A Ton of Snow

Task

Eric and Julianne are shoveling snow. After an hour of hard work, Eric remarks ''I bet we have shoveled more than a ton of snow.''

1. Explain what measurements Eric and Julianne can make to verify whether or not they have shoveled more than a ton of snow.
2. Is Eric's statement reasonable? Explain.

IM Commentary

The goal of this task is to examine a mathematical statement about the mass of snow, hopefully providing some stimulating thought to go along with the very arduous and demanding physical exercise. Ideally, this task should be done at a time and place where students can experiment with snow and make the relevant measurements. With guidance, students could profitably work on this problem even if they live in a climate which does not normally support snow but they would likely require significant guidance in this case. One recommendation would be to use some other substances such as rocks or sand in order to understand how to approach the problem. The context of the problem can readily be changed accordingly.

Depending on the method employed, vital information students will need to think about for this problem includes:

• How large was the area that Eric and Julianne shoveled?
• What was the average depth of the snow?
• What was the approximate density of the snow?
• At what rate would you need to shovel in order to move a ton of snow?

For the first point, students can use numbers that they think are reasonable by estimating the size of their walkways and driveways. For the third point, there is a very wide variation in the density of snow and so answers may vary greatly. For the numbers used in the solution, the author used snow in New Mexico which tends to be very fluffy and hence low in density.

There are a couple of viable approaches to this problem. Students might focus on how feasible it is to move a ton of material in this amount of time (independent of whether or not the material is snow) or they might focus on how much space this amount of snow would occupy and examine whether or not it would be reasonable to shovel this volume of snow in an hour. The focus of the second method is on understanding the density of snow. The first method also requires an estimate for the mass of a given volume of snow (one shovel full) but it does not need to be precise. With guidance, the first method is appropriate for middle school as an illustration of ratios and proportional reasoning.

This task is intended to support MP4, Model with Mathematics. The context is relatively simple but little guidance has been provided and so students will have to decide which assumptions to make and which questions to ask. Like many modeling tasks, group work should be encouraged and if the task is done in the winter when there is heavy snowfall, physical experimentation is a must to complement and inform mathematical work.

Solutions

Solution: 1 Using arithmetic and measurement

1. One ton of snow is 2000 pounds. If Eric and Julianne each shovel the same amount of snow this would be 1000 pounds each. There are 60 minutes in an hour so if they shovel at a constant rate, this would mean that they would each need to shovel 1000 $\div$ 60 $\approx$ 17 pounds of snow each minute. If they are able to move 4 shovels full each minute this would be a little more than 4 pounds per shovel.

2. In order to test this, students could use snow, shovels, and a scale. As long as the snow is deep enough, 4 pounds per shovel full is certainly possible. Students also might experiment to see if the pace of 4 shovels full each minute (with a weight of 4 pounds or more) is sustainable. On the whole, this direct approach would indicate that Eric's statement is very plausible.

Solution: 2 Estimating a rate for shoveling

1. One way to estimate how much snow Eric and Julianne are shovelling is to estimate how many snow shovels per minute they are able to move and the mass of each load. Over a period of a full hour, 3 shovels full per minute seems reasonable. The mass of the snow will depend on many factors including how deep the snow and how wet it is. Light fluffy snow might have a mass of only 2 pounds per shovel whereas heavier snow could be between 5 and 10 pounds per shovel full.

   Concerning the rate of shovelling, this could be measured by counting over a few one minute time intervals and taking an average. As for the mass of each shovel full, this could be done by bringing one shovel full inside and allowing it to melt and it could then be measured using measuring cups or using a scale and the density of water.

2. At 3 shovels per minute each, Eric and Julianne woul deach move 60 $\times$ 3 = 180 shovles full of snow in an hour. This is 360 shovels full of snow total. In order for this to have a mass of one ton or 2000 pounds, an average shovel full of snow would need to weigh at least $$ 2000 \div 360 \approx 5\frac{1}{2} \text{ pounds}. $$ This is definitely reasonable if the snow is relatively deep and/or dense. It is not reasonable if it is only a small amount of snow or if the snow is very fluffy. To the extent that Eric is less likely to make this statement had the work been easy, the claim is reasonable.

Solution: 3 Calculating the density of snow

1. In this approach, we will measure the density of the snow and the approximate volume of the snow which Eric and Julianne shoveled. For the density, we can take a certain volume of snow and then allow it to melt, applying the appropriate scale factor to the (known) density of water. For the approximate volume of snow Eric and Julianne moved, we can give an estimate for the surface area and depth of snow and then model the snow with a rectangular prism.

   This approach requires some experience. The surface area for a sidewalk and driveway can be calculated modeling these shapes with rectangles (in most cases). The depth of the snow can be estimated with a tape measure. Without experience, however, it is difficult to have a sense of the kind of area it takes an hour to clear (and under what conditions). So one way to proceed here would be to ''work backward.'' We can make an estimate of the surface area to be cleared and then determine how deep a ton of snow (of given density) would be, covering the area. If this depth would be possible to clear in an hour then Eric's statement is reasonable.

2. We filled a measuring cup with snow and then let it melt. The melted snow was about 2 tablespoons. Since there are 16 tablespoons in a cup this means that 1 cup of snow contains about $\frac{1}{8}$ cup of water. For the surface area, we need to take account of the sidewalk and driveway (assuming that Eric and Julianne are shoveling snow at a home). We use metric units as this will make it easier to find the total mass of water since 1000 cubic centimeters of water has a mass of 1 kilogram. We make an estimate of $1\frac{1}{2}$ meters for the width of the sidewalk and 50 meters for the length of sidewalk. This is a total of 75 square meters. For the driveway, we use an estimate of 6 meters for the width and 10 meters for the length, adding an additional 60 square meters. This makes a total of 135 square meters.

   We will assume that the snow is relatively deep, 15 centimeters for example. This would mean that Eric and Julianne would need to clear about 135 $\times$ 0.15 cubic meters of snow. This is about 20 cubic meters. We have estimated that the volume of snow is eight times the volume of water, so 20 cubic meters of snow would contain about 20 $\div$ 8 or about 2$\frac{1}{2}$ cubic meters of water. In order to find the mass of 2$\frac{1}{2}$ cubic meters of water note that 1 liter of water occupies 1000 cm$^3$ and has a mass of 1 kilogram. One cubic meter is 1,000,000 cubic centimeters and so a cubic meter of water would weigh 1000 kilograms, a metric ton. According to our estimates, if Eric and Julianne could clear this area of 6 inches of snow in one hour then they would move about two and a half tons of snow, making Eric's claim accurate.

   Some experience or experimentation would be necessary to judge whether or not two people could clear 6 inches of snow from the sidewalk and driveway of a home of the size indicated. As an order of magnitude estimate, one ton is very reasonable compared to 10 tons or 1/10 of a ton.