mr barton maths

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Algebraic Proportion

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Foundational skills
Understand the proportionality symbol
\[ y \propto x \text{ means?} \]
Interpret the ∝ symbol for proportionality.
Convert proportionality to equation
\[ y \propto x \rightarrow y = kx \]
Convert a proportionality statement to an equation with constant k.
Identify direct proportion equation
\[ y = 3x \text{ or } y = x + 3? \]
Recognise the algebraic form of direct proportion.
Identify inverse proportion equation
\[ y = \frac{5}{x} \text{ or } y = 5x? \]
Recognise the algebraic form of inverse proportion.
Direct proportion equations
Find k from values (direct)
\[ x = 4, y = 20 \Rightarrow k = \square \]
Calculate the constant of proportionality for direct proportion.
Write the equation (direct)
\[ x = 3, y = 12 \Rightarrow y = \square x \]
Find k and write the complete equation for direct proportion.
Use equation to find y (direct)
\[ y = 6x, \; x = 7 \Rightarrow y = \square \]
Substitute into a direct proportion equation to find y.
Use equation to find x (direct)
\[ y = 5x, \; y = 35 \Rightarrow x = \square \]
Rearrange and substitute to find x in direct proportion.
Find missing value (direct)
\[ (6, 15) \rightarrow (10, \square) \]
Use ratio methods or find k to solve direct proportion problems.
Verify direct proportion
\[ \text{Is } \frac{y}{x} \text{ constant?} \]
Check if given values satisfy direct proportion.
Inverse proportion equations
Find k from values (inverse)
\[ x = 4, y = 6 \Rightarrow k = xy \]
Calculate the constant for inverse proportion.
Write the equation (inverse)
\[ x = 3, y = 8 \Rightarrow y = \frac{\square}{x} \]
Find k and write the complete equation for inverse proportion.
Use equation to find y (inverse)
\[ y = \frac{36}{x}, \; x = 9 \Rightarrow y = \square \]
Substitute into an inverse proportion equation to find y.
Use equation to find x (inverse)
\[ y = \frac{48}{x}, \; y = 6 \Rightarrow x = \square \]
Rearrange and substitute to find x in inverse proportion.
Find missing value (inverse)
\[ (5, 12) \rightarrow (10, \square) \]
Use constant product or find k to solve inverse proportion problems.
Verify inverse proportion
\[ \text{Is } xy \text{ constant?} \]
Check if given values satisfy inverse proportion.
Graphical representation
Identify direct proportion graph
\[ \text{Straight line through origin} \]
Recognise that direct proportion graphs pass through (0,0).
Identify inverse proportion graph
\[ \text{Curve (hyperbola)} \]
Recognise that inverse proportion graphs are curves.
Find k from direct graph
\[ (4, 12) \Rightarrow k = \frac{y}{x} \]
Read values from a graph to find k for direct proportion.
Find k from inverse graph
\[ (6, 5) \Rightarrow k = xy \]
Read values from a graph to find k for inverse proportion.
Sketch direct proportion graph
\[ y = 2x \text{ (describe)} \]
Draw a straight line through the origin with correct gradient.
Identify proportion from a graph
\[ \text{Direct, Inverse, or Neither?} \]
Determine if a graph shows direct, inverse, or neither.
Proportion with powers
Direct proportion with squares
\[ y \propto x^2 \Rightarrow y = kx^2 \]
Handle direct proportion with squared terms.
Find y using proportion with squares
\[ y = 2x^2, \; x = 5 \Rightarrow y = \square \]
Calculate y values using squared proportion.
Direct proportion with cubes
\[ y \propto x^3 \Rightarrow y = kx^3 \]
Handle direct proportion with cubed terms.
Direct proportion with √x
\[ y \propto \sqrt{x} \Rightarrow y = k\sqrt{x} \]
Handle direct proportion with square root terms.
Inverse proportion with squares
\[ y \propto \frac{1}{x^2} \Rightarrow y = \frac{k}{x^2} \]
Handle inverse proportion with squared terms.
Find y using inverse square
\[ y = \frac{72}{x^2}, \; x = 3 \Rightarrow y = \square \]
Calculate y values using inverse square proportion.
Problem solving with algebra
Set up and solve proportion equation
\[ C \propto W \text{ context} \]
Use algebraic proportion to solve contextual problems.
Compare proportional relationships
\[ y = 3x \text{ vs } y = 5x \]
Compare equations with different constants of proportionality.
Find when values are equal
\[ 2x = \frac{18}{x} \Rightarrow x = \square \]
Solve equations to find where direct and inverse are equal.
Determine proportion type from context
\[ C \propto W \Rightarrow C = kW \]
Translate contextual proportion into algebraic form.
Special cases
Recognise no proportion
\[ \text{Neither direct nor inverse?} \]
Identify when values do not follow either proportion type.
Proportion with k = 1
\[ x = 7, y = 7 \Rightarrow y = x \]
Handle the special case where k = 1.
Distinguish y = k/x from y = x/k
\[ y = \frac{12}{x} \text{ vs } y = \frac{x}{12} \]
Recognise that y = x/k is direct proportion with constant 1/k.
Multiple proportionality
\[ z \propto x, \; z \propto \frac{1}{y} \Rightarrow z = \frac{kx}{y} \]
Express combined proportionality relationships.
Timer (Optional)
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