🌱 START HERE — Math from zero
Hi Faustina 👋 — this booklet is just for you.
If you feel lost, that's totally OK. This starts at the very beginning.
Read slowly, top to bottom. Don't skip. Each tiny step builds the next one.
You don't need to be "good at math". You just need to follow small steps in order. Anyone can do that. 💪
Respirá tranquila — no estás sola en esto, yo te ayudo. Leé de a poquito, paso a paso, y vas a ver que es mucho más fácil de lo que parece. Confío en vos más que en nadie. — Papá
- I will read each part slowly, even if it looks easy.
- When I see "Now you try", I'll actually try it on paper before looking at the answer.
- If a part confuses me, I'll re-read it once, then keep going — it often clicks on the next example.
Level 1 — the very basics
1 A letter is just a mystery number
In math, a letter like x is not scary. It's just a box hiding a number we don't know yet. Our job is usually to find out what number is hiding.
What number plus 3 makes 5? 2. So x = 2. You just did algebra. 🎉
Una letra es un número escondido. Resolver = descubrir ese número.
Real life
You already do this. "I had some money, spent $3, now I have $5 — how much did I start with?" Your brain says $8. That "some money" is x. Math just writes it with letters.
2 Solving baby equations (the balance rule)
An equation is like a balance scale ⚖️. The = sign means both sides weigh the same. The golden rule:
Our goal: get x alone on one side. We do that by "undoing" what's around it.
Example A — undo a +
Example B — undo a −
Example C — undo a × (times)
You always do the opposite operation, to both sides.
✏️ Now you try (do these on paper first!)
1) x + 6 = 10 2) x − 2 = 7 3) 3x = 12 4) x + 5 = 1
See answers
1) x = 4 (subtract 6)
2) x = 9 (add 2)
3) x = 4 (divide by 3)
4) x = −4 (subtract 5: 1−5 = −4 — yes, negatives are allowed!)
3 What "squared" (x²) and "square root" mean
The little ² just means "multiply the thing by itself." That's all.
So 3² = 3 · 3 = 9. 5² = 5 · 5 = 25. 10² = 100.
Why it's called "squared" ⬛
A square that is 3 long and 3 wide has an area of 3×3 = 9 little boxes. "3 squared" = the area of a 3-by-3 square. That's literally where the name comes from.
(−3)² = (−3)·(−3) = +9. Two negatives multiplied make a positive.
So both +3 and −3, when squared, give 9. (Remember this — it's why curves later have two answers.)
Square root √ = the reverse of squared
It asks: "what number, times itself, gives this?" √25 = 5, √100 = 10.
And because both +3 and −3 square to 9, the full answer is often written ±3 (plus-or-minus). (± significa "más o menos": dos respuestas)
✏️ Now you try
1) 4² = ? 2) (−5)² = ? 3) √16 = ? 4) √49 = ?
See answers
1) 16 2) +25 (minus×minus=plus) 3) 4 4) 7
Level 2 — pictures
4 The graph grid (how we draw math)
To draw math, we use a grid made of two number lines that cross:
The dashed orange path shows how to reach the point (3, 2): 3 across, then 2 up.
To plot (3, 2): start at the center, walk 3 to the right, then 2 up. Make a dot.
(x = cuánto camino de costado · y = cuánto subo o bajo)
A few points
(0, 0) = the center · (2, 0) = 2 right, 0 up (sits on the flat line) · (0, 4) = 0 right, 4 up (sits on the up line) · (−1, −3) = 1 left, 3 down.
• the flat sideways line = the x-axis (here y is 0). (eje x)
• the straight up-down line = the y-axis (here x is 0). (eje y)
5 A function is a black box: put x in, get y out
You already know this one 😉 — a function is a black box (a little machine). You drop a number x into it, something happens inside the box, and a number y comes out the other side. The rule (like y = x + 1) is just what the box does inside.
Drop the number 2 in the left, the box runs its rule x + 1 inside, and 3 comes out the right. 🎬
Drop numbers into the box y = x + 1
| put in x | box does x + 1 | out comes y | the point |
|---|---|---|---|
| 0 | 0 + 1 | 1 | (0, 1) |
| 1 | 1 + 1 | 2 | (1, 2) |
| 2 | 2 + 1 | 3 | (2, 3) |
Now plot those three dots and connect them — you get a straight line. That's the "graph" of the function.
6 Now the star of the exam: the U-curve (parabola)
Everything above was practice for this. Let's use a black box whose inside rule is y = x² and see what shape comes out.
Drop numbers into the box y = x² (the box squares whatever you put in)
| x | x · x | y | point |
|---|---|---|---|
| −2 | (−2)(−2) | 4 | (−2, 4) |
| −1 | (−1)(−1) | 1 | (−1, 1) |
| 0 | 0·0 | 0 | (0, 0) |
| 1 | 1·1 | 1 | (1, 1) |
| 2 | 2·2 | 4 | (2, 4) |
Plot the 5 dots from the table, join them smoothly → you get a U. That U is a parabola. 🎉
You've seen this shape everywhere 🏀⛲
A thrown ball, a water fountain, a smiley mouth — they all draw this U (or upside-down U). The exam is all about reading and describing this curve.
- Vertex = the turning point (the very bottom or top). (vértice)
- Zeros / roots = where the U touches the flat ground line (x-axis). (raíces)
- Y-intercept = where the U crosses the up-down line. (ordenada al origen)
Level 3 — the one skill you need for solving
7 Multiplying brackets, and going backwards (factoring)
Quadratics often come as two brackets multiplied, like (x + 2)(x + 3). You need to (a) open them up, and (b) go backwards. Let's learn both — slowly.
(a) Opening brackets — "everybody shakes hands"
Cada término del primer paréntesis multiplica a cada uno del segundo. (Como un "todos con todos".)
✏️ Now you try: open (x + 1)(x + 4)
See answer
x·x=x², x·4=4x, 1·x=x, 1·4=4 → x² + 4x + x + 4 = x² + 5x + 4
(b) Factoring — the reverse (this is the magic trick for solving)
The trick: find two numbers that multiply to the last number and add to the middle number.
Factor x² + 5x + 6
✏️ Now you try: factor x² + 7x + 10
See answer
Two numbers that multiply to 10 and add to 7 → 2 and 5. So (x + 2)(x + 5).
If two things multiply to ZERO, one of them must be zero. A · B = 0 → A = 0 or B = 0
So (x+2)(x+3) = 0 means x+2 = 0 (→ x = −2) or x+3 = 0 (→ x = −3). Those are the two answers! 🎉
Si dos cosas multiplicadas dan 0, una tiene que ser 0. Por eso factoreamos.
⚽ Pausa de Papá (River ❤️🤍)
Factorizar es como el mediocampo de River: ordena todo para que el gol —la respuesta— salga solo. Los bosteros, en cambio, todavía están factorizando cómo ganarle a River en una final 🤣. Vos seguí así, crack.
8 ✅ That's the foundation — you're ready
If you followed everything here, you now know the building blocks the whole exam is made of:
- A letter is a hidden number, and we solve by doing the opposite operation to both sides.
- x² means x·x, and minus×minus = plus (so squares give two answers via ±).
- A graph is dots (x, y) on a grid; a function is a black box turning x into y.
- y = x² draws a U (parabola), the star of the exam.
- We open brackets by "all-with-all", and factor backwards to solve using the zero rule.