Maths for Romanian classes 6–9 (ages roughly 12 to 16)
Functions, graphs and the quadratic equation, explained with a ball, a balance scale and some tiles
You try every idea with your own hand: throw a ball, weigh on a balance, drag a point on a map. Along the way you meet the unknown, the function, the line, the parabola and the quadratic equation. The steps follow the school order; the first seven each have a prediction, a toy and a check at the end.
Hit the flag
Speed is measured in metres per second (m/s). Pick the speed and the angle, or drag the yellow handle.
A thrown ball draws a parabola; the distance at which it lands is the solution of a quadratic equation.
The drawing, in words
A thrown ball draws a parabola; the distance at which it lands is the solution of a quadratic equation.
Here g = 10 m/s², not 9.81, to keep the numbers simple; the shape of the curve is the same. The same value is used throughout the page.
Try it now: hit the flag. The ball flies along a curve called a parabola, and the distance at which it lands is found by solving an equation. You will solve that equation in step 7.
The lesson plan: seven steps, in order, and a recap
- Who it is for
- Pupils in Romanian classes 6 to 9 (ages about 12 to 16) who want to reach the quadratic equation with no gaps on the way, and anyone helping them.
- What you need beforehand
- The four operations with whole numbers, negative ones included.
- By the end
- You solve a quadratic equation on your own in three ways (with tiles, with the formula, with the sum and the product) and know what its solutions mean on the graph.
Each step stands on the one before:
- Equality and the unknown
- Coordinates
- The function
- The line
- Squares and roots
- The parabola
- The quadratic equation
Squares (step 5) prepare the tiles used to complete the square in step 7.
- 1Equality and the unknownWhich number is missing? · classes 6–7
- 2CoordinatesHow do you say exactly where a point is? · class 7
- 3The functionHow do you write once a calculation you keep repeating? · class 8
- 4The lineWhat does a quantity that always grows by the same amount look like? · class 8
- 5Squares and rootsFrom a square's area, how long is the side? · classes 7–8
- 6The parabolaWhat does a quantity that grows ever faster look like? · class 9
- 7The quadratic equationWhen does the ball reach a given height? · classes 8–9
- 8Nine common mistakesHow do you catch a wrong rule? · all classes
Each step starts with a twenty-second question: if you know it, move on. It ends with a check; when the answer is wrong, the page says why and which step to go back to. What you tick is kept on this device only.
A route by class: class 6: step 1; class 7: steps 1, 2 and 5; class 8: 3, 4, 5 and 7; class 9: 6 and 7. Steps 1–5 are the foundation for the last two.
Class numbers follow Romania's national curriculum (class 6 is for 12–13 year olds, class 9 for 15–16 year olds).
Step 1 of 8 · classes 6–7
What is an unknown, and how do you solve an equation?
The problem
Three equally heavy bags of apples and another 2 kg weigh 14 kg together. How heavy is one bag?
Why does this idea exist? To find a number you cannot see, starting from what you know about it. The equation is the sentence that says what you know; the unknown is the number you are after.
An unknown is a number you do not know yet, shut in a box. An equation is a balance at rest: one pan holds the box and whatever sits beside it, the other pan holds a weight. To find what is in the box, you make the same move on both pans until the box stands alone.
Predict first
The left pan holds three equal boxes and a 2 kg weight, the right pan 14 kg; the balance is level. You take the 2 kg weight off the left pan only. What does the balance do?
Why? To keep the balance level you take 2 kg off the right as well: three boxes weigh 12 kg, so one box is 4 kg. That is where the rule of every equation comes from: whatever you do to one side, you do to the other. In the balance below, pick “the left pan only” and see what happens.
Whatever you do to one pan, you do to the other.
The balance
Take off the weight, then split. Also try the wrong move, on one pan only, to see what happens.
A balance with boxes of a hidden number on one pan and a weight on the other, where you take the same weight off both pans.
The drawing, in words
A balance with boxes of a hidden number on one pan and a weight on the other, where you take the same weight off both pans.
The equals sign means “weigh the same”, not “the answer comes next” as on a calculator key. The same balance will solve equations with x² in step 7.
Got it? Check
What do you do first to solve 5x + 3 = 18?
Why? 5x + 3 = 18, so 5x = 15, so x = 3. Check: 5 · 3 + 3 = 18.
Step 2 of 8 · class 7 (graphs: 7–8)
How do you read a point on the Cartesian plane?
The problem
How do you tell someone on the phone exactly where you put a dot on a sheet of squared paper?
Why does this idea exist? Two numbers fix any spot on the sheet. With them a calculation can be drawn and a drawing can be calculated; every later step rests on this.
The Cartesian plane is a map with two streets: a horizontal one, the x axis (Ox), and a vertical one, the y axis (Oy). They meet at the point (0, 0), called the origin. Every point has an address made of two numbers: (3, 2) means “walk three squares along the street, then climb two”.
Predict first
Are (2, 5) and (5, 2) the same point?
Why? (2, 5) means 2 along the street and 5 up; (5, 2) means 5 along the street and 2 up. Swapping them is the mistake the boat game below catches.
First x, then y: first along the street, then up the stairs. Minus means left or down.
Sink the ship
The ship sits on a hidden crossing. Call out its address.
A Cartesian plane with a ship hidden on a crossing; you call out x and y.
The drawing, in words
A Cartesian plane with a ship hidden on a crossing; you call out x and y.
Draw the boat
Place the points in the order asked; the lines appear by themselves.
A Cartesian plane on which you draw a boat point by point from the coordinates asked.
The drawing, in words
A Cartesian plane on which you draw a boat point by point from the coordinates asked.
A graph is not a drawing of a route. It is a map of the relation between two quantities, here time and distance. Reading a graph means looking at where it climbs, where it stays flat and where it falls.
Ana's walk
Drag the time slider and see where Ana is on the street. Then answer the questions.
A graph of distance from home over time for a walk to school, with a stop and a turn back.
The drawing, in words
A graph of distance from home over time for a walk to school, with a stop and a turn back.
A flat line means the distance does not change. The steeper the line climbs, the faster Ana walks. The graph does not say which streets she took, only how far from home she was each minute.
Got it? Check
A point sits on the vertical axis, Oy. What do you know for sure about it?
Why? Every point on Oy has the address (0, something): (0, 3), (0, −2). In step 4, the line f(x) = ax + b will cut Oy at the point (0, b).
Step 3 of 8 · class 8
What is a function?
The problem
At a copy shop a page costs 2 lei and the envelope 1 leu. What do you pay for 4 pages? For 10? For 37?
Why does this idea exist? When you do the same calculation with a different number each time, you write the calculation once, as a rule: 2 · x + 1. That rule is called a function.
A function is a machine with a rule. You put a number in, a number comes out, and the same input always gives the same output. The rule can be told in three ways: as a formula, f(x) = 2x + 1, as a table of values or as a drawing, the graph. These are three languages for one thing. The notation f(x) is read “f of x” and means the number that comes out when x goes in; on the graph it is the height y.
Predict first
The machine f(x) = x² takes 3, then −3. What comes out?
Why? Two different inputs may give the same output; a machine may not give two outputs for the same input. The drawing “Is the curve a machine?” below shows the difference.
Guess the machine's rule
Put in numbers, watch the table and the points, then pick the rule.
A machine with a hidden rule: you put numbers in, a table of pairs comes out, and you guess the rule.
The drawing, in words
A machine with a hidden rule: you put numbers in, a table of pairs comes out, and you guess the rule.
One number is not enough: several rules can give the same result for it. That is why it pays to try odd numbers too, such as 0 or a negative number.
Is the curve a machine?
Move the vertical line and count how many points it cuts the curve in.
Four curves and a vertical line that shows how many y values belong to each x.
The drawing, in words
Four curves and a vertical line that shows how many y values belong to each x.
A machine may not give two outputs for the same input. On the drawing, that means no vertical line cuts the curve in more than one point.
From a table to a drawing
Work out f(x) for each x. Then decide how many points to join.
A table with x from −3 to 3 that you fill in; every row puts a point on the plane.
The drawing, in words
A table with x from −3 to 3 that you fill in; every row puts a point on the plane.
How many points do you need to have the graph? With seven points you have a broken line, with many points a smooth curve. The true graph contains all the points, not only those from the table.
Got it? Check
A machine's graph is a straight line through the points given by 1 → 5, 2 → 7, 3 → 9. What comes out for 10?
Why? The rule is f(x) = 2x + 3, so f(10) = 2 · 10 + 3 = 23. Step by step gives the same: 9 + 7 · 2 = 23. At the copy shop from the start of this step, the rule 2x + 1 gives 9 lei for 4 pages, 21 lei for 10 and 75 lei for 37.
Step 4 of 8 · class 8
What does the linear function f(x) = ax + b show?
The problem
Two taxis charge differently: one asks more to start, the other more per kilometre. Which is cheaper for your ride?
Why does this idea exist? Many quantities grow by the same amount at every step: the fare grows by the same sum each kilometre, the water in a bath by the same amount each second while the tap runs steadily. Their graph is a straight line, and two lines show at a glance where they become equal.
A taxi charges a starting fee, b, and a price for each kilometre, a. The price for x kilometres is f(x) = a·x + b. The graph is a line: it starts at height b and climbs a lei with every kilometre. The number a is called the slope.
Predict first
Taxi A: 10 lei to start and 2 lei per km. Taxi B: 4 lei to start and 3 lei per km. Which is cheaper for 8 km?
Why? A: 10 + 2 · 8 = 26 lei. B: 4 + 3 · 8 = 28 lei. Their lines meet at 6 km; before that B is cheaper, after it A. On a long ride the slope matters, not the starting fee.
The taxi
Change the starting fee and the price per kilometre. The blue and red stairs show the slope.
The graph of a taxi fare: a line that starts at the starting fee and climbs by the price per kilometre. Prices in lei, the currency of Romania.
The drawing, in words
The graph of a taxi fare: a line that starts at the starting fee and climbs by the price per kilometre. Prices in lei, the currency of Romania.
The line through two points
Drag points A and B. The page works out the slope and the equation.
A line through two movable points, with its slope, its equation and the point where it cuts the horizontal axis.
The drawing, in words
A line through two movable points, with its slope, its equation and the point where it cuts the horizontal axis.
The zero of a function. The zero of a function is the value of x for which f(x) = 0; on the graph it matches the point where the curve meets the Ox axis. For f(x) = ax + b, with a not 0, the zero is x = −b/a. It is exactly the solution of the equation ax + b = 0 from step 1, seen on the map: solving an equation means looking for where the graph touches the axis.
Got it? Check
The line f(x) = 3x − 6 cuts the Ox axis at the point where x equals:
Why? 3x − 6 = 0, so 3x = 6, so x = 2. The point is (2, 0). Looking for where the graph touches the Ox axis means solving an equation.
Step 5 of 8 · classes 7–8
What is a square root, and what does (a + b)² mean?
The problem
You have 36 square tiles and want to make a square out of them. How many tiles go along one side?
Why does this idea exist? Squaring turns a side into an area; the square root goes back the other way. Without it you cannot get x out of x² = 25, and the equation in step 7 has an x².
A square with side 5 has area 25. The square root goes the other way: from the area it finds the side, √25 = 5. An equation with x² usually has two solutions though, as the comparison below shows.
x + 3 = 7
Only one number works: 4.
x² = 9
Two numbers work: 3 and −3, because 3 · 3 = 9 and (−3) · (−3) = 9.
Predict first
The square with side 3 + 2 = 5 has area 25. What is 3² + 2²?
Why? The sum 3² + 2² leaves out the two 3 by 2 rectangles inside the big square: 13 + 6 + 6 = 25. That is why (a + b)² = a² + 2ab + b². In the drawing “(a + b)², with tiles” below, change a and b and watch the two rectangles.
Tiles, areas and roots
Pick how many tiles you have. They fall into a square, if they can.
Tiles laid out in a square whose side is the square root of their number, and the two solutions of x² = n on the number line.
The drawing, in words
Tiles laid out in a square whose side is the square root of their number, and the two solutions of x² = n on the number line.
(a + b)², with tiles
Change a and b and compare the whole square with a² + b².
A square with side a + b cut into a², b² and two rectangles ab.
The drawing, in words
A square with side a + b cut into a², b² and two rectangles ab.
The tiles show why (a + b)² is not a² + b²: the two rectangles ab are missing. The mistake is common because it looks like the rule (a·b)² = a²·b²[5]. The same tiles will solve the quadratic equation further down.
Got it? Check
How many solutions does x² = 16 have?
Why? The side of the square with area 16 is √16 = 4. The equation x² = 16 has one more solution, −4: (−4) · (−4) also gives 16, even though a side cannot be negative. With the 36 tiles from the start of this step you put 6 along each side, because 6 · 6 = 36.
Step 6 of 8 · class 9
Why is the graph of a quadratic function a parabola?
The problem
A ball thrown upwards climbs ever more slowly, stops for a moment and falls ever faster. What drawing does its height make over time?
Why does this idea exist? The line describes quantities that grow the same at every step. The area of a square or the height of a ball changes by ever bigger or ever smaller steps. The simplest curve for them is the parabola.
The graph of every function ax² + bx + c, with a not 0, is a curve called a parabola, wider or narrower, higher or lower. The two tables below show why it bends.
The line: f(x) = 2x
| x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| 2x | 0 | 2 | 4 | 6 | 8 |
| step | +2 | +2 | +2 | +2 |
The step stays +2: the graph climbs straight.
The parabola: f(x) = x²
| x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| x² | 0 | 1 | 4 | 9 | 16 |
| step | +1 | +3 | +5 | +7 |
The step grows, +1, +3, +5, +7: the graph bends.
Predict first
In f(x) = a · x², a goes from 1 to −1. What does the parabola do?
Why? x² is never negative. Multiplied by −1, everything that was above the axis goes below it, as in a mirror. Try it with the a slider in “Your parabola” below.
The staircase of odd numbers
Increase x and watch the new L-shaped layer.
Squares of side 1, 2, 3 nested, with a table of x² values and their points on the plane.
The drawing, in words
Squares of side 1, 2, 3 nested, with a table of x² values and their points on the plane.
Your parabola: a, b and c
Move the sliders and watch the drawing and the algebra. Then hit the flag.
Three sliders, a, b and c, that change a parabola on the plane, with its vertex, roots and crossing with the vertical axis.
The drawing, in words
Three sliders, a, b and c, that change a parabola on the plane, with its vertex, roots and crossing with the vertical axis.
The number a says which way the parabola opens and how narrow it is. The number c is the height where it cuts the vertical axis. The x of the vertex is −b/(2a); its y comes from putting that value into the function. Δ = b² − 4ac, explained below, says whether the parabola reaches the Ox axis. The parabola is symmetric about the vertical line through the vertex: the mirror.
The ball thrown straight up
Start the ball and watch the ball and its graph at the same time.
A ball thrown straight up goes up and down along a vertical line; the graph of height against time is a parabola.
The drawing, in words
A ball thrown straight up goes up and down along a vertical line; the graph of height against time is a parabola.
With g = 10 m/s², as in the first toy, h(t) = h₀ + v·t − 5·t².
The ball goes up and down along one vertical line, yet its graph is a curve, because the horizontal axis is time. The graph does not show the shape of the route[8]. If the ball were thrown horizontally, with no air resistance, its route would be a semi-parabola; Galileo showed in 1638 that a projectile moving this way follows a semi-parabola[11].
Bonus: the dish
Change the depth of the dish and the number of rays.
A parabola that gathers parallel rays into a single point, the focus.
The drawing, in words
A parabola that gathers parallel rays into a single point, the focus.
All rays arriving parallel to the axis reflect to the same point. That is why many satellite dishes and older car headlights have a parabolic cross-section.
Got it? Check
A parabola cuts the Ox axis at x = 1 and x = 5. Where is its axis of symmetry (the mirror), the vertical line through the vertex?
Why? The two points have the same height, 0, so they mirror each other: (1 + 5) ÷ 2 = 3. The vertex formula gives the same for y = x² − 6x + 5: −b/(2a) = 6/2 = 3. The height of the ball thrown upwards, from the start of this step, makes a parabola too, opening downwards with its vertex at the highest point.
Step 7 of 8 · classes 8–9
How do you solve a quadratic equation?
The problem
The ball leaves the ground upwards at 20 m/s. After how many seconds is it at 15 m?
Why does this idea exist? Guessing only works for nice numbers. The quadratic equation has a method that works every time, completing the square, and the formula is that same method written once and for all.
The quadratic equation has the form ax² + bx + c = 0, with a not 0, and asks: for which x is the function ax² + bx + c equal to 0? On the graph you look for the points where the parabola touches the horizontal axis. The values of x at those points are the solutions of the equation, also called its roots, or the zeros of the function: three names for the same numbers. Real solutions are the ones that sit on the number line.
Predict first
The height of the ball is h = 20t − 5t². How many times does it pass a window at 15 m?
Why? 20t − 5t² = 15 has two solutions, t = 1 s and t = 3 s. At 20 m, the top, there would be one solution, t = 2 s; at 25 m, none. A quadratic equation can have two solutions, one or none, depending on how the line cuts the parabola. Move the window in “The ball and the window” below.
When does the ball reach the ground, or a window?
Throw the ball straight up and ask when it reaches a given height: the equation is quadratic in time. Move the window and see that the equation can have two solutions, one, or none. Let the ball start from a balcony, with the window below the starting point, and a negative solution appears: a moment before the throw.
The ball and the window
Choose where the ball starts, its speed and the height of the window.
The graph of the ball's height over time and a window at a chosen height; the solutions of the equation are the points where the window line cuts the parabola.
The drawing, in words
The graph of the ball's height over time and a window at a chosen height; the solutions of the equation are the points where the window line cuts the parabola.
Δ: does it reach, or not?
The number under the root in the formula, Δ = b² − 4ac, is called the discriminant, and it tells whether the parabola reaches the axis. Δ positive: the parabola cuts the axis at two points. Δ equal to 0: it touches it at one point. Δ negative: it never reaches it, so the equation has no real solutions.
Three parabolas, three values of Δ
Three parabolas: y = x² − 4, y = x² and y = x² + 4, with Δ = 16, Δ = 0 and Δ = −16.
The drawing, in words
Three parabolas: y = x² − 4, y = x² and y = x² + 4, with Δ = 16, Δ = 0 and Δ = −16.
Completing the square, with tiles
Clay tablets from Mesopotamia, about 3,800 years old (around 1800 BC), hold problems whose steps match this method. Later, in Baghdad under Caliph al-Ma'mun (813–833), al-Khwarizmi explained equations like these with geometric figures[9][10]. The idea: x² + bx is almost a square; a corner is missing.
The tiles that complete the square
Pick b and c, then press “Next step”.
An x by x square, two strips of width b/2 and a corner of (b/2)², which together make a big square; from it come the solutions of x² + bx = c.
The drawing, in words
An x by x square, two strips of width b/2 and a corner of (b/2)², which together make a big square; from it come the solutions of x² + bx = c.
Where the formula comes from
The formula is no magic. The same steps as with the tiles, written with letters instead of numbers, give the formula of the solutions. After the steps below you can try the formula on any equation.
The formula, step by step
Press “Next step” and read what was done.
Six lines of algebra, from ax² + bx + c = 0 to x = (−b ± √Δ)/2a, each with an explanation.
The drawing, in words
Six lines of algebra, from ax² + bx + c = 0 to x = (−b ± √Δ)/2a, each with an explanation.
Viète's relations: the sum and the product
Dividing the equation ax² + bx + c = 0 by a gives x² + (b/a)x + c/a = 0. If the solutions are x₁ and x₂, the same equation is (x − x₁)(x − x₂) = 0, that is x² − (x₁ + x₂)x + x₁·x₂ = 0. Comparing coefficients, the sum of the solutions is −b/a and their product is c/a. You can solve without the formula too: look for two numbers with the right sum and product.
The sum and the product
Move the solutions and watch the equation. Then solve the puzzle.
A parabola through two solutions you choose, with their sum, product and its equation; and a puzzle with a given sum and product.
The drawing, in words
A parabola through two solutions you choose, with their sum, product and its equation; and a puzzle with a given sum and product.
Practise: a worked example first, then you
If you do not know where to start, press “Show the working” before you try. Looking at a worked example is not cheating. The equations are built backwards from their solutions, so the answers come out clean.
Solve equations
Choose a level and solve. You can ask for a hint or the working.
Quadratic equations generated on demand, with checking, a hint and a step-by-step working.
The drawing, in words
Quadratic equations generated on demand, with checking, a hint and a step-by-step working.
Got it? Check
For x² + 2x + 5 = 0, Δ = 2² − 4 · 1 · 5 = −16. What does that mean?
Why? x² + 2x + 5 = (x + 1)² + 4. The lowest point of the parabola is at height 4, above the axis, so no value of x gives 0.
Step 8 of 8 · all classes
Nine common mistakes with functions and equations, and how to catch them
A mistake is easier to catch with an example than with a rule. Every card holds a claim that sounds reasonable. Put in numbers and see where it breaks, or press “Find an example that breaks it”.
Nine wrong claims, each with an example that breaks it: (a + b)² = a² + b², √(x²) = x, the points (a, b) and (b, a), dividing by x, −x² and (−x)², how many solutions a quadratic has, the graph as a drawing of the route, the equals sign and the slope.
For teachers and parents: what the page does and why
The page is a lesson for pupils in Romanian classes 6 to 9, in curriculum order: equality and the unknown, coordinates, the function, the line, squares, the parabola, the quadratic equation. Every step follows the same order: the steps it depends on, a familiar problem, a prediction with its explanation, a concrete scene (a ball, a balance, tiles), its drawing on the plane, the symbols, and a check that, on a wrong answer, names the mistake and points to the missing step. The choices below rest on studies of how pupils learn; the page itself has not been tested with pupils.
- A review of the research proposes that moving by stages from the concrete scene, through a picture, to symbols may help more than symbols alone or the scene alone; that is why every step goes from the scene, through a picture, to formulas[3].
- In a study of 76 high-school pupils, equations in symbols did nearly 20% worse than matching problems in words, and teachers and researchers predicted the opposite[2]; that is why the story comes before the equation.
- The equals sign as a balance, not as a “calculate” key. In a study of 177 American pupils aged 11 to 14, those who understood the sign as balance solved equations better, even when maths scores were accounted for (in a group of 65 pupils)[1].
- The graph as a map of a relation, not a drawing of the route: college physics students have had trouble telling the shape of a graph from the path of the motion, and younger pupils often take the highest point for the steepest[8]; that is why Ana's walk and the ball thrown straight up show the route and the graph side by side.
- (a + b)² with tiles: pupils stretch the visible rule (xy)² = x²y² to sums[5].
- Letters are not objects: in Küchemann's Chelsea test (about 3,000 pupils, 1976), 52% of the third-year pupils (mean age 14.3) read “8c + 6t” as eight cabbages and six turnips[6]; the sliders show a letter as a number that moves.
- The two axis crossings can be mixed up: in a study of 18 ninth-grade pupils, several first took the x-intercept for “b”[7]; both are marked on the drawing.
- “Try first” helps on average (a synthesis of 53 studies), but younger pupils (grades 2–5 in the studies, about ages 7 to 11) learned better when taught first[4]; that is why the tries here are short and skippable.
- For most mistakes in the gallery the page cites no study; only four have a source here: (a + b)², the graph taken for the route, the equals sign and the mix-up of the two axis crossings. The rest are mistakes described in teacher materials, not research results.
Where each step sits in the national curriculum
The table summarises what the official curricula list. A new curriculum for classes 9 and 10 (common core: the humanities, technological and vocational tracks) was published in the Official Gazette on 8 January 2026 and applies progressively from the 2026–2027 school year; the curriculum for the mathematics-intensive tracks is not summarised here. In the common-core text, the quadratic function is studied through its graph, with emphasis on vertex and symmetry.
| Step | Class | What the curriculum lists | Source |
|---|---|---|---|
| 1. The box and the balance | 6–7 | Equations x + a = b, ax + b = c (class 6); ax + b = 0, unknown, coefficients, free term (class 7) | [13] |
| 2. The map | 7 | Orthogonal axes, plotting points, reading dependences from graphs | [13] |
| 3. The machine | 8 | Functions given by diagrams, tables, formulas; the graph of a function | [13] |
| 4. The line | 8 | Functions f(x) = ax + b, geometric meaning, crossings with the axes | [13] |
| 5. Squares and roots | 7–8 | Square root, x² = a (class 7); the formula (a ± b)² (class 8) | [13] |
| 6. The parabola | 9 | The quadratic function, its graph, vertex, symmetry; the new curriculum suggests linking it to the path of a ball | [14][15] |
| 7. The quadratic equation | 8–9 | ax² + bx + c = 0 solved with standard algebraic identities (class 8); Viète's relations, the vertex (class 9) | [13][14] |
The class 8 curriculum solves the quadratic equation by factoring and the standard algebraic identities; the 2009 class 9 curriculum (the four-hour version) lists Viète's relations and the vertex of the parabola. The new common-core text names the vertex and symmetry, but not Viète's relations. The discriminant is a tool that works on any quadratic equation; it does not appear under that name in the curriculum texts cited here.[13][14]
How many pupils fall behind?
In PISA 2025, 48% of Romania's 15-year-olds reached at least level 2 in mathematics, the test's baseline, against an OECD average of 65%[12]. In PISA 2022 the share was 51%, against 69%; the OECD describes Romania's average mathematics score in 2025 as about the same as in 2022. PISA measures pupils, not textbooks: it shows how many are below the baseline, not why.
Limits of this page
The page uses g = 10 m/s², not 9.81, to keep the numbers whole. The tiles for completing the square work only for positive b; for negative b the algebra carries on. The page does not replace a textbook and needs no account; what you tick is kept on this device only.
Sources
- Knuth, Stephens, McNeil & Alibali (2006), Does understanding the equal sign matter? Evidence from solving equations, Journal for Research in Mathematics Education 37(4), 297–312. open
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