Problem Solving and the Common Core

Appendix

Example Problems

Key to this unit is the selection of problems, of which I’ve specified four classes below. Estimation problems and decomposition problems each have two levels that are differentiated by the instructions supplied and what the students are asked to do.

One-Step Estimation Problems

Estimate the order of magnitude of the requested quantity. Justify your estimate in a sentence or two describing any observations, assumptions or prior knowledge you used.

Estimate the requested quantity by setting upper and lower bounds and finding the geometric mean. Justify your estimate by describing any observations, assumptions or prior knowledge you used.

1. What is the height of the school in meters?
2. How many skittles come in a bag?
3. How much does the teacher weigh?
4. How many jellybeans are in a jar?
5. How high above the street is a traffic light?
6. How many seconds have you been alive?
7. How fast do you walk? One mile per hour is about 0.5 m/s and about 1.5 ft/s.
8. How much does a stack of pancakes weigh?
9. If you woke up late, what’s the fastest you could get ready and get to school?
10. How much water could this (oddly shaped real-life) bowl hold?

Unit Conversion Problems

Express the requested quantity in terms of the specified unit. Show the conversion calculation and justify your answer in a sentence or two.

One Step:

1. How many minutes is 10³ seconds?
2. How many centimeters are in a kilometer?
3. How many feet are in 60 miles?
4. A man weighed 180 lbs. What is his weight in kilograms?
5. A large coffee is 20 oz. How many liters is it?
6. Six nanometers is how many millimeters?
7. If an average double spaced essay has 250 words per page, how many pages is a 5000 word double spaced essay?
8. Everest is 8.8km tall. How many feet is that?
9. How many meters tall is the 973 ft. tall Comcast Center in Philadelphia?
10. A typical housefly can live up to 30 days. How many hours is that?

Multi-Step:

1. Human hair grows at a rate of about 6 inches per year. How fast is that in meters per second? How many meters per second are all the hairs on your head growing in combined length?
2. An NBA court is 94 ft. long and a standard basketball is 25 cm in diameter. How many basketballs long is a basketball court?
3. How many days is 10⁶ seconds? How many years is 10⁹ seconds?
4. The speed limit on US roads is 65 mi/hr. What is that in meters per second?
5. The longest game in Major League Baseball history was 8 hours 25 minutes. How many seconds long was it?
6. The Suzuki Hayabusa was once the fastest production motorcycle, traveling at 194 miles per hour. What is this speed in meters per second?
7. A “stone” is a unit of measure equal to 14 lbs. If a half-ton pickup can transport a half-ton of cargo, how many stones can it transport?
8. Abraham Simpson drives a car that gets 40 rods to the hogshead. If a hogshead is 63 gallons and a rod is 16.5 feet, how many miles per gallon did his car get?
9. A man is 5’10”. How many meters tall is he?
10. If a car tire is 60cm in diameter, how many times does it rotate every mile?
11. Compare the population densities of India, the United States, China and Canada.

Decomposition Problems

Specify the unit on the requested quantity. Then, decompose each quantity into its constituent quantities and specify their units.

Estimate the answer, showing the constituent quantities used to arrive at your estimate. Justify your answer.

One Step:

1. What is the orbital speed of the Earth as it travels around the sun?
2. What is the volume of the Earth?
3. If you turned it all into one-dollar bills and stacked it, how tall would a billion dollars be?
4. How many people are eating dinner right now?
5. How long would it take to drive to the moon?
6. How many hairs are on your head?
7. What percentage of Earth’s total mass is people?³
8. How many functioning ballpoint pens are in the school right now?
9. How much would it cost to survive on ramen noodles for one year?⁴
10. How many blades of grass are on a football field?

Multi Step:

1. How many football fields could you cover with all the pizza the school cafeteria serves in a year?
2. What’s the weight of the atmosphere?
3. How massive is a mole of tennis balls? How does this compare to the mass of the Earth?
4. How many cubic meters of trash does Philadelphia generate in a year?
5. During a big California earthquake, two million books fell off the shelves at a university library. How many students would need to be hired to reshelf all of the books in three weeks?⁵
6. Picture a yellow school bus filled with high school students. By what factor do the weight of just the students and the weight of just the bus differ?
7. How much would the ocean’s surface rise if the Antarctic ice sheet melted?
8. How much land area would be needed for solar panels to provide the United States with all of its power needs?
9. If you were to convert Wikipedia to a physical encyclopedia, how thick would it be?
10. How long will it take until the surface of the Earth is entirely covered in gravestones?
11. If everyone in the world went swimming in Lake Michigan, how much would the water level rise?
12. Compare the daily increase in the human population with the total population of various animals – e.g., lions, tigers, rhinoceroses, elephants and bison.

Content Standards

The Next Generation Science Standards call for a shift from “skills” to “practices” that are germane to science and engineering and should be engaged in by students across all grade bands⁶. Eight practices are outlined in the NGSS, of which this unit aids development in four. By solving these problems, students engage in the first practice of asking questions by not only asking themselves what they need to know, but also by asking what the answers might mean. Students will also use the fifth practice, using mathematics and computational thinking, increasing their fluency in the types of calculations that are common in science and engineering. By having students justify, report on and discuss their calculations, they will be engaging in the seventh and eighth practices of engaging in argument from evidence and obtaining, evaluating and communicating information.

For a mathematics classroom implementing the Common Core Standards, this unit addresses two High School Standards explicitly. Under the High School standards for Number and Quantity, the Quantities heading⁷ specifies that students reason quantitatively and use units to solve problems. The very approach outlined has students using units as a way to understand problems and guide the solution of multi-step problems. The Common Core also asks that students be able to define appropriate quantities as well as choose a level of accuracy appropriate to limitations on measurement when reporting quantities. All of these are integral to the approach students will take when solving these problems.

Modeling gets its own heading in the High School Common Core standards⁸, and their description of the skill and its application embodies what students are being asked to do in this unit almost exactly. The Common Core emphasizes choices, assumptions and approximations, which I’ve set out to focus on and develop explicitly within the problems for this unit. Modeling goes beyond what this unit asks students to do, however, to draw in graphical and statistical tools. It’s my hope that solving these problems will provide students with a solid foundation that will allow them to incorporate and understand these other tools with more depth at a later date.

The final notes on standards come from the English and Language Arts. Under the Common Core, there is an upgraded emphasis on argumentative writing⁹. Teachers of all subject areas are using these standards to improve student communication and writing of non-fiction and technical texts. Having students write justifications for their reasoning sounds small, but forces them to document their thought process for others in a way that they more than likely don’t frequently practice. The Common Core also contains a set of Speaking and Listening Standards¹⁰ that ask for participation in discussions and giving presentations in a way that I find extremely productive and helpful to developing and cementing student understanding of their work.