# Birds' Eggs

## Task

This scatter diagram shows the lengths and widths of the eggs of some American birds.

- A biologist measured a sample of one hundred Mallard duck eggs and found they had an average length of $57.8$ millimeters and average width of $41.6$ millimeters. Use an X to mark a point that represents this on the scatter diagram.
- What does the graph show about the relationship between the lengths of birds' eggs and their widths?
- Another sample of eggs from similar birds has an average length of $35$ millimeters. If these bird eggs follow the trend in the scatter plot, about what width would you expect these eggs to have, on average?
- Describe the differences in shape of the two eggs corresponding to the data points marked C and D in the plot.
- Which of the eggs A, B, C, D, and E has the greatest ratio of length to width? Explain how you decided.

## IM Commentary

This task asks students to glean contextual information about bird eggs from a collection of measurements of said eggs organized in a scatter plot. In particular, students are asked to identify a correlation and use it to make interpolative predictions, and reason about the properties of specific eggs via the graphical presentation of the data.

This task is based on a task developed by the MARS/ Shell Centre team Mathematics Assessment Resource Service. The task is shared with the with attribution, non-commercial, share-alike Creative Commons License.

## Solution

- There seems to be a positive linear relationship between the length and width of the eggs.
- The line below appears to fit the data fairly well: Since it passes through $(0,0)$ and $(50,36)$, its slope is $\frac{36}{50} = 0.72$, so the equation of the line is $$y=0.72x$$ If $x=35$, then our line would predict that $y=0.72\cdot 35 = 25.2$. So we would expect the width of these eggs to be, on average, about 25 mm. Answers using different lines can vary up to 1 mm in either direction.
- Without reading off precise numerical values from the plot, we can see that eggs $C$ and $D$ have very nearly the same width, but egg $D$ is about 12 millimeters longer than egg $C$.
- First we note that egg $E$ certainly has a higher length-to-width ratio than $C$ or $D$, since it is both longer and narrower. Similarly, $E$ has a higher ratio than $B$ because it is significantly longer, and only a tad wider. It is harder to visually identify the difference between $A$ and $E$, we compute their respective length-to-width ratios numerically, which turn out to be approximately $1.3$ for $A$ and $1.6$ for $E$. So $E$ has the greatest ratio of length to width.

## Birds' Eggs

This scatter diagram shows the lengths and widths of the eggs of some American birds.

- A biologist measured a sample of one hundred Mallard duck eggs and found they had an average length of $57.8$ millimeters and average width of $41.6$ millimeters. Use an X to mark a point that represents this on the scatter diagram.
- What does the graph show about the relationship between the lengths of birds' eggs and their widths?
- Another sample of eggs from similar birds has an average length of $35$ millimeters. If these bird eggs follow the trend in the scatter plot, about what width would you expect these eggs to have, on average?
- Describe the differences in shape of the two eggs corresponding to the data points marked C and D in the plot.
- Which of the eggs A, B, C, D, and E has the greatest ratio of length to width? Explain how you decided.