Factorise simple and harder quadratics, including quadratics with common factors
S.O.S.! Ships navigating the treacherous Bermuda Rectangle are falling prey to bizarre disasters. Factorise quadratic expressions to answer the ships’ distress calls and save their stricken crews!
In this maths game you are the chief of rescue operations for the Bermuda Rectangle. Your task is to protect the ships crossing this notorious body of water, keeping watch for any vessels that fall prey to its strange and terrifying phenomena.
Stricken ships will transmit their search area coordinates encoded as a quadratic expression. You must decipher the code before the ship sinks, and dispatch a rescue chopper to locate life rafts and save the crew.
It’s a tough job but we know you can handle it, Chief!
How To Play
To start a game, click ‘Play’ and then click ‘OK’ to accept the default settings on the two menu screens that follow.
Ships will sail into view from either side of the screen. Randomly ships will fall victim to the terrors of the Bermuda Rectangle and start to sink. Your task is to intercept these ships and rescue their crews before they finally slip below the surface.
When a ship starts sinking it transmits its search area encoded as a quadratic expression. First, you decipher the signal by factorising the quadratic. Next, you ‘draw’ a search area around the ship using your mouse, making sure the lengths of the area’s sides correspond directly to the quadratic’s factors.
If you factorise the quadratic correctly and draw the search area accurately then a rescue chopper is dispatched to scour the search area for life rafts and rescue the survivors. Your rewards are...
- Points added to your score.
- Seconds added to the timer.
- A lifering added to the Multiplier bar (in the top-left corner of the screen). If you fill this bar then your Multiplier increases. This Multiplier is applied to any points you earn, and can really help you score big – see if you can build up a massive ×5 Multiplier!
If you factorise the quadratic incorrectly and/or draw the search area inaccurately then a warning sounds and the ship continues sinking. Your penalties are:
- Seconds deducted from the timer.
- Three liferings deducted from the Multiplier bar (in the top-left corner of the screen). If the bar drops to or beyond empty then your Multiplier decreases.
When the timer falls to zero your game ends, and you are shown a summary of the ships you saved and how many points you earned.
This game is controlled entirely with the mouse. As you move the mouse pointer over the grid, a red ‘x’ cursor tracks its movement.
To draw a search area:
- Click and hold down the mouse button - the ‘x’ cursor turns green and locks in place.
- Drag the mouse - a green search area forms.
- Release the mouse button to ‘confirm’ your search area input -
- If the factors are correct then a ‘success’ tone sounds and the green search area remains visible for a short period.
- If the factors are incorrect or the search area contains no ships, then a ‘fail’ tone sounds and the search area flashes red and disappears.
When you draw a search area, the lengths of the two sides of the rectangle correspond directly to the number terms in the quadratic’s two factors.
Also, after locking the ‘x’ cursor, the direction in which you drag the mouse determines whether the number terms are positive, negative or a mix of both.
- If both number terms are positive, you drag up and to the right.
- If both number terms are negative, you drag down and to the left.
- If one number term is positive and the other is negative, you drag up and to the left or down and to the right.
In Modes 3 & 4, the x coefficient in one of the factors may be greater than 1, e.g. (2x + 3). To allow this input, there are two rows of buttons along the top and to the side of the grid. When clicked, these buttons set the value of the x coefficient in the horizontal or vertical factor to the indicated value.
Note that only one of these buttons may be selected at any given time. If the horizontal factor is ‘2x’ or greater and you set the vertical factor to ‘2x’ or greater then the horizontal factor will reset automatically to ‘x’, and vice versa.
For example: If you click on the ‘2x’ button along the top row, when you draw a search area the horizontal factor will be (2x + ?) and the vertical factor with be (x + ?). If you now click on the ‘4x’ button in the side row, when you draw a search area the vertical factor will be (4x + ?) and the horizontal factor will be (x + ?).
In Mode 4, some quadratics may have common factors that need to be removed before you factorise them (as the game does not consider an input valid unless the quadratic is fully factorised). To remove factors, first click on the signal bubble. Four smaller bubbles will appear, each displaying a different factor value, e.g. ‘÷4’.
Click on the factor you want to remove. If the factor is valid, you will see it pulled out of the quadratic expression. If the factor is invalid, you lose a few second from the timer.
‘Select Mode’ Options
On this menu screen you choose which type of quadratic factorisation problems to face. Click on the Mode you want to play, then click ‘OK’ (by default, the game highlights the highest Mode available).
When you play for the first time, only the easiest Mode will be available. To unlock the next Mode, you must earn enough points for a Bronze Medal or better.
A score modifier is applied according to the Mode’s difficulty – tackle the toughest maths to earn the biggest points!
|Mode||Maths Content||Score Modifier|
|1||Quadratic factorising like (x + 2)(x + 3)||All points × 1|
|2||Quadratic factorising like (x – 3)(x + 2)||All points × 2|
|3||Quadratic factorising like (2x + 1)(x – 3)||All points × 3|
|4||Quadratic factorising like 2(3x – 1)(x + 2)||All points × 4|
‘Select Assist’ Options
On this menu screen you choose how much help with the maths that the game gives you. Click on the Assist you require, then click ‘OK’ (by default, the game highlights ‘Standard Assist’).
A score modifier is applied according to the Assist you choose. The less help you use, the more points you earn – try beating Mode 4 with Minimum Assist to really rack up a high score!
|Assist Type||Maths Help Provided||Score Modifier|
|Maximum||Display the factors, the search area size and the quadratic created.||All points × ½|
|Standard||Display the factors and the search area size.||All points × 1|
|Minimum||Display the factors only.||All points × 2|
The total number of points earned for each rescued ship is calculated as:
There are five different types of ship sailing the Bermuda Rectangle. Each moves at a different speed and sinks at a different rate, and so is worth a different ‘base score’ if rescued:
|Ship Type||Frequency||Sailing Speed||Sinking Rate||Base Score|
||Very Common||Very Slow||Very Slow||1000 points|
||Very Uncommon||Fast||Fast||5000 points|
||Rare||Very Fast||Very Fast||8000 points|
The base score is ‘modified’ according to the Mode and Assists selected before the game (as outlined in previous sections), and is further boosted by the Multiplier bar’s value at the time of rescue. In addition, a special Link Bonus is applied if you manage to overlap (or ‘link’) two or more search areas before they fade away. The more search areas that are linked, the more points you win. You also earn a bigger time bonus and extra liferings, so you can grow your Multiplier far more quickly:
|Search Areas ‘Linked’||Score Multiplier||Time Bonus||Liferings|
I am playing Mode 3 (score modifier ×3) with Minimum Assists (score modifier ×2). My Multiplier bar has grown to ×3 when I rescue a Liner (base score = 5,000). I didn’t manage to overlap the search area with that of another ship, so there’s no Link Bonus. So my total score for rescuing the ship is:
Total Score = 5,000 × 3 × 2 × 3 × 1 = 90,000 points
I am playing Mode 2 (score modifier ×2) with Maximum Assists (score modifier ×½). My Multiplier bar has grown to ×2 when I rescue a Tanker (base score = 1,000). I overlap the search area with that of two other ships, so the Link Bonus is ×2.25. So my total score for rescuing the ship is:
Total Score = 1,000 × 2 × 0.5 × 2 × 2.25 = 4,500 points
Improve your score
- Before drawing a search area, think carefully about where to place the cursor:
- If the correct factorisation is (x + 3)(x + 2) then place the cursor to the bottom left of the ship, as the search area will be created by dragging up and right.
- If the correct factorisation is (x – 3)(x – 2) then place the cursor to the top right of the ship, as the search area will be created by dragging down and left.
- If the correct factorisation is (x + 3)(x – 2) then place the cursor to the top left or bottom right of the ship, as the search area will be created by dragging up and left or down and right.
- The bigger the number terms in the correct factorisation, the bigger the search area. Be careful: count squares to ensure that the search area fits the grid from your chosen start point.
- The smaller the number terms in the correct factorisation, the smaller the search area. Make sure your start point is close enough to the ship that the search area contains the ship.
- Look out for special cases. The search areas for factorisations like x2 – 9 = (x + 3)(x – 3) are especially easy to draw: it’s square and you drag either down and right or up and left.
- Choose carefully which ship to save next. For each ship you need to figure out how long it’s been waiting for rescue, and how difficult its factorisation is. You should be able to decide quickly how difficult a factorisation is by looking at the numbers in the quadratic. Sometimes it’s worth leaving an ‘easy’ factorisation (even if the ship has been waiting for longer) and focus on a difficult one. Once you save the difficult ship, you can return to the easier one.
- Each ship type is worth a different amount of points and sinks at a different rate. Focus on saving the fastest sinking ship types first for the biggest scores.
- Once you save a ship, it will stop sinking. Use this fact to wait until you have a lot of ships sinking in the same area, then save them all as quickly as possible while ensuring their search areas overlap – this can generate huge bonus points!
- If the number term is positive and the coefficient of x is also positive, for example in the quadratic x2 + 5x + 6, then the number terms in the factorisation are both positive, e.g. (x + 3)(x + 2). Start drawing your search area below and to the left of the ship.
- If the number term is positive and the coefficient of x negative, like in x2 – 5x + 6, then the number terms in the factorisation are both negative, e.g. (x – 3)(x – 2). Start drawing your search area above and to the right of the ship.
- If the number term is negative then one of the number terms in the factorisation is positive and the other is negative. You need to think more carefully about these cases; take your time and try to get it right first time. Start drawing your search area above and to the left or below and to the right of the ship.
- To be good at The Wrecks Factor you need to be able to spot quickly pairs of numbers that multiply to give a particular number. For example: these pairs – 1 and 12, 2 and 6, 3 and 4 – all multiply together to give 12.
- In Mode 4, if the quadratic has a common factor then make sure you pull out the largest common factor. For example: 2 is a common factor of the numbers in the quadratic 4x2 + 12x + 8. But so is 4, which is a bigger number. As 4 is the largest common factor, pull out the factor of 4 and not 2.