Thinking Maths by Topic
Each topic below is a complete problem-type family: how to recognise the type, fully worked examples, and an animated lesson your child can learn from right away. Start with the type they find hardest.
Operations & Algebraic Thinking
- Finding Two Numbers (Sum & Difference, Sum & Ratio, Difference & Ratio)Grades 4, 5 · 4 lessons · 30 worked solutionsFinding two numbers is a foundation family of Grade 4 and Grade 5 thinking maths: the problem never states the two numbers directly, only how they relate — a sum and a difference, a sum and a ratio, a difference and a ratio — or, in age problems, how that relation shifts over time. Every member of the family is unlocked by the same tool, the bar model: once the bars are drawn, it becomes visible whether the sum, the difference or the ratio must be split, and into how many parts. This page gathers all four lessons of the family, with fully worked examples, so parents can coach at home.
- Tree-Planting and Interval ProblemsGrades 5 · 1 lessons · 8 worked solutionsTree-planting problems are a small topic with a big trap in Grades 4–5: planting along a road, staking out a plot, or spacing objects evenly along a line — the answer is always off by exactly one from a straight division, and which way it leans depends on whether one end, both ends, or a closed loop is planted. There is one key: count the gaps first and derive the number of trees second. This page teaches the type on an animated road that counts every gap and every tree, so children stop guessing whether to add or subtract one.
- The Assumption MethodGrades 5 · 1 lessons · 8 worked solutionsThe assumption method is a method, not just a problem type: whenever a problem mixes two kinds of objects at two value levels — chickens and rabbits, two- and three-wheeled vehicles, two ticket prices — assume everything is one kind, measure how far that lands from the given total, then swap one object at a time until it matches. The Grade 5 chickens-and-rabbits problem is the classic doorway into the method, and this page presents it with animation so children can see what each swap does to the total.
- Working BackwardsA favourite of olympiad and entrance exams: given the final result, undo each operation in reverse order to find the starting number — then check by replaying the story forwards. Free.
- Summing Arithmetic SequencesAdd 1 + 2 + … + 20 without adding one by one: fold the sequence and pair first with last — every pair is equal. A sequence type common in olympiad papers and entrance exams, shown step by step.
Number & Operations in Base Ten
Number & Operations — Fractions
Measurement & Data
Geometry
- Area of a TriangleNo rote formula: cut along the midline and flip the top pieces down — the triangle becomes a rectangle with half the height, and the ÷2 appears by itself. Geometry is core to olympiad and entrance exams.
- Area of a TrapezoidClone the trapezoid, rotate it half a turn and join the two: a parallelogram whose base is the sum of both bases. See the formula being built step by step instead of memorising it. Free.
- Circle Circumference & AreaWrap a string around a circle and straighten it: just over 3 diameters — that is 3.14. Then cut the circle into 16 wedges and rearrange them toward a rectangle to see the area formula. Grade 5, visual.
Ratios & Proportional Relationships
- Transfer RatioThe trickiest advanced ratio type: pouring 10 litres across changes the ratio but never the total. Animated tanks make the invariant obvious before any calculation. Step by step, free.
- Map ScaleReading 6 cm on a 1 : 500 000 map as 6 km is the trap: each paper centimetre hides 5 real kilometres. Learn to read map scale with visual measurement — the applied-ratio type of grades 4–5. Free.
- Two Taps Filling a TankThe classic work-rate trap: taps of 4 hours and 6 hours do not take 10 hours together. Add each tap's hourly share instead — an animated tank shows why the answer is 2.4 hours.