Answer key · Unit 1

Numbers.

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 20 minutes

Negative numbers on the number line

Adding and subtracting as jumps along the number line, which way you go when you subtract from a negative number, what subtracting a negative does, and which negative number is greater.

Open the lesson

The 5 steps and their answers

Step 1 of 5
Open this step
A jump along the line
Practice · Predict, then look
Question
Start at $-2$ ⁠ and jump $5$ ⁠ steps to the right. Where do you land? In other words, what is $-2 + 5$ ⁠?

Correct answer
3 ± 0.01
A number prediction counts as right within ± 0.01
Setup, choices and feedback
The Scene at this step
Start
-2fixed
Operation
Add +fixed
Number
5fixed
The start / After the jump
The startfixed
Once the prediction is locked in, the Scene changes
The start / After the jump
After the jumpfixed
The Scene then shows: 3 · Line: where the last jump lands/nodes/board/outputs/landing
A prediction is never marked as a failure: the learner sees the outcome and the explanation either way.
Feedback for a right answer
$-2 + 5 = 3$ ⁠: two steps to zero and three more past it. The addition moved right, as it always does, and the minus belongs to the $2$ ⁠ alone.

Hint
Two steps from $-2$ ⁠ bring you to zero. How many steps are left?

Worked solution
1. Start at $-2$ ⁠, two steps left of zero.
2. Adding $5$ ⁠ is a jump of $5$ ⁠ steps to the right.
3. The first two steps reach zero: $-2 + 2 = 0$ ⁠.
4. $5 - 2 = 3$ ⁠ steps are left, so $-2 + 5 = 3$ ⁠.
If you got $-7$ ⁠, the minus was carried over the whole sum, as if it read $-(2 + 5)$ ⁠. It belongs to the $2$ ⁠ alone.
Step 2 of 5
Open this step
Subtracting from a negative number
Practice · Predict, then look
Question
In the morning it was $-3$ ⁠ degrees, and by the evening the temperature had dropped by $5$ ⁠ degrees. How cold was it in the evening? In other words, what is $-3 - 5$ ⁠?

Correct answer
- Correct $-8$ ⁠
Setup, choices and feedback
The Scene at this step
Start
-3fixed
Operation
Subtract −fixed
Number
5fixed
The start / After the jump
The startfixed
Choices
- Correct $-8$ ⁠
- Wrong $-2$ ⁠
  Misconception: Thinks −3 − 5 = −2: subtracts the smaller digit from the larger, moving toward zero.
  Feedback from the catalogue
  Subtracting is always a jump to the left, even when you start at a negative number. From $-3$ ⁠ you jump $5$ ⁠ steps left and land on $-8$ ⁠, further from zero. Think of a temperature: $3$ ⁠ degrees below zero, then it drops another $5$ ⁠ degrees.
- Wrong $8$ ⁠
  Feedback
  A drop in temperature cannot take you from $-3$ ⁠ to above zero: subtracting moves left, further below zero.
- Wrong $2$ ⁠
  Feedback
  $2$ ⁠ is $-3 + 5$ ⁠, a jump to the right. Here you subtract, so the jump goes left.
Once the prediction is locked in, the Scene changes
The start / After the jump
After the jumpfixed
The Scene then shows: -8 · Line: where the last jump lands/nodes/board/outputs/landing
A prediction is never marked as a failure: the learner sees the outcome and the explanation either way.
Feedback for a right answer
From $-3$ ⁠ you jump $5$ ⁠ steps left and land on $-8$ ⁠. Notice that the answer is further from zero than the start: subtracting from a negative number makes it more negative still.

Worked solution
1. Start at $-3$ ⁠.
2. Subtracting $5$ ⁠ is a jump of $5$ ⁠ steps to the left.
3. To the left of $-3$ ⁠ the numbers only move further from zero: $-4, -5, -6, -7, -8$ ⁠.
4. So $-3 - 5 = -8$ ⁠: in the evening it was $8$ ⁠ degrees below zero.
If you chose $-2$ ⁠, you worked out $5 - 3$ ⁠ and kept the minus. But the temperature dropped, so it cannot have moved closer to zero.
Step 3 of 5
Open this step
Subtracting a negative number
Practice · Calculate
Question
What is $-3 - (-5)$ ⁠?

Correct answer
2 ± 0.01
Setup, choices and feedback
The Scene at this step
Start
-3fixed
Operation
Subtract −fixed
Number
-5fixed
The start / After the jump
The start
Accepted within
± 0.01
Computed from the Scene at this step’s setting
Line: where the last jump lands/nodes/board/outputs/landing
Wrong answers it recognises
- -8 ± 0.01
  Misconception: Thinks subtracting a negative number moves left, like subtracting a positive one: −3 − (−5) = −8.
  Feedback from the catalogue
  Subtracting a negative number undoes adding it. Adding $-5$ ⁠ moves $5$ ⁠ to the left, so subtracting $-5$ ⁠ moves $5$ ⁠ to the right: $-3 - (-5) = -3 + 5 = 2$ ⁠. Two minus signs side by side become a plus.
Feedback for a right answer
$-3 - (-5) = -3 + 5 = 2$ ⁠. Subtracting a negative number moves right, like adding the positive number. Drag Look inside to see the arc jump to the right.

Feedback for a wrong answer
Think of subtracting as the opposite of adding: where would adding $-5$ ⁠ take you, and where does the opposite move go?

Hint
Adding $-5$ ⁠ would move $5$ ⁠ to the left. Subtracting is the opposite move.

Worked solution
1. Two minus signs side by side, subtracting a negative number, become a plus: $-(-5) = +5$ ⁠.
2. The expression becomes $-3 + 5$ ⁠.
3. From $-3$ ⁠ jump $5$ ⁠ steps to the right: three to zero and two more.
4. So $-3 - (-5) = 2$ ⁠.
If you got $-8$ ⁠, you jumped left as if subtracting $5$ ⁠. Subtracting $-5$ ⁠ undoes adding $-5$ ⁠, so it moves right.
Step 4 of 5
Open this step
Adding a negative number
Practice · Reach the target
Question
Operation is locked to Add and the start to $4$ ⁠. Choose Number so that you land on the ring at $-2$ ⁠, then press Check.

Goal
- -2 ± 0.01
  At the start of the step: 4
Setup, choices and feedback
The Scene at this step
Start
4fixed
Operation
Add +fixed
Number
0
The start / After the jump
After the jump
Other settings
Target/nodes/board/props/target = -2
Read from the Scene at
Line: where the last jump lands/nodes/board/outputs/landing
One setting that reaches it
Number
-6
Feedback for a right answer
$4 + (-6) = -2$ ⁠. Adding a negative number moves left, exactly like subtracting the positive number: $4 + (-6) = 4 - 6$ ⁠.

Feedback for a wrong answer
Not there yet. A positive number moves right, so here you need a negative one.

Hint
You need to move $6$ ⁠ steps to the left, using only addition. Which number moves you left when you add it?

Worked solution
1. You start at $4$ ⁠ and need to reach $-2$ ⁠: a move of $-2 - 4 = -6$ ⁠, that is $6$ ⁠ steps to the left.
2. Operation is locked to Add, so you need a number that moves you $6$ ⁠ to the left when added: $-6$ ⁠.
3. Set Number to $-6$ ⁠: $4 + (-6) = 4 - 6 = -2$ ⁠.
A positive number such as $6$ ⁠ would move you right, to $10$ ⁠.
Step 5 of 5
Open this step
Which is greater?
Practice · Choose
Question
The line marks $-7$ ⁠ and $-2$ ⁠. Which of them is greater?

Correct answer
- Correct $-2$ ⁠
Setup, choices and feedback
The Scene at this step
Start
0hidden
Operation
No jumphidden
Number
0hidden
The start / After the jump
After the jump
Other settings
Marks/nodes/board/props/marks = [-7,-2]
Choices
Shown in a random order.
- Correct $-2$ ⁠
- Wrong $-7$ ⁠, because $7$ ⁠ is greater than $2$ ⁠
  Misconception: Thinks −7 is greater than −2 because 7 is greater than 2.
  Feedback from the catalogue
  On the number line a greater number sits further to the right. $-7$ ⁠ is to the left of $-2$ ⁠, so it is smaller: $-7 < -2$ ⁠. Just as with temperatures: $-7$ ⁠ degrees is colder than $-2$ ⁠ degrees.
- Wrong They are equal, because both are negative
  Feedback
  Two different numbers sit at different places on the line, and one of them is always further right, which means greater.
Feedback for a right answer
$-2$ ⁠ is to the right of $-7$ ⁠, so $-2 > -7$ ⁠. Among negative numbers, the one closer to zero is the greater one.

Worked solution
1. On the number line a greater number sits further to the right.
2. $-7$ ⁠ is $7$ ⁠ steps left of zero, and $-2$ ⁠ only $2$ ⁠ steps left of it.
3. $-2$ ⁠ is to the right of $-7$ ⁠, so $-2 > -7$ ⁠.
If you chose $-7$ ⁠, you compared $7$ ⁠ and $2$ ⁠ without the sign. On the negative side it works the other way: the further from zero, the smaller.

Lesson 2 of 4 · About 20 minutes

Fractions and decimals

What a numerator and a denominator say, how a fraction becomes a decimal, why fractions are added only over a common denominator, and how to compare decimals.

Open the lesson

The 5 steps and their answers

Step 1 of 5
Open this step
Parts of a whole
Practice · Calculate
Question
The bar shows $\frac{3}{4}$ ⁠. Write this fraction as a decimal.

Correct answer
0.75 ± 0.001
Setup, choices and feedback
The Scene at this step
Numerator
3fixed
Denominator
4fixed
Second numerator
1defaulthidden
As cut / Common denominator
As cut
Accepted within
± 0.001
Computed from the Scene at this step’s setting
Bars: n1/d1/nodes/board/outputs/value1
Wrong answers it recognises
- 3.4 ± 0.001
  Misconception: Writes 3/4 as 3.4 or 0.34, copying the digits instead of dividing.
  Feedback from the catalogue
  The fraction bar means divide: $\frac{3}{4} = 3 \div 4$ ⁠. A quarter is $0.25$ ⁠, and three quarters are $0.75$ ⁠. $3.4$ ⁠ is more than $3$ ⁠ wholes, far more than the whole bar.
- 0.34 ± 0.001
  Misconception: Writes 3/4 as 3.4 or 0.34, copying the digits instead of dividing.
  Feedback from the catalogue
  The fraction bar means divide: $\frac{3}{4} = 3 \div 4$ ⁠. A quarter is $0.25$ ⁠, and three quarters are $0.75$ ⁠. $3.4$ ⁠ is more than $3$ ⁠ wholes, far more than the whole bar.
Feedback for a right answer
A quarter is $0.25$ ⁠, so three quarters are $0.75$ ⁠, that is $75$ ⁠ hundredths, or $75\%$ ⁠. Drag Look inside to compare.

Hint
$\frac{3}{4} = 3 \div 4$ ⁠. What is one quarter as a decimal?

Worked solution
1. The fraction bar is a division: $\frac{3}{4} = 3 \div 4$ ⁠.
2. An easy way: scale it up to hundredths. $4 \times 25 = 100$ ⁠, so $\frac{3}{4} = \frac{3 \times 25}{4 \times 25} = \frac{75}{100}$ ⁠.
3. $75$ ⁠ hundredths are written $0.75$ ⁠.
If you wrote $3.4$ ⁠ or $0.34$ ⁠, you copied the digits of the numerator and the denominator. But $\frac{3}{4}$ ⁠ is less than one whole, and $0.34$ ⁠ is only about a third of the bar.
Step 2 of 5
Open this step
Same part, different names
Practice · Reach the target
Question
The top bar shows $\frac{1}{2}$ ⁠, and the bottom one is cut into sixths. Choose Second numerator so that both bars are shaded to the same length, then press Check.

Goal
- 0.5 ± 0.001
  At the start of the step: 0.1667
Setup, choices and feedback
The Scene at this step
Numerator
1fixed
Denominator
2fixed
Second numerator
1
As cut / Common denominator
As cut
Other settings
D2/nodes/board/props/d2 = 6
Other settings
Bars/nodes/board/props/bars = "two"
Read from the Scene at
Bars: n2/d2/nodes/board/outputs/value2
One setting that reaches it
Second numerator
3
Feedback for a right answer
$\frac{3}{6} = \frac{1}{2}$ ⁠: three sixths are exactly a half. Multiply the numerator and the denominator by the same number, here $3$ ⁠, and the fraction stays the same: $\frac{1 \cdot 3}{2 \cdot 3} = \frac{3}{6}$ ⁠.

Feedback for a wrong answer
Not yet. Compare where the shading ends on the two bars.

Hint
How many sixths fit into half a bar?

Worked solution
1. The denominator goes from $2$ ⁠ to $6$ ⁠, that is, it is multiplied by $3$ ⁠: $2 \times 3 = 6$ ⁠.
2. For the fraction to stay the same, multiply the numerator by $3$ ⁠ too: $1 \times 3 = 3$ ⁠.
3. Set Second numerator to $3$ ⁠: $\frac{3}{6} = \frac{1}{2}$ ⁠, and both bars are shaded to the same point.
Step 3 of 5
Open this step
Adding fractions
Practice · Predict, then look
Question
What is $\frac{1}{2} + \frac{1}{3}$ ⁠?

Correct answer
- Correct $\frac{5}{6}$ ⁠
Setup, choices and feedback
The Scene at this step
Numerator
1fixed
Denominator
2fixed
Second numerator
1fixed
As cut / Common denominator
As cutfixed
Other settings
D2/nodes/board/props/d2 = 3
Other settings
Bars/nodes/board/props/bars = "sum"
Choices
- Correct $\frac{5}{6}$ ⁠
- Wrong $\frac{2}{5}$ ⁠
  Misconception: Adds fractions by adding the numerators and the denominators: 1/2 + 1/3 = 2/5.
  Feedback from the catalogue
  A denominator says how big each piece is, so denominators cannot be added. $\frac{2}{5}$ ⁠ is less than $\frac{1}{2}$ ⁠ on its own. First cut both bars into pieces of the same size, sixths: $\frac{1}{2} = \frac{3}{6}$ ⁠ and $\frac{1}{3} = \frac{2}{6}$ ⁠, and only then count: $\frac{5}{6}$ ⁠.
- Wrong $\frac{2}{6}$ ⁠
  Feedback
  The denominator $6$ ⁠ is right, but the numerators change too: a half is three sixths and a third is two.
- Wrong $\frac{1}{6}$ ⁠
  Feedback
  $\frac{1}{6}$ ⁠ is the difference $\frac{1}{2} - \frac{1}{3}$ ⁠. A sum of two parts is bigger than either of them.
Once the prediction is locked in, the Scene changes
As cut / Common denominator
Common denominatorfixed
A prediction is never marked as a failure: the learner sees the outcome and the explanation either way.
Feedback for a right answer
Cut both bars into sixths, the smallest piece both can be split into: $\frac{1}{2} = \frac{3}{6}$ ⁠ and $\frac{1}{3} = \frac{2}{6}$ ⁠. Now all the pieces are the same size, and you count: $\frac{3}{6} + \frac{2}{6} = \frac{5}{6}$ ⁠.

Worked solution
1. The denominators $2$ ⁠ and $3$ ⁠ differ, so first find a common denominator: the smallest number both divide into is $6$ ⁠.
2. Rewrite: $\frac{1}{2} = \frac{3}{6}$ ⁠ and $\frac{1}{3} = \frac{2}{6}$ ⁠.
3. Add the numerators, and the denominator stays: $\frac{3}{6} + \frac{2}{6} = \frac{5}{6}$ ⁠.
If you chose $\frac{2}{5}$ ⁠, you added the denominators too. But $\frac{2}{5}$ ⁠ is less than $\frac{1}{2}$ ⁠, and a sum cannot be less than one of its parts.
Step 4 of 5
Open this step
A common denominator
Practice · Calculate
Question
$\frac{1}{4} + \frac{1}{6} = \frac{?}{12}$ ⁠. What is the numerator?

Correct answer
5 ± 0.01
Setup, choices and feedback
The Scene at this step
Numerator
1fixed
Denominator
4fixed
Second numerator
1fixed
As cut / Common denominator
As cut
Other settings
D2/nodes/board/props/d2 = 6
Other settings
Bars/nodes/board/props/bars = "sum"
Accepted within
± 0.01
Computed from the Scene at this step’s setting
Bars: the sum's numerator over the common denominator/nodes/board/outputs/sumNumerator
Wrong answers it recognises
- 2 ± 0.01
  Misconception: Finds the common denominator but keeps the old numerators: 1/4 + 1/6 = 2/12.
  Feedback from the catalogue
  When the denominator changes, the numerator changes with it: one quarter is three twelfths, because $\frac{1}{4} = \frac{1 \cdot 3}{4 \cdot 3} = \frac{3}{12}$ ⁠. In the same way $\frac{1}{6} = \frac{2}{12}$ ⁠, and the sum is $\frac{5}{12}$ ⁠.
Feedback for a right answer
$\frac{1}{4} = \frac{3}{12}$ ⁠ and $\frac{1}{6} = \frac{2}{12}$ ⁠, so the sum is $\frac{5}{12}$ ⁠. $12$ ⁠ is the smallest number both denominators divide into. Drag Look inside to check.

Hint
How many twelfths are there in a quarter? And in a sixth?

Worked solution
1. The common denominator is $12$ ⁠: $4 \times 3 = 12$ ⁠ and $6 \times 2 = 12$ ⁠.
2. Scale each fraction by the same number as its denominator: $\frac{1}{4} = \frac{1 \times 3}{12} = \frac{3}{12}$ ⁠ and $\frac{1}{6} = \frac{1 \times 2}{12} = \frac{2}{12}$ ⁠.
3. Add: $\frac{3}{12} + \frac{2}{12} = \frac{5}{12}$ ⁠. The numerator is $5$ ⁠.
If you got $2$ ⁠, you changed the denominators to $12$ ⁠ but kept the numerators $1$ ⁠ and $1$ ⁠.
Step 5 of 5
Open this step
Comparing decimals
Practice · Choose
Question
Which number is greater: $0.5$ ⁠ or $0.25$ ⁠?

Correct answer
- Correct $0.5$ ⁠
Setup, choices and feedback
The Scene at this step
Numerator
5fixed
Denominator
10fixed
Second numerator
25fixed
As cut / Common denominator
As cut
Other settings
D2/nodes/board/props/d2 = 100
Other settings
Bars/nodes/board/props/bars = "two"
Choices
Shown in a random order.
- Correct $0.5$ ⁠
- Wrong $0.25$ ⁠, because $25$ ⁠ is greater than $5$ ⁠
  Misconception: Thinks 0.25 > 0.5 because 25 > 5, reading the digits after the point as a whole number.
  Feedback from the catalogue
  The digits after the point are parts, not a whole number: the first digit counts tenths, the second hundredths. $0.5$ ⁠ is five tenths, that is $0.50$ ⁠, fifty hundredths, and $0.25$ ⁠ is only twenty-five hundredths. Compare digit by digit from the point onward: $5 > 2$ ⁠.
- Wrong They are equal
  Feedback
  Look at the bars: the shading on the top one is twice as long as on the bottom one.
Feedback for a right answer
$0.5 = \frac{5}{10} = \frac{50}{100}$ ⁠, and $0.25 = \frac{25}{100}$ ⁠. In hundredths you see at once that $50 > 25$ ⁠. You can also compare digit by digit from the point: in the tenths $5 > 2$ ⁠, and that settles it.

Worked solution
1. Give both numbers the same number of digits after the point: $0.5 = 0.50$ ⁠.
2. Now both are in hundredths: $0.50 = \frac{50}{100}$ ⁠ and $0.25 = \frac{25}{100}$ ⁠.
3. $50 > 25$ ⁠, so $0.5 > 0.25$ ⁠.
If you chose $0.25$ ⁠, you read the digits after the point as a whole number, $25$ ⁠ against $5$ ⁠. But the $5$ ⁠ in $0.5$ ⁠ counts tenths, and so does the $2$ ⁠ in $0.25$ ⁠: $5 > 2$ ⁠.

Lesson 3 of 4 · About 20 minutes

Percentages

A percentage is a part per hundred: how to find a percentage of an amount, why a discount is a multiplication, and why a fifty per cent rise and a fifty per cent fall do not bring you back to the start.

Open the lesson

The 5 steps and their answers

Step 1 of 5
Open this step
Out of a hundred
Practice · Reach the target
Question
Use Shaded squares to shade three quarters of the grid, then press Check.

Goal
- 0.75 ± 0.001
  At the start of the step: 0
Setup, choices and feedback
The Scene at this step
Shaded squares
0
First change
50 %defaulthidden
Second change
-50 %defaulthidden
Before / Worked through
Worked through
Read from the Scene at
Grid: shaded ÷ cells, as a decimal/nodes/board/outputs/shadedFraction
One setting that reaches it
Shaded squares
75
Feedback for a right answer
$\frac{3}{4} = \frac{75}{100} = 75\%$ ⁠. Per cent means “out of a hundred”, so on a grid of $100$ ⁠ squares each square is one per cent.

Feedback for a wrong answer
Not yet. A quarter is $25$ ⁠ squares, and three quarters are three times that.

Hint
The grid has $100$ ⁠ squares. How many are a quarter of them?

Worked solution
1. The grid has $100$ ⁠ squares, so each square is $1\%$ ⁠.
2. A quarter of $100$ ⁠ is $100 \div 4 = 25$ ⁠ squares.
3. Three quarters are $3 \times 25 = 75$ ⁠ squares.
4. Set Shaded squares to $75$ ⁠: $\frac{3}{4} = \frac{75}{100} = 0.75 = 75\%$ ⁠.
Step 2 of 5
Open this step
A percentage of an amount
Practice · Calculate
Question
The whole grid stands for $80$ ⁠ shekels, and $30$ ⁠ squares are shaded. What is $30\%$ ⁠ of $80$ ⁠?

Correct answer
24 ± 0.01
Setup, choices and feedback
The Scene at this step
Shaded squares
30fixed
First change
50 %defaulthidden
Second change
-50 %defaulthidden
Before / Worked through
Before
Other settings
Whole/nodes/board/props/whole = 80
Accepted within
± 0.01
Computed from the Scene at this step’s setting
Grid: the shaded part of `whole` (NaN without one)/nodes/board/outputs/amount
Wrong answers it recognises
- 50 ± 0.01
  Misconception: Treats a percentage as a number of units: 30% of 80 is 80 − 30, or 20% off 200 is 180.
  Feedback from the catalogue
  A percentage is a part per hundred of the amount, not a number of units. $1\%$ ⁠ of $80$ ⁠ is $0.8$ ⁠, so $30\%$ ⁠ is $30 \cdot 0.8 = 24$ ⁠. In the same way $20\%$ ⁠ of $200$ ⁠ is $40$ ⁠, not $20$ ⁠.
Feedback for a right answer
$1\%$ ⁠ of $80$ ⁠ is $0.8$ ⁠, so $30\% = 30 \cdot 0.8 = 24$ ⁠. The same as a decimal: $0.3 \cdot 80 = 24$ ⁠.

Hint
Each square is $1\%$ ⁠, that is $\frac{80}{100} = 0.8$ ⁠ shekels. What are $30$ ⁠ squares worth?

Worked solution
1. One per cent of $80$ ⁠ is $80 \div 100 = 0.8$ ⁠.
2. $30\%$ ⁠ is $30$ ⁠ times that: $30 \times 0.8 = 24$ ⁠.
3. In one step: $30\% = 0.3$ ⁠, and $0.3 \times 80 = 24$ ⁠.
If you got $50$ ⁠, you worked out $80 - 30$ ⁠, as if $30\%$ ⁠ were $30$ ⁠ shekels. A percentage is a part of the amount, not a number of units.
Step 3 of 5
Open this step
A rise, then a fall
Practice · Predict, then look
Question
A price of $100$ ⁠ shekels rose by $50\%$ ⁠, then fell by $50\%$ ⁠. What is the price at the end?

Correct answer
- Correct $75$ ⁠
Setup, choices and feedback
The Scene at this step
Shaded squares
37defaulthidden
First change
50 %fixed
Second change
-50 %fixed
Before / Worked through
Beforefixed
Other settings
Mode/nodes/board/props/mode = "change"
Other settings
Base/nodes/board/props/base = 100
Choices
- Correct $75$ ⁠
- Wrong $100$ ⁠: the rise and the fall cancel out
  Misconception: Thinks a 50% rise followed by a 50% fall returns to the starting amount.
  Feedback from the catalogue
  Each percentage is worked out from the amount there at that moment. The rise is half of $100$ ⁠, but the fall is half of $150$ ⁠, which is more: $100 \cdot 1.5 \cdot 0.5 = 75$ ⁠. A percentage change is a multiplication, and $1.5 \cdot 0.5 = 0.75$ ⁠, not $1$ ⁠.
- Wrong $50$ ⁠
  Feedback
  $50$ ⁠ would be half of $100$ ⁠. But before the fall the price is already $150$ ⁠.
- Wrong $125$ ⁠
  Feedback
  The fall is bigger than the rise, not smaller: half of $150$ ⁠ is $75$ ⁠ shekels, while half of $100$ ⁠ is only $50$ ⁠.
Once the prediction is locked in, the Scene changes
Before / Worked through
Worked throughfixed
The Scene then shows: 75 · Change: the amount after every change/nodes/board/outputs/final
A prediction is never marked as a failure: the learner sees the outcome and the explanation either way.
Feedback for a right answer
The rise is half of $100$ ⁠: the price becomes $150$ ⁠. The fall is half of $150$ ⁠, which is $75$ ⁠ shekels, so $75$ ⁠ is left. Every percentage change is a multiplication: $100 \cdot 1.5 \cdot 0.5 = 75$ ⁠, an overall fall of $25\%$ ⁠.

Worked solution
1. A $50\%$ ⁠ rise from $100$ ⁠: $50\%$ ⁠ of $100$ ⁠ is $50$ ⁠, and the price becomes $100 + 50 = 150$ ⁠.
2. A $50\%$ ⁠ fall from $150$ ⁠: $50\%$ ⁠ of $150$ ⁠ is $75$ ⁠, and the price becomes $150 - 75 = 75$ ⁠.
3. As multipliers: $100 \times 1.5 \times 0.5 = 100 \times 0.75 = 75$ ⁠, an overall fall of $25\%$ ⁠.
If you chose $100$ ⁠, you assumed the two $50\%$ ⁠ cancel out. But the fall is worked out from $150$ ⁠, so it is bigger than the rise.
Step 4 of 5
Open this step
A discount is a multiplication
Practice · Calculate
Question
A coat that costs $200$ ⁠ shekels is reduced by $20\%$ ⁠. What does it cost after the discount?

Correct answer
160 ± 0.01
Setup, choices and feedback
The Scene at this step
Shaded squares
37defaulthidden
First change
-20 %fixed
Second change
-50 %defaulthidden
Before / Worked through
Before
Other settings
Mode/nodes/board/props/mode = "change"
Other settings
Base/nodes/board/props/base = 200
Other settings
Changes/nodes/board/props/changes = 1
Accepted within
± 0.01
Computed from the Scene at this step’s setting
Change: the amount after every change/nodes/board/outputs/final
Wrong answers it recognises
- 180 ± 0.01
  Misconception: Treats a percentage as a number of units: 30% of 80 is 80 − 30, or 20% off 200 is 180.
  Feedback from the catalogue
  A percentage is a part per hundred of the amount, not a number of units. $1\%$ ⁠ of $80$ ⁠ is $0.8$ ⁠, so $30\%$ ⁠ is $30 \cdot 0.8 = 24$ ⁠. In the same way $20\%$ ⁠ of $200$ ⁠ is $40$ ⁠, not $20$ ⁠.
Feedback for a right answer
$20\%$ ⁠ of $200$ ⁠ is $40$ ⁠ shekels, and $200 - 40 = 160$ ⁠. In one step: $200 \cdot 0.8 = 160$ ⁠. Drag Look inside to watch the bar shrink.

Hint
A $20\%$ ⁠ discount leaves $80\%$ ⁠ of the price: multiply by $0.8$ ⁠.

Worked solution
1. A $20\%$ ⁠ discount leaves $100\% - 20\% = 80\%$ ⁠ of the price.
2. $80\% = 0.8$ ⁠, so the price after the discount is $200 \times 0.8 = 160$ ⁠ shekels.
3. Check: the discount itself is $0.2 \times 200 = 40$ ⁠ shekels, and $200 - 40 = 160$ ⁠.
If you got $180$ ⁠, you took off $20$ ⁠ shekels instead of $20\%$ ⁠ of $200$ ⁠.
Step 5 of 5
Open this step
Undoing a rise
Practice · Reach the target
Question
The price rose by $25\%$ ⁠, from $100$ ⁠ to $125$ ⁠. Choose a Second change that brings it back to exactly $100$ ⁠, then press Check.

Goal
- 100 ± 0.01
  At the start of the step: 125
Setup, choices and feedback
The Scene at this step
Shaded squares
37defaulthidden
First change
25 %fixed
Second change
0 %
Before / Worked through
Worked through
Other settings
Mode/nodes/board/props/mode = "change"
Other settings
Base/nodes/board/props/base = 100
Read from the Scene at
Change: the amount after every change/nodes/board/outputs/final
One setting that reaches it
Second change
-20 %
Feedback for a right answer
$-20\%$ ⁠: the fall is worked out from $125$ ⁠, and $20\%$ ⁠ of $125$ ⁠ is exactly $25$ ⁠. As multipliers: $1.25 \cdot 0.8 = 1$ ⁠. To undo a $25\%$ ⁠ rise, a $20\%$ ⁠ fall is enough.

Feedback for a wrong answer
Not $100$ ⁠ yet. Remember that the second change is worked out from $125$ ⁠, not from $100$ ⁠.

Hint
You need to take $25$ ⁠ shekels off $125$ ⁠. What part of $125$ ⁠ is $25$ ⁠?

Worked solution
1. After the rise the price is $125$ ⁠, and you need to get back to $100$ ⁠: take off $25$ ⁠ shekels.
2. The second change is worked out from $125$ ⁠: $\frac{25}{125} = \frac{1}{5} = 0.2 = 20\%$ ⁠.
3. Set Second change to $-20\%$ ⁠. Check: $125 \times 0.8 = 100$ ⁠.
A $25\%$ ⁠ fall from $125$ ⁠ would take off $31.25$ ⁠ shekels, down to $93.75$ ⁠.

Lesson 4 of 4 · About 25 minutes

Order of operations, powers and roots

The order in which an expression is worked out: brackets, powers and roots, multiplication and division, and last addition and subtraction. What a square and a square root are, and why minus three squared is minus nine.

Open the lesson

The 6 steps and their answers

Step 1 of 6
Open this step
What comes first?
Practice · Predict, then look
Question
What is $2 + 3 \times 4$ ⁠?

Correct answer
- Correct $14$ ⁠
Setup, choices and feedback
The Scene at this step
a
2fixed
b
3fixed
c
4fixed
As written / Worked out
As writtenfixed
Choices
- Correct $14$ ⁠
- Wrong $20$ ⁠
  Misconception: Works an expression strictly left to right, ignoring that × and ÷ come before + and −.
  Feedback from the catalogue
  Multiplication and division come before addition and subtraction, even when they are written after them. In $2 + 3 \times 4$ ⁠ you work out $3 \times 4 = 12$ ⁠ first, then $2 + 12 = 14$ ⁠. To add first, you write brackets: $(2 + 3) \times 4 = 20$ ⁠.
- Wrong $24$ ⁠
  Feedback
  $24$ ⁠ is $2 \times 3 \times 4$ ⁠. Here there is one plus and one times.
Once the prediction is locked in, the Scene changes
As written / Worked out
Worked outfixed
The Scene then shows: 14 · Expression: its value in the order of operations (NaN when it cannot be read)/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
Multiplication comes before addition: $3 \times 4 = 12$ ⁠, then $2 + 12 = 14$ ⁠. The convention lets you write a sum of products without brackets, such as the price of $3$ ⁠ notebooks at $4$ ⁠ shekels plus a pen at $2$ ⁠.

Worked solution
1. The expression $2 + 3 \times 4$ ⁠ has an addition and a multiplication. Multiplication comes first.
2. $3 \times 4 = 12$ ⁠, and the expression becomes $2 + 12$ ⁠.
3. $2 + 12 = 14$ ⁠.
If you chose $20$ ⁠, you worked left to right: $2 + 3 = 5$ ⁠, then $5 \times 4$ ⁠. That is the answer to $(2 + 3) \times 4$ ⁠, with brackets.
Step 2 of 6
Open this step
Brackets
Practice · Calculate
Question
What is $(2 + 3) \times 4$ ⁠?

Correct answer
20 ± 0.01
Setup, choices and feedback
The Scene at this step
a
2fixed
b
3fixed
c
4fixed
As written / Worked out
As written
Other settings
Expression/nodes/board/props/expression = "(a + b) × c"
Accepted within
± 0.01
Computed from the Scene at this step’s setting
Expression: its value in the order of operations (NaN when it cannot be read)/nodes/board/outputs/value
Wrong answers it recognises
- 14 ± 0.01
  Misconception: Ignores brackets and applies the usual precedence: (2 + 3) × 4 = 14.
  Feedback from the catalogue
  Brackets come before everything: what is inside them is worked out first, which is how they change the usual order. $(2 + 3) \times 4 = 5 \times 4 = 20$ ⁠. Without the brackets it would come out as $14$ ⁠, which is why they are written.
Feedback for a right answer
Brackets first: $2 + 3 = 5$ ⁠, then $5 \times 4 = 20$ ⁠. Drag Look inside to see the lines of working.

Worked solution
1. Brackets come first: $2 + 3 = 5$ ⁠.
2. The expression becomes $5 \times 4$ ⁠.
3. $5 \times 4 = 20$ ⁠.
If you got $14$ ⁠, you multiplied $3 \times 4$ ⁠ first, as if there were no brackets. The brackets are written exactly so that the addition is done first.
Step 3 of 6
Open this step
Powers
Practice · Calculate
Question
What is $2 + 3 \times 4^2$ ⁠?

Correct answer
50 ± 0.01
Setup, choices and feedback
The Scene at this step
a
2fixed
b
3fixed
c
4fixed
As written / Worked out
As written
Other settings
Expression/nodes/board/props/expression = "a + b × c²"
Accepted within
± 0.01
Computed from the Scene at this step’s setting
Expression: its value in the order of operations (NaN when it cannot be read)/nodes/board/outputs/value
Wrong answers it recognises
- 80 ± 0.01
  Misconception: Works an expression strictly left to right, ignoring that × and ÷ come before + and −.
  Feedback
  $80$ ⁠ is what you get working left to right: $(2 + 3) \times 16$ ⁠. But multiplication comes before addition, even after the power has been worked out: $3 \times 16 = 48$ ⁠, and only then $2 + 48 = 50$ ⁠.
- 146 ± 0.01
  Misconception: Applies a power to the whole product before it: 3 × 4² read as (3 × 4)².
  Feedback from the catalogue
  A power applies only to the number right under it: in $3 \times 4^2$ ⁠ only the $4$ ⁠ is squared. $4^2 = 16$ ⁠, then $3 \times 16 = 48$ ⁠. To square the whole product you write $(3 \times 4)^2 = 144$ ⁠.
Feedback for a right answer
The power first: $4^2 = 16$ ⁠. Then the multiplication: $3 \times 16 = 48$ ⁠. Last the addition: $2 + 48 = 50$ ⁠.

Hint
The order: powers, then multiplication, and addition last.

Worked solution
1. There are no brackets, so start with the power: $4^2 = 4 \times 4 = 16$ ⁠. The expression becomes $2 + 3 \times 16$ ⁠.
2. Then the multiplication: $3 \times 16 = 48$ ⁠. The expression becomes $2 + 48$ ⁠.
3. Last the addition: $2 + 48 = 50$ ⁠.
If you got $80$ ⁠, you added $2 + 3$ ⁠ before multiplying. If you got $146$ ⁠, you squared the whole of $3 \times 4$ ⁠.
Step 4 of 6
Open this step
Squares
Practice · Calculate
Question
The board shows a square with sides of $6$ ⁠ cells. How many cells does it have? In other words, what is $6^2$ ⁠?

Correct answer
36 ± 0.01
Setup, choices and feedback
The Scene at this step
a
2defaulthidden
b
3defaulthidden
c
4defaulthidden
As written / Worked out
As written
Other settings
Mode/nodes/board/props/mode = "grid"
Other settings
Show/nodes/board/props/show = "area"
Other settings
Rows/nodes/board/props/rows = 6
Other settings
Cols/nodes/board/props/cols = 6
Accepted within
± 0.01
Computed from the Scene at this step’s setting
Grid: rows × cols/nodes/board/outputs/cells
Wrong answers it recognises
- 12 ± 0.01
  Misconception: Reads 6² as 6 × 2 = 12 instead of 6 × 6 = 36.
  Feedback from the catalogue
  The power $6^2$ ⁠ is $6 \times 6$ ⁠, a square of side $6$ ⁠ with $36$ ⁠ cells. $6 \times 2 = 12$ ⁠ is only two rows of six. The exponent says how many times the number is multiplied by itself.
Feedback for a right answer
$6^2 = 6 \times 6 = 36$ ⁠: six rows of six cells. And the other way round, the root $\sqrt{36} = 6$ ⁠ is the side of a square of area $36$ ⁠. Drag Look inside to watch the cells being counted.

Worked solution
1. $6^2$ ⁠ means $6$ ⁠ times itself: $6 \times 6$ ⁠.
2. A square of side $6$ ⁠ has $6$ ⁠ rows of $6$ ⁠ cells.
3. $6 \times 6 = 36$ ⁠.
If you got $12$ ⁠, you worked out $6 \times 2$ ⁠. The exponent $2$ ⁠ says how many times $6$ ⁠ is multiplied by itself, not what to multiply it by.
Step 5 of 6
Open this step
Roots
Practice · Calculate
Question
What is $\sqrt{9 + 16}$ ⁠?

Correct answer
5 ± 0.01
Setup, choices and feedback
The Scene at this step
a
9fixed
b
16fixed
c
4defaulthidden
As written / Worked out
As written
Other settings
Expression/nodes/board/props/expression = "√(a + b)"
Accepted within
± 0.01
Computed from the Scene at this step’s setting
Expression: its value in the order of operations (NaN when it cannot be read)/nodes/board/outputs/value
Wrong answers it recognises
- 7 ± 0.01
  Misconception: Takes the root of each term of a sum: √(9 + 16) = √9 + √16 = 7.
  Feedback from the catalogue
  The root of a sum is not the sum of the roots. Brackets first: $9 + 16 = 25$ ⁠, then $\sqrt{25} = 5$ ⁠. Check: $7^2 = 49$ ⁠, not $25$ ⁠. A $3 \times 3$ ⁠ square and a $4 \times 4$ ⁠ square together make $25$ ⁠ cells, and one square of $25$ ⁠ cells has side $5$ ⁠, not $7$ ⁠.
Feedback for a right answer
$9 + 16 = 25$ ⁠, and $\sqrt{25} = 5$ ⁠, because $5^2 = 25$ ⁠. Notice that $\sqrt{9} + \sqrt{16} = 7$ ⁠, which is a different number.

Hint
The root sign works like brackets: first work out what is under it.

Worked solution
1. The root sign works like brackets, so first work out what is under it: $9 + 16 = 25$ ⁠.
2. Now $\sqrt{25}$ ⁠: which number squared gives $25$ ⁠? $5 \times 5 = 25$ ⁠.
3. So $\sqrt{9 + 16} = 5$ ⁠.
If you got $7$ ⁠, you took the root of each term separately: $\sqrt{9} + \sqrt{16} = 3 + 4$ ⁠. Check: $7^2 = 49$ ⁠, not $25$ ⁠.
Step 6 of 6
Open this step
Minus, squared
Practice · Predict, then look
Question
What is $-3^2$ ⁠?

Correct answer
- Correct $-9$ ⁠
Setup, choices and feedback
The Scene at this step
a
3fixed
b
3defaulthidden
c
4defaulthidden
As written / Worked out
As writtenfixed
Other settings
Expression/nodes/board/props/expression = "−a²"
Choices
- Correct $-9$ ⁠
- Wrong $9$ ⁠
  Misconception: Reads −3² as (−3)² = 9.
  Feedback from the catalogue
  The power comes before the minus in front of the number: $-3^2$ ⁠ means $-(3^2) = -9$ ⁠. Only when the minus is inside brackets is it squared along with the number: $(-3)^2 = (-3) \times (-3) = 9$ ⁠. A calculator reads it the same way.
- Wrong $-6$ ⁠
  Misconception: Reads 6² as 6 × 2 = 12 instead of 6 × 6 = 36.
  Feedback from the catalogue
  The power $6^2$ ⁠ is $6 \times 6$ ⁠, a square of side $6$ ⁠ with $36$ ⁠ cells. $6 \times 2 = 12$ ⁠ is only two rows of six. The exponent says how many times the number is multiplied by itself.
Once the prediction is locked in, the Scene changes
As written / Worked out
Worked outfixed
The Scene then shows: -9 · Expression: its value in the order of operations (NaN when it cannot be read)/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
The power applies to the $3$ ⁠ alone: $3^2 = 9$ ⁠, and the minus stays in front: $-9$ ⁠. To square $-3$ ⁠ you write brackets: $(-3)^2 = 9$ ⁠.

Worked solution
1. In $-3^2$ ⁠ the power comes before the minus, and it applies to the $3$ ⁠ alone: $3^2 = 9$ ⁠.
2. The minus stays in front of the result: $-3^2 = -(3^2) = -9$ ⁠.
If you chose $9$ ⁠, you squared all of $-3$ ⁠. That is written with brackets: $(-3)^2 = (-3) \times (-3) = 9$ ⁠. If you chose $-6$ ⁠, you worked out $3 \times 2$ ⁠ instead of $3 \times 3$ ⁠.

Across the unit · 16 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

Negative numbers on the number line

A jump along the line

Question

Correct answer

The Scene at this step

Once the prediction is locked in, the Scene changes

Hint

Worked solution

Subtracting from a negative number

Question

Correct answer

The Scene at this step

Choices

Once the prediction is locked in, the Scene changes

Worked solution

Subtracting a negative number

Question

Correct answer

The Scene at this step

Wrong answers it recognises

Hint

Worked solution

Adding a negative number

Question

Goal

The Scene at this step

One setting that reaches it

Hint

Worked solution

Which is greater?

Question

Correct answer

The Scene at this step

Choices

Worked solution

Fractions and decimals

Parts of a whole

Question

Correct answer

The Scene at this step

Wrong answers it recognises

Hint

Worked solution

Same part, different names

Question

Goal

The Scene at this step

One setting that reaches it

Hint

Worked solution

Adding fractions

Question

Correct answer

The Scene at this step

Choices

Once the prediction is locked in, the Scene changes

Worked solution

A common denominator

Question

Correct answer

The Scene at this step

Wrong answers it recognises

Hint

Worked solution

Comparing decimals

Question

Correct answer

The Scene at this step

Choices

Worked solution

Percentages

Out of a hundred

Question

Goal

The Scene at this step

One setting that reaches it

Hint

Worked solution

A percentage of an amount

Question