Mathematics 9

Adding a negative number moves you left on the number line (or down); subtracting a negative number is the same as adding its positive — the two negatives 'flip' into an addition.

A power a^n means the base a multiplied by itself n times — the exponent counts factors, it does not multiply the base.

Line up decimals by place value and find common denominators for fractions before combining, keeping track of the sign the whole time.

The base can be a rational number (a fraction or decimal), and the same repeated-multiplication meaning still applies.

Same signs multiply or divide to a positive result; different signs give a negative result — predict the sign before you calculate the size.

Only LIKE terms (same variable and same exponent) can be combined — 3x and 5x combine to 8x, but 3x and 5x^2 cannot combine.

Subtracting a polynomial means distributing a negative sign across every term inside the brackets before combining like terms.

The distributive property means the monomial multiplies EVERY term inside the brackets, one at a time.

Multiply the numbers together and add the exponents of matching variables, following the exponent laws from Unit 1.

Dividing a polynomial by a monomial means dividing EVERY term of the polynomial by that same monomial, one at a time.

Choose several input values, calculate the matching output using the relation's rule, and record each pair as a row in a table.

Plot each (input, output) pair as a point; if the relation is linear, the points line up in a straight line.

Interpolating means estimating a value that falls BETWEEN two points you already have — read straight up/across from the graph, or use the relation's rule directly.

An interpolated estimate is usually reliable because it stays inside the range of data you already trust.

Extrapolating means predicting a value OUTSIDE the range you plotted — extend the line (or the rule) beyond the known points.

Collect all variable terms onto one side and all constant terms onto the other, using inverse operations, before isolating the variable.

Whatever you do to one side of the equation, you must do to the other side too, to keep it balanced.

Expand (distribute) any brackets first, multiplying the term outside by every term inside, before collecting like terms.

After expanding, the equation becomes a regular multi-step equation solved with the same inverse-operation method.

Identify what the unknown quantity is, assign it a variable, and translate the words of the problem into an equation term by term.

The scale factor is the ratio between a length on the diagram (or model) and the matching real length — it can enlarge (>1) or reduce (<1).

Once you know the scale factor, multiply any diagram length by it to find the real length, or divide the real length by it to find the diagram length.

Similar triangles have equal corresponding angles and corresponding sides that are all in the same proportion (the same scale factor).

Set up a proportion between a known pair of corresponding sides and the pair containing the unknown, then solve for the unknown.

Real problems (maps, floor plans, models, shadows) often require choosing whether to use a scale factor or a similar-triangle proportion, then setting it up correctly.

A sample should represent the whole population fairly; a biased sample over- or under-represents some group, which skews the result.

Common bias sources: who was asked (convenience sampling), how the question was worded (leading questions), and when/where the survey happened.

Before trusting a reported statistic, ask: how was the data collected, how big and how representative was the sample, and does the graph's scale exaggerate the difference?

A graph with a cut or stretched axis can make a small difference look huge — read the axis values carefully, not just the shape of the bars.

A budget lists all expected income (money coming in) and expenses (money going out) over a set period, so you can see if you'll have money left or run short.

If total expenses exceed total income, the budget needs adjusting — either increase income, cut expenses, or both.

Each transaction (deposit, withdrawal, purchase) either adds to or subtracts from the current balance — track it one transaction at a time in order.

A running balance should be checked after every transaction to make sure it never goes negative unexpectedly.