Problem solver finale · Code with Martin

Everything you can do now

The thirtieth lesson. Stop a moment and look back — because what you have behind you is the whole of year 9 maths.

Algebra: algebraic fractions with conditions, equations with x in the denominator, simultaneous equations by two methods and word problems you translate into them. Functions: a number machine, a straight line, a parabola, a hyperbola and reading graphs. Geometry: similarity with tests AA, SAS, SSS, a shadow that measures trees, and the basics of trigonometry. Solids: a pyramid, a cone, a sphere and kits from them. Money and data: simple and compound interest, charts that do not lie, and probability. And with that entrance-exam prep: calculating, geometry, word problems, strategy.

Today nothing new. The finale is a big mix — you go across everything and find weak spots while there is still time to patch them.

Take one thing beyond formulae: you have learned to work reliably. A method on paper, a check, a sense check, an answer to the question. That is a skill for a whole life — formulae can be looked up, this cannot.

The whole gang sits in Sofie’s garden over a pile of written papers from the whole year. “Remember how simultaneous equations felt like black magic?” Filip laughs. Tereza flips through a notebook: “And now we solve them by two methods and even pick which is faster.” Jonáš lifts a calculator: “Compound interest, probability, a cone… we really calculated the whole of year 9.” Ema raises a lemonade: “To Problem solver. And to entrance exams not surprising us.” Adam adds: “We will surprise them.”

💡

You recognise a weak spot by hesitation: a question where you do not know how to START belongs on the recap list. A question where you only look for a formula is fine — the formula will be found.

A big map of Problem solver

Go through the map and at each stop answer: can I start a question?

Algebraic fractions (1–3): condition denominator ≠ 0, cancelling through factoring out, equations with a check.

Simultaneous equations (4–7): substitution (make it the subject and substitute with a bracket), elimination (cancel an unknown), word problems with a key and a check against the question.

Functions (8–12): y = ax + b is a straight line (a slope, b start), x² makes a parabola, k/x a hyperbola with a constant product. Read graphs by the axes.

Similarity and trigonometry (13–16): coefficient k from a ratio of sides, area grows with k², tests AA/SAS/SSS, the shadow method, sin/cos/tan as ratios of sides.

Solids (17–20): a pyramid and a cone = a third of a prism/cylinder, a sphere (4/3)πr³ and 4πr², do not mix up two heights, cut composite solids, surface only visible.

Money, data, chance (21–24): interest = deposit · rate/100, compound interest multiplies by (1 + p/100) each year, charts with an honest axis, probability = favourable/all.

Entrance exams (25–28): a method on paper, a sketch, translating word problems, four phases of a test, the two-minute rule.

Where you hesitated, go back — the lesson numbers are in brackets.

Example 1: across algebra

Question: Solve x + y = 11 and x − y = 3, then check that the solution satisfies the equation 12/(x − 4) = y − 1 (and do not forget the condition).

Method: Elimination: 2x = 14, so x = 7 and y = 4.

Second part — condition: x ≠ 4. Seven passes.

I substitute: 12/(7 − 4) = 12/3 = 4. Right side: 4 − 1 = 3. And 4 ≠ 3 — the solution does NOT satisfy the equation.

Answer: The pair of equations gives [7; 4], but the checked equation does not hold for them. Even “it does not match” is a full result of a check — that is exactly how mathematical honesty works.

Example 2: across geometry

Question: A tent has the shape of a square pyramid 4 × 4 m with height 3 m. A stick next to the tent casts a shadow 2 m and measures 1 m, the tent’s shadow (from the centre of the base) measures 6 m. Does the tent’s height match the shadow method?

Method: Shadow method: coefficient k = 6/2 = 3, height = 1 · 3 = 3 m. It matches the given height.

Bonus volume: V = (4 · 4 · 3)/3 = 16 m³.

Answer: The shadow confirms height 3 m and the tent holds 16 m³ of air. Similarity and solids in one question — that is exactly how a test mixes too.

Example 3: across money and chance

Question: You get 5 000 Kč. You put it away for 2 years at 10 % a year (compound). From the resulting amount you then want to buy a lottery ticket for 55 Kč, where 1 ticket in 5 wins. How much do you have after two years and what is the chance of a win?

Method: Saving: 5 000 · 1.1 = 5 500 Kč after the first year, 5 500 · 1.1 = 6 050 Kč after the second.

Ticket: probability of a win = favourable/all = 1/5, so 20 %.

Answer: After two years you have 6 050 Kč and the ticket wins with probability 1/5. Free advice: interest is sure, a ticket is not — 4 of 5 tickets is just more expensive paper.

⚠️

Do not recap only what you are good at. The natural pull is to practise favourite questions — they are pleasant, because they come out. But marks are added by training weak spots. From the finale mix write down every hesitation and give it one targeted afternoon. Strong spots will not betray you, weak ones can be caught up — but only if you know about them.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. Start with ten. When it goes well, add more.

A closing mix like in entrance exams — equations, percents, fractions, Pythagoras, areas. Write fractions as a/b, decimals with a comma. Do a series at least twice.

Practise the questions

Finale set on paper

One question from each big area. Method, check, answer — like in a real test.

Algebra: solve 2x + y = 16 and x − y = 2 (method of your choice) and do the check in both equations.
Functions: the function y = 3x − 2. Build a table for x = 0, 1, 2, 3, say where the line crosses the y-axis, and for which x is y = 13.
Solids: a cone with radius 3 cm and height 4 cm — volume, slant height (Pythagoras) and surface, π ≈ 3.14.
Money and chance: 8 000 Kč for 2 years at 10 % compound; from the result calculate how much is left after a purchase of 1 680 Kč. And the probability that from 3 red and 5 blue beads you pull a red one (as a fraction).

Now you 💪

From each big area (algebra, functions, geometry, solids, money, chance) you solve at least one question without a hint.
You have a written list of hesitations — topics for targeted recap.
You keep the ritual: a method on paper, a check, a sense check, an answer to the question.

✅

Done when: The finale set is solved across the whole level, a list of weak spots written and mix training gave at least two series of five. Problem solver finished — and let the entrance exams look forward to you.

What to take from this lesson

You have done the whole of year 9: expressions, simultaneous equations, functions, similarity, solids, money, data and probability.
The biggest win is not a formula, but a work ritual: method, check, sense, answer.
Look for weak spots by the hesitation “how to start” — and train exactly those.
Entrance exams are a game of sure marks, time and calm. You have a plan for all three.
Maths does not end with a test — budgets, interest and graphs go with you further. Now you understand them.