This task is one that dropped into my lap at just the right time. I had planned to teach a lesson in a 5th grade class for two reasons: 1) to keep my math teacher muscles toned and 2) to get some student work to share about “What if…?” for an upcoming conference. The problem was perfect because it is problem-based, feels like a puzzle, and involves multiplying a whole number by a fraction (the content the teacher said they were working on at the time.
The Task:
The original task comes from NCTM – I received it in a monthly email blast they send out. I liked it, but I didn’t want to just give it to students as is. I wanted to make it more like a “live” 3-act Task. So, I got a jar (see above) and filled it with $3.52 made up of one-fifth dimes, one-sixth nickels, one-tenth quarters, and the rest pennies (the reveal below, tells all).
The Launch:
I’ve been collecting coins since I was about your age. Whenever I come home and have change in my pocket, I drop it in a large jar just inside my front door. It looks like this:
But, sometimes, I forget to empty my pockets before I go upstairs at night and I have to empty my change on my dresser. That used to be a problem, because the change would pile up and just get in the way. So, I decided I would bring a small jar to put change in and then empty it into the larger jar when it gets full.
And this is my smaller jar right now. What do you notice? What do you wonder?
Notices and Wonders:
| Notice | Wonder |
| There are silver and copper coins | How much money is in there? |
| It looks like there is more than a dollar or maybe two dollars. | How many coins is that? |
| The jar is about one-quarter or one-fifth full. | What kinds of coins are in there? |
| It’s a glass jar. | Why does it have a black lid? |
| The lid screws on. | Is there a half-dollar in there? |
| It has more air than coins. | How many of each coin are in there? |
I brought this in to challenge you to figure out how many of each coin are in the jar. I have some clues I’m going to give you. It may help if you and your partner write them down.
- One-fifth of the coins are dimes.
- One-sixth of the coins are nickels.
- One-tenth of the coins are quarters.
- The rest of the coins in the jar are pennies.
Teacher: Talk with your partner. What information would be helpful right now.
Students: The total number of coins in the jar.
Teacher: Well, I can’t give you the total number of coins, but I can tell you the total amount of money and maybe you could estimate the total number of coins just by looking at it. The total amount of money is $3.52.
The Thinking/Work Session:
Some things I overheard:
- Student: I think 30 coins because 30 works with all of the fractions.
- Student: I think 120 coins because 120 works with all of the fractions. But, it doesn’t look like 120 coins. Hmm.
- Student: I think 70…
- Student: 70 only works with one-fifth, and one-tenth…
Questions for Supporting Students
- What are you trying to figure out? The number of pennies, nickels, dimes, and quarters in your jar.
- What does the $3.52 tell you? The amount of money in the jar.
- What do you know?
- How many coins do you think might be in the jar?
- What do the fractions tell you about the number of coins in the jar?
- Looking at the number of coins in the jar, which of those numbers seems reasonable?
The students persevered and solved the problem. Many of them solving it incorrectly at first. More than a few groups of students used 30 as the number of coins, but when they tested to see if the coins totaled $3.52, they were a bit short. Some groups chose to try 60 next, since it worked with the fractions nicely. They recalculated everything and solve the problem. One group decided to try 60 coins and really looked at their work for 30 coins and thought that everything should be doubled since 60 is double 30. This was a very efficient move.
After sharing solutions and then counting each of the different coins in the jar, I asked students to ask a “What if…” question to extend this problem to something more they would like to learn.
This is now my favorite part of the lesson. I think it is for students, too, because they get to ask any question related to the problem we just solved. They get to stay curious and think creatively.
Here are some of their What ifs:
One group wanted to know how much the coins in my jar weighed. This quiet group in the back, asked their teacher for the scale, weighed one of each of the coins from the jar on the digital scale in grams. Then, they multiplied that by the numbers in their solution to find the weight. When we weighed all of the coins, their solution was off by only 4 grams! Then, they kept going to find out how much it might weigh if the jar was full – even thought they couldn’t possibly check it.
At this point, students have purchased their tickets to ride the exhilarating math train. They’ve boarded. And I am more than happy to punch their ticket!