A tank gains nine liters in three minutes, then ten liters in five minutes. The second gain is larger, but it takes longer. Divide each change by its own elapsed time: three liters per minute in the first interval and two in the second. A rate compares changes per unit input, not merely the heights of successive table entries.
Predict
Predict whether equal row spacing in a printed table means equal time spacing. It does not; read the input column. Predict whether a straight line between observed points proves that the real process followed a constant rate. It only does so if the line is declared part of the model. Otherwise it is a visual interpolation choice.
Experiment
Graph R. The listed piecewise-linear segments are the model, so their landmark coordinates are exact.
Graph R is an explicitly supplied piecewise-linear volume model, with coordinates listed for accessibility. Its first segment joins (0,10) to (4,30); its second joins (4,30) to (10,40). Time is in minutes and volume in liters. Horizontal tick steps are two minutes and vertical tick steps are ten liters.
Time, minutes
Volume, liters
0
10
4
30
10
40
Read the change from 4 to 10 minutes directly from the graph, then verify using the coordinate table. The table states exact model landmarks; readings from an unlabeled curve would instead be estimates.
Understand
For an output q at input t, the average rate over t₁ to t₂ is [q(t₂)−q(t₁)]/(t₂−t₁), with t₂≠t₁. Read this as output change divided by input change. Its units are output units per input unit. A negative rate means the output decreases over that interval; it does not automatically mean the quantity itself is negative.
In Graph R, the first segment has rate (30−10)/(4−0)=5 L/min. The second has rate (40−30)/(10−4)=5/3 L/min. Over the whole ten minutes, the rate is (40−10)/10=3 L/min. This equals the duration-weighted mean of the two segment rates: four minutes at 5 and six minutes at 5/3. An unweighted average would give the two intervals equal influence despite their unequal durations.
The secant line joins two points on a graph. Its slope is the average rate over those endpoints. The chosen axes determine the numerical rise and run, so counting grid squares without converting their scales can give the wrong slope. A graph’s visual angle also changes if its axes are stretched, while the rate in liters per minute stays the same.
When values are estimated from a graph, report suitable precision. If each endpoint volume may be read one liter high or low, their difference may vary by up to two liters. Divide that difference range by the elapsed time to obtain a compatible rate range. This is graph-reading uncertainty, distinct from uncertainty in the original measurements or a statistical model.
Apply
Write a context sentence for each computed rate: between which inputs, what output changed, by how much per input unit, and whether the value is exact under a model or estimated from a plot. Then produce a varying flow schedule with the same endpoint volumes. This shows why average rate alone does not reveal every intermediate event.
Continue
Unequal input spacing is common in experiments and records. Keep the denominator visible before comparing changes. Next, compare a supplied graph with a formula and a table on a stated common domain, using the same feature and units for every representation.
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
A tank table gives (time minutes, volume liters): (0,12),(3,21),(8,31). What is the average rate from 3 to 8 minutes?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
In supplied Graph R, read values at 4 and 10 minutes and calculate the average rate between them in liters per minute.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
A learner calls the second interval in the tank table faster because volume rises by 10 rather than 9 liters. Repair the claim.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For Graph R, compute average rate from 0 to 10 minutes and explain why averaging the two segment rates without weights is wrong.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
On a plotted graph the vertical reading uncertainty is about ±1 L at each endpoint, and endpoint times 2 and 7 minutes are exact. The estimated volumes are 14 and 24 L. Give a compatible interval for the secant rate.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Can a table showing equal average rates on two recorded intervals prove constant instantaneous filling? Give a compatible alternative.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
