NCEA Pāngarau Co-requisite (NZQA Standard 32412) key terms

The study of number, shape, measurement, pattern and chance, and the skills used to solve problems with them. In this standard it names the subject area being assessed.

A value used to count, measure or label, such as 7, 0.25, -3 or 2/3. Numbers can be whole, fractional, decimal, negative or irrational.

A whole number with no fractional part, including zero and negatives, for example -4, 0, 12. Integers are used for counts and for positions on a number line.

A number less than zero, written with a minus sign, such as -8. It can represent a debt, a temperature below zero, or a position left or down from a starting point.

A number written using place value with a decimal point, such as 3.75. The digits after the point show tenths, hundredths and so on, and rounding is often needed in answers.

A way of writing part of a whole as one number over another, such as 3/4. The bottom number shows how many equal parts there are and the top shows how many are taken.

Fractions that name the same amount even though the digits differ, for example 1/2, 2/4 and 5/10. Multiplying or dividing top and bottom by the same number keeps the value unchanged.

A fraction expressed out of 100, so 35% means 35 out of every 100, or 0.35. Percentages are used for discounts, interest, GST, increases and comparisons of amounts.

The size of an increase or decrease written as a percentage of the original amount. It is found by dividing the change by the starting value and multiplying by 100.

A comparison of two or more quantities by division, written like 2:5 or 3 to 1. Ratios are used to scale recipes, mix materials and share amounts in given proportions.

Dividing a total into parts that follow a given ratio, such as sharing $120 in the ratio 2:3 to give $48 and $72. Each share is one part of the total number of parts.

A comparison of two quantities with different units, such as 80 km per hour or $6.50 per kilogram. Rates let you predict one quantity from another by multiplying or dividing.

The branch of mathematics that uses letters and symbols to stand for numbers, allowing general rules and unknown values to be written and solved, for example 3x + 5 = 20.

A letter or symbol that stands for a number that may change or is not yet known, such as x in 2x + 1. It lets one rule describe many cases at once.

A combination of numbers, variables and operations without an equals sign, such as 4n - 7. Expressions can be simplified by collecting like terms and expanding brackets.

A statement that two expressions are equal, such as 5x - 3 = 17. Solving it means finding the value or values of the variable that make the statement true.

The agreed order for calculating: brackets first, then powers, then multiplication and division, then addition and subtraction. Following it ensures everyone gets the same answer from one expression.

A number that, when multiplied by itself, gives the stated value, so the square root of 49 is 7. Square roots are used with areas, Pythagoras and reversing a squared quantity.

The standard quantity used for measuring, such as metre, kilogram, litre or minute. Recording units with numbers keeps measurements meaningful and allows conversion between scales.

Changing a measurement from one unit to another using a multiplier, for example 2.5 km to 2500 m. Knowing that 1 km equals 1000 m and 1 hour equals 60 minutes makes this reliable.

The total distance around the outside of a flat shape, found by adding the lengths of all its sides. It is measured in units such as metres and is used for fencing and edging.

The amount of flat surface a shape covers, measured in square units such as cm². For a rectangle it is length times width; other shapes use their own formulas.

The amount of space a solid object occupies, measured in cubic units such as m³ or in litres. For a cuboid it is length times width times height.

How much a container can hold, usually measured in litres or millilitres. Capacity differs from volume only in that it describes the inside space available for a liquid or gas.

The amount of turn between two lines meeting at a point, measured in degrees. Angles are classified as acute, right, obtuse, straight or reflex and are used in shape and direction problems.

A flat shape with three straight sides and three angles. Its interior angles always add to 180°, and triangles are classified as equilateral, isosceles or scalene by their sides.

In a right-angled triangle, the square of the longest side equals the sum of the squares of the other two sides, written a² + b² = c². It finds an unknown side length.

The fixed relationship between a drawing or model and the real object, such as 1:200. Scale factors let you convert measured lengths on a plan into real distances and back again.

Replacing a number with a nearby simpler one, such as 4.67 to 4.7 to one decimal place. Look at the next digit: 5 or more rounds up, otherwise round down.

Entering calculations on a device correctly, using brackets and the correct order of operations, then interpreting the display sensibly. Answers should be checked for reasonableness and rounded appropriately.

The steps a solver takes to understand a problem, choose a strategy, carry it out and check the answer. Processes include drawing a diagram, working backwards and trial and improvement.

A chosen plan for solving a problem, such as making a table, drawing a diagram, guessing and checking, or breaking the task into smaller parts. Choosing well often matters more than speed.

Explaining why a result is true using definitions, rules and evidence rather than only stating an answer. It includes justifying each step and checking whether a conclusion follows logically.

Giving reasons or evidence that support an answer or method, for example showing the calculation, quoting a rule, or testing with a counter-example. Justified answers are required for full credit.

Finding an approximate answer by rounding numbers to convenient values before calculating, such as 19.8 × 5 ≈ 20 × 5 = 100. It helps check whether an exact answer is sensible.

The real-life situation a mathematics question is set in, such as a marae hui, a shop sale, a building project or a sports game. The context decides which numbers and operations make sense.

Explaining what a calculated number means in the situation, including units and whether it is realistic. For example, 3.6 buses needed means 4 buses must be booked.

Information collected through counting, measuring or asking, such as heights, test scores or survey responses. Data can be numerical or categorical and is organised before being displayed or analysed.

The kind of information collected: numerical (discrete counts or continuous measurements) or categorical (named groups such as colours or yes/no). The type decides which graph and average are suitable.

Gathering information by survey, measurement, observation or existing records. Good collection uses a clear question and a sample that fairly represents the group being studied.

A smaller group selected from a larger population to represent it. A random sample reduces bias, and larger samples usually give more reliable conclusions about the whole group.

A slant in data that makes results unrepresentative, caused by choosing a one-sided sample, asking leading questions or measuring carelessly. Bias makes conclusions unreliable even when numbers are accurate.

The average found by adding all values and dividing by how many there are. The mean uses every value, so a very high or low value can pull it away from the middle.

The middle value when data is placed in order from smallest to largest. With an even number of values, the median is the mean of the two middle ones.

The value or category that appears most often in a data set. A set may have no mode, one mode or several; the mode suits categorical data such as favourite colours.

A visual display of data such as a bar chart, line graph, pie chart or histogram. The right graph depends on the data type and what pattern you want to show.

A measure from 0 to 1 of how likely an event is, where 0 means impossible and 1 means certain. It can be found from equally likely outcomes or estimated from trials.

One possible result of an activity involving chance, such as rolling a 4 or drawing a red counter. Listing all outcomes helps count how many favour a particular event.

The probability worked out from counting equally likely outcomes, as favourable outcomes divided by total outcomes. For a fair six-sided die, the probability of a 3 is 1/6.

The probability estimated from actually carrying out trials, calculated as the number of successful results divided by the number of trials. More trials usually bring it closer to the theoretical value.