Answer key · Unit 3

Algebraic expressions.

Each lesson’s practice questions, their correct answers and the mistakes they spot. Answers read off the Scene use the same setting the learner sees.

Misconception catalogue

Answer key 4

Lesson 1 of 4 · About 25 minutes

Letters and like terms

A letter that stands for a number, how to put a number into an expression, and why three x plus two x can be added, but three x plus two y cannot.

Open the lesson

The 5 steps and their answers

Step 1 of 5
Open this step
A letter for a number
Practice · Calculate
Question
The board shows $3x + 2$ ⁠: three strips of length x and two squares. Here $x = 4$ ⁠. What is $3x + 2$ ⁠?

Correct answer
14 ± 0.01
Setup, choices and feedback
The Scene at this step
x
4fixed
y
3defaulthidden
As written / Sorted and worked out
As written
Accepted within
± 0.01
Computed from the Scene at this step’s setting
Product: the area, left × top, with the letters at their values; tiles: the expression's value/nodes/board/outputs/value
Wrong answers it recognises
- 36 ± 0.01
  Misconception: Puts the number beside the coefficient as a digit: 3x with x = 4 read as 34.
  Feedback from the catalogue
  A number written next to a letter hides a multiplication: $3x$ ⁠ means $3 \times x$ ⁠. With $x = 4$ ⁠ you get $3 \times 4 = 12$ ⁠, not $34$ ⁠. So $3x + 2 = 12 + 2 = 14$ ⁠.
- 9 ± 0.01
  Misconception: Reads 3x as 3 + x.
  Feedback from the catalogue
  $3x$ ⁠ is three strips of length x, that is $x + x + x = 3 \times x$ ⁠. With $x = 4$ ⁠ that is $12$ ⁠, not $3 + 4 = 7$ ⁠.
Feedback for a right answer
$3x + 2 = 3 \times 4 + 2 = 12 + 2 = 14$ ⁠. Choose Sorted and worked out under Look inside to see the working under the board.

Feedback for a wrong answer
Put $4$ ⁠ in place of x, and remember that a multiplication hides between the $3$ ⁠ and the x.

Hint
$3x$ ⁠ means $3 \times x$ ⁠. What is $3 \times 4$ ⁠?

Worked solution
1. The letter x stands for a number, and here $x = 4$ ⁠.
2. $3x$ ⁠ means $3 \times x$ ⁠, so $3x = 3 \times 4 = 12$ ⁠.
3. Add the $2$ ⁠: $12 + 2 = 14$ ⁠.
If you got $36$ ⁠, you wrote the $4$ ⁠ next to the $3$ ⁠ and read $34$ ⁠. If you got $9$ ⁠, you added $3 + 4$ ⁠ instead of multiplying.
Step 2 of 5
Open this step
A variable varies
Practice · Reach the target
Question
Move x until $3x + 2$ ⁠ equals $20$ ⁠, then press Check.

Goal
- 20 ± 0.01
  At the start of the step: 5
Setup, choices and feedback
The Scene at this step
x
1
y
3defaulthidden
As written / Sorted and worked out
Sorted and worked out
Read from the Scene at
Product: the area, left × top, with the letters at their values; tiles: the expression's value/nodes/board/outputs/value
One setting that reaches it
x
6
Feedback for a right answer
With $x = 6$ ⁠: $3 \times 6 + 2 = 18 + 2 = 20$ ⁠. The same expression gives a different value for every x, and that is what a variable means.

Feedback for a wrong answer
Not yet. If the value under the board is less than $20$ ⁠, make x bigger; if it is more, make it smaller.

Hint
Without the two squares, the three strips must make $18$ ⁠. How long is each strip?

Worked solution
1. You want $3x + 2 = 20$ ⁠.
2. Without the $2$ ⁠, the three strips must make $20 - 2 = 18$ ⁠.
3. Three equal strips that make $18$ ⁠ together: each is $18 \div 3 = 6$ ⁠.
4. Set $x = 6$ ⁠ and check: $3 \times 6 + 2 = 20$ ⁠.
If you stopped at $x = 7$ ⁠, you got $3 \times 7 + 2 = 23$ ⁠: the two squares count towards the total too.
Step 3 of 5
Open this step
Like terms
Practice · Predict, then look
Question
The board shows $2x + 3 + 4x + 1$ ⁠. Predict: how can it be written more shortly?

Correct answer
- Correct $6x + 4$ ⁠
Setup, choices and feedback
The Scene at this step
x
4
y
3defaulthidden
As written / Sorted and worked out
As writtenfixed
Other settings
Expression/nodes/board/props/expression = "2x + 3 + 4x + 1"
Choices
- Correct $6x + 4$ ⁠
- Wrong $10x$ ⁠
  Misconception: Adds unlike terms into one: 3x + 2y = 5xy, or 2x + 3 + 4x + 1 = 10x.
  Feedback from the catalogue
  You can only add terms of the same kind: x strips with x strips, y strips with y strips, and squares with squares. A y strip has a different length from an x strip, and a square is not a strip, so they cannot be counted together. $3x + 2y$ ⁠ stays two terms, and $2x + 3 + 4x + 1 = 6x + 4$ ⁠.
- Wrong $6x^2 + 4$ ⁠
  Misconception: Multiplies letters when adding terms: x + x = x², 2x + 4x = 6x².
  Feedback from the catalogue
  $x + x$ ⁠ is two strips of length x, that is $2x$ ⁠. $x^2$ ⁠ is $x \times x$ ⁠, a square with side x. With $x = 4$ ⁠: $x + x = 8$ ⁠, but $x^2 = 16$ ⁠. Adding counts strips; the letter stays the same.
- Wrong $7x + 3$ ⁠
  Feedback
  The $1$ ⁠ is a square, not a strip: it joins the $3$ ⁠, not the x terms.
Once the prediction is locked in, the Scene changes
As written / Sorted and worked out
Sorted and worked outfixed
The Scene then shows: 6 · The coefficient of x after collecting like terms/nodes/board/outputs/coefX
A prediction is never marked as a failure: the learner sees the outcome and the explanation either way.
Feedback for a right answer
Strips join strips, $2x + 4x = 6x$ ⁠, and squares join squares, $3 + 1 = 4$ ⁠. Together $6x + 4$ ⁠. Move x and see that it holds for every x.

Worked solution
1. The terms are $2x$ ⁠, $3$ ⁠, $4x$ ⁠ and $1$ ⁠.
2. Like terms have the same letters: $2x$ ⁠ and $4x$ ⁠; and also $3$ ⁠ and $1$ ⁠, which have no letter.
3. $2x + 4x = 6x$ ⁠, like two strips and four more strips.
4. $3 + 1 = 4$ ⁠.
5. The result: $6x + 4$ ⁠.
If you chose $10x$ ⁠, you counted the squares in with the strips. If you chose $6x^2 + 4$ ⁠, you multiplied the letters instead of counting strips.
Step 4 of 5
Open this step
Unlike terms
Practice · Choose
Question
The board shows $3x + 2y$ ⁠. Move x and y. Can $3x + 2y$ ⁠ be written more shortly?

Correct answer
- Correct No: $3x + 2y$ ⁠ is already as short as it gets
Setup, choices and feedback
The Scene at this step
x
4
y
3
As written / Sorted and worked out
Sorted and worked out
Other settings
Expression/nodes/board/props/expression = "3x + 2y"
Choices
Shown in a random order.
- Correct No: $3x + 2y$ ⁠ is already as short as it gets
- Wrong $5xy$ ⁠
  Misconception: Adds unlike terms into one: 3x + 2y = 5xy, or 2x + 3 + 4x + 1 = 10x.
  Feedback from the catalogue
  You can only add terms of the same kind: x strips with x strips, y strips with y strips, and squares with squares. A y strip has a different length from an x strip, and a square is not a strip, so they cannot be counted together. $3x + 2y$ ⁠ stays two terms, and $2x + 3 + 4x + 1 = 6x + 4$ ⁠.
- Wrong $5x$ ⁠
  Feedback
  The y strips are not x strips: move y and see that they change length on their own.
- Wrong $6xy$ ⁠
  Feedback
  $6xy$ ⁠ is $3x \cdot 2y$ ⁠, a product. Here there is an addition, of two different terms.
Feedback for a right answer
The x strips and the y strips have different lengths, so they cannot be counted together: $3x + 2y$ ⁠ stays two terms. For example, with $x = 4$ ⁠ and $y = 3$ ⁠ it is $12 + 6 = 18$ ⁠, while $5xy$ ⁠ would be $60$ ⁠.

Worked solution
1. Like terms need the same letters: $3x$ ⁠ has x and $2y$ ⁠ has y, so they are not alike.
2. On the board the x strips are as long as x, and the y strips as long as y. There is no one name for all five strips.
3. So $3x + 2y$ ⁠ is already in its shortest form.
4. Check with numbers, $x = 4$ ⁠ and $y = 3$ ⁠: $3x + 2y = 12 + 6 = 18$ ⁠, but $5xy = 5 \cdot 4 \cdot 3 = 60$ ⁠.
If you chose $5xy$ ⁠, you added the numbers and joined the letters. That only works when the letters are the same.
Step 5 of 5
Open this step
A minus in front of a term
Practice · Calculate
Question
The board shows $5x + 4 - 2x - 1$ ⁠. Collect like terms: how many x are left? In other words, what number stands before x in the result?

Correct answer
3 ± 0.01
Setup, choices and feedback
The Scene at this step
x
4
y
3defaulthidden
As written / Sorted and worked out
As written
Other settings
Expression/nodes/board/props/expression = "5x + 4 − 2x − 1"
Accepted within
± 0.01
Computed from the Scene at this step’s setting
The coefficient of x after collecting like terms/nodes/board/outputs/coefX
Wrong answers it recognises
- 7 ± 0.01
  Misconception: Loses the minus sign in front of a term, as if it were a plus: 5x − 2x = 7x, 4(2x − 3) = 8x + 12.
  Feedback from the catalogue
  The sign in front of a term belongs to that term. In $5x + 4 - 2x - 1$ ⁠ the terms are $5x$ ⁠, $4$ ⁠, $-2x$ ⁠ and $-1$ ⁠, so $5x - 2x = 3x$ ⁠. And when you expand brackets, the number outside multiplies each term with its sign: $4 \cdot (-3) = -12$ ⁠.
Feedback for a right answer
$5x - 2x = 3x$ ⁠ and $4 - 1 = 3$ ⁠, so $5x + 4 - 2x - 1 = 3x + 3$ ⁠. Choose Sorted and worked out under Look inside: each pale strip cancels one solid strip.

Feedback for a wrong answer
Count the solid strips, and take one away for each pale strip.

Hint
The minus in front of $2x$ ⁠ belongs to the $2x$ ⁠. The pale strips are strips taken away.

Worked solution
1. The terms, each with the sign in front of it: $5x$ ⁠, $4$ ⁠, $-2x$ ⁠ and $-1$ ⁠.
2. The x terms: $5x - 2x = 3x$ ⁠. On the board, two pale strips cancel two of the five solid strips.
3. The numbers: $4 - 1 = 3$ ⁠.
4. The result is $3x + 3$ ⁠, and the number before x is $3$ ⁠.
If you got $7$ ⁠, you added $5x + 2x$ ⁠: the minus in front of $2x$ ⁠ got lost.

Lesson 2 of 4 · About 25 minutes

Expanding brackets

Why a number before brackets multiplies every term inside: a product is the area of a rectangle, and a rectangle whose side is split is cut into parts. And what a minus before brackets does.

Open the lesson

The 5 steps and their answers

Step 1 of 5
Open this step
Cutting a rectangle
Practice · Predict, then look
Question
The rectangle is $7 \times 13$ ⁠, and the side $13$ ⁠ is cut into $10 + 3$ ⁠. Predict: what is $7 \times 13$ ⁠?

Correct answer
91 ± 0.01
A number prediction counts as right within ± 0.01
Setup, choices and feedback
The Scene at this step
x
4defaulthidden
The whole rectangle / Split into parts
The whole rectanglefixed
Other settings
Left/nodes/board/props/left = "7"
Other settings
Top/nodes/board/props/top = "10 + 3"
Other settings
Units/nodes/board/props/units = true
Other settings
Span/nodes/board/props/span = [13,7]
Once the prediction is locked in, the Scene changes
The whole rectangle / Split into parts
Split into partsfixed
The Scene then shows: 91 · Product: the area, left × top, with the letters at their values; tiles: the expression's value/nodes/board/outputs/value
A prediction is never marked as a failure: the learner sees the outcome and the explanation either way.
Feedback for a right answer
$7 \times 13 = 7 \times 10 + 7 \times 3 = 70 + 21 = 91$ ⁠. Cutting a side in two cuts the rectangle into two rectangles, and their areas add up.

Hint
The left part is $7 \times 10$ ⁠, and the right part $7 \times 3$ ⁠.

Worked solution
1. Write $13 = 10 + 3$ ⁠, so $7 \times 13 = 7(10 + 3)$ ⁠.
2. The rectangle splits in two: $7 \times 10 = 70$ ⁠ and $7 \times 3 = 21$ ⁠.
3. The whole area is the sum of the parts: $70 + 21 = 91$ ⁠.
If you got $73$ ⁠, you multiplied only the $10$ ⁠ and added the $3$ ⁠. The narrow part is $7$ ⁠ high too.
Step 2 of 5
Open this step
A number before brackets
Practice · Predict, then look
Question
The left side is $3$ ⁠, and the top side $x + 2$ ⁠. Predict: what is $3(x + 2)$ ⁠ without brackets?

Correct answer
- Correct $3x + 6$ ⁠
Setup, choices and feedback
The Scene at this step
x
4
The whole rectangle / Split into parts
The whole rectanglefixed
Choices
- Correct $3x + 6$ ⁠
- Wrong $3x + 2$ ⁠
  Misconception: Multiplies only the first term in the brackets: 3(x + 2) = 3x + 2.
  Feedback from the catalogue
  The number in front of the brackets multiplies every term inside. The rectangle shows why: the outside side runs along all the parts. For example, in $3(x + 2)$ ⁠ the side $3$ ⁠ runs along $x$ ⁠ and also along $2$ ⁠, so there are two areas, $3x$ ⁠ and $6$ ⁠: $3(x + 2) = 3x + 6$ ⁠.
- Wrong $5x$ ⁠
  Feedback
  The rectangle has two separate parts, $3 \cdot x$ ⁠ and $3 \cdot 2$ ⁠, and the right part has no x at all.
- Wrong $x + 6$ ⁠
  Feedback
  The x is multiplied by $3$ ⁠ too: the side $3$ ⁠ runs along both parts.
Once the prediction is locked in, the Scene changes
The whole rectangle / Split into parts
Split into partsfixed
A prediction is never marked as a failure: the learner sees the outcome and the explanation either way.
Feedback for a right answer
The side $3$ ⁠ runs along both parts, so it multiplies both: $3 \cdot x = 3x$ ⁠ and $3 \cdot 2 = 6$ ⁠. $3(x + 2) = 3x + 6$ ⁠. This is the distributive law.

Worked solution
1. $3(x + 2)$ ⁠ is the area of a rectangle with sides $3$ ⁠ and $x + 2$ ⁠.
2. The top side is split into $x$ ⁠ and $2$ ⁠, so the rectangle is cut into two parts.
3. The left part is $3 \cdot x = 3x$ ⁠, and the right part $3 \cdot 2 = 6$ ⁠.
4. Add them: $3(x + 2) = 3x + 6$ ⁠.
If you chose $3x + 2$ ⁠, you multiplied only the first term. The $3$ ⁠ multiplies every term inside the brackets.
Step 3 of 5
Open this step
Checking with a number
Practice · Calculate
Question
Now $x = 5$ ⁠. Work out $3(x + 2)$ ⁠, and check that $3x + 6$ ⁠ gives the same number.

Correct answer
21 ± 0.01
Setup, choices and feedback
The Scene at this step
x
5fixed
The whole rectangle / Split into parts
The whole rectangle
Accepted within
± 0.01
Computed from the Scene at this step’s setting
Product: the area, left × top, with the letters at their values; tiles: the expression's value/nodes/board/outputs/value
Wrong answers it recognises
- 17 ± 0.01
  Misconception: Multiplies only the first term in the brackets: 3(x + 2) = 3x + 2.
  Feedback from the catalogue
  The number in front of the brackets multiplies every term inside. The rectangle shows why: the outside side runs along all the parts. For example, in $3(x + 2)$ ⁠ the side $3$ ⁠ runs along $x$ ⁠ and also along $2$ ⁠, so there are two areas, $3x$ ⁠ and $6$ ⁠: $3(x + 2) = 3x + 6$ ⁠.
Feedback for a right answer
$3(5 + 2) = 3 \times 7 = 21$ ⁠, and also $3 \cdot 5 + 6 = 15 + 6 = 21$ ⁠. The two forms are equal for every x, because they are two names for the same area.

Feedback for a wrong answer
Work out what is inside the brackets first, then multiply by $3$ ⁠.

Hint
Brackets first: $5 + 2 = 7$ ⁠. Then $3 \times 7$ ⁠.

Worked solution
1. Put in $x = 5$ ⁠: $3(5 + 2)$ ⁠.
2. Brackets first: $5 + 2 = 7$ ⁠, and $3 \times 7 = 21$ ⁠.
3. Check with the expanded form: $3x + 6 = 3 \cdot 5 + 6 = 15 + 6 = 21$ ⁠. The same number.
If you got $17$ ⁠, you worked out $3 \cdot 5 + 2$ ⁠: the $3$ ⁠ multiplied only the x.
Step 4 of 5
Open this step
A minus before brackets
Practice · Predict, then look
Question
A minus in front of brackets is a multiplication by $-1$ ⁠. The board’s sides are $-1$ ⁠ and $x - 4$ ⁠. Predict: what is $-(x - 4)$ ⁠ without brackets?

Correct answer
- Correct $-x + 4$ ⁠
Setup, choices and feedback
The Scene at this step
x
6
The whole rectangle / Split into parts
The whole rectanglefixed
Other settings
Left/nodes/board/props/left = "−1"
Other settings
Top/nodes/board/props/top = "x − 4"
Other settings
Span/nodes/board/props/span = [14,3]
Choices
- Correct $-x + 4$ ⁠
- Wrong $-x - 4$ ⁠
  Misconception: Changes only the first sign after a minus before brackets: −(x − 4) = −x − 4.
  Feedback from the catalogue
  A minus in front of brackets is a multiplication by $-1$ ⁠, and it meets every term inside: $-1 \cdot x = -x$ ⁠, and also $-1 \cdot (-4) = 4$ ⁠, just as $-(-4) = 4$ ⁠. So $-(x - 4) = -x + 4$ ⁠: every sign inside the brackets flips.
- Wrong $x - 4$ ⁠
  Feedback
  The minus in front of the brackets does not disappear: it flips the sign of every term.
- Wrong $x + 4$ ⁠
  Feedback
  The x gets the minus too: $-1 \cdot x = -x$ ⁠.
Once the prediction is locked in, the Scene changes
The whole rectangle / Split into parts
Split into partsfixed
A prediction is never marked as a failure: the learner sees the outcome and the explanation either way.
Feedback for a right answer
$-1 \cdot x = -x$ ⁠ and $-1 \cdot (-4) = 4$ ⁠: every sign in the brackets flips, and $-(x - 4) = -x + 4$ ⁠. The pale part on the board is $-x$ ⁠, an area taken away.

Worked solution
1. $-(x - 4)$ ⁠ means $-1 \cdot (x - 4)$ ⁠.
2. Multiply each term: $-1 \cdot x = -x$ ⁠.
3. $-1 \cdot (-4)$ ⁠ flips the sign of $-4$ ⁠ and gives $4$ ⁠, just as $-(-4) = 4$ ⁠ in the previous Unit.
4. The result: $-x + 4$ ⁠.
If you chose $-x - 4$ ⁠, you flipped only the first sign. The minus meets the $-4$ ⁠ too.
Step 5 of 5
Open this step
A minus inside the brackets
Practice · Calculate
Question
Expand the brackets in $4(2x - 3)$ ⁠. What is the term with no x?

Correct answer
-12 ± 0.01
Setup, choices and feedback
The Scene at this step
x
5
The whole rectangle / Split into parts
The whole rectangle
Other settings
Left/nodes/board/props/left = "4"
Other settings
Top/nodes/board/props/top = "2x − 3"
Other settings
Span/nodes/board/props/span = [23,4]
Accepted within
± 0.01
Computed from the Scene at this step’s setting
The number without a letter after collecting like terms/nodes/board/outputs/constant
Wrong answers it recognises
- -3 ± 0.01
  Misconception: Multiplies only the first term in the brackets: 3(x + 2) = 3x + 2.
  Feedback from the catalogue
  The number in front of the brackets multiplies every term inside. The rectangle shows why: the outside side runs along all the parts. For example, in $3(x + 2)$ ⁠ the side $3$ ⁠ runs along $x$ ⁠ and also along $2$ ⁠, so there are two areas, $3x$ ⁠ and $6$ ⁠: $3(x + 2) = 3x + 6$ ⁠.
- 12 ± 0.01
  Misconception: Loses the minus sign in front of a term, as if it were a plus: 5x − 2x = 7x, 4(2x − 3) = 8x + 12.
  Feedback from the catalogue
  The sign in front of a term belongs to that term. In $5x + 4 - 2x - 1$ ⁠ the terms are $5x$ ⁠, $4$ ⁠, $-2x$ ⁠ and $-1$ ⁠, so $5x - 2x = 3x$ ⁠. And when you expand brackets, the number outside multiplies each term with its sign: $4 \cdot (-3) = -12$ ⁠.
Feedback for a right answer
$4(2x - 3) = 4 \cdot 2x + 4 \cdot (-3) = 8x - 12$ ⁠. The term with no x is $-12$ ⁠, with the minus that stood before the $3$ ⁠.

Feedback for a wrong answer
Multiply the $4$ ⁠ by every term in the brackets, together with the sign in front of it.

Hint
The $4$ ⁠ multiplies the $-3$ ⁠ too, together with its sign.

Worked solution
1. The $4$ ⁠ multiplies each term in the brackets: $4 \cdot 2x$ ⁠ and $4 \cdot (-3)$ ⁠.
2. $4 \cdot 2x = 8x$ ⁠.
3. $4 \cdot (-3) = -12$ ⁠.
4. $4(2x - 3) = 8x - 12$ ⁠, and the term with no x is $-12$ ⁠.
If you wrote $-3$ ⁠, the $4$ ⁠ multiplied only the $2x$ ⁠. If you wrote $12$ ⁠, the minus in front of the $3$ ⁠ got lost.

Lesson 3 of 4 · About 25 minutes

Multiplying two brackets

A product of two sums is a rectangle with four parts: where the middle term comes from, why the square of a sum is more than the sum of the squares, and what happens when a side has a minus.

Open the lesson

The 5 steps and their answers

Step 1 of 5
Open this step
Four parts
Practice · Predict, then look
Question
The sides are $x + 3$ ⁠ and $x + 2$ ⁠. Predict: what is $(x + 3)(x + 2)$ ⁠ without brackets?

Correct answer
- Correct $x^2 + 5x + 6$ ⁠
Setup, choices and feedback
The Scene at this step
x
4
a
5defaulthidden
b
3defaulthidden
The whole rectangle / Split into parts
The whole rectanglefixed
Choices
- Correct $x^2 + 5x + 6$ ⁠
- Wrong $x^2 + 6$ ⁠
  Misconception: Multiplies only first by first and last by last: (x + 3)(x + 2) = x² + 6.
  Feedback from the catalogue
  The rectangle has four parts: each term of one side times each term of the other, and the two middle parts are x terms. For example, in $(x + 3)(x + 2)$ ⁠ the parts are $x^2$ ⁠, $2x$ ⁠, $3x$ ⁠ and $6$ ⁠, and the middle ones together make $5x$ ⁠: $(x + 3)(x + 2) = x^2 + 5x + 6$ ⁠.
- Wrong $x^2 + 6x + 5$ ⁠
  Misconception: Multiplies the two numbers to get the middle coefficient: (x + 4)(x + 3) = x² + 12x + 12.
  Feedback from the catalogue
  In $(x + p)(x + q)$ ⁠ the product of the numbers, $p \cdot q$ ⁠, is the corner part, the number with no x. The middle is made of two rectangles, $qx$ ⁠ and $px$ ⁠, together $(p + q)x$ ⁠. For example, $(x + 4)(x + 3) = x^2 + 7x + 12$ ⁠: the sum $4 + 3$ ⁠ gives the middle, and the product $4 \cdot 3$ ⁠ the corner.
- Wrong $2x + 5$ ⁠
  Feedback
  $2x + 5$ ⁠ is the sum of the two sides. A product is the area inside the rectangle.
Once the prediction is locked in, the Scene changes
The whole rectangle / Split into parts
Split into partsfixed
The Scene then shows: 5 · The coefficient of x after collecting like terms/nodes/board/outputs/coefX
A prediction is never marked as a failure: the learner sees the outcome and the explanation either way.
Feedback for a right answer
Four parts: $x^2$ ⁠, $2x$ ⁠, $3x$ ⁠ and $6$ ⁠. The two middle parts have the same colour because they are like terms, together $5x$ ⁠. $(x + 3)(x + 2) = x^2 + 5x + 6$ ⁠.

Worked solution
1. Each term of one side times each term of the other, so there are four parts.
2. $x \cdot x = x^2$ ⁠, $x \cdot 2 = 2x$ ⁠, $3 \cdot x = 3x$ ⁠ and $3 \cdot 2 = 6$ ⁠.
3. Collect like terms: $2x + 3x = 5x$ ⁠.
4. $(x + 3)(x + 2) = x^2 + 5x + 6$ ⁠.
If you chose $x^2 + 6$ ⁠, you took only the two corners. The two middle rectangles are part of the area.
Step 2 of 5
Open this step
The middle term
Practice · Calculate
Question
The sides are now $x + 4$ ⁠ and $x + 3$ ⁠. After collecting like terms, $(x + 4)(x + 3) = x^2 + kx + 12$ ⁠. What is $k$ ⁠?

Correct answer
7 ± 0.01
Setup, choices and feedback
The Scene at this step
x
4
a
5defaulthidden
b
3defaulthidden
The whole rectangle / Split into parts
The whole rectangle
Other settings
Left/nodes/board/props/left = "x + 4"
Other settings
Top/nodes/board/props/top = "x + 3"
Accepted within
± 0.01
Computed from the Scene at this step’s setting
The coefficient of x after collecting like terms/nodes/board/outputs/coefX
Wrong answers it recognises
- 12 ± 0.01
  Misconception: Multiplies the two numbers to get the middle coefficient: (x + 4)(x + 3) = x² + 12x + 12.
  Feedback from the catalogue
  In $(x + p)(x + q)$ ⁠ the product of the numbers, $p \cdot q$ ⁠, is the corner part, the number with no x. The middle is made of two rectangles, $qx$ ⁠ and $px$ ⁠, together $(p + q)x$ ⁠. For example, $(x + 4)(x + 3) = x^2 + 7x + 12$ ⁠: the sum $4 + 3$ ⁠ gives the middle, and the product $4 \cdot 3$ ⁠ the corner.
- 0 ± 0.01
  Misconception: Multiplies only first by first and last by last: (x + 3)(x + 2) = x² + 6.
  Feedback from the catalogue
  The rectangle has four parts: each term of one side times each term of the other, and the two middle parts are x terms. For example, in $(x + 3)(x + 2)$ ⁠ the parts are $x^2$ ⁠, $2x$ ⁠, $3x$ ⁠ and $6$ ⁠, and the middle ones together make $5x$ ⁠: $(x + 3)(x + 2) = x^2 + 5x + 6$ ⁠.
Feedback for a right answer
$3x + 4x = 7x$ ⁠, so $k = 7$ ⁠. The middle is the sum of the numbers, $4 + 3$ ⁠, and the corner is their product, $4 \cdot 3 = 12$ ⁠.

Feedback for a wrong answer
Choose Split into parts under Look inside and add the two middle parts.

Hint
The two middle rectangles are $x \cdot 3$ ⁠ and $4 \cdot x$ ⁠.

Worked solution
1. The four parts: $x \cdot x = x^2$ ⁠, $x \cdot 3 = 3x$ ⁠, $4 \cdot x = 4x$ ⁠ and $4 \cdot 3 = 12$ ⁠.
2. The like terms: $3x + 4x = 7x$ ⁠.
3. $(x + 4)(x + 3) = x^2 + 7x + 12$ ⁠, so $k = 7$ ⁠.
If you wrote $12$ ⁠, you multiplied $4 \cdot 3$ ⁠ for the middle too. If you wrote $0$ ⁠, you forgot the two middle rectangles.
Step 3 of 5
Open this step
The square of a sum
Practice · Predict, then look
Question
Now both sides are $a + b$ ⁠, so this is a square: $(a + b)^2$ ⁠. Predict: what is it equal to?

Correct answer
- Correct $a^2 + 2ab + b^2$ ⁠
Setup, choices and feedback
The Scene at this step
x
4defaulthidden
a
5
b
3
The whole rectangle / Split into parts
The whole rectanglefixed
Other settings
Left/nodes/board/props/left = "a + b"
Other settings
Top/nodes/board/props/top = "a + b"
Choices
- Correct $a^2 + 2ab + b^2$ ⁠
- Wrong $a^2 + b^2$ ⁠
  Misconception: Squares each term on its own: (a + b)² = a² + b², (x − 3)² = x² − 9.
  Feedback from the catalogue
  $(a + b)^2$ ⁠ is a square with side $a + b$ ⁠, and it has four parts: the squares $a^2$ ⁠ and $b^2$ ⁠, and two rectangles $ab$ ⁠ that are easy to forget. So $(a + b)^2 = a^2 + 2ab + b^2$ ⁠. With $a = 5$ ⁠ and $b = 3$ ⁠: $8^2 = 64$ ⁠, not $25 + 9 = 34$ ⁠. The same goes for a difference: $(x - 3)^2 = x^2 - 6x + 9$ ⁠, not $x^2 - 9$ ⁠.
- Wrong $a^2 + ab + b^2$ ⁠
  Feedback
  The square has two rectangles $ab$ ⁠, one in each of the other corners. Count the parts once they are shown.
- Wrong $2a + 2b$ ⁠
  Misconception: Squares by doubling: (a + b)² = 2(a + b).
  Feedback from the catalogue
  Squared means times itself, not times $2$ ⁠: $(a + b)^2 = (a + b)(a + b)$ ⁠. When $a + b = 8$ ⁠, that is $8 \cdot 8 = 64$ ⁠, not $2 \cdot 8 = 16$ ⁠.
Once the prediction is locked in, the Scene changes
The whole rectangle / Split into parts
Split into partsfixed
The Scene then shows: 2 · The coefficient of ab after collecting like terms/nodes/board/outputs/coefAB
A prediction is never marked as a failure: the learner sees the outcome and the explanation either way.
Feedback for a right answer
A square $a^2$ ⁠, a square $b^2$ ⁠, and two rectangles $a \times b$ ⁠: $(a + b)^2 = a^2 + 2ab + b^2$ ⁠. Move a and b: the sizes change, and the pattern stays.

Worked solution
1. $(a + b)^2 = (a + b)(a + b)$ ⁠: a square with side $a + b$ ⁠.
2. Each side is split into $a$ ⁠ and $b$ ⁠, so there are four parts.
3. $a \cdot a = a^2$ ⁠, $a \cdot b = ab$ ⁠, $b \cdot a = ab$ ⁠ and $b \cdot b = b^2$ ⁠.
4. The two $ab$ ⁠ are like terms: $(a + b)^2 = a^2 + 2ab + b^2$ ⁠.
If you chose $a^2 + b^2$ ⁠, you counted only the two squares. The two rectangles $ab$ ⁠ are the missing area.
Step 4 of 5
Open this step
The square in numbers
Practice · Calculate
Question
$a = 5$ ⁠ and $b = 3$ ⁠. What is $(a + b)^2$ ⁠?

Correct answer
64 ± 0.01
Setup, choices and feedback
The Scene at this step
x
4defaulthidden
a
5fixed
b
3fixed
The whole rectangle / Split into parts
The whole rectangle
Other settings
Left/nodes/board/props/left = "a + b"
Other settings
Top/nodes/board/props/top = "a + b"
Accepted within
± 0.01
Computed from the Scene at this step’s setting
Product: the area, left × top, with the letters at their values; tiles: the expression's value/nodes/board/outputs/value
Wrong answers it recognises
- 34 ± 0.01
  Misconception: Squares each term on its own: (a + b)² = a² + b², (x − 3)² = x² − 9.
  Feedback from the catalogue
  $(a + b)^2$ ⁠ is a square with side $a + b$ ⁠, and it has four parts: the squares $a^2$ ⁠ and $b^2$ ⁠, and two rectangles $ab$ ⁠ that are easy to forget. So $(a + b)^2 = a^2 + 2ab + b^2$ ⁠. With $a = 5$ ⁠ and $b = 3$ ⁠: $8^2 = 64$ ⁠, not $25 + 9 = 34$ ⁠. The same goes for a difference: $(x - 3)^2 = x^2 - 6x + 9$ ⁠, not $x^2 - 9$ ⁠.
- 16 ± 0.01
  Misconception: Squares by doubling: (a + b)² = 2(a + b).
  Feedback from the catalogue
  Squared means times itself, not times $2$ ⁠: $(a + b)^2 = (a + b)(a + b)$ ⁠. When $a + b = 8$ ⁠, that is $8 \cdot 8 = 64$ ⁠, not $2 \cdot 8 = 16$ ⁠.
Feedback for a right answer
$(5 + 3)^2 = 8^2 = 64$ ⁠. And in parts: $25 + 15 + 15 + 9 = 64$ ⁠. Without the two rectangles of $15$ ⁠ you would get $34$ ⁠.

Feedback for a wrong answer
Work out the brackets first, then square.

Hint
Brackets first: $a + b = 8$ ⁠. What is $8^2$ ⁠?

Worked solution
1. Brackets first: $a + b = 5 + 3 = 8$ ⁠.
2. $8^2 = 8 \cdot 8 = 64$ ⁠.
3. Check with the parts: $a^2 + 2ab + b^2 = 25 + 2 \cdot 15 + 9 = 64$ ⁠.
If you got $34$ ⁠, you added only $5^2 + 3^2$ ⁠. If you got $16$ ⁠, you multiplied by $2$ ⁠ instead of squaring.
Step 5 of 5
Open this step
The square of a difference
Practice · Predict, then look
Question
Both sides are $x - 3$ ⁠. A part with a minus is drawn pale: its area is taken away. Predict: what is $(x - 3)^2$ ⁠ equal to?

Correct answer
- Correct $x^2 - 6x + 9$ ⁠
Setup, choices and feedback
The Scene at this step
x
8
a
5defaulthidden
b
3defaulthidden
The whole rectangle / Split into parts
The whole rectanglefixed
Other settings
Left/nodes/board/props/left = "x − 3"
Other settings
Top/nodes/board/props/top = "x − 3"
Choices
- Correct $x^2 - 6x + 9$ ⁠
- Wrong $x^2 - 9$ ⁠
  Misconception: Squares each term on its own: (a + b)² = a² + b², (x − 3)² = x² − 9.
  Feedback from the catalogue
  $(a + b)^2$ ⁠ is a square with side $a + b$ ⁠, and it has four parts: the squares $a^2$ ⁠ and $b^2$ ⁠, and two rectangles $ab$ ⁠ that are easy to forget. So $(a + b)^2 = a^2 + 2ab + b^2$ ⁠. With $a = 5$ ⁠ and $b = 3$ ⁠: $8^2 = 64$ ⁠, not $25 + 9 = 34$ ⁠. The same goes for a difference: $(x - 3)^2 = x^2 - 6x + 9$ ⁠, not $x^2 - 9$ ⁠.
- Wrong $x^2 - 6x - 9$ ⁠
  Misconception: Thinks a negative times a negative is negative: (−3)(−3) = −9.
  Feedback from the catalogue
  Multiplying by $-3$ ⁠ is multiplying by $3$ ⁠ and flipping the sign. $(-3) \cdot 3 = -9$ ⁠, and flipping the sign gives $(-3)(-3) = 9$ ⁠. So the corner of $(x - 3)^2$ ⁠ is $+9$ ⁠, and $(x - 3)^2 = x^2 - 6x + 9$ ⁠.
- Wrong $x^2 + 9$ ⁠
  Feedback
  Here too there are two middle parts, $-3x$ ⁠ each, and they do not disappear.
Once the prediction is locked in, the Scene changes
The whole rectangle / Split into parts
Split into partsfixed
The Scene then shows: 9 · The number without a letter after collecting like terms/nodes/board/outputs/constant
A prediction is never marked as a failure: the learner sees the outcome and the explanation either way.
Feedback for a right answer
$x \cdot x = x^2$ ⁠, two parts of $x \cdot (-3) = -3x$ ⁠, and a corner $(-3)(-3) = 9$ ⁠. Together $(x - 3)^2 = x^2 - 6x + 9$ ⁠.

Worked solution
1. $(x - 3)^2 = (x - 3)(x - 3)$ ⁠, and each side has the terms $x$ ⁠ and $-3$ ⁠.
2. Four parts: $x \cdot x = x^2$ ⁠, $x \cdot (-3) = -3x$ ⁠, $(-3) \cdot x = -3x$ ⁠ and $(-3)(-3) = 9$ ⁠.
3. Collect: $-3x - 3x = -6x$ ⁠.
4. $(x - 3)^2 = x^2 - 6x + 9$ ⁠.
5. Check with $x = 8$ ⁠: $5^2 = 25$ ⁠, and also $64 - 48 + 9 = 25$ ⁠.
If you chose $x^2 - 9$ ⁠, you squared each term on its own. If you chose $x^2 - 6x - 9$ ⁠, you forgot that a minus times a minus is a plus.

Lesson 4 of 4 · About 25 minutes

Factorising

Expanding brackets the other way round: when the areas are known, look for the sides. A common factor, a letter in common, sum and product, and the difference of two squares.

Open the lesson

The 5 steps and their answers

Step 1 of 5
Open this step
A common factor
Practice · Predict, then look
Question
The board has two parts, $6x$ ⁠ and $15$ ⁠, and the sides are hidden. Predict: which product equals $6x + 15$ ⁠?

Correct answer
- Correct $3(2x + 5)$ ⁠
Setup, choices and feedback
The Scene at this step
x
4
p
1defaulthidden
q
1defaulthidden
Parts only / Sides named
Parts onlyfixed
Choices
- Correct $3(2x + 5)$ ⁠
- Wrong $3(2x + 15)$ ⁠
  Misconception: Divides only one term by the common factor: 6x + 15 = 3(2x + 15).
  Feedback from the catalogue
  The common factor comes out of every term: divide each term by it. Check by expanding: $3(2x + 15) = 6x + 45$ ⁠, not $6x + 15$ ⁠. The right way: $6x = 3 \cdot 2x$ ⁠ and $15 = 3 \cdot 5$ ⁠, so $6x + 15 = 3(2x + 5)$ ⁠. In the same way $x^2 + 5x = x(x + 5)$ ⁠, because $5x \div x = 5$ ⁠.
- Wrong $3(6x + 15)$ ⁠
  Feedback
  Expand to check: $3(6x + 15) = 18x + 45$ ⁠. What stays in the brackets must be divided by $3$ ⁠.
- Wrong $6(x + 15)$ ⁠
  Feedback
  Expand to check: $6(x + 15) = 6x + 90$ ⁠. $6$ ⁠ does not divide $15$ ⁠.
Once the prediction is locked in, the Scene changes
Parts only / Sides named
Sides namedfixed
A prediction is never marked as a failure: the learner sees the outcome and the explanation either way.
Feedback for a right answer
$3$ ⁠ divides both $6x$ ⁠ and $15$ ⁠: $6x = 3 \cdot 2x$ ⁠ and $15 = 3 \cdot 5$ ⁠. So $6x + 15 = 3(2x + 5)$ ⁠. This is factorising: expanding brackets the other way round.

Worked solution
1. Look for a number that divides both terms: $3$ ⁠ divides $6$ ⁠ and also $15$ ⁠.
2. Divide each term by $3$ ⁠: $6x \div 3 = 2x$ ⁠ and $15 \div 3 = 5$ ⁠.
3. The common side is $3$ ⁠, and the other side $2x + 5$ ⁠: $6x + 15 = 3(2x + 5)$ ⁠.
4. Check by expanding: $3 \cdot 2x + 3 \cdot 5 = 6x + 15$ ⁠.
If you chose $3(2x + 15)$ ⁠, you divided only the first term by $3$ ⁠.
Step 2 of 5
Open this step
Finding the factor
Practice · Reach the target
Question
Choose p and q so that $p(x + q)$ ⁠ is $4x + 12$ ⁠, then press Check.

Goal
- 4 ± 0.01
  At the start of the step: 1
- 12 ± 0.01
  At the start of the step: 1
Setup, choices and feedback
The Scene at this step
x
4defaulthidden
p
1
q
1
Parts only / Sides named
Sides named
Other settings
Given/nodes/board/props/given = "sides"
Other settings
Left/nodes/board/props/left = "p"
Other settings
Top/nodes/board/props/top = "x + q"
Other settings
Target/nodes/board/props/target = "4x + 12"
Read from the Scene at
The coefficient of x after collecting like terms/nodes/board/outputs/coefX
Read from the Scene at
The number without a letter after collecting like terms/nodes/board/outputs/constant
One setting that reaches it
p
4
q
3
Feedback for a right answer
$4(x + 3) = 4x + 12$ ⁠: the side $4$ ⁠ times x gives $4x$ ⁠, and times $3$ ⁠ gives $12$ ⁠.

Feedback for a wrong answer
Not yet. Compare the part with x to $4x$ ⁠, and the part without x to $12$ ⁠.

Hint
The x stands alone in the brackets, so the part with x is $p \cdot x$ ⁠. What must p be? And then: how many times does p go into $12$ ⁠?

Worked solution
1. $p(x + q) = px + pq$ ⁠: the parts are $px$ ⁠ and $pq$ ⁠.
2. The part with x must be $4x$ ⁠, so $p = 4$ ⁠.
3. The part without x must be $12$ ⁠: $4q = 12$ ⁠, so $q = 3$ ⁠.
4. Check: $4(x + 3) = 4x + 12$ ⁠.
If you chose $q = 12$ ⁠, you forgot that the $12$ ⁠ is made by multiplying by p too: $4(x + 12) = 4x + 48$ ⁠.
Step 3 of 5
Open this step
A letter in common
Practice · Predict, then look
Question
The parts are $x^2$ ⁠ and $5x$ ⁠, and the sides are hidden. Predict: how does $x^2 + 5x$ ⁠ factorise?

Correct answer
- Correct $x(x + 5)$ ⁠
Setup, choices and feedback
The Scene at this step
x
4
p
1defaulthidden
q
1defaulthidden
Parts only / Sides named
Parts onlyfixed
Other settings
Left/nodes/board/props/left = "x"
Other settings
Top/nodes/board/props/top = "x + 5"
Choices
- Correct $x(x + 5)$ ⁠
- Wrong $x(x + 5x)$ ⁠
  Misconception: Divides only one term by the common factor: 6x + 15 = 3(2x + 15).
  Feedback from the catalogue
  The common factor comes out of every term: divide each term by it. Check by expanding: $3(2x + 15) = 6x + 45$ ⁠, not $6x + 15$ ⁠. The right way: $6x = 3 \cdot 2x$ ⁠ and $15 = 3 \cdot 5$ ⁠, so $6x + 15 = 3(2x + 5)$ ⁠. In the same way $x^2 + 5x = x(x + 5)$ ⁠, because $5x \div x = 5$ ⁠.
- Wrong $5x(x + 1)$ ⁠
  Feedback
  Expand to check: $5x(x + 1) = 5x^2 + 5x$ ⁠. The big part is $x^2$ ⁠, not $5x^2$ ⁠.
- Wrong $x^2(1 + 5x)$ ⁠
  Feedback
  Expand to check: $x^2(1 + 5x) = x^2 + 5x^3$ ⁠. $x^2$ ⁠ does not divide $5x$ ⁠.
Once the prediction is locked in, the Scene changes
Parts only / Sides named
Sides namedfixed
A prediction is never marked as a failure: the learner sees the outcome and the explanation either way.
Feedback for a right answer
x divides both terms: $x^2 = x \cdot x$ ⁠ and $5x = x \cdot 5$ ⁠. The common side is x, and the other side $x + 5$ ⁠: $x^2 + 5x = x(x + 5)$ ⁠.

Worked solution
1. What do the two terms share? Both have an x.
2. Divide each term by x: $x^2 \div x = x$ ⁠ and $5x \div x = 5$ ⁠.
3. $x^2 + 5x = x(x + 5)$ ⁠.
4. Check: $x \cdot x + x \cdot 5 = x^2 + 5x$ ⁠.
If you chose $x(x + 5x)$ ⁠, you divided only the first term by x.
Step 4 of 5
Open this step
Sum and product
Practice · Reach the target
Question
Choose p and q so that $(x + p)(x + q) = x^2 + 7x + 10$ ⁠, then press Check.

Goal
- 7 ± 0.01
  At the start of the step: 2
- 10 ± 0.01
  At the start of the step: 1
Setup, choices and feedback
The Scene at this step
x
4defaulthidden
p
1
q
1
Parts only / Sides named
Sides named
Other settings
Given/nodes/board/props/given = "sides"
Other settings
Left/nodes/board/props/left = "x + p"
Other settings
Top/nodes/board/props/top = "x + q"
Other settings
Target/nodes/board/props/target = "x² + 7x + 10"
Read from the Scene at
The coefficient of x after collecting like terms/nodes/board/outputs/coefX
Read from the Scene at
The number without a letter after collecting like terms/nodes/board/outputs/constant
One setting that reaches it
p
5
q
2
Feedback for a right answer
$(x + 2)(x + 5) = x^2 + 5x + 2x + 10 = x^2 + 7x + 10$ ⁠: the sum $2 + 5 = 7$ ⁠ and the product $2 \cdot 5 = 10$ ⁠.

Feedback for a wrong answer
Not yet. Check both: $p + q$ ⁠ must be $7$ ⁠, and $p \cdot q$ ⁠ must be $10$ ⁠.

Hint
The corner is $p \cdot q$ ⁠ and the middle is $p + q$ ⁠. Which two numbers make $10$ ⁠ when multiplied and $7$ ⁠ when added?

Worked solution
1. $(x + p)(x + q) = x^2 + qx + px + pq = x^2 + (p + q)x + pq$ ⁠.
2. You need $p \cdot q = 10$ ⁠ and also $p + q = 7$ ⁠.
3. Pairs whose product is $10$ ⁠: $1 \cdot 10$ ⁠ and $2 \cdot 5$ ⁠. Only $2$ ⁠ and $5$ ⁠ add up to $7$ ⁠.
4. $p = 2$ ⁠ and $q = 5$ ⁠ (or the other way round): $(x + 2)(x + 5) = x^2 + 7x + 10$ ⁠.
If you chose $1$ ⁠ and $6$ ⁠, the sum $7$ ⁠ is right, but the corner comes out $6$ ⁠, not $10$ ⁠.
Step 5 of 5
Open this step
The difference of two squares
Practice · Predict, then look
Question
The board has four parts: $x^2$ ⁠, $-3x$ ⁠, $3x$ ⁠ and $-9$ ⁠. The two middle ones cancel, leaving $x^2 - 9$ ⁠. Predict: what are the sides?

Correct answer
- Correct $(x + 3)(x - 3)$ ⁠
Setup, choices and feedback
The Scene at this step
x
6
p
1defaulthidden
q
1defaulthidden
Parts only / Sides named
Parts onlyfixed
Other settings
Left/nodes/board/props/left = "x + 3"
Other settings
Top/nodes/board/props/top = "x − 3"
Choices
- Correct $(x + 3)(x - 3)$ ⁠
- Wrong $(x - 3)^2$ ⁠
  Misconception: Squares each term on its own: (a + b)² = a² + b², (x − 3)² = x² − 9.
  Feedback from the catalogue
  $(a + b)^2$ ⁠ is a square with side $a + b$ ⁠, and it has four parts: the squares $a^2$ ⁠ and $b^2$ ⁠, and two rectangles $ab$ ⁠ that are easy to forget. So $(a + b)^2 = a^2 + 2ab + b^2$ ⁠. With $a = 5$ ⁠ and $b = 3$ ⁠: $8^2 = 64$ ⁠, not $25 + 9 = 34$ ⁠. The same goes for a difference: $(x - 3)^2 = x^2 - 6x + 9$ ⁠, not $x^2 - 9$ ⁠.
- Wrong $(x - 9)(x + 1)$ ⁠
  Feedback
  Expand to check: $x^2 + x - 9x - 9 = x^2 - 8x - 9$ ⁠. The middle parts do not cancel.
- Wrong $x^2 - 9$ ⁠ cannot be factorised
  Feedback
  It can: the corners are $x^2 = x \cdot x$ ⁠ and $-9 = 3 \cdot (-3)$ ⁠, and the middle parts cancel.
Once the prediction is locked in, the Scene changes
Parts only / Sides named
Sides namedfixed
A prediction is never marked as a failure: the learner sees the outcome and the explanation either way.
Feedback for a right answer
$(x + 3)(x - 3) = x^2 - 3x + 3x - 9 = x^2 - 9$ ⁠. When the sides differ only in a sign, the middle parts cancel. This is the difference of two squares: $a^2 - b^2 = (a + b)(a - b)$ ⁠.

Worked solution
1. $x^2 = x \cdot x$ ⁠, so each side has an x.
2. $-9$ ⁠ is the product of the numbers in the sides, and the x parts must cancel: the numbers are opposites, $3$ ⁠ and $-3$ ⁠.
3. $(x + 3)(x - 3) = x^2 - 3x + 3x - 9 = x^2 - 9$ ⁠.
If you chose $(x - 3)^2$ ⁠, expand it: $x^2 - 6x + 9$ ⁠. There the two middle parts add up instead of cancelling.

Across the unit · 13 misconceptions

Misconception catalogue

Each common mistake this unit recognises, in a few words. Open one for what the learner reads when an answer matches it, and where it is asked.

Index

Letters and like terms

A letter for a number

Question

Correct answer

The Scene at this step

Wrong answers it recognises

Hint

Worked solution

A variable varies

Question

Goal

The Scene at this step

One setting that reaches it

Hint

Worked solution

Like terms

Question

Correct answer

The Scene at this step

Choices

Once the prediction is locked in, the Scene changes

Worked solution

Unlike terms

Question

Correct answer

The Scene at this step

Choices

Worked solution

A minus in front of a term

Question

Correct answer

The Scene at this step

Wrong answers it recognises

Hint

Worked solution

Expanding brackets

Cutting a rectangle

Question

Correct answer

The Scene at this step

Once the prediction is locked in, the Scene changes

Hint

Worked solution

A number before brackets

Question

Correct answer

The Scene at this step

Choices

Once the prediction is locked in, the Scene changes

Worked solution

Checking with a number

Question

Correct answer

The Scene at this step

Wrong answers it recognises

Hint

Worked solution

A minus before brackets

Question

Correct answer

The Scene at this step

Choices

Once the prediction is locked in, the Scene changes

Worked solution

A minus inside the brackets

Question

Correct answer

The Scene at this step

Wrong answers it recognises

Hint

Worked solution

Multiplying two brackets

Four parts

Question

Correct answer

The Scene at this step

Choices

Once the prediction is locked in, the Scene changes

Worked solution

The middle term