Everything you can do now
The thirtieth lesson. Stop for a moment and look back.
At the start of the level 7² was just a weird notation. Today you can do powers and square roots, you write a billion without counting zeros, you find a side of a triangle nobody measured. You calculate with letters, you expand, you factor out, you know the formulae.
And most of all: you solve equations. Scales in balance, rearrangements on both sides, a check with L and R. Equations with brackets, with fractions, word problems, motion, mixtures. That is the skill the rest of maths stands on.
Plus geometry — circumference, π, area of a circle, a cylinder, Thales — and statistics, thanks to which you will not be fooled by a mean.
Today no new topic. The finale is a review circuit: a few cleverly chosen problems that join more lessons at once. Go through them honestly — and find what holds and where to go back.
The gang made a finale quiz in the park. Ema prepared slips with problems, everyone draws. Adam draws a ladder against a wall — he draws and calculates Pythagoras. Tereza draws an equation with brackets, mutters “expand, tidy, check”. Jonáš gets a pizza: area of a circle and a decision, large or two small. Sofie calculates the median of the whole gang’s pocket money. Filip draws the last slip: “Explain to a younger sibling why (a + b)² is not a² + b².” The hardest problem of the day — explaining is a higher level than calculating. He manages it.
The best test of whether you can do something: explain it out loud to someone else. Where you get stuck, there is a hole — and that is exactly where you go back.
A map of the level: what belongs with what
Powers and square roots (lessons 1–5): n² = n · n, a square root is a power backwards, rules for calculating, writing big numbers through 10ⁿ.
Pythagoras (6–7): a² + b² = c² in a right-angled triangle. Hypotenuse = square root of the sum, shorter side = square root of the difference. A sketch always.
Expressions and polynomials (8–13): a letter is a box for a number. Substituting with brackets, collecting like terms, expanding each with each, factoring out, formulae (a ± b)² and a² − b².
Equations (14–20): scales in balance. Rearrangements on both sides, expand brackets, multiply fractions by the denominator, check L = R, turning formulae, translating word problems.
Applications of equations (21–22): motion (s = v · t, towards each other I add distances, catching up = the same distances) and mixtures (you add crowns and rates, not prices and times).
Geometry (23–27): circumference vs disc, o = πd, S = πr², cylinder V = πr²v, Thales’ circle for right angles.
Statistics (28): mean, median, mode — and why an extreme fools only the first one.
See the seams? Square roots feed Pythagoras, expressions feed equations, equations feed geometry of formulae. None of it is a separate island.
Finale problem 1: an aerial on the roof
An aerial 6 m tall is held by a steel rope anchored 8 m from the foot. How long is the rope — and how much do three ropes cost at 45 Kč a metre?
Pythagoras: rope² = 6² + 8² = 36 + 64 = 100 → rope = 10 m.
Three ropes: 30 m. Price: equation c = 30 · 45 = 1350 Kč.
Two lessons in one problem: a triangle and multiplying as an expression. That is what tests look like — topics mix.
Finale problem 2: an equation with the full show
Solve: 3(x − 2) + x/2 = 15.
I expand: 3x − 6 + x/2 = 15.
Fraction gone — times 2 everything: 6x − 12 + x = 30.
I collect: 7x − 12 = 30 → 7x = 42 → x = 6.
Check: L = 3 · (6 − 2) + 6/2 = 12 + 3 = 15 = R. It fits.
Brackets, a fraction, collecting, a check — four lessons in five lines. If you can do this example on your own, the equations of the level hold.
Finale problem 3: a pool in the garden
A round pool has diameter 3 m and depth 0.9 m. How many litres of water fit into it? And is a garden hose that gives on average 12 litres a minute enough to fill it in an afternoon?
Cylinder: r = 1.5 m. V = 3.14 · 1.5² · 0.9 = 3.14 · 2.25 · 0.9 ≈ 6.36 m³ = 6360 litres.
Time: 6360 : 12 = 530 minutes ≈ 8.8 hours.
An afternoon will not be enough — fill overnight. A cylinder, converting units, a mean as statistics and plain sense. The whole Equation kid in one pool.
The finale is not a race. If some problem will not go, it is not shame — it is an arrow. Go back to the matching lesson, go through it again and try the problem in a day. Holes close by returning, not by skipping.
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 final round of equations. Solve and write only x.
Final round on paper
Four problems, each from a different corner of the level. Write working as if it were a test.
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Simplify and calculate: (x + 3)² − x² for x = 10. (A formula, expanding, substituting — three lessons at once.)
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Solve the equation 2(x + 5) − 4 = x + 13 with a check L = R.
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A round cake has diameter 26 cm. Calculate its area and decide whether a bigger portion is an eighth of this cake, or a quarter of a cake with diameter 18 cm.
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Invent and solve one word problem on an equation for a younger sibling or a friend — and let them check your solution with a check.
Now you 💪
- I can handle a problem that mixes two or more lessons together.
- I know which chapters hold for me and where I need to go back.
- I can explain at least one thing from the level to someone else.
Done when: You have the final round done with all the working, both practice runs, and a list of chapters for a possible return. Equation kid is yours.
What to take from this lesson
- Equations are scales: rearrangements on both sides and a check at the end.
- Pythagoras, circles and cylinders = geometry you checked at home with a ruler.
- Expressions and formulae are a language — equations speak in it.
- The mean can be fooled by an extreme; the median cannot.
- A hole in the topic is not shame. Returning to the lesson is the fastest way on.
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