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An applied example of the use of two points on a graph to develop a straight line function, then use the t estimate and predict. MCQ's are also used to develop student understanding of the uses of gradient and intercepts as well as the limitations of prediction.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

{person['name']} is given a sunflower seedling for {person['pronouns']['their']} 30th birthday (day $0$) and observes its height. {capitalise(person['pronouns']['they'])} make{s} the following observations later that week:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

Observation	A	B
Day	$\\var{xa}$	$\\var{xb}$
height (cm)	$\\var{ya}$	$\\var{yb}$

{person['name']} plots the 2 points:

{plotPoints()}

", "advice": "

a)

The gradient is the ratio of vertical change ($y_2-y_1$) to horizontal change ($x_2-x_1$).

\\[ m = \\frac{y_2-y_1}{x_2-x_1}=\\frac{\\simplify[!collectNumbers]{{yb}-{ya}}}{\\simplify[!collectNumbers]{{xb}-{xa}}}=\\frac{\\simplify{{yb}-{ya}}}{\\simplify{{xb}-{xa}}}=\\simplify[simplifyFractions,unitDenominator]{({yb-ya})/({xb-xa})}\\text{.}\\]

b)

Rearranging the equation $y=mx+c$ for $c$ and using point A:

\\[ c = y_1-mx_1 = \\var{ya}-\\var{m}\\times\\var{xa}=\\simplify{{ya-m*xa}}\\text{.}\\]

We then check this against point $B$:

\\[ y_2 = mx_2 + c = \\simplify[fractionNumbers]{{m}{xb}+{c}}=\\simplify{{m}*{xb}+{c}}\\text{.}\\]

b)

We now substitute the values for $m$ and $c$ into the equation of a straight line, $y=mx+c$,

\\[y=\\simplify[!noLeadingMinus,unitFactor]{{m} x+ {c}}\\text{.}\\]

{advicePoints()}

c)

The gradient represents the vertical change (height in cm) per unit of the horizontal axis (days): the change in height of the sunflower per day.

d)

Substituting $x=\\var{d}$ into the straight line equation, the height $y$ after $\\var{d}$ days is

\\begin{align}
y&=\\simplify{{m}}x+\\var{c}\\\\
&=\\simplify[]{{m}{d}}+\\var{c}\\\\
&=\\var{m*d+c}\\text{cm.}
\\end{align}

e)

Substituting $x=\\var{1826}$ into the straight line equation, the height $y$ after $1826$ days is

\\begin{align}
y&=\\simplify{{m}}x+\\var{c}\\\\
&=\\simplify[]{{m}1826} + \\var{c}\\\\
&=\\var{m*1826+c}\\text{cm.}
\\end{align}

Note that this is $\\var{(m*1826+c)/100}$ metres. In 2014, a sunflower of $9.17$ metres was entered into the Guinness World Records as tallest sunflower.

f)

Possible reasons that the prediction will not be accurate are:

from everyday observations and basic biology that a sunflower cannot keep growing forever, but our straight line equation would predict otherwise.
the set of observations are limited in their time frame. In particular another relationship could easily fit the 2 data points.

Invalid reasons that the prediction will not be accurate are:

There can only be one straight line between two points; only one equation describes this.
Based on common sense, we know that sunflowers do grow over time.

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He makes, they make.

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y coordinate of point B

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The gradient

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y coordinate of point A

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A random person

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x coordinate of point B

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x coordinate of point a

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The intercept

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A number of days after receiving the seedling, on which the height is estimated

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What is the gradient, $m$, of the straight line between the two points?

$m =$ [[0]]

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Use the gradient and the coordinates of the two points to find the height of the sunflower when {person['name']} received it.

[[0]] cm.

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Let $y$ be the sunflower height and $x$ the time, in days, since {person['name']} received the sunflower. What is the equation of the straight line between the points?

$y(x) = $ [[0]]

Use the preview on {person['name']}'s plot to check your answer.

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What does the gradient represent?

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The length of time taken in days for the sunflower to grow $1$ cm

", "

The change in height (in cm) of the sunflower over $1$ day

", "

The width of the ruler used to measure the sunflower

", "

All of the above

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{person['name']} uses the straight line equation to predict the future height of the sunflower. What will the height be on day $\\var{d}$?

[[0]] cm

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{person['name']} wonders if {person['pronouns']['they']} can guess what the height of the sunflower will be on {person['pronouns']['their']} 35th birthday. {capitalise(person['pronouns']['they'])} work{s} out that this is day 1826. Using the straight line equation, what would the height be on day 1826?

[[0]] cm

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{person['name']} doubts {person['pronouns']['their']} result. Which of the following reason(s) may mean that the height on {person['pronouns']['their']} 35th birthday is not accurate?

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Sunflower height as a function of time may not have a straight linear relationship.

", "

The observations only span a very limited time range.

", "

There are multiple straight linear relationships that could be obtained using the same $2$ data points.

", "

Sunflower height never actually increases over time.

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