Here are 200 mathematics words and established terms for school-age learners in the IB Primary Years Programme. Start with the meanings and examples below, then choose an early, secure or stretch activity. This original eduKate selection supports understanding and explanation; it is not an official IB vocabulary list or a requirement to memorise 200 words.
The 200-word list: meanings and learning examples
1–20: Number and place value · 21–40: Comparing and calculating · 41–60: Groups, factors and patterns · 61–80: Fractions, decimals and proportions · 81–100: Measurement and units · 101–120: Flat shapes and their properties · 121–140: Solids, angles and transformations · 141–160: Time, money and practical plans · 161–180: Data and chance · 181–200: Reasoning and showing understanding
Assessment requirements · Starting check · Fencing Method · Practice and answers
1–20. Number and place value
- number — An idea used to express an amount, a measurement or a position. Example: The number twelve tells us how many counters we have.
- numeral — A written symbol or group of symbols representing a number. Example: The numeral 12 represents the same number as the word twelve.
- digit — One of the ten symbols 0 to 9 in decimal notation. Example: The digit 7 has a value of seventy in seventy-three.
- zero — The number representing none, and a placeholder in place-value notation. Example: Zero counters remain after all six counters have been shared.
- whole number — One of the numbers 0, 1, 2, 3 and so on. Example: Seven is a whole number, but seven and a half is not.
- integer — A whole number, its negative, or zero. Example: The integer −3 is three steps left of zero.
- ones — The place representing individual units in a whole number. Example: In 246, the ones digit is 6.
- tens — The place representing groups of ten in a whole number. Example: The 4 in 246 represents four tens, or forty.
- hundreds — The place representing groups of one hundred in a whole number. Example: The hundreds digit in 246 is 2, representing two hundred.
- thousands — The place representing groups of one thousand in a whole number. Example: The 3 in 3,482 represents three thousands.
- place value — The value a digit has because of its position. Example: Place value explains why 50 is ten times as much as 5.
- partition — To split a number into parts that add to the original. Example: We partition 68 into 60 and 8 to make addition easier.
- compose — To combine number parts to make a whole number or amount. Example: We compose 346 from three hundreds, four tens and six ones.
- consecutive — Following one another without a gap in a stated sequence. Example: The consecutive whole numbers 8, 9 and 10 each differ by one.
- odd — Describing an integer that is not divisible by two exactly. Example: Thirteen is odd because grouping thirteen counters in pairs leaves one.
- even — Describing an integer that is divisible by two exactly. Example: Zero is even because zero equals two multiplied by zero.
- ordinal number — A number naming a position in an ordered list. Example: The ordinal number third identifies the runner after second place.
- estimate — To find an approximate amount using sensible information or reasoning. Example: We estimate about thirty shells before counting all twenty-eight.
- round — To replace a number with a nearby value at a stated place. Example: We round 47 to the nearest ten and obtain 50.
- number line — A line on which positions represent numbers using a consistent scale. Example: On this number line, each equal interval represents two units.
21–40. Comparing and calculating
- equal — Having the same value, quantity or measure in the stated comparison. Example: The expressions 7 + 3 and 12 − 2 are equal.
- greater than — Having a larger numerical value than another number. Example: Nine is greater than six, so we write 9 > 6.
- less than — Having a smaller numerical value than another number. Example: Negative two is less than one on the number line.
- at least — Equal to or more than a stated amount. Example: We need at least eight chairs; eight chairs will be enough.
- at most — Equal to or less than a stated amount. Example: The box holds at most ten books, so eleven will not fit.
- compare — To identify similarities, differences or relative sizes using a stated feature. Example: Compare the bags by mass, rather than by how large they look.
- order — To arrange items according to a stated rule or comparison. Example: Order these lengths from shortest to longest: 2 cm, 5 cm, 9 cm.
- ascending — Arranged from a smaller value to a larger value. Example: The numbers 3, 7 and 12 are in ascending order.
- descending — Arranged from a larger value to a smaller value. Example: The scores 18, 12 and 5 are in descending order.
- add — To combine numerical amounts to find their total. Example: Add six counters to the original four to make ten.
- subtract — To take away an amount or find a numerical difference. Example: Subtract five from twelve to find the seven counters remaining.
- multiply — To find a product; with whole numbers, combine equal groups. Example: Multiply four by three to count three groups of four.
- divide — To find how much in each equal group, or how many groups. Example: Divide twelve counters into three equal groups to give four in each.
- sum — The result of adding two or more numbers. Example: The sum of 8 and 6 is 14.
- difference — The result of subtracting one number from another. Example: The difference between eleven and seven is four.
- product — The result of multiplying numbers. Example: The product of six and five is thirty.
- quotient — The result of dividing one number by another non-zero number. Example: The quotient of twenty divided by four is five.
- remainder — What is left after whole-number division into complete equal groups. Example: Dividing fourteen counters into groups of four leaves a remainder of two.
- inverse — An operation that reverses another operation in the relevant situation. Example: Subtraction is the inverse of addition: 9 + 4 − 4 = 9.
- operation — A mathematical action, such as addition, performed on numbers or expressions. Example: The operation in 6 × 4 is multiplication.
41–60. Groups, factors and patterns
- pair — A group containing exactly two items. Example: Each pair of gloves contains one left glove and one right glove.
- dozen — A group of twelve items. Example: One dozen pencils shared among four children gives three pencils each.
- array — An arrangement of items in equal rows and columns. Example: An array with three rows of five dots contains fifteen dots.
- row — A horizontal line of items in an arrangement. Example: Each row of the array contains five counters.
- column — A vertical line of items in an arrangement. Example: Each column contains three counters, one from each row.
- multiple — A number obtained by multiplying a given number by an integer. Example: Twenty is a multiple of five because 5 × 4 = 20.
- factor — Here, a positive whole number that divides another whole number exactly. Example: Four is a factor of twenty because 20 ÷ 4 = 5.
- prime number — A whole number greater than one with exactly two positive factors. Example: Seven is a prime number because its only positive factors are 1 and 7.
- composite number — A whole number greater than one with more than two positive factors. Example: Twelve is a composite number, with factors 1, 2, 3, 4, 6 and 12.
- square number — A number made by multiplying a whole number by itself. Example: Thirty-six is a square number because 6 × 6 = 36.
- cube number — A number made by multiplying a whole number by itself twice. Example: Twenty-seven is a cube number because 3 × 3 × 3 = 27.
- common factor — A positive whole number that divides each of the given whole numbers exactly. Example: Three is a common factor of twelve and eighteen.
- common multiple — A number that is a multiple of each given number. Example: Twenty-four is a common multiple of six and eight.
- pattern — A regularity or relationship that can be described and investigated. Example: The pattern red, blue, red, blue repeats a two-colour unit.
- sequence — An ordered list of numbers or objects. Example: In the sequence 2, 5, 8, 11, each number increases by three.
- term — An individual number or expression in a sequence. Example: The fourth term of 2, 5, 8, 11 is eleven.
- rule — A stated instruction or relationship governing a mathematical process. Example: The rule add three turns an input of four into seven.
- input — A value put into a rule, calculation or mathematical model. Example: With the rule double, an input of six gives twelve.
- output — A value produced by applying a rule, calculation or model. Example: The output is twelve when the rule doubles six.
- variable — A symbol representing a quantity that can take different values. Example: In y = x + 3, the variable x can take different values.
61–80. Fractions, decimals and proportions
- fraction — A number written as one integer divided by another non-zero integer. Example: The fraction 3/4 can represent three of four equal parts.
- numerator — The number above the line in a fraction. Example: In 3/5, the numerator 3 counts three fifths.
- denominator — The non-zero number below the line in a fraction. Example: The denominator 5 in 3/5 names equal fifth-sized parts.
- unit fraction — A fraction with numerator one and a positive whole-number denominator. Example: The unit fraction 1/8 represents one of eight equal parts.
- proper fraction — Here, a fraction with a positive numerator smaller than its positive denominator. Example: The proper fraction 3/5 is less than one.
- improper fraction — Here, a fraction with a positive denominator and numerator at least that denominator. Example: The improper fraction 7/4 equals one and three quarters.
- mixed number — A whole-number part combined with a proper fractional part. Example: The mixed number 2 1/3 means two plus one third.
- equivalent fraction — A fraction with the same value as another fraction. Example: An equivalent fraction for 1/2 is 3/6.
- simplify — Here, rewrite a fraction with the same value using smaller whole-number numerator and denominator. Example: We simplify 6/8 to 3/4 by dividing both numbers by two.
- half — One of two equal parts of a whole, or the number 1/2. Example: Half of fourteen counters is seven counters.
- quarter — One of four equal parts of a whole, or the number 1/4. Example: A quarter of twenty counters is five counters.
- third — One of three equal parts of a whole, or the number 1/3. Example: One third of twelve counters is four counters.
- tenth — One of ten equal parts of a whole, or the number 1/10. Example: One tenth of a metre is ten centimetres.
- hundredth — One of one hundred equal parts, or the number 1/100. Example: One hundredth of a metre is one centimetre.
- decimal — Here, a base-ten number written using a decimal point. Example: The decimal 0.25 represents twenty-five hundredths.
- decimal point — The mark separating the whole-number part from fractional place values. Example: In 3.45, the decimal point comes before the tenths digit.
- percentage — An amount expressed in parts per hundred. Example: A percentage of 25% means twenty-five out of every hundred.
- ratio — A comparison of quantities by relative size, often written using a colon. Example: Three red counters and two blue counters give a red-to-blue ratio of 3:2.
- proportion — Here, the fraction of the whole belonging to a particular part. Example: With three red counters among five, the red proportion is 3/5.
- scale factor — For positive factors here, the multiplier connecting corresponding lengths in an enlargement or reduction. Example: A scale factor of two changes a 3 cm length to 6 cm.
81–100. Measurement and units
- measure — To compare a quantity with a chosen unit to find its size. Example: Measure the ribbon in centimetres with the ruler starting at zero.
- length — The measurement of how far something extends along a specified direction. Example: The length of this ribbon is thirty centimetres.
- width — The measurement across an object in a specified direction. Example: The drawing labels the rectangle's width as four centimetres.
- height — The measurement from a stated base to a stated top. Example: The tower's height is eight centimetres from tabletop to highest block.
- distance — The length of a route or separation between specified points. Example: The distance along the marked path is two hundred metres.
- perimeter — The total length of the boundary of a flat shape. Example: A rectangle measuring 6 cm by 4 cm has a perimeter of 20 cm.
- area — The amount of surface covered, measured in square units. Example: A 6 cm by 4 cm rectangle has an area of 24 cm².
- volume — The amount of three-dimensional space occupied or enclosed. Example: A cube with 2 cm edges has a volume of 8 cm³.
- capacity — The amount a container can hold when filled to a stated limit. Example: The jug's capacity is one litre when filled to its marked line.
- mass — A measure of the amount of matter in an object. Example: The flour has a mass of five hundred grams.
- weight — The force of gravity acting on an object. Example: An object's weight changes when the gravitational pull on it changes.
- unit — An agreed amount used as a reference when measuring. Example: The metre is a unit of length.
- centimetre — A metric length unit equal to one hundredth of a metre. Example: The square's side measures one centimetre, written 1 cm.
- metre — The SI unit of length, equal to one hundred centimetres. Example: A ribbon one metre long can be divided into one hundred centimetres.
- kilometre — A metric length unit equal to one thousand metres. Example: A route of one kilometre is one thousand metres long.
- gram — A metric mass unit equal to one thousandth of a kilogram. Example: A one-gram mass is written 1 g.
- kilogram — The SI unit of mass, equal to one thousand grams. Example: A bag containing two thousand grams has a mass of two kilograms.
- litre — A volume unit equal to one cubic decimetre or one thousand millilitres. Example: A one-litre jug holds one thousand millilitres at its stated capacity.
- millilitre — A volume unit equal to one thousandth of a litre. Example: Five hundred millilitres is half a litre.
- convert — To express the same quantity in a different unit or form. Example: Convert two metres to centimetres: 2 m = 200 cm.
101–120. Flat shapes and their properties
- shape — The form of an object or figure, described by its properties. Example: We identify the shape from its straight sides and three vertices.
- polygon — A closed flat shape with straight sides that do not cross. Example: A triangle is a polygon, but a circle is not.
- side — A straight boundary segment of a polygon. Example: Each side of this square is four centimetres long.
- vertex — A corner where polygon sides or polyhedron edges meet. Example: Each vertex of a triangle joins two sides.
- triangle — A polygon with three sides and three vertices. Example: The triangle has three sides even when we rotate the drawing.
- quadrilateral — A polygon with four sides. Example: A square is one kind of quadrilateral.
- pentagon — A polygon with five sides. Example: The pentagon has five sides, although their lengths need not match.
- hexagon — A polygon with six sides. Example: This hexagon has six sides and six vertices.
- octagon — A polygon with eight sides. Example: We count eight straight sides around the octagon.
- rectangle — A quadrilateral with four right angles. Example: A square is a rectangle because it has four right angles.
- square — A quadrilateral with four equal sides and four right angles. Example: The square remains a square when turned onto one vertex.
- rhombus — A quadrilateral with four equal sides. Example: A square is a rhombus with four right angles.
- parallelogram — A quadrilateral with both pairs of opposite sides parallel. Example: A rectangle is a parallelogram because its opposite sides are parallel.
- circle — A flat curve whose points are all equally distant from its centre. Example: Every point on the circle is 3 cm from its centre.
- radius — A line segment from a circle's centre to its boundary, or its length. Example: A circle with radius 3 cm has diameter 6 cm.
- diameter — A segment across a circle through its centre, or its length. Example: The diameter is twice the radius of the same circle.
- circumference — The distance around a circle. Example: We compare the circumference with the straight distance across the circle.
- diagonal — A segment joining two non-adjacent vertices of a polygon. Example: A diagonal of a rectangle joins opposite corners.
- symmetry — A property where a shape matches itself under a reflection or rotation. Example: This square has reflection symmetry across a line through opposite side midpoints.
- congruent — Having the same shape and size, even after moving, turning or reflecting. Example: Two squares with 4 cm sides are congruent.
121–140. Solids, angles and transformations
- cube — A solid with six equal square faces. Example: A cube has twelve edges, each the same length.
- cuboid — A solid with six rectangular faces meeting at right angles. Example: A cube is a cuboid whose edges all have equal lengths.
- sphere — A three-dimensional surface whose points are equally distant from its centre. Example: A mathematical sphere models the rounded surface of a ball.
- cylinder — Here, a solid with two equal parallel circular bases and constant circular cross-section. Example: The cylinder has a circular base at each end.
- cone — Here, a solid with one circular base tapering to a single vertex. Example: The cone narrows from its circular base to its vertex.
- pyramid — A solid with a polygon base and triangular faces meeting at one vertex. Example: A square-based pyramid has four triangular side faces.
- prism — A solid with congruent parallel polygon bases joined by parallelogram faces. Example: A triangular prism has two triangular bases and three side faces.
- face — Here, one flat surface forming part of a polyhedron's boundary. Example: Each face of a cube is a square.
- edge — Here, the line segment where two faces of a polyhedron meet. Example: Two square faces meet along each edge of a cube.
- net — A flat arrangement of connected shapes that folds into a solid's surface. Example: Six suitably arranged squares make a net of a cube.
- angle — The measure of turn between two rays meeting at a common point. Example: The angle between these perpendicular rays measures ninety degrees.
- right angle — An angle measuring ninety degrees, or one quarter of a full turn. Example: Each corner of a rectangle forms a right angle.
- acute angle — An angle greater than zero degrees and smaller than ninety degrees. Example: An angle of forty degrees is an acute angle.
- obtuse angle — An angle greater than ninety degrees and smaller than one hundred and eighty. Example: An angle of one hundred and twenty degrees is an obtuse angle.
- parallel — Describing distinct lines in the same plane that never meet. Example: The two opposite edges of this rectangular page are parallel.
- perpendicular — Describing lines that meet at a right angle. Example: A horizontal line and a vertical line on this page are perpendicular.
- horizontal — Running from side to side, level with the chosen reference. Example: On this upright chart, the horizontal axis runs from left to right.
- vertical — Running straight up and down in the chosen reference. Example: On this upright chart, the vertical axis runs from bottom to top.
- rotation — A turn about a fixed point in a plane. Example: A clockwise quarter-turn rotation moves the arrow from pointing up to pointing right.
- reflection — A transformation producing a mirror image across a stated line. Example: A right-pointing arrow points left after reflection across a vertical line.
141–160. Time, money and practical plans
- second — A unit of time equal to one sixtieth of a minute. Example: The timer advances by one second with each displayed count.
- minute — A unit of time equal to sixty seconds. Example: A task lasting one minute lasts sixty seconds.
- hour — A unit of time equal to sixty minutes. Example: A lesson lasting half an hour takes thirty minutes.
- day — Here, a period of twenty-four hours. Example: A duration of two days equals forty-eight hours.
- week — A period of seven days. Example: Two weeks contain fourteen days.
- month — One of the twelve named divisions of the calendar year used here. Example: June is a month with thirty days.
- year — Here, a calendar period of twelve months. Example: One calendar year runs from January through December.
- duration — The length of time something lasts. Example: The duration from 09:10 to 09:35 is twenty-five minutes.
- interval — The gap between two stated positions, values or times. Example: The interval between bus departures at 10:00 and 10:15 is fifteen minutes.
- timetable — A schedule showing when planned events occur. Example: The timetable lists the library visit at 10:30.
- deadline — The latest stated time for completing an action. Example: The deadline is noon, so the work must finish by then.
- cost — The amount spent or needed to obtain something. Example: The total cost of three two-dollar notebooks is six dollars.
- price — The amount charged for an item or service. Example: Each notebook has a price of two dollars.
- budget — A plan or limit for how much can be spent. Example: Our ten-dollar budget covers three notebooks costing two dollars each.
- change — The money returned when payment is greater than the amount owed. Example: Paying ten dollars for a six-dollar purchase gives four dollars change.
- discount — A reduction from an original price. Example: A two-dollar discount reduces a ten-dollar price to eight dollars.
- profit — The amount by which income exceeds all costs included in the calculation. Example: With income of twelve dollars and total costs of eight, profit is four dollars.
- loss — The amount by which included costs exceed income. Example: With costs of nine dollars and income of six, the loss is three dollars.
- balance — The amount remaining after stated additions and deductions. Example: Starting with ten dollars and spending six leaves a balance of four dollars.
- receipt — A record confirming payment and usually listing the purchased items. Example: The receipt shows three notebooks and a total payment of six dollars.
161–180. Data and chance
- data — Recorded information, such as counts, measurements, observations or categories. Example: Our data list how many books each child borrowed.
- category — A group defined by a shared characteristic. Example: We place apples in the fruit category.
- sort — To arrange items into groups using stated properties or rules. Example: Sort the shapes into groups with three sides and four sides.
- tally — A mark used to record one counted item or occurrence. Example: We add one tally for each child choosing the blue card.
- frequency — The number of times a value or category occurs. Example: If seven children choose blue, the frequency for blue is seven.
- table — Information arranged in labelled rows and columns. Example: The table pairs each colour with the number of votes it received.
- bar chart — A graph using separate bars to show quantities for categories. Example: The tallest bar in this bar chart represents the greatest frequency.
- pictogram — A display using pictures or symbols to represent quantities. Example: In this pictogram, each book picture represents two borrowed books.
- key — An explanation of what symbols or colours represent in a display. Example: The key states that one star represents five votes.
- axis — A reference line used to locate values or organise a graph. Example: The vertical axis shows the number of children.
- scale — A system of marks and values used to read a measurement or graph. Example: The scale increases by two children at each marked interval.
- mean — For non-empty numerical data, the sum of values divided by their count. Example: The mean of 2, 4 and 9 is 15 ÷ 3 = 5.
- median — The middle ordered value, or the mean of the two middle values. Example: The median of 2, 4, 7 and 9 is 5.5.
- mode — A value occurring most often in a data set. Example: The mode of 1, 1 and 4 is 1.
- range — For numerical data, the largest value minus the smallest value. Example: The range of 3, 7 and 10 is 10 − 3 = 7.
- survey — A planned collection of answers to questions from people. Example: Our survey asks each classmate which reading corner they prefer.
- sample — The selected people, objects or observations studied from a larger group. Example: Ten pupils form our sample from a school of two hundred.
- outcome — One possible result of a chance process. Example: Rolling a four is one outcome when rolling a six-sided die.
- event — A set of outcomes of a chance process. Example: Rolling an even number is the event containing outcomes 2, 4 and 6.
- probability — A number from zero to one describing likelihood within a stated model. Example: For a fair six-sided die, the probability of rolling two is 1/6.
181–200. Reasoning and showing understanding
- problem — A question or situation requiring mathematical thinking to resolve. Example: The problem asks how to share twelve counters equally among three trays.
- information — Facts or details available for interpreting and solving a task. Example: The information gives the rectangle's length and width in centimetres.
- unknown — A quantity whose value has not yet been found in the problem. Example: The unknown in 8 + □ = 13 is five.
- equation — A statement that two mathematical expressions are equal. Example: The equation 7 + 5 = 12 shows equal values on both sides.
- expression — A number, symbol or combination joined by operations, without a relation sign. Example: The expression 3 × n represents three multiplied by n.
- symbol — A mark with an agreed mathematical meaning. Example: The symbol > means greater than.
- represent — To show a mathematical idea using numbers, objects, words or drawings. Example: We represent twelve as three equal rows of four counters.
- model — A simplified representation used to explore or explain a situation. Example: Our model uses counters to represent the children on each bus.
- diagram — A drawing showing relevant features or relationships. Example: The diagram labels both sides needed to calculate the rectangle's area.
- explain — To make mathematical thinking clear through steps, relationships or examples. Example: Explain why six pairs contain twelve counters by showing the groups.
- justify — To support an answer or claim with relevant mathematical reasons. Example: Justify the choice by checking that every stated condition is met.
- reason — A fact or logical connection that supports a conclusion. Example: Our reason is that 12 divides by 3 without a remainder.
- check — To examine whether work satisfies the mathematics and the problem's conditions. Example: Check 18 ÷ 3 = 6 by multiplying 6 × 3.
- assumption — A condition accepted for a calculation or model and stated explicitly. Example: Our assumption is that every counter is equally likely to be selected.
- solution — An answer or arrangement that meets all the problem's requirements. Example: Four counters in each tray is a solution to the equal-sharing problem.
- strategy — A planned way to approach or solve a mathematical problem. Example: Our strategy is to draw equal groups before writing the division equation.
- efficient — Using a suitable method with little unnecessary work. Example: Adding thirty and subtracting one is an efficient way to add twenty-nine.
- accurate — Correct in the calculation or representation being considered. Example: The accurate calculation is 8 × 7 = 56, not 54.
- reasonable — Making sense when checked against the problem's size, units and context. Example: A three-centimetre pencil would be short, so we check whether that measurement is reasonable.
- counterexample — One example showing that a claim about every case is false. Example: Nine is a counterexample to the claim that all odd numbers are prime.
List check: 200 distinct entries in 10 groups of 20. This is a reference bank for learning and application, not a quota for one response.
Use the bank as a reference while counting, drawing, measuring and solving problems. A child who can show equal groups with counters may understand division before being ready to write a full explanation. The teaching sections below help you identify that starting point, add precise mathematical language and check independent understanding.
Choose your next useful step
Choose the part of the guide that helps with your current task.
Guide contents
Sections 1–4: Understand the stage and use words accurately
Sections 5–6: Worked examples and practice
Sections 7–11: Review, revise and continue
The IB Primary Years Programme covers ages 3–12. This pack concentrates on school-age primary learning, approximately ages 5–11. School Years 1–6 are an illustrative navigation aid, not official IB programme-year labels. A school’s starting age and year names may differ.
The PYP is a transdisciplinary framework. Schools develop their own programme of inquiry, connecting subject knowledge with meaningful questions. Accordingly, these ten vocabulary groups are an eduKate teaching selection. They are not an IB-prescribed sequence or a universal set of annual mathematics standards.
There is no formal IB PYP assessment. Schools still assess learning and provide feedback. The IB assessment overview distinguishes PYP feedback and reflection from formal assessment in later programmes. A school may set mathematics tasks or tests; follow its instructions, content and success criteria. There is no official requirement to use a specified number of these words in an answer.
| Route | Useful starting work | Evidence to look for |
|---|---|---|
| Early | Count, compare, make groups and describe familiar shapes. | Show the idea accurately with objects, speech or drawings. |
| Secure | Connect place value, operations, fractions and measurement. | Choose a representation and explain how it supports an answer. |
| Stretch | Investigate ratios, data, variables and conditions. | Justify a claim, check assumptions and test another case. |
Start here: can the child show equal sharing?
Use twelve counters and three empty trays, or draw twelve dots and three circles. In this invented classroom task, a child puts five counters in the first tray, four in the second and three in the third. The child says, “There are twelve altogether, so the sharing is equal.” Decide whether that statement is correct. Show what you would change if every tray must contain the same number and all counters must be used. Then explain your decision using objects, a drawing or spoken words. If you are ready, record the calculation in symbols. Your five target words are equal, divide, array, equation and check; choose the ones that help your explanation. Finally, try fifteen counters with the same three trays. Work out the new arrangement independently and show how you know it meets both conditions.
Answer guidance and a useful diagnosis
Five, four and three total twelve, but the groups are unequal. Moving one counter from the first tray to the third gives four in each. The equation is 12 ÷ 3 = 4; three rows of four form a matching array. Check by adding 4 + 4 + 4. With fifteen counters, each tray receives five.
If the child preserves the total but accepts unequal groups, revisit what equal means. If the arrangement is correct but the explanation is difficult, provide a sentence opening: “Each tray has …”. Accept gesture, a familiar-language explanation or adult transcription. Record mathematical understanding separately from spelling and sentence control. A copied sentence alone does not show that the child can repeat the reasoning with new numbers.
Learn words in useful relationships. “Subtract five from twelve” means 12 − 5, while “five less than twelve” describes seven. The order of words can change the calculation. Read the sentence aloud, represent it, then write symbols.
| Pattern | Meaning to preserve | Example |
|---|---|---|
| Equal to | Both sides have the same value. | 8 + 4 = 6 + 6. |
| At least / at most | The stated boundary is included. | At most ten allows ten, but excludes eleven. |
| A factor of / a multiple of | Keep the relationship in the correct direction. | Three is a factor of twelve; twelve is a multiple of three. |
| Ratio / proportion | State what each quantity refers to. A ratio may compare parts or a part with the whole; proportion here means the fraction of the whole. | Three red and two blue give ratio 3:2 and red proportion 3/5. |
| One vertex / several vertices | The plural changes without changing the property. | A triangle has three vertices. |
Attach units to quantities: a perimeter of 20 cm and an area of 24 cm² answer different questions. NIST’s SI guidance treats unit symbols as symbols, not abbreviations: write 4 cm, not 4 cms. Use a space between the number and symbol. For capacity, 1 L = 1,000 mL; NIST explains the volume relationships.
Keep mass and weight distinct in scientific explanations. Mass is recorded in grams or kilograms. Weight, as a force, uses newtons. Everyday speech often calls mass “weight”; NIST explains both usages. Identify which quantity the task means before choosing a unit.
The Fencing Method is eduKate’s routine for building and checking sentences. This mathematical adaptation is a teaching aid, not an IB answer formula. Begin with a correct idea; add only information that helps another person follow it.
- Correct core: for a rectangle with length 6 cm and width 4 cm, write, “The perimeter is 20 cm.”
- One useful addition: add, “because 6 + 4 + 6 + 4 = 20.” The reason accounts for all four sides.
- Check grammar, meaning and evidence: does perimeter mean boundary length here? Do the four lengths match the stated rectangle? Is cm the appropriate unit?
- Repair: if the draft says “The perimeter is 24 cm²,” separate the two ideas. Its area is 24 cm², but its perimeter is 20 cm.
- Independent transfer: give a fresh rectangle measuring 5 cm by 3 cm. Ask for a perimeter statement and reason before showing a model answer.
The transfer answer is 16 cm because 5 + 3 + 5 + 3 = 16; its area is 15 cm². An early learner can use the same routine orally: “There are twelve counters” becomes “There are twelve counters because three groups of four make twelve.” Add a second sentence only when the first idea is secure.
SECTION 5 OF 11
5. Worked investigation: which display card satisfies every condition?
This invented planning task uses three rectangular cards. Measurements are exact for the model. The labelled width must be at most 4 cm, the area must be at least 24 cm², and the border must use at most 20 cm of ribbon. Assume the ribbon follows the complete boundary once, with no overlap or waste. Choose a card that meets all three conditions.
| Card | Length | Labelled width |
|---|---|---|
| A | 6 cm | 4 cm |
| B | 8 cm | 3 cm |
| C | 5 cm | 5 cm |
Represent and calculate
Draw each rectangle and label its sides. For these rectangles, area equals length multiplied by width; the ribbon requirement equals perimeter, found by adding all four sides. An early learner can draw a grid of unit squares and count the squares and outer edges. A learner ready for multiplication can record the calculations directly.
| Card | Area | Perimeter | Decision |
|---|---|---|---|
| A | 6 × 4 = 24 cm² | 6 + 4 + 6 + 4 = 20 cm | All three conditions met. |
| B | 8 × 3 = 24 cm² | 8 + 3 + 8 + 3 = 22 cm | Needs 2 cm more ribbon than allowed. |
| C | 5 × 5 = 25 cm² | 5 + 5 + 5 + 5 = 20 cm | Width exceeds the limit by 1 cm. |
Justify the decision
“Choose card A. Its width equals the maximum allowed, its area equals the minimum required, and its perimeter equals the ribbon limit. Card B needs too much ribbon. Card C is too wide.” This explanation addresses every condition rather than listing calculations without a decision.
Cards A and B have equal areas but different perimeters. That comparison is a counterexample to the claim that equal area always means equal perimeter. If real ribbon needed an extra 1 cm for joining, A would require 21 cm and fail the original ribbon limit. State the changed assumption before changing the conclusion.
These are original, invented practice situations. Attempt the question before reading its answer. Use the early, secure or stretch label to select a suitable challenge; the labels are teaching choices, not official IB levels.
1. Early route: share whole counters
Task: Share eleven counters between two trays. Put as many as possible in each tray, keep the groups equal, and leave any unused counters outside. Counters cannot be cut. Show the result.
Answer: Each tray receives five counters, with one left outside. The check is 5 + 5 + 1 = 11. In whole-number division, 11 ÷ 2 has quotient 5 and remainder 1. Six in one tray and five in the other uses every counter but fails the equal-groups condition.
2. Secure route: compare parts of the same whole
Task: Use the same collection of twelve counters as the whole. Find one third and one quarter. Which is larger? Explain why a larger denominator does not give a larger unit fraction here.
Answer: One third contains four counters because 12 ÷ 3 = 4. One quarter contains three because 12 ÷ 4 = 3. Therefore 1/3 > 1/4. Splitting the same whole into more equal groups makes each group smaller.
3. Secure route: manage time and a budget
Task: A pretend stall opens at 09:35 and closes at 10:10 on the same day. How long is it open? Three cards cost $2.40 each. Find the change from $10; assume no additional charges.
Answer: It opens for 35 minutes: 25 minutes to 10:00, then 10 more. The cards cost 3 × $2.40 = $7.20, so the change is $2.80. Check that $7.20 + $2.80 = $10.
4. Stretch route: describe a complete data set
Task: Five fictional reading records show 2, 4, 4, 6 and 9 books. Find the mean, median, mode and range. Must the mean appear in the records?
Answer: The sum is 25, so the mean is 25 ÷ 5 = 5. The ordered middle value is 4, also the most frequent value, so median and mode are both 4. The range is 9 − 2 = 7. The mean need not occur as a recorded value: no record here equals 5.
5. Stretch route: state the chance model
Task: A bag contains three red tokens and two blue tokens. Assume each individual token has the same chance of selection. Find the red-to-blue ratio and the probability of red. Remove one red token and leave it out. What is the new probability of red?
Answer: The ratio is 3:2, but the probability is 3/5 because five tokens are available. After removal, two red and two blue remain. Under the same equal-chance assumption, the probability becomes 2/4 = 1/2.
Use the child’s representation to locate the difficulty. The suggested next task is a proposed check, not evidence that the repair has already worked.
| Draft or action | Repair the meaning | Fresh check |
|---|---|---|
| “The equals sign means write the answer next.” | Both sides must have the same value. | Complete 7 + 5 = 9 + □. |
| “Subtract five from twelve” becomes 5 − 12. | Start with twelve and take away five. | Show “subtract three from ten” using counters. |
| “One quarter is bigger than one third.” | Compare equal parts of the same-sized whole. | Fold equal paper strips into thirds and quarters. |
| “The perimeter is 18 cm².” | Boundary length uses length units; area uses square units. | Find both quantities for a 6 cm by 3 cm rectangle. |
| “Every odd number is prime.” | Nine is odd but has factors 1, 3 and 9. | Find another odd composite number and justify it. |
| “The median is the biggest number.” | Order the values and locate the middle. | Find the median of 8, 1 and 6. |
Check answers: the missing value is 3; 10 − 3 = 7; 1/3 > 1/4; the rectangle has perimeter 18 cm and area 18 cm²; 15 is one suitable odd composite number; the ordered data 1, 6, 8 have median 6.
Select four to six connected terms, such as array, row, column, multiply and product. Use short sessions that fit the learner’s concentration and school work. The following schedule is optional; move on when the representation and explanation are secure.
| Session | Suggested activity | Record |
|---|---|---|
| 1: Explore | Build an array and name its features together. | Which ideas can the child already show? |
| 2: Retrieve | Rebuild a different array without copying. | Which terms need a prompt? |
| 3: Explain | Use the Fencing Method for one calculation and reason. | Does the sentence match the model? |
| 4: Transfer | Apply the idea to seats, tiles or a supplied diagram. | Which steps are independent? |
Revisit a difficult relationship with smaller numbers or clearer objects. Repeating a definition more loudly does not reveal why a calculation or representation is wrong.
Choose a new context that preserves the mathematical relationship: move from counters in trays to books on equal shelves, or from a card border to a rectangular garden drawing. Supply all necessary measurements. Do not add a reading puzzle when the purpose is to check multiplication or perimeter.
Student check: “Can I show the idea, name the useful quantities and explain why my answer fits?” Parent check: note what the child did before receiving a hint. Teacher check: distinguish a mathematical misconception from unfamiliar English, recording demands or a misunderstood instruction.
A simple progress record can show the date, task, chosen words, representation, help given and next task. Mark work as modelled, prompted or independent. One correct response is a useful observation, not proof of permanent mastery. A later task with changed numbers or conditions provides further evidence. The PYP learning and teaching framework connects inquiry with reflection and ongoing assessment.
Should an early primary learner study all 200 entries?
No. Choose words that name ideas the child is exploring. A small set used accurately is a sensible starting point. Keep ratio, variable and formal data summaries available for later use.
Does a longer answer earn more credit?
Follow the school’s task. A clear diagram and one justified calculation may communicate more than several sentences of unused terminology. Include enough reasoning for the requested explanation.
What if the child understands the mathematics in another language?
Let the child explain that way, then connect the idea with the English term and symbol. Ask for a new example to check the concept separately from English recall.
Why is a square also a rectangle?
This pack uses inclusive classification: a rectangle has four right angles, and a square satisfies that definition. Extra properties can place one shape in several families. Discuss the properties shown, not only the drawing’s orientation.
What if a data set has two middle values?
Order the values and find the mean of those two middle values. For 2, 4, 7 and 9, the median is (4 + 7) ÷ 2 = 5.5. See OpenStax’s explanation of measures of centre.
Programme information and technical references were checked on 9 November 2026, Singapore time. Official IB sources linked above support the programme and assessment statements; NIST supports unit conventions, and OpenStax supports data-summary terminology. The word choices, definitions, mathematical scenarios and answer explanations are original eduKate teaching material.
For further mathematical exploration, see the University of Cambridge’s NRICH primary activities. Its curriculum navigation is not an IB year map. This article follows the immediate-reference approach of eduKate’s Top 200 vocabulary spine, with subject-specific explanations and tasks. Use your school’s current programme and teacher instructions to choose the next learning step.
Continue with vocabulary for your stage
The IB programme overview places PYP at ages 3–12, MYP at 11–16 and DP at 16–19. Schools use different year and grade labels. In this eduKate series, School Years 1–13 are an illustrative school-year route; your school’s programme placement takes priority.
Age 12 vocabulary reference · Fencing Method for vocabulary · Vocabulary Learning Hub
Related subject Essential Word Packs
- PYP Mathematics — 200 words — this guide
- PYP Science — 200 words
- MYP Mathematics — 200 words
- MYP Sciences — 200 words
General Essential Word Packs by school year
School Year 1 · School Year 2 · School Year 3 · School Year 4 · School Year 5 · School Year 6