Mathematics, 11–13 years — 127 topics · MyLeoNes™ Kuks
127 mathematics topics written for 11–13 years — not a older text simplified. In teaching order, each a deck of five cards: the idea, why it exists, a worked example, the common trap and where you meet it.
Mathematics · 11–13 years
The stages of solving a problem
Interpret, choose and carry out a strategy, then check the result in context — and see that one problem can be solved in several ways.
Conjecture, test and justify
Spot a pattern, state a conjecture, and learn that testing cases is not the same as validating — with counterexamples and other ways to justify.
Computational thinking: split, spot patterns, write an algorithm
Keep the essential, break a problem into easier steps, reuse patterns, write a step-by-step algorithm and debug it — on real numbers.
Communicate, represent and connect maths to the world
Say the same idea in words, diagrams, numbers and symbols, explain your thinking, link topics together and use maths as a model of real situations.
Multiples and divisors of a natural number
Divisors and multiples and how they relate: remainder zero, finitely many divisors, endless multiples.
Prime numbers below 100
Primes are the numbers with only two divisors: find the 25 below 100 and use them to solve problems.
Powers: base, exponent and powers of 10
A power as a product of equal factors: base and exponent, powers of 10, and why repeated multiplication grows so fast.
Equivalent fractions and percentages
Two fractions can name the same amount, and a percentage is just a fraction with denominator 100.
Comparing and ordering fractions and decimals
Pick a smart strategy to say which number is greater, and place numbers on the number line.
Approximations and rounding to hundredths
Approximating a decimal to hundredths: from below, from above and by rounding.
Adding and subtracting fractions
Adding and subtracting fractions when one denominator is a multiple of the other, with mental strategies and estimates.
Multiplying a fraction by a natural number
Multiplying a fraction by a natural number is adding that fraction again and again, or taking that part of a quantity.
Multiplying decimals
Multiply as if there were no decimal points, then put the point back: the product has as many decimal places as both factors together.
Dividing decimals
Dividing by 0.1, 0.01 or 0.001 means multiplying by 10, 100 or 1000, and other divisions with decimals turn into divisions of natural numbers.
Mental calculation with decimals and reasonable answers
Choose the shortest path to calculate decimals in your head, and use an estimate to see whether a result makes sense.
Sequences and rules
A rule can build a list of numbers or figures, and the best rules jump straight from a term's position to the term.
Describing relations with letters
A letter stands for a number, so one short expression can say what a whole set of numbers or a whole relation has in common.
Values of expressions and equivalent expressions
Replace the letter by a number to find the value, and see that two different-looking expressions can tell the same story.
From question to data: what, where and how to collect
A statistical question has answers that vary; before collecting, you decide what data you need, where to get them, whom to ask and whether people answer in public or in secret.
Questionnaires and frequency tables
Build a questionnaire with closed questions, then organise the answers in a table with absolute and relative frequencies (as percentages).
Bar charts and pie charts
Turn a table of frequencies into bars or a pie chart with an honest scale, a title, a source and a legend, and compare two groups side by side.
Reading media graphs critically
The same data can look very different depending on the scale and the type of chart; check the axis, the source and whether the chart fits the story.
The mean: sharing the total equally
The mean is what each value would be if the total were shared out equally, and it reacts to every single value.
Mode or mean? Summarising data
Categories can only have a mode; numbers can have a mode and a mean, and the question of the study decides which to use.
Reading data, drawing conclusions, telling the study
Read a graph carefully, conclude only what the data support, ask the next question and tell the story on a digital poster.
Probability: how convinced are we?
A probability tells how strongly we believe something will happen, from 0% to 100%, and it can be estimated with a relative frequency.
Lines, rays, line segments and how lines sit relative to each other
Tell a line from a ray and a segment, and name two lines as parallel, intersecting, perpendicular or oblique.
Angles: amplitude, degrees, protractor and construction
Measure an angle in degrees with a protractor, classify it, estimate it from 45°, 90° and 180°, and construct one.
Triangles: types, sides and angles, construction and congruence
Classify triangles by sides and angles, link sides to angles, and know when a triangle can be built and when two are congruent.
Equivalent figures and the area of a parallelogram
Same area, different shape: cut and move pieces to see why a parallelogram has area base × height.
The area of a triangle and its heights
A triangle is half a parallelogram: area = base × height ÷ 2, with three heights and a position that depends on the type of triangle.
Prisms: parallel and perpendicular faces
Find the parallel and perpendicular faces of a prism, and see why every cube is also a rectangular cuboid and a right prism.
Patterns in the faces, edges and vertices of polyhedra
Count, spot a pattern, write it with a letter, test it and explain why it works.
Nets of polyhedra
Unfold a solid flat, fold it back, and see that one solid can have several nets.
Solving a problem in stages
The same stages, now with fractions of a whole you cannot see: interpret, compare strategies, evaluate the result — and pose problems of your own.
From conjecture to justification
Classify, spot a regularity, state a conjecture and tell testing from validating — with counterexamples, exhaustion and generic examples.
Decompose, spot patterns, debug
Keep the essential, split into steps, reuse a pattern, write an algorithm — and hunt the bug with the awkward case.
Communicate, represent and connect
One idea in words, diagram, table and symbols; explain your thinking and answer an argument — and see the same maths in juice, rose windows and surveys.
Prime factorisation: every number in prime bricks
Break a number into prime factors, by tree or by successive divisions, and see why there is only one way to do it.
Lowest common multiple and highest common factor
Find the LCM and HCF of two numbers with lists or prime factors, choose the quickest method and use them to solve problems.
Multiplying and dividing powers: the rules
Same base: add or subtract the exponents. Same exponent: multiply or divide the bases. And why counting factors proves it.
A fraction as a measure, and the irreducible fraction
A fraction measures a piece against a unit, and among all its equivalent names the irreducible one is the shortest.
Adding and subtracting fractions with a common denominator
When neither denominator divides the other, find a new size of part that fits both, then add or subtract the numerators.
Multiplying fractions and powers of a fraction
A fraction of a fraction is a product, and repeating the same fraction as a factor gives a power such as (2/3)³.
Inverse numbers and dividing fractions
Two numbers are inverses when their product is 1, and dividing by a fraction is multiplying by its inverse.
Numerical expressions: from a situation to its value
Write a situation as an expression, then calculate it with parentheses, powers and the order of operations.
Properties of operations: calculating faster
Swap, regroup and split numbers with the commutative, associative and distributive properties, and judge which strategy is simplest.
Mental calculation with fractions
Add and multiply fractions faster by swapping and grouping terms, without always reducing to a common denominator.
Decreasing sequences and their rules
In a decreasing sequence every term is smaller than the one before, and the rule shows up when you look at how each term comes from the previous one or from its position.
Direct proportionality: a multiplicative relation
Two quantities are directly proportional when multiplying one by a number multiplies the other by the same number, so one known pair unlocks all the others.
Ratio, proportion and constant of proportionality
A ratio compares two quantities with a fraction, a proportion says two ratios are equal, and the constant of proportionality is the ratio that never changes.
Properties of operations written with letters
The properties of addition and multiplication work for any numbers, so an expression with letters says them all at once and lets you complete equalities.
Measuring instead of counting: questions and collecting continuous data
Heights, times and weights are measured, not counted: learn to frame the question, decide what, where and whom to collect from, and keep every measured value in a list.
Classes of equal width and the frequency table
When almost every measured value appears only once, you group them into classes [a, b) of equal width and organise them in a table of absolute and relative frequencies.
The histogram and the line graph
The histogram shows how values grouped in classes are distributed; the line graph shows how a value changes over time. Both need a scale, a title, a source and a legend.
Reading graphs critically and choosing the right graph
The same data can give different pictures: learn to compare graphs from the media, to see what the choices hide and to justify the graph you choose.
The modal class and choosing a summary measure
When data are grouped in classes the mode becomes the modal class, and the nature of the data decides which summary measure is honest.
From data to conclusion, report and infographic
Read the distribution, draw a conclusion backed by numbers, decide, ask more, and tell it all in a report and in a rigorous infographic.
Equal chances, equal probabilities
Not every random situation has equally possible results; when it does, their probabilities are equal, and an experiment helps to decide.
Polygons: concave or convex, regular or irregular
Tell concave from convex and regular from irregular polygons, and use the perimeter to solve problems with them.
Circle: circumference, π and area
See that the perimeter of a circle is directly proportional to its diameter, with constant π, and use P = π × d and A = π × r² in problems.
Complementary and supplementary angles
Two angles that add up to 90° are complementary and two that add up to 180° are supplementary; the sum stays the same even when the parts change.
Triangles: the sum of interior and exterior angles
Conjecture from cut-outs that the interior angles add up to 180° and the exterior ones to 360°, explain why, and use it in triangle problems.
Volume: what it is and how we measure it
Volume is the space an object takes up: you measure it by counting units, and the unit you pick changes the number.
Volume of a cuboid and of a cube
Fill a box layer by layer and count the unit cubes: length × width × height, and for a cube, edge × edge × edge.
Volume of a cylinder and of solids made of parts
A cylinder is a stack of equal circles, so its volume is the area of the base times the height; then add the parts of a bigger solid.
Turning a point and a shape around a centre
Turning a point, a triangle or a quadrilateral about a centre through a given angle and direction, using ruler, compass and protractor.
Rosettes: rotational symmetry and axes
Finding the minimum rotation angle of a rosette, linking it to the number of axes of symmetry, and building a rosette by successive rotations.
Solving a problem: from the table to the equation
The same three stages, now with an unknown you can name with a letter: compare a table with an equation, and always check the answer against the situation.
Conjecture, test and justify with letters
From a pattern to a statement: test it, tell testing from validating, and justify it with letters or knock it down with a counterexample.
Computational thinking with sequences
Abstract, decompose, spot a pattern, write an algorithm and debug it, all on one question: is this number a term of the sequence?
Representing, modelling and connecting
One relation told in words, table, graph and symbols; a model built from measurements; and a rectangle that connects geometry, numbers and algebra.
Integers: negative numbers on the number line
See negative numbers as the other side of zero, place them on the number line, order them and meet ℤ.
Absolute value and the opposite of a number
Measure how far a number is from zero, and find the number on the other side at the same distance.
Adding and subtracting integers
Add by moving on the number line, turn every subtraction into an addition, and know which properties hold.
Numerical expressions with integers, and problems
Write, simplify and calculate expressions with parentheses and minus signs, choose an efficient path and solve problems.
What a rational number is, and the set ℚ
A rational number is an integer over a non-zero integer, positive or negative; ℚ holds the integers and the fractions and decimals you know, now with a sign.
Rational numbers on the number line: comparing and ordering
Every rational number has its place on the number line, and the further right it is, the greater it is: that is how fractions, decimals and negatives are compared and ordered together.
Adding and subtracting rational numbers
Rational numbers add and subtract with the rules of integers, as fractions or decimals; subtracting is adding the opposite, and the properties of addition make mental calculation shorter.
Percentages: the part, the whole and the percentage
Three numbers are linked — part, whole and percentage — and knowing any two lets you find the third.
Percentage problems: discounts, increases and more than 100%
Turn a discount or an increase into a single percentage of the starting value, and explain your method so others can check it.
Scientific notation: very large numbers, written short
Write a big number as a number from 1 to under 10 times a power of 10, then compare and calculate with it.
Sequences of rational numbers and the general term
With negatives, decimals and fractions in the list, the rule still comes from the jump between terms, and the general term gives any term straight from its position.
First-degree equations: recognising and translating
An equation asks for which value of the unknown an equality is true, and writing a situation as an equation is the first step to solving it.
Solving first-degree equations and problems
Solving means undoing operations on both sides so the solution never changes, and then checking that the answer makes sense in the story.
Functions: every object has exactly one image
A function is a correspondence where each object of the starting set is linked to exactly one image, and the same function can be shown as arrows, a table, an expression or a graph.
Graphs: describing and modelling situations
Reading a graph from left to right tells the story of two variables; modelling goes the other way, from a situation to the function that describes it.
Direct proportionality as a function
A direct-proportionality function is y = k × x with a domain: its table, its graph (points on a line through the origin) and its expression tell the same story.
What kind of variable is it? Nominal, ordinal, discrete or continuous
Sort what varies in a statistical question into four kinds, and choose a question and a precision that give the data you need.
Population and sample: who do you ask, and how do you choose them?
Tell the whole group you want to know about from the part you actually study, and plan a draw that makes the sample represent it.
Define, collect and clean the data
Decide exactly what to record and how, collect it from a credible source, then sort the values and check the unexpected ones before you use them.
Grouping counted data in classes: the frequency table
When counted values are very spread out and almost none repeats, group them in classes of equal width and organise them in a titled frequency table.
Two sets of data in one graph
Compare two sets of data with a line graph (time on one axis) or overlapping bars, on a fair scale, with title, source and legend, and justify the graph you choose.
From the study to the news: conclude, communicate, question
Finish a study with a conclusion, a decision and a new question, tell its story to the right audience without misleading, and ask the same questions of a study reported in the news.
Median and range: where the centre is, how spread out the data are
The median says where the centre of the data is, the range says how spread out they are; learn to compute both, to see how an outlier moves them, and to pick the measure that fits the data.
Probability of an event with several outcomes
Given a probability model, the probability of an event made of several outcomes is the sum of the probabilities of those outcomes, even when the outcomes are not equally likely.
Interior and exterior angles of a convex polygon
Where the sums (n − 2) × 180° and 360° come from, and how to use them to find a missing angle.
Alternate interior angles and vertically opposite angles
Which angles are equal when two lines cross and when a transversal cuts two parallel lines, and why.
The diagonals of a quadrilateral and their properties
What a diagonal is and what each kind of quadrilateral does with its two: bisect each other, be equal, cross at a right angle.
Classifying quadrilaterals in a hierarchy, and explaining why
Trapezoid, parallelogram, rectangle, rhombus, square and kite as families inside families, and how to say why a figure belongs.
Areas of the trapezium, the rhombus and the kite
Derive the area formulas for the trapezium, rhombus and kite from triangles and parallelograms you already know.
Similar figures, ratio of similarity and homothety
Recognise figures with the same shape, find the ratio of similarity, build enlargements and reductions by homothety and read map scales.
Similarity criteria for triangles
Find out that two angles are enough to make triangles similar, meet the other two criteria, avoid misusing them and use them to work out measurements.
Ratios of perimeters and areas of similar figures
Perimeter is multiplied by k and area by k²: find out why and use it with plans, enlargements and problems.
Regular and irregular polyhedra: models and nets
Two tests tell a regular polyhedron from an irregular one; then go from a net to the solid and back.
Why there are only five regular polyhedra
Add up the angles at a corner and see, polygon by polygon, which regular polyhedra can exist.
Faces, edges, vertices and Euler's formula
Count many polyhedra, find what never changes, and use it to check your own counts.
Fractions as numbers
See a fraction as a number with a place on the number line, not only as a piece of a cake.
Ratios
Compare quantities fairly by keeping track of how many parts of one thing match another.
Percentages
A percentage gives a quantity as parts out of one hundred, making comparisons easier.
Linear equations
Use an unknown number and preserve balance to find what a statement is really saying.
Negative numbers
Numbers below zero help describe amounts that go in the opposite direction from ordinary counting.
Powers and square roots
Powers repeat multiplication, while square roots undo a square.
Pythagorean theorem
In a right-angled triangle, the three side lengths follow a dependable square relationship.
Probability
Probability measures how likely an event is when the possible outcomes are understood.
The coordinate plane
A way to give every point on a flat surface an exact address using two numbers.
Angles in triangles
The three inside angles of every triangle always add up to a straight angle.
The area of a triangle
How to measure the amount of flat space inside a triangle.
The mean average
One number that represents a set of values by sharing their total equally.
Decimal place value
Understand what each digit means after the decimal point, and use that meaning to compare, round and calculate with decimal numbers.
Perimeter
Find the distance around a shape by adding its side lengths, and understand why different shapes can have the same perimeter.
Volume of rectangular prisms
Measure how much space a box-shaped solid contains by counting equal layers of cubes and multiplying its three dimensions.
Scale drawings
Use a fixed relationship between a drawing and the real object to read maps, plans and models without losing the real distances.
Prime factorisation
Prime factorisation breaks a whole number into the prime numbers that multiply to make it. It gives each number a useful fingerprint for comparing, simplifying and organising numbers.
Algebraic expressions
An algebraic expression uses letters and numbers to describe a calculation whose value may change. Learning to read and simplify one makes a changing situation easier to calculate and explain.
Inequalities
An inequality compares amounts when they are not necessarily equal. It helps describe limits, ranges and choices such as a bag weighing at most 7 kilograms or a game score above 100.
Circumference of a circle
The circumference is the distance all the way around a circle. It is connected to the circle’s diameter by the constant π, so you can find the outside distance without measuring every point of the curve.
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