Free math worksheets

- Match the direction of the fitted line to the direction of the cloud. - Inspect the sign of each candidate’s slope.

1. A line summarizing a negative association should decrease from left to right. 2. Only \(y=-2x+18\) has a negative slope. 3. It is the only candidate with the correct direction.

\(y=-2x+18\).

- Compare each \(x\)-value with the smallest and largest observed \(x\)-values. - Boundary values are still within the data range.

1. \(12\) lies inside the observed range, so a) is interpolation. 2. \(20\) lies outside the range, so b) is extrapolation. 3. \(4\) is an observed boundary value, so c) is interpolation.

a) Interpolation b) Extrapolation c) Interpolation

- Check whether the line would miss different regions in the same direction. - A clear curve should not be forced into one straight trend.

1. The points follow a clear U-shaped pattern. 2. Any straight line would miss the low middle and the high ends in a systematic way. 3. A straight linear fit is not appropriate for this nonlinear data.

A straight line is not useful because the data follow a strong U-shaped nonlinear pattern.

- Compare each point’s \(y\)-coordinate with the line’s \(y\)-value at the same \(x\)-coordinate. - Use vertical distance, not the straight-line distance to the graph.

1. Compute the fitted values: at \(x=1, 2, 3, 4, 5\), the line gives \(2, 3, 4, 5, 6\). 2. The vertical differences for the points are \(0, 1, 1, 5, 1\). 3. The point \((4, 10)\) has the largest vertical difference.

\((4, 10)\).

- A useful line should reflect the direction of the association. - The center height alone is not enough when the cloud has a trend.

1. A horizontal line ignores the downward relationship. 2. Line \(b\) matches both the direction and central location of the cloud. 3. Line \(b\) is the better fit.

Line \(b\).

- Compare observed minus predicted when the prediction is too low. - Use the \(y\)-variable’s units for the difference.

1. Underprediction means the observed value is above the fitted value. 2. Since \(47-42=5\), the line underpredicts by \(5\) units.

\(5\) units.

- Distinguish the model line from the observations it summarizes. - Choose coordinates that can be read exactly from the line.

1. The equation must describe the drawn line itself. 2. Data points near the line may not lie exactly on it. 3. Clear points on the line give an accurate slope and intercept for that line.

Use clear points on the fit line because nearby data points may not lie on the line and can give the wrong equation.

- Estimate the line’s \(y\)-coordinate at the requested \(x\)-value. - Choose the option consistent with the model rather than an individual extreme point.

1. Predictions from an informal fit should be near the line’s value. 2. The line’s value is about \(28\). 3. \(27\) is close to \(28\), while \(41\) is not.

\(27\) is more defensible.

- Identify the variable on the vertical axis. - Translate “above the line” into observed versus predicted language.

1. Vertical distance is measured in the \(y\)-variable’s units, which are score points here. 2. The observed score is \(6\) points higher than the fitted prediction for that study time.

The student’s score is \(6\) points above the model’s prediction.

- Distinguish a model from a tracing of the raw observations. - Ask whether the display supports simple prediction between and beyond individual points.

1. A fit summarizes the overall relationship rather than passing through every observation. 2. The zigzag follows local noise and is not one straight line. 3. It is not an informal linear model for the data.

The zigzag overfits individual points instead of summarizing the overall linear tendency.

- First match the direction and approximate steepness of the point cloud. - A reasonable fit should pass through the middle of the data rather than above or below most points. - Check whether the line represents both the left and right sides of the plot.

1. The point cloud has a positive, approximately linear pattern. 2. In panel a), the line follows the direction of the cloud, passes through its center, and leaves points on both sides. 3. In panel b), the line is too high for most of the data. In panel c), the line is too flat and misses the pattern at both ends. 4. Therefore, line a) is the most reasonable informal line of fit.

Line a), because it follows the positive trend and passes through the center of the point cloud with points on both sides.

- Read from the fitted line rather than choosing the nearest data point. - Compare the observed \(y\)-coordinate with the fitted line’s value at the same \(x\)-coordinate.

1. The fit line follows \(y=2x+1\). 2. At \(x=7\), the predicted value is \(2\cdot 7+1=15\). 3. The plotted observation is \((7, 16)\). 4. Since \(16-15=1\), the observation is \(1\) unit above the prediction.

The prediction is \(y=15\). The observed value is \(y=16\), which is \(1\) unit above the prediction.

- Work backward from the \(y\)-value to the line. - Find the horizontal coordinate of the point on the fitted line.

1. Find the point on the fit line where \(y=18\). 2. The line follows \(y=3x\). 3. Solve \(18=3x\) to get \(x=6\).

\(x=6\).

- Compare the errors at small and large \(x\)-values. - A consistent vertical error suggests changing position rather than tilt.

1. Similar vertical misses across the \(x\)-range indicate that the slope is reasonable. 2. The line is consistently too high. 3. Shift the line downward while keeping approximately the same slope.

Shift the line downward without substantially changing its slope.

- Compare the line with the cloud at both ends of the \(x\)-range. - Decide whether the line rises faster or slower than the data trend.

1. The line starts too high relative to the cloud. 2. It ends too low relative to the cloud. 3. It rises too slowly, so it is too shallow and needs a larger slope.

The line is too shallow; its slope should be increased.

- Inspect opposite ends of the \(x\)-range. - A line that overtakes the cloud rises too quickly.

1. The line begins too low and finishes too high. 2. It rises faster than the data trend. 3. Decrease the slope, rotating the line to be less steep.

Decrease the slope; the line is too steep.

- Use points on the drawn line, even if they are not observed data points. - Find the line’s rate of change before locating its vertical starting value.

1. The slope is \(m=\frac{17-5}{8-2}=\frac{12}{6}=2\). 2. Substitute \((2, 5)\) into \(y=2x+b\): \(5=4+b\), so \(b=1\). 3. The equation is \(y=2x+1\).

\(y=2x+1\).

- Ask whether the two chosen points are typical of the data. - Consider what information about the middle points the method ignores.

1. The endpoint observations may be unusual and may not represent the center of the cloud. 2. A good informal fit considers all points and balances the overall pattern. 3. Informal fitting can produce more than one reasonable line.

The endpoint points may be unrepresentative, and a line of fit should summarize the whole cloud rather than be determined by only two observations.

- Estimate the overall rise and run of the cloud. - Compare both the slope and vertical placement of each candidate.

1. The cloud rises by about \(8\) units while \(x\) rises by about \(7\) units. 2. This suggests a slope near \(1\), not \(2\). 3. The equation \(y=x+2\) better matches the center and rate of the cloud.

\(y=x+2\).

- Consider how the line must tilt to move closer to the new point. - The point is influential because it is far from the center in the \(x\)-direction.

1. The new point has both a large \(x\)-value and a \(y\)-value above the current line. 2. A line adjusted toward that point will rise more over the full \(x\)-range. 3. The redrawn fit will likely have a larger slope.

The slope will likely increase.

- Compare each requested \(x\)-value with the range actually observed. - A fitted trend may not continue far beyond the data.

1. The value \(x=22\) lies inside the observed range. 2. The value \(x=90\) lies far outside the observed range. 3. The prediction at \(x=22\) is more defensible because it is interpolation rather than distant extrapolation.

The prediction at \(x=22\) is more defensible.

- In each panel, compare how well the lines match the cloud’s center and direction. - Parallel lines have equal slopes, so vertical separation comes from different \(y\)-intercepts. - Lines with the same \(y\)-intercept must be compared by their slopes.

1. In panel a), lines \(p\) and \(q\) are parallel, so they have equal slopes. Line \(p\) is centered, while line \(q\) is too high. 2. Choose line \(p\); the lines differ in their \(y\)-intercepts. 3. In panel b), lines \(r\) and \(s\) share a \(y\)-intercept. Line \(r\) follows the cloud, while line \(s\) rises too quickly. 4. Choose line \(r\); the lines differ in their slopes.

a) Line \(p\); the \(y\)-intercepts differ. b) Line \(r\); the slopes differ.

- Compare distances as well as the number of points on each side. - A line can split the count evenly and still miss the center of the cloud.

1. Equal counts alone do not measure how far the points lie from the line. 2. The large one-sided vertical misses show poor centrality. 3. The line should be adjusted to better balance the sizes of the misses.

No. A good informal fit should consider the sizes and pattern of the vertical misses, not only their counts.

- Estimate the overall rise and run of the cloud. - Check the order of magnitude of the proposed rate.

1. The cloud’s overall rise-to-run ratio is about \(\frac{15}{5}=3\). 2. A slope of \(0.3\) is ten times smaller. 3. It is not plausible for the displayed trend.

No. A plausible slope is near \(3\), not \(0.3\).

- Use representative regions of the cloud rather than isolated extreme points. - Estimate the overall rise divided by the overall run.

1. Representative centers are near \((2, 7)\) on the left and \((10, 23)\) on the right. 2. The rise is \(23-7=16\), and the run is \(10-2=8\). 3. A reasonable slope is \(\frac{16}{8}=2\).

A reasonable slope is about \(2\).

- Decide whether the questionable point represents real variation or a known mistake. - A model should summarize the valid data.

1. A known error does not represent the intended data. 2. Including it could pull the fit away from the genuine cloud. 3. Correct the value first, then fit the corrected data.

Correct the data-entry error before drawing the line of fit.

- A single shared point gives no information about tilt. - Compare how each line behaves away from the center.

1. One point does not determine how a line follows the rest of the cloud. 2. The slope must match the overall direction and rate of change. 3. Line \(a\) tracks the cloud across the \(x\)-range, while line \(b\) is much too steep.

Line \(a\) is the better fit. Passing through one central point is not sufficient; the slope must also match the overall cloud.

- Use points on the line that fall exactly on readable grid locations. - Compute vertical change over horizontal change in left-to-right order.

1. Two clear points on the line are \((0, 12)\) and \((6, 3)\). 2. The slope is \(\frac{3-12}{6-0}=\frac{-9}{6}=-1.5\). 3. The fit line has slope \(-1.5\).

The slope is \(-1.5\).

- Update the range of \(x\)-values using the new observations. - A former extrapolation can become interpolation when data are added.

1. The new points provide observed evidence beyond \(x=8\). 2. They continue the same pattern through \(x=12\). 3. Predictions through \(x=12\) are now supported by interpolation within the expanded data range.

The data-supported range expands to \(2\le x\le 12\).

- Separate slope errors from vertical-position errors. - Prefer the candidate that needs the smaller structural change.

1. Line \(a\) already matches the cloud’s rate of change. 2. Its main problem is a small vertical placement error. 3. Shift line \(a\) upward; line \(b\) would require a larger slope correction.

Line \(a\); shift it upward while keeping its slope.

- Identify what makes a line summarize data rather than what makes its equation look simple. - Ask how each candidate behaves across the entire \(x\)-range.

1. The numerical size of coefficients does not determine fit quality. 2. Counts alone may also be insufficient. 3. The comparison needs the lines' placement relative to the whole cloud, including direction, centrality, and vertical misses.

The students need evidence about how each line follows the cloud’s direction and center and how large the vertical misses are.

- Examine whether the misses are randomly mixed or arranged by \(x\)-region. - A repeated directional pattern in the misses often signals a shape problem.

1. The line misses the data in an above-below-above pattern. 2. A simple shift or rotation would not remove all three regions of error. 3. The data likely have a curved relationship, so one straight line is unsuitable.

The systematic pattern suggests nonlinearity rather than a well-fitting straight line.

- Ask whether the combined line passes through regions containing actual data. - Consider whether the groups may follow different relationships.

1. One combined line can fall between the clusters where few or no observations occur. 2. It can hide differences between the machine types. 3. Separate fits or separate descriptions may better represent the two groups.

One combined line may describe neither machine type well; the clusters should be examined separately.

- Distinguish an error from a genuine extreme observation. - Consider both mathematical influence and the reason the data were collected.

1. First verify that the point is accurate and belongs to the same population. 2. Compare fits with and without the point because its extreme \(x\)-value can strongly affect the line. 3. Retain or separate it based on context, not solely because it is visually extreme.

Check the point’s validity and context, then examine how much it changes the fit; do not automatically keep or delete it.

- Look for the interval where the rate of change is roughly steady. - A model can be useful locally even when it fails over the full range.

1. The first interval has an approximately linear pattern. 2. The second interval changes form by leveling off. 3. A straight fit is most useful on \(0\le x\le 6\), not across the full range.

The interval \(0\le x\le 6\).