Math problem set Grade 8 Math
Umbercote Creek Sediment: Comparing Three Unit Rates
A grade 8 problem set where three fictional tributaries carry sediment at proportional rates given as a table, an equation, and a point on a graph.
Objectives
- Students will find the unit rate of a proportional relationship given as a table, an equation, or a point.
- Students will graph proportional relationships and compare their slopes on one grid.
- Students will use a combined unit rate to find an elapsed time from a total amount.
Vocabulary
- Tributary
- A tributary is a smaller stream or river that flows into a bigger one. Many small tributaries join together to build a large river.
Also: unit rate, slope, proportional relationship, suspended sediment
Teacher overview
Suspended sediment is soil carried along in moving water, and too much of it smothers the gravel that fish spawn in, so watershed groups measure how many pounds each tributary delivers during a storm. Three tributaries feed the fictional Umbercote Creek, and the monitoring group recorded each one in a different form: a table, an equation, and a single point read off a graph. Every problem in this set draws on those same three records. The misconception this set targets is comparing totals instead of rates, since a larger total identifies the faster stream only when both totals are read at the same hour of the storm.
Problems
In our scenario, all three tributaries of the fictional Umbercote Creek begin carrying sediment at hour 0 of a storm, and each relationship is proportional. Let t stand for hours of storm flow.
Neap Brook was recorded as a table.
| Hours of storm flow | Pounds of sediment |
|---|---|
| 2 | 340 |
| 5 | 850 |
| 8 | 1,360 |
Hookwell Brook was recorded as the equation s = 215t, where s is pounds of sediment.
Pikestone Ditch was recorded as a graph. Its line passes through the origin and through the point with 4 hours and 560 pounds.
- Find the unit rate for Neap Brook, in pounds of sediment per hour.
- Graph all three relationships on one grid for hours 0 through 8 and label each line. Describe how you can tell which line is steepest without computing anything.
- Find the unit rate for Hookwell Brook, in pounds of sediment per hour.
- Find the unit rate for Pikestone Ditch, in pounds of sediment per hour.
- Explain how the Neap Brook table shows that the relationship is proportional.
- After 6 hours, how many more pounds of sediment has Hookwell Brook carried than Neap Brook?
- Find the combined rate of all three tributaries, in pounds of sediment per hour.
- The gage on the fictional Umbercote Creek recorded 10,500 pounds of sediment from the three tributaries. Using the combined rate, how many hours of storm flow does that represent?
- All three graphs pass through the origin. Explain why that makes sense for this situation.
From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.
Answer Key
- Answer: 170 pounds per hour. Divide any pair from the table: 340 divided by 2 is 170, 850 divided by 5 is 170, and 1,360 divided by 8 is 170, so the unit rate is the same throughout.
- Answer: The steepest line is the one that rises most for the same horizontal distance. Reading all three lines at hour 4 gives Hookwell Brook 860 pounds, Neap Brook 680 pounds, and Pikestone Ditch 560 pounds, so Hookwell Brook is steepest. Graphs should show three straight lines from the origin, with Hookwell Brook highest at every hour after 0.
- Answer: 215 pounds per hour. In the equation s = 215t, the coefficient of t is the unit rate, which is also the slope of the line.
- Answer: 140 pounds per hour. Divide the pounds by the hours at the given point: 560 divided by 4 equals 140 pounds per hour.
- Answer: Every pair of values in the table gives the same quotient, 170 pounds per hour, so pounds and hours change by the same factor together. A relationship with a constant ratio and a value of 0 pounds at hour 0 is proportional.
- Answer: 270 pounds. Hookwell Brook carries 215 times 6, or 1,290 pounds, and Neap Brook carries 170 times 6, or 1,020 pounds. Subtracting gives 1,290 minus 1,020, which is 270 pounds. The same result comes from the difference in rates, since 215 minus 170 is 45, and 45 times 6 is 270.
- Answer: 525 pounds per hour. Add the three unit rates: 170 plus 215 plus 140 equals 525 pounds per hour.
- Answer: 20 hours. Divide the total by the combined rate: 10,500 divided by 525 equals 20 hours of storm flow.
- Answer: At hour 0 the storm has just begun, so no sediment has been carried past the gage yet.
The ordered pair with 0 hours and 0 pounds belongs to every one of the three relationships, which is exactly what passing through the origin means.
Differentiation
Support
Give these students graph paper with both axes already scaled for the fictional Umbercote Creek data and the Neap Brook line already drawn. Assign problems 1, 3, 4, and 7, and provide a ratio table template with the pounds column over the hours column so the division step is visible before the answer is written.
Extension
Ask students to find a fourth tributary's unit rate that would make the combined rate exactly 600 pounds per hour, then write its equation and describe where its line would fall relative to the other three. Have them also find how long that four-stream system would take to deliver 10,500 pounds and compare the result with problem 8.
ELL supports
Pre-teach tributary, sediment, and gage with a simple map sketch showing three small streams joining one creek and a measuring station at the junction. Post the phrase "pounds per hour" beside a division bar with pounds on top. Provide the sentence frame for problem 5: "The table is proportional because every pair gives ___, and at hour 0 the amount is ___."
Extensions
Have students look up how their own local watershed group reports sediment or turbidity and compare the units used there with the pounds per hour in this set. Ask them to write two sentences about why a rate is more useful than a single total when two streams are being compared, using the fictional Umbercote Creek records as their example.