mr barton maths

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Direct Proportion

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Foundational skills
Recognise a directly proportional relationship
\[ \text{Which is directly proportional?} \]
Identify when two quantities are in direct proportion.
Identify the constant of proportionality
\[ \text{£2 each} \rightarrow k = 2 \]
Find the multiplier that relates two proportional quantities.
Complete a proportion table (find multiplier)
\[ 3 \rightarrow 6 \text{, multiplier} = \square \]
Find the scale factor between corresponding values.
Use the unitary method setup
\[ 8 \text{ cost £}12 \rightarrow 1 \text{ costs } \square \]
Find the value of one unit.
Unitary method
Find cost of multiple items
\[ 5 \text{ cost £}30 \rightarrow 8 \text{ cost } \square \]
Use unitary method to find cost of different quantities.
Find quantity from cost
\[ \text{£}0.40 \text{ each, £}3.20 \rightarrow \square \]
Work backwards from total cost to find quantity.
Find time using unitary method
\[ 12 \text{ in } 4\text{min} \rightarrow 30 \text{ in } \square \]
Apply unitary method to time-based problems.
Recipe scaling
Scale a recipe up
\[ 4 \text{ people} \rightarrow 12 \text{ people} \]
Scale recipe quantities when increasing servings.
Scale a recipe down
\[ 8 \text{ people} \rightarrow 2 \text{ people} \]
Scale recipe quantities when decreasing servings.
Scale a recipe to a non-multiple
\[ 6 \text{ serves} \rightarrow 10 \text{ serves} \]
Scale when servings aren’t simple multiples.
Find how many servings from ingredients
\[ 150\text{g for } 6 \rightarrow 400\text{g} = \square \]
Work backwards from available ingredients.
Currency and exchange
Convert using exchange rate (multiply)
\[ \text{£}1 = \text{€}1.20, \text{£}50 = \square \]
Convert currency using multiplication.
Convert using exchange rate (divide)
\[ \text{£}1 = \$1.25, \$100 = \square \]
Convert currency using division.
Two-step currency problem
\[ \text{Convert then spend} \]
Apply conversion then perform further calculation.
Best buy problems
Compare unit prices (same units)
\[ \text{6 for £}1.80 \text{ vs } 8 \text{ for £}2 \]
Calculate and compare unit prices.
Compare unit prices (different units)
\[ 500\text{ml vs } 1.2\text{L} \]
Compare prices with different units.
Find the cost per unit to compare
\[ \text{Pack A: }12 \text{ for £}6 \rightarrow \square\text{/each} \]
Calculate cost per unit for comparison.
Compare using scaling
\[ 3 \text{ for £}1.20 \text{ vs } 5 \text{ for £}1.75 \]
Use scaling to compare prices fairly.
Problem solving
Proportion with rates
\[ 15\text{L/min}, 180\text{L} \rightarrow \square\text{min} \]
Apply direct proportion to rate problems.
Proportion with wages
\[ \text{£}12\text{/hr} \times 7.5\text{hr} = \square \]
Apply direct proportion to calculate earnings.
Proportion involving speed
\[ 60\text{mph} \times 2.5\text{hr} = \square \]
Apply proportion to speed-distance-time.
Two-stage proportion problem
\[ \text{Workers} \times \text{Days} \times \text{Rate} \]
Solve problems requiring multiple steps.
Special cases
Recognise when proportion doesn’t apply
\[ \text{Walking together} = \square\text{min?} \]
Identify when direct proportion does not apply.
Proportion starting from a value
\[ \text{£}5 + \text{£}2\text{/ride} \rightarrow \text{proportional?} \]
Identify when fixed cost breaks proportion.
Verify a proportion relationship
\[ \text{Is Amy correct?} \]
Check if values are consistent with proportion.
Set up but not solve a proportion
\[ (36 \div 8) \times 13 \]
Express a proportion as a calculation.
Timer (Optional)
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