit after the point is TENTHS and the second is HUNDREDTHS — like dimes and pennies in a dollar. $0.3$ means 3 tenths; $0.37$ means 3 tenths and 7 hundredths.

Stack the numbers with ones under ones, tens under tens, and work right to left.

A factor divides a number evenly, with no remainder — factors come in PAIRS: $12 = 1\times12 = 2\times6 = 3\times4$.

Break a factor apart by place value and multiply the pieces: $24 \times 9$$= (20 \times 9) + (4 \times 9)$$= 180 + 36$$= 216$.

Splitting the same whole into more, smaller pieces doesn't change how much you have: $\frac{1}{2} = \frac{2}{4} = \frac{3}{6}$.

The equal sign is a balance: whatever is on the left must be worth the same as whatever is on the right.

An equation can have operations on BOTH sides: $7 + 5 = 3 + 9$ is true because both sides make 12.

To find the unknown, ask 'what number makes both sides equal?' — in $8 + \square = 15$, the box is 7 because $8 + 7 = 15$.

You can represent equality in more than one way — a box, a question mark, or a letter can all stand for the unknown.

Sort shapes by properties: number of sides, whether sides are equal, whether corners are right angles, and whether sides are parallel.

The same shapes can be sorted more than one way — a square fits under 'has 4 right angles' AND 'all sides equal'.

By their sides, triangles are equilateral (three equal sides), isosceles (two equal sides) or scalene (no equal sides). Measure with a ruler to be sure.

By their angles, triangles are right (one 90° angle), obtuse (one angle bigger than 90°) or acute (all three angles smaller than 90°). Every triangle has a name from each list, like a right isosceles triangle.

Two angles that together make 90° are complementary, like 30° and 60°. Two angles that together make 180°, a straight line, are supplementary, like 110° and 70°.

Area is the number of equal unit squares needed to cover a flat shape, written in square units like $\text{cm}^2$.

An angle measures how much you turn, in DEGREES. A full turn is $360°$, a half turn (straight line) is $180°$, and a right angle (square corner) is $90°$.

You can find area by covering the shape and counting the squares — no gaps and no overlaps.

A bigger opening means a bigger angle — the length of the arms doesn't matter.

A rectangle's area is rows × columns of unit squares — that's just length × width: a $5 \times 3$ rectangle has area 15 squares.

An arithmetic sequence ADDS (or subtracts) the same amount each step — that fixed amount is the constant step.

Find the step by subtracting one term from the next, then keep adding it to extend the pattern: $3, 7, 11, 15,\dots$ adds 4.

A geometric sequence MULTIPLIES by the same amount each step: $2, 4, 8, 16,\dots$ doubles (× 2) every time.

Test which kind it is: if you ADD the same amount it's arithmetic; if you MULTIPLY by the same amount it's geometric.

In one hour the minute hand goes once around the clock, a whole circle. 30 minutes is half the circle, 15 minutes is a quarter and 45 minutes is three quarters.

60 minutes split into 3 equal parts of 20, so 20 minutes is one third of the circle and 40 minutes is two thirds.

We can say the time of day with fractions: 3:15 is quarter past 3, 3:30 is half past 3, and 3:45 is quarter to 4, because a quarter of the hour is left before 4:00.

Time units nest inside each other: 60 seconds = 1 minute, 60 minutes = 1 hour, 24 hours = 1 day, 7 days = 1 week.

To go from a bigger unit to a smaller one you multiply (2 hours = $2 \times 60 = 120$ minutes); the other way, you divide.

The scale (or key) tells you how much each square, gridline, or picture stands for — count by the scale, not by 1.

To read a bar: find where it ends and multiply by the scale; for a pictograph, each symbol is worth the key amount.

A good scale fits the data — big numbers need a bigger scale (count by 10s or 100s) so the graph isn't impossibly tall.

The SAME data can look very different: a small scale makes bars look tall and dramatic; a large scale makes them look short and flat.