A company named Hammermill manufactures copy paper that it claims to be “99.99% jam free.”
The Challenge(s)
How many paper jams would you expect to have happen?
How many paper jams would you expect to have happen in one ream?
How many paper jams would you expect to have happen in two reams?
How many paper jams would you expect to have happen in one case?
How many paper jams would you expect to have happen in 40 cases?
On average, how many sheets of paper would you expect to use before having your first paper jam?
Question(s) To Ask
These questions may be useful in helping students down the problem solving path:
What is a guess that you know is too little?
What is a guess that you know is too high?
How can we figure out what it means when we have an answer that is less than one paper jam but more than zero paper jams?
Consider This
This problem gives students an opportunity to make sense of what “99.99%” means using the context of paper jams. There are many great opportunities for misconceptions to come out and be addressed. Each of “The Challenge” questions listed above have a specific purpose so let’s analyze them one at a time:
How many paper jams would you expect to have happen?
This question intentionally does not list an amount of paper like the other questions do. As such, there is no way to answer this yet I have feeling that many students will have no problem giving a numeric value. What I want to ultimately come from this question is the understanding that to determine a quantity of paper jams from a percentage, we must know the amount of paper being used.
How many paper jams would you expect to have happen in one ream?
This is the first problem that students can actually answer. Students may not know that 500 sheets of paper are in one ream so this will likely be information they will need to ask you for. To figure out the answer, students need to realize that if 99.99% of the sheets will be jam free, 0.01% will cause a paper jam. 0.01% of 500 sheets of paper is 0.05 paper jams. Since you cannot have 0.05 paper jams, there will likely be 0 paper jams. Try to have a discussion with students where they debate whether the answer should be 0.05 paper jams or 0 paper jams. This question, combined with the next two questions, is hopefully creating a misconception that will be addressed in the final challenge question.
How many paper jams would you expect to have happen in two reams?
Two reams of paper will have 1000 sheets. 0.01% of 1000 sheets of paper is 0.1 paper jams. Since you cannot have 0.1 paper jams, there will likely be 0 paper jams. An alternative, but misleading, method for finding the answer is to take the previous answer of 0 paper jams per ream and double it since there are now two reams. This method will eventually lead to a misconception. Again, try to have a discussion with students where they debate whether the answer should be 0.1 paper jams or 0 paper jams.
How many paper jams would you expect to have happen in one case?
One regular case of paper will have 10 reams for a total of 5000 sheets. 0.01% of 5000 sheets of paper is 0.5 paper jams. Since you cannot have 0.5 paper jams, there will likely be 0 paper jams. Using the alternative method, students may say that there are 0 paper jams because there have been 0 paper jams in each ream so 10 reams should also be 0 paper jams. Again, try to have a discussion with students where they debate whether the answer should be 0.5 paper jams or 0 paper jams.
How many paper jams would you expect to have happen in 40 cases?
The image above shows a palette holding 40 cases of paper (5 layers of 8 cases of paper). 40 cases of paper would hold 200,000 sheets of paper. 0.01% of 200,000 sheets of paper is 20 paper jams. Finally, we should expect paper jams. However, I wonder if some students will try to use the conveniently flawed reasoning from the previous questions and also assume that there are 0 paper jams since each ream has 0 paper jams and 400 reams must also have o paper jams.
On average, how many sheets of paper would you expect to use before having your first paper jam?
There are many ways to figure this answer out but ultimately we want students to reason that there will be 1 paper jam, on average, for every 10,000 sheets of paper since 0.01% of 10,000 is one. This is also the same as 1 paper jam for every 2 regular cases of paper. The image below comes from the Hammermill website and confirms that this is the correct answer.
CCSS 6.RP.3c – Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
CCSS 7.RP.3 – Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.
