- A small square is 1 (gold). A long thin tile is x (green): it is 1 wide and x long, and x is not a whole number of small squares. A big square is x² (blue). In Simplify you can also use y (orange), y² (pink) and xy (teal).
- Every negative tile is red and says so (−x). A tile and its negative make a zero pair: together they are worth 0.
- ⊕ Simplify: tap the tiles in the tray to build an expression, or type one. ⇄ Group like terms sorts them, ⊘ Zero pairs cancels them, and x = … shows what every tile is worth for a value of x. Tap a tile on the board to flip, copy or remove it.
- ▦ Expand: set the two brackets with the steppers (or type (x + 2)(x − 3)). The first bracket runs down the side and the second across the top. Tap each ? block to fill it, or press Space. A red tile in the middle means the two tiles that made it had different signs.
- ▭ Factorise: you are given some tiles. Change the sides of the rectangle with the steppers until it uses exactly those tiles. The checklist says what still doesn’t match. □ Make a square starts completing the square.
- ⚖ Solve: whatever you do to one side, you do to the other. Add tiles, take them away, cancel zero pairs, then share into equal groups. Every step is written down under the mat.
🎲 New question makes one at the chosen level. ▶ Next step (Space) does one step for you. Type your own answer next to ✏ My answer and press Check: you get told which part is wrong without being given the answer. 🙈 on (the usual) keeps answers hidden until you tap ?.
Shortcuts only work after you’ve clicked on the tool, so they never get in the way of the rest of the page.
Press 📋 Questions (or Q) to pick a question, even in full screen. Set it up loads it (↩ brings back what was there). 🖨 prints this board, or 8 questions like it with answers on a second page. 🔗 copies a link that opens the tool as it is now. Nothing is sent anywhere, and nothing is stored on this device.
Investigation questions
64 questions. Work through these with the tool above. Set it up puts the tiles on the board for you (↩ undo brings back what was there), and Show on board puts the question on screen, full size.
First experiments
The board shows 3x + 2 − x + 4. How many tiles of each kind are there? Predict the simplified expression, then tap the ? to check.
Teacher notes
2x + 6. Three green x tiles and one red −x tile leave 2x; the ones make 2 + 4 = 6. Watch for 8x from students who add every number they can see.
Tap ⇄ Group like terms. Where does the red −x tile go? Then tap ⊘ Zero pairs. Has the value of the expression changed?
Teacher notes
The −x tile sits under an x tile and the two are ringed as a zero pair. Cancelling it removes 2 tiles but leaves 2x + 6: a zero pair is worth 0, so taking it away changes nothing.
Switch to ▦ Expand. The board shows (x + 2)(x + 3). Predict how many x tiles the rectangle will need, then press ▶ Next step until it is full.
Teacher notes
5 x tiles: 3 from x × 3 and 2 from 2 × x, with 1 x² and 6 ones, so x² + 5x + 6. Each block is one of the four multiplications.
Switch to ▭ Factorise. You have 1 x², 5 x tiles and 6 ones. Use the steppers to change the sides of the rectangle until all three checks are ticks.
Teacher notes
(x + 2)(x + 3), in either order. 2 and 3 work because they multiply to 6 (the ones in the corner) and add to 5 (the x tiles along the two edges).
Switch to ⚖ Solve. The mat shows 2x + 3 = 11. What would you take away from both sides first? Use Take from both sides, then ÷ Share into groups.
Teacher notes
x = 4. Take 3 ones from both sides (2x = 8), then share each side into 2 equal groups: one x tile matches 4 ones.
Collecting like terms
4x + 3 + 2x + 5: type the simplified expression next to ✏ My answer before you group the tiles. Then press Check.
Teacher notes
6x + 8. The x tiles and the ones can’t be joined: they are different sizes, so the answer keeps two terms.
Build 2x + 3x yourself from the tray. Is it 5x or 5x²? Look at what the tiles make, then compare them with an x² tile.
Teacher notes
5x. Five long tiles side by side are still x tiles. x² is the big square, x × x, not x + x.
x² + 3x + 2x² + x + 4: group the like terms. Why can’t the x² tiles and the x tiles be collected together?
Teacher notes
3x² + 4x + 4. An x² tile is a square with side x; an x tile is 1 by x. They are different areas, so they are different terms.
Switch on y tiles. 3x + 2y + x − y: which tiles can go together, and which never can?
Teacher notes
4x + y. x tiles and y tiles are different lengths, so 4x + y can’t be simplified any further. Students who write 5xy have joined unlike terms.
2xy + 3x + xy − y: how many terms does the simplified answer have?
Teacher notes
Three: 3xy + 3x − y. The xy tile is a rectangle with sides x and y, which is a different shape from both the x tile and the y tile.
Choose level 3 (x², x and 1) and press 🎲 New question. Type your simplified answer next to ✏ My answer before you touch the tiles. Do five.
Teacher notes
Each question has 3 to 5 terms in x², x and 1, some negative. The feedback names the kind of tile that is wrong without giving the answer.
Type 5 + x + 3x − 2 + x² and build it. Simplify it. Does the order the terms are written in matter?
Teacher notes
x² + 4x + 3. The order doesn’t change the value, because addition can be done in any order; writing x² first, then x, then the number is a convention.
Zero pairs and negatives
Build x − x with one green and one red tile. What is it worth? Now try 3 − 3 and x² − x².
Teacher notes
0 every time: a tile and its negative make a zero pair.
5x − 2x: before you tap ⇄ Group like terms, how many zero pairs will there be?
Teacher notes
2 zero pairs, leaving 3x. Subtracting 2x is the same as adding 2 red x tiles.
2x − 5x: what is left once the zero pairs cancel? Does the answer surprise anyone?
Teacher notes
−3x: three red tiles. Students often write 3x because they take the smaller number from the larger.
−x + 4 − 3x − 6: group the tiles, cancel the zero pairs, then simplify.
Teacher notes
−4x − 2. There are no green x tiles, so all 4 red ones stay: −4x. The ones leave 2 red.
Build an expression of exactly 10 tiles that is worth 0. Can you use three different kinds of tile? Why must the number of tiles be even?
Teacher notes
For example x² + 2x + 2 − x² − 2x − 2 (10 tiles). An expression worth 0 is made only of zero pairs, and each pair is 2 tiles, so the count is always even.
Substitution
Tap x = …. With x = 3, what is each tile worth? Predict the value of 2x + 5 before you tap the ?.
Teacher notes
11. Each x tile shows 3 and each 1 shows 1, so 3 + 3 + 5 = 11. Watch for 2x read as “23”.
The same expression with x = −2. What does an x tile show now? What is 2x + 5 worth?
Teacher notes
1. Each x tile shows −2, so −2 + (−2) + 5 = 1.
3x + 2 − x with x = 4: work out the value as written, then the value of the simplified expression. Why must they be the same?
Teacher notes
10 both times: 12 + 2 − 4 = 10 and 2 × 4 + 2 = 10. Simplifying never changes the value, whatever x is.
x² + 2x with x = −3: what does the x² tile show? What is the expression worth?
Teacher notes
The x² tile shows 9, because (−3) × (−3) is positive, and 2x is −6, so x² + 2x = 3. Students who type −3² into a calculator get −9.
Find a value of x that makes 2x + 5 and 4x − 1 worth the same. Use x = … on one, then build the other.
Teacher notes
x = 3: both are worth 11. This is the equation 2x + 5 = 4x − 1, and ⚖ Solve gives the same answer.
Expanding single brackets
3(x + 4): predict the expansion, then tap each ? block to fill it.
Teacher notes
3x + 12. Each of the 3 rows is a copy of x + 4: three x tiles and 3 × 4 = 12 ones.
A student writes 3(x + 4) = 3x + 4. Their answer has been checked next to ✏ My answer. What does the rectangle show they missed?
Teacher notes
They multiplied the x but not the 4: the corner holds 12 ones, not 4. 3(x + 4) = 3x + 12.
2(x − 3): what colour will the ones in the rectangle be? Fill it and see.
Teacher notes
Red: 2 × (−3) = −6, so 2(x − 3) = 2x − 6.
x(x + 5): what goes in the big block, and why is it a square?
Teacher notes
x × x = x², a square with side x. x(x + 5) = x² + 5x.
−2(x − 4): predict the colour of every tile in the rectangle before you fill it.
Teacher notes
−2 × x makes 2 red x tiles and −2 × (−4) makes 8 green ones: −2x + 8. The usual slip is −2x − 8.
Expanding double brackets
(x + 1)(x + 4): ▦ Label blocks is on. Fill the rectangle. Which two blocks hold x tiles?
Teacher notes
The two thin blocks: x × 4 = 4x and 1 × x = x, so x² + 5x + 4. The labelled blocks are the grid method.
A student says (x + 3)² = x² + 9. Their answer has been checked. Fill the rectangle: what is missing?
Teacher notes
x² + 6x + 9. The two blocks of 3 x tiles are what x² + 9 leaves out.
(x + 2)(x − 3): before you fill anything, which blocks will be red?
Teacher notes
The x × (−3) block (3 red x tiles) and the 2 × (−3) corner (6 red ones). The 2 green x tiles cancel 2 red ones: x² − x − 6.
(x − 2)(x − 3): why is the corner green when both numbers are negative?
Teacher notes
(−2) × (−3) = +6, so x² − 5x + 6. Students who think “two negatives make a negative” predict −6.
(x + 4)(x − 4): fill the rectangle. What happens to the x tiles?
Teacher notes
4 green and 4 red x tiles make 4 zero pairs, leaving x² − 16: the difference of two squares.
(2x + 1)(x + 3): how many x² tiles will there be? Predict the whole answer, then fill it.
Teacher notes
2x² + 7x + 3: 2 x² tiles, 6 + 1 = 7 x tiles and 3 ones.
Choose level 4 (With negatives) and press 🎲 New question. Type each answer next to ✏ My answer before you fill the rectangle.
Teacher notes
The feedback says which part is wrong: the x² block, the x blocks or the sign of the corner, without giving the answer.
Factorising
3x + 6: you have 3 x tiles and 6 ones. Make a rectangle with all of them. What goes down the side?
Teacher notes
3 down the side and x + 2 across the top: 3(x + 2). Taking out a common factor is a rectangle with a plain number on one side.
x² + 7x + 12: which pairs of numbers multiply to 12? Which pair also adds to 7? Use 💡 Hint if you need it.
Teacher notes
3 and 4: (x + 3)(x + 4). The other pairs, 1 and 12 and 2 and 6, add to 13 and 8.
x² − 5x + 6: the x tiles are red but the ones are gold. What signs must the two numbers in the brackets have?
Teacher notes
Both negative: (x − 2)(x − 3). They must multiply to +6 and add to −5.
x² + x − 6: there is only one x tile, but the rectangle needs more. Build it anyway. How many zero pairs of x tiles did you need?
Teacher notes
(x + 3)(x − 2): 3 green and 2 red x tiles make +1, using 2 zero pairs. Adding a zero pair doesn’t change the value, which is why it is allowed.
x² − 9: there are no x tiles at all. Can the tiles still make a rectangle?
Teacher notes
Yes: (x + 3)(x − 3). 3 green and 3 red x tiles are 3 zero pairs: the difference of two squares.
2x² + 7x + 3: put the two x² tiles in a column. Where do the ones go?
Teacher notes
(2x + 1)(x + 3): the three ones fill the corner as 1 × 3. The other way round, (2x + 3)(x + 1), gives 2x² + 5x + 3.
x² + 4x + 5: try every rectangle you can. Then tap the ? next to “Can these tiles make a rectangle?”.
Teacher notes
No. The pairs that multiply to 5 are 1 and 5 (sum 6) and −1 and −5 (sum −6); neither adds to 4. It needs completing the square: (x + 2)² + 1.
x² + 2x: there are no ones. What are the sides of the rectangle?
Teacher notes
x and x + 2: x(x + 2). The common factor is x itself.
Completing the square
x² + 6x + 4 can’t make a rectangle. Tap □ Make a square. How many ones does the square need, and how many are missing?
Teacher notes
(x + 3)² needs 9 ones and there are 4, so 5 are missing: x² + 6x + 4 = (x + 3)² − 5.
x² + 4x + 7: make a square. This time there are too many ones. What happens to the extra ones?
Teacher notes
(x + 2)² needs 4 ones, so 3 are left over: x² + 4x + 7 = (x + 2)² + 3.
x² − 8x + 10: make a square. Why are the ones in the corner gold when the x tiles are red?
Teacher notes
(x − 4)² needs (−4) × (−4) = 16 ones, and they are positive. 10 − 16 = −6, so x² − 8x + 10 = (x − 4)² − 6.
x² + 5x + 1: try □ Make a square. Why does the tool refuse?
Teacher notes
5 x tiles can’t be shared equally between the two sides without cutting a tile in half. On paper it is (x + 5/2)² − 21/4.
Solving equations
x + 5 = 12: what single move leaves x on its own?
Teacher notes
Take 5 ones from both sides: x = 7.
3x + 2 = 14: solve it, saying each move out loud before you tap it. The working is written under the mat.
Teacher notes
Take 2 from both sides (3x = 12), share into 3 groups and keep one: x = 4.
2x − 3 = 7: there are red ones on the left and none on the right. Try Take from both sides − (−1). Why does the tool refuse? What can you do instead?
Teacher notes
The right side has no −1 tiles to take away. Add 3 ones to both sides instead; the left then has 3 zero pairs to cancel, giving 2x = 10 and x = 5.
5x + 2 = 2x + 11: there are x tiles on both sides. What should you take away first?
Teacher notes
Take 2x from both sides: 3x + 2 = 11, then 3x = 9 and x = 3. Trying to take 5x would need 5 x tiles on the right.
3 − 2x = 9: solve it. What does ± Change every sign do, and when do you need it?
Teacher notes
x = −3. After taking 3 from both sides, −2x = 6; sharing gives −x = 3, and changing every sign gives x = −3. Changing every sign is multiplying both sides by −1.
2x = 7: share into groups. What goes wrong, and what is x?
Teacher notes
7 ones can’t be shared into 2 equal groups of whole tiles: 1 is left over, so x = 7/2 = 3½. The tiles can only show whole-number answers.
3x + 1 = 2x + 5. Before solving, type 4 next to ✏ My answer: x = and press Check. Then try 3. What does the tool tell you each time?
Teacher notes
x = 4 works: both sides are 13. x = 3 makes the sides 10 and 11, so the left is too small. Substituting checks an answer without solving.
Misconceptions to expose
A student simplifies 2x + 3 to 5x. Their answer has been checked. Use the tiles to explain why it is wrong.
Teacher notes
The ones and the x tiles are different sizes, so they can’t be counted together. 2x + 3 is already as simple as it gets.
x × x or x + x: which is x² and which is 2x? Fill this rectangle, then build x + x in ⊕ Simplify.
Teacher notes
x × x is the square, x²; x + x is two thin tiles, 2x. They are only equal when x = 0 or x = 2.
(x + 5)² = x² + 25? The student’s answer has been checked. Fill the rectangle to see why.
Teacher notes
x² + 10x + 25. Squaring a sum isn’t squaring each part: two strips of 5 x tiles are missing.
A student writes −3(x − 2) = −3x − 6. What colour should the corner tiles be?
Teacher notes
Green: (−3) × (−2) = +6, so −3(x − 2) = −3x + 6.
2x + 4 = 10: a student takes 4 from the left side only and writes 2x = 10. Why doesn’t the tool let you do that?
Teacher notes
The two sides would stop being equal. Every move in the tool changes both sides, so every line of the working is still true: 2x = 6, then x = 3.
x² + 5x + 6: a student says (x + 5)(x + 6). Their answer has been checked. What were they thinking, and what is wrong?
Teacher notes
(x + 5)(x + 6) = x² + 11x + 30. The numbers in the brackets must multiply to the ones (6) and add to the x tiles (5): 2 and 3.
Open investigations
Find every expression x² + bx + 12 that makes a rectangle, with b positive. How do you know you have them all?
Teacher notes
b = 7, 8 or 13, from the factor pairs 3 × 4, 2 × 6 and 1 × 12. Every rectangle comes from a factor pair of 12, and 12 has three.
Now x² + bx − 12. How many values of b work?
Teacher notes
Six: b = ±1, ±4 and ±11. Each factor pair of 12 can have its minus sign on either number.
Which is bigger, (x + 1)(x + 5) or (x + 3)²? Expand both. Does the answer depend on x?
Teacher notes
(x + 3)² = x² + 6x + 9 is always 4 more than (x + 1)(x + 5) = x² + 6x + 5, whatever x is. The square has the bigger area.
Make your own equation with x on both sides whose answer is negative. Solve it with the tiles, then check your answer next to ✏ My answer.
Teacher notes
For example 2x + 5 = 4x + 9 gives x = −2. One way to make one is to choose the answer first and build both sides to match.
Design an expression with at least 8 tiles that simplifies to 2x − 1. Swap with a partner: can they simplify yours?
Teacher notes
Any expression with the right totals works, for example 4x + 3 − 2x − 4 (13 tiles). Zero pairs can make it as long as you like.
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