Ratio and scale.

The Scene at this step

Marker

600fixed

One pair / Every pair

One pairfixed

Once the prediction is locked in, the Scene changes

One pair / Every pair

Every pairfixed

The Scene then shows: 900 · Ratio: the matching value on the bottom line, at × ratioB ÷ ratioA/nodes/board/outputs/bottom

A prediction is never marked as a failure: the learner sees the outcome and the explanation either way.

Hint

Worked solution

The Scene at this step

Marker

200

One pair / Every pair

One pair

Choices

Shown in a random order.

• Correct $2 : 3$⁠

• Wrong $3 : 2$⁠

  Misconception: Writes a ratio in the wrong order: milk to flour (3 : 2) for flour to milk.

  Feedback from the catalogue

  The order of a ratio follows the words: in the ratio of flour to milk the flour comes first, $2 : 3$⁠. The ratio $3 : 2$⁠ says there is more flour than milk, which is a different recipe.

• Wrong $20 : 30$⁠

  Feedback

  $20 : 30$⁠ is equal to $200 : 300$⁠, but it can still be divided by $10$⁠.

• Wrong $100 : 200$⁠

  Misconception: Keeps a ratio by adding the same amount to both quantities instead of multiplying both by the same number: 200 : 300 → 400 : 500.

  Feedback from the catalogue

  In a ratio both amounts grow by the same factor, not by the same amount. If the flour doubles, from $200$⁠ to $400$⁠, the milk doubles too, from $300$⁠ to $600$⁠. Adding $200$⁠ to both gives $400 : 500 = 4 : 5$⁠, a different ratio and a thicker batter.

Worked solution

The Scene at this step

Marker

400fixed

One pair / Every pair

One pairfixed

Choices

• Correct No: 400 g of flour needs 600 ml of milk

• Wrong Yes: the same was added to both amounts

  Misconception: Keeps a ratio by adding the same amount to both quantities instead of multiplying both by the same number: 200 : 300 → 400 : 500.

  Feedback from the catalogue

  In a ratio both amounts grow by the same factor, not by the same amount. If the flour doubles, from $200$⁠ to $400$⁠, the milk doubles too, from $300$⁠ to $600$⁠. Adding $200$⁠ to both gives $400 : 500 = 4 : 5$⁠, a different ratio and a thicker batter.

• Wrong No: it needs 700 ml of milk

  Feedback

  $700$⁠ is $500 + 200$⁠: adding again. The flour doubled, so the milk doubles too: $2 \times 300 = 600$⁠.

Once the prediction is locked in, the Scene changes

One pair / Every pair

Every pairfixed

The Scene then shows: 600 · Ratio: the matching value on the bottom line, at × ratioB ÷ ratioA/nodes/board/outputs/bottom

A prediction is never marked as a failure: the learner sees the outcome and the explanation either way.

Worked solution

The Scene at this step

Marker

200

One pair / Every pair

Every pair

Other settings

Target/nodes/board/props/target = 750

Read from the Scene at

Ratio: the matching value on the bottom line, at × ratioB ÷ ratioA/nodes/board/outputs/bottom

One setting that reaches it

Marker

500

Hint

Worked solution

The Scene at this step

Marker

350fixed

One pair / Every pair

One pair

Accepted within

± 0.5

Computed from the Scene at this step’s setting

Ratio: the matching value on the bottom line, at × ratioB ÷ ratioA/nodes/board/outputs/bottom

Wrong answers it recognises

• 450 ± 0.5

  Misconception: Keeps a ratio by adding the same amount to both quantities instead of multiplying both by the same number: 200 : 300 → 400 : 500.

  Feedback from the catalogue

  In a ratio both amounts grow by the same factor, not by the same amount. If the flour doubles, from $200$⁠ to $400$⁠, the milk doubles too, from $300$⁠ to $600$⁠. Adding $200$⁠ to both gives $400 : 500 = 4 : 5$⁠, a different ratio and a thicker batter.

• 233.3 ± 0.5

  Misconception: Applies a ratio upside down: multiplies by 2/3 where 3/2 is needed (350 g of flour → 233 ml of milk).

  Feedback from the catalogue

  The recipe has more milk than flour, so $350$⁠ grams of flour need more than $350$⁠ millilitres of milk too. Each gram of flour takes $\frac{300}{200} = 1.5$$\frac{300}{200} =$$1.5$⁠ millilitres, and $350 \times 1.5 = 525$$350 \times 1.5 =$$525$⁠. Always check which amount should come out bigger.

Hint

Worked solution

The Scene at this step

Marker

2.5fixed

Before / Converted

Before

Accepted within

± 0.5

Computed from the Scene at this step’s setting

Ratio: the matching value on the bottom line, at × ratioB ÷ ratioA/nodes/board/outputs/bottom

Wrong answers it recognises

• 0.0025 ± 0.0001

  Misconception: Converts in the wrong direction, dividing where the factor multiplies (2.5 km = 0.0025 m) or the reverse.

  Feedback from the catalogue

  Ask first: is the new unit smaller or bigger? A metre is smaller than a kilometre, so the same length is many metres: $2.5 \times 1000 = 2500$$2.5 \times 1000 =$$2500$⁠. To a smaller unit you multiply, to a bigger unit you divide.

• 250 ± 0.5

  Misconception: Uses the wrong factor for a metric prefix, such as 100 m in a kilometre.

  Feedback from the catalogue

  Kilo means a thousand: a kilometre is $1000$⁠ metres and a kilogram is $1000$⁠ grams. Centi means a hundredth: a metre has $100$⁠ centimetres. And milli means a thousandth: a metre has $1000$⁠ millimetres.

Hint

Worked solution

The Scene at this step

Marker

1

Before / Converted

Converted

Other settings

Ratio b/nodes/board/props/ratioB = 100

Other settings

Label a/nodes/board/props/labelA = "m"

Other settings

Label b/nodes/board/props/labelB = "cm"

Other settings

Target/nodes/board/props/target = 350

Read from the Scene at

Ratio: the matching value on the bottom line, at × ratioB ÷ ratioA/nodes/board/outputs/bottom

One setting that reaches it

Marker

3.5

Hint

Worked solution

The Scene at this step

Marker

1.25fixed

Before / Converted

Before

Other settings

Ratio b/nodes/board/props/ratioB = 60

Other settings

Label a/nodes/board/props/labelA = "h"

Other settings

Label b/nodes/board/props/labelB = "min"

Accepted within

± 0.01

Computed from the Scene at this step’s setting

Ratio: the matching value on the bottom line, at × ratioB ÷ ratioA/nodes/board/outputs/bottom

Wrong answers it recognises

• 125 ± 0.01

  Misconception: Reads the decimal part of hours as minutes: 1.25 h as 1 h 25 min (85 or 125 minutes).

  Feedback from the catalogue

  An hour has $60$⁠ minutes, not $100$⁠, so the digits after the point are a part of an hour. $0.25$⁠ of an hour is a quarter of an hour, $15$⁠ minutes, and $1.25$⁠ hours are $75$⁠ minutes. Multiply by $60$⁠: $1.25 \times 60 = 75$⁠.

• 85 ± 0.01

  Misconception: Reads the decimal part of hours as minutes: 1.25 h as 1 h 25 min (85 or 125 minutes).

  Feedback from the catalogue

  An hour has $60$⁠ minutes, not $100$⁠, so the digits after the point are a part of an hour. $0.25$⁠ of an hour is a quarter of an hour, $15$⁠ minutes, and $1.25$⁠ hours are $75$⁠ minutes. Multiply by $60$⁠: $1.25 \times 60 = 75$⁠.

Hint

Worked solution

The Scene at this step

Marker

1.5fixed

Before / Converted

Before

Other settings

Ratio b/nodes/board/props/ratioB = 3.6

Other settings

Label a/nodes/board/props/labelA = "m/s"

Other settings

Label b/nodes/board/props/labelB = "km/h"

Accepted within

± 0.01

Computed from the Scene at this step’s setting

Ratio: the matching value on the bottom line, at × ratioB ÷ ratioA/nodes/board/outputs/bottom

Wrong answers it recognises

• 5400 ± 1

  Misconception: Converts only one of the two units of a speed: 1.5 m/s × 3600 = 5400, still in metres per hour.

  Feedback from the catalogue

  A speed has two units, and both change. Seconds to hours: multiply by $3600$⁠, which gives $5400$⁠ metres per hour. Metres to kilometres: divide by $1000$⁠, which leaves $5.4$⁠ kilometres per hour. The two steps together are a multiplication by $3.6$⁠.

• 0.4167 ± 0.01

  Misconception: Converts in the wrong direction, dividing where the factor multiplies (2.5 km = 0.0025 m) or the reverse.

  Feedback from the catalogue

  Ask first: is the new unit smaller or bigger? A metre is smaller than a kilometre, so the same length is many metres: $2.5 \times 1000 = 2500$$2.5 \times 1000 =$$2500$⁠. To a smaller unit you multiply, to a bigger unit you divide.

Hint

Worked solution

The Scene at this step

Marker

1defaulthidden

Before / Converted

Converted

Other settings

Mode/nodes/board/props/mode = "grid"

Choices

Shown in a random order.

• Wrong $100$⁠

  Misconception: Converts an area with the length factor: 1 m² = 100 cm².

  Feedback from the catalogue

  An area is a length times a width, so it changes twice. The side of a metre is $100$⁠ centimetres, and a square metre holds $100$⁠ rows of $100$⁠ squares: $100 \times 100 = 10{,}000$$100 \times 100 =$$10{,}000$⁠. The board shows the same: a side of $10$⁠ gives $10 \times 10 = 100$⁠ squares.

• Wrong $1000$⁠

  Feedback

  $1000 = 10 \times 10 \times 10$$1000 =$$10 \times 10 \times 10$⁠ counts cubes in a volume, three dimensions. An area has two: $100 \times 100$⁠.

• Correct $10{,}000$⁠

Worked solution

The Scene at this step

Marker

3fixed

Before / Converted

Before

Other settings

Ratio b/nodes/board/props/ratioB = 0.5

Other settings

Label a/nodes/board/props/labelA = "cm"

Other settings

Label b/nodes/board/props/labelB = "km"

Accepted within

± 0.01

Computed from the Scene at this step’s setting

Ratio: the matching value on the bottom line, at × ratioB ÷ ratioA/nodes/board/outputs/bottom

Wrong answers it recognises

• 150000 ± 1

  Misconception: Multiplies by a map scale but leaves the distance in centimetres or metres instead of the unit asked for.

  Feedback from the catalogue

  A scale of $1 : 50{,}000$⁠ says the length on the ground is $50{,}000$⁠ times bigger in the same unit: $3$⁠ centimetres on the map are $150{,}000$⁠ centimetres on the ground. That still needs converting: $150{,}000$⁠ centimetres are $1500$⁠ metres, which are $1.5$⁠ kilometres.

• 1500 ± 1

  Misconception: Multiplies by a map scale but leaves the distance in centimetres or metres instead of the unit asked for.

  Feedback from the catalogue

  A scale of $1 : 50{,}000$⁠ says the length on the ground is $50{,}000$⁠ times bigger in the same unit: $3$⁠ centimetres on the map are $150{,}000$⁠ centimetres on the ground. That still needs converting: $150{,}000$⁠ centimetres are $1500$⁠ metres, which are $1.5$⁠ kilometres.

Hint

Worked solution

The Scene at this step

Number in front

1fixed

Power of ten

2

As given / Written both ways

Written both ways

Other settings

Target/nodes/board/props/target = 1000000

Read from the Scene at

Powers: mantissa × 10^exponent/nodes/board/outputs/scientific

One setting that reaches it

Power of ten

6

Hint

Worked solution

The Scene at this step

Number in front

1.4fixed

Power of ten

6fixed

As given / Written both ways

As given

Other settings

Given/nodes/board/props/given = "decimal"

Accepted within

± 0.01

Computed from the Scene at this step’s setting

Powers: the exponent once the number is written with 1 ≤ mantissa < 10, its order of magnitude/nodes/board/outputs/order

Wrong answers it recognises

• 5 ± 0.01

  Misconception: Takes the exponent to be the number of zeros: 1,400,000 = 1.4 × 10⁵, 0.00025 = 2.5 × 10⁻³, 10⁻³ = 0.0001.

  Feedback from the catalogue

  The power counts how many places the decimal point moves, not how many zeros there are. $1{,}400{,}000$⁠ has five zeros, but the $4$⁠ takes up a place too: from $1.4$⁠ the point moves six places, $1.4 \times 10^6$⁠. In the same way $0.001 = 10^{-3}$⁠, because the $1$⁠ sits in the third place after the point.

• 7 ± 0.01

  Misconception: Takes the exponent to be the number of digits: 1,400,000 (seven digits) = 1.4 × 10⁷, 0.00025 = 2.5 × 10⁻⁵.

  Feedback from the catalogue

  The number $1$⁠ itself is $10^0$⁠: one digit and a power of $0$⁠. So a number with seven whole digits is of order of magnitude $6$⁠: $1{,}400{,}000 = 1.4 \times 10^6$$1{,}400{,}000 =$$1.4 \times 10^6$⁠. In a small number you also count steps, not digits: $0.00025$⁠ has five digits after the point, but the point moves only four places to reach $2.5$⁠.

Hint

Worked solution

The Scene at this step

Number in front

1fixed

Power of ten

-3fixed

As given / Written both ways

As givenfixed

Choices

• Correct $0.001$⁠

• Wrong $-1000$⁠

  Misconception: Thinks a negative exponent makes the number negative: 10⁻³ = −1000.

  Feedback from the catalogue

  The minus in the power is not the number’s sign but a direction: a positive power multiplies by $10$⁠, a negative one divides. $10^{-3} = \frac{1}{10^3} = 0.001$$10^{-3} =$$\frac{1}{10^3} =$$0.001$⁠, a small positive number. A negative number needs a minus in front of the $10$⁠: $-10^3 = -1000$⁠.

• Wrong $-0.001$⁠

  Feedback

  The number is small but not negative: $10^{-3} = \frac{1}{1000}$$10^{-3} =$$\frac{1}{1000}$⁠, one part in a thousand, which is a positive number.

• Wrong $0.0001$⁠

  Misconception: Takes the exponent to be the number of zeros: 1,400,000 = 1.4 × 10⁵, 0.00025 = 2.5 × 10⁻³, 10⁻³ = 0.0001.

  Feedback from the catalogue

  The power counts how many places the decimal point moves, not how many zeros there are. $1{,}400{,}000$⁠ has five zeros, but the $4$⁠ takes up a place too: from $1.4$⁠ the point moves six places, $1.4 \times 10^6$⁠. In the same way $0.001 = 10^{-3}$⁠, because the $1$⁠ sits in the third place after the point.

Once the prediction is locked in, the Scene changes

As given / Written both ways

Written both waysfixed

The Scene then shows: 0.001 · Powers: mantissa × 10^exponent/nodes/board/outputs/scientific

A prediction is never marked as a failure: the learner sees the outcome and the explanation either way.

Worked solution

The Scene at this step

Number in front

2.5fixed

Power of ten

-4fixed

As given / Written both ways

As given

Other settings

Given/nodes/board/props/given = "decimal"

Accepted within

± 0.01

Computed from the Scene at this step’s setting

Powers: the exponent once the number is written with 1 ≤ mantissa < 10, its order of magnitude/nodes/board/outputs/order

Wrong answers it recognises

• -3 ± 0.01

  Misconception: Takes the exponent to be the number of zeros: 1,400,000 = 1.4 × 10⁵, 0.00025 = 2.5 × 10⁻³, 10⁻³ = 0.0001.

  Feedback from the catalogue

  The power counts how many places the decimal point moves, not how many zeros there are. $1{,}400{,}000$⁠ has five zeros, but the $4$⁠ takes up a place too: from $1.4$⁠ the point moves six places, $1.4 \times 10^6$⁠. In the same way $0.001 = 10^{-3}$⁠, because the $1$⁠ sits in the third place after the point.

• -5 ± 0.01

  Misconception: Takes the exponent to be the number of digits: 1,400,000 (seven digits) = 1.4 × 10⁷, 0.00025 = 2.5 × 10⁻⁵.

  Feedback from the catalogue

  The number $1$⁠ itself is $10^0$⁠: one digit and a power of $0$⁠. So a number with seven whole digits is of order of magnitude $6$⁠: $1{,}400{,}000 = 1.4 \times 10^6$$1{,}400{,}000 =$$1.4 \times 10^6$⁠. In a small number you also count steps, not digits: $0.00025$⁠ has five digits after the point, but the point moves only four places to reach $2.5$⁠.

• 4 ± 0.01

  Misconception: Gives a number smaller than 1 a positive exponent: 0.00025 = 2.5 × 10⁴.

  Feedback from the catalogue

  A positive power of $10$⁠ is bigger than $1$⁠: $2.5 \times 10^4 = 25{,}000$$2.5 \times 10^4 =$$25{,}000$⁠. A number smaller than $1$⁠, such as $0.00025$⁠, gets a negative power, because to reach it from $2.5$⁠ you divide by $10$⁠ four times: $2.5 \times 10^{-4}$⁠.

Hint

Worked solution

The Scene at this step

Number in front

32fixed

Power of ten

4fixed

As given / Written both ways

As given

Accepted within

± 0.01

Computed from the Scene at this step’s setting

Powers: the exponent once the number is written with 1 ≤ mantissa < 10, its order of magnitude/nodes/board/outputs/order

Wrong answers it recognises

• 3 ± 0.01

  Misconception: Shrinks the mantissa and the exponent together: 32 × 10⁴ = 3.2 × 10³.

  Feedback from the catalogue

  When the number in front gets $10$⁠ times smaller, the power must get $10$⁠ times bigger so that the value stays the same. $32 = 3.2 \times 10$⁠, so $32 \times 10^4 = 3.2 \times 10^5$$32 \times 10^4 =$$3.2 \times 10^5$⁠. Check: $3.2 \times 10^3 = 3200$$3.2 \times 10^3 =$$3200$⁠, far less than $320{,}000$⁠.

Hint

Worked solution

The Scene at this step

Solar System clock

20 days/sdefaulthidden

Zoom out

Atom

Accepted within

± 0.01

A fixed value in the question

Wrong answers it recognises

• 10 ± 0.01

  Misconception: Gives a number smaller than 1 a positive exponent: 0.00025 = 2.5 × 10⁴.

  Feedback from the catalogue

  A positive power of $10$⁠ is bigger than $1$⁠: $2.5 \times 10^4 = 25{,}000$$2.5 \times 10^4 =$$25{,}000$⁠. A number smaller than $1$⁠, such as $0.00025$⁠, gets a negative power, because to reach it from $2.5$⁠ you divide by $10$⁠ four times: $2.5 \times 10^{-4}$⁠.

Hint

Worked solution

The Scene at this step

Solar System clock

20 days/sdefaulthidden

Zoom out

Atomfixed

Choices

• Correct $10^5$⁠ times, that is $100{,}000$⁠ times

• Wrong $5$⁠ times, because the powers differ by $5$⁠

  Misconception: Reads a difference of five orders of magnitude as five times bigger.

  Feedback from the catalogue

  Each order of magnitude is a multiplication by $10$⁠, and five orders of magnitude are five multiplications: $10 \times 10 \times 10 \times 10 \times 10 = 10^5 = 100{,}000$$10 \times 10 \times 10 \times 10 \times 10 =$$10^5 =$$100{,}000$⁠. The difference of the exponents says how many times you multiply by $10$⁠, not how many times bigger.

• Wrong $2$⁠ times, because $(-10) \div (-5) = 2$⁠

  Misconception: Compares two powers of ten by dividing their exponents: 10⁻⁵ is (−10) ÷ (−5) = 2 times 10⁻¹⁰.

  Feedback from the catalogue

  When you divide powers of $10$⁠ you subtract the exponents: $\frac{10^{-5}}{10^{-10}} = 10^{-5-(-10)} = 10^5$$\frac{10^{-5}}{10^{-10}} =$$10^{-5-(-10)} =$$10^5$⁠. Dividing one exponent by the other says nothing about the numbers: $10^6$⁠ is not $2$⁠ times $10^3$⁠ but $1000$⁠ times.

• Wrong About $100$⁠ times, as in the picture

  Feedback

  Each stop in the scene is drawn at a comfortable size for the screen, so the picture does not show the real ratio. The comparison comes from the numbers.

Once the prediction is locked in, the Scene changes

Zoom out

Cellfixed

A prediction is never marked as a failure: the learner sees the outcome and the explanation either way.

Worked solution

The Scene at this step

Solar System clock

20 days/sdefaulthidden

Zoom out

Sun

Accepted within

± 0.01

A fixed value in the question

Wrong answers it recognises

• 6 ± 0.01

  Misconception: Converts km to m by keeping the exponent: 1.4 × 10⁶ km = 1.4 × 10⁶ m.

  Feedback from the catalogue

  A kilometre is $10^3$⁠ metres, so going to metres adds $3$⁠ to the power: $1.4 \times 10^6$⁠ kilometres are $1.4 \times 10^9$⁠ metres. Without the conversion the number is a thousand times too small.

• 18 ± 0.01

  Misconception: Multiplies the exponents when multiplying powers of ten: 10⁵ × 10¹⁶ = 10⁸⁰.

  Feedback from the catalogue

  When you multiply powers of $10$⁠ you add the exponents: $10^5 \times 10^{16} = 10^{21}$$10^5 \times 10^{16} =$$10^{21}$⁠. Think of the zeros: $100{,}000 \times 10^{16}$⁠ puts five zeros after sixteen, twenty-one in all. Multiplying exponents belongs to a power of a power, such as $(10^5)^{16}$⁠.

Hint

Worked solution

The Scene at this step

Solar System clock

20 days/sdefaulthidden

Zoom out

Milky Way

Accepted within

± 0.01

A fixed value in the question

Wrong answers it recognises

• 80 ± 0.01

  Misconception: Multiplies the exponents when multiplying powers of ten: 10⁵ × 10¹⁶ = 10⁸⁰.

  Feedback from the catalogue

  When you multiply powers of $10$⁠ you add the exponents: $10^5 \times 10^{16} = 10^{21}$$10^5 \times 10^{16} =$$10^{21}$⁠. Think of the zeros: $100{,}000 \times 10^{16}$⁠ puts five zeros after sixteen, twenty-one in all. Multiplying exponents belongs to a power of a power, such as $(10^5)^{16}$⁠.

Hint

Worked solution

The Scene at this step

Solar System clock

20 days/sdefaulthidden

Zoom out

Milky Way

Accepted within

± 0.01

A fixed value in the question

Wrong answers it recognises

• 11 ± 0.01

  Misconception: Subtracts exponents without their signs: from 10⁻¹⁰ to 10²¹ is 21 − 10 = 11 orders of magnitude.

  Feedback from the catalogue

  The difference is $21 - (-10)$⁠, and subtracting a negative number moves right, like adding: $21 + 10 = 31$⁠. On the ladder: ten steps from $10^{-10}$⁠ up to one metre, and twenty-one more up to $10^{21}$⁠.

Hint

Worked solution