How Much Money in the Bowl?

I just returned from doing some work with some amazing 6th teachers in Monterey County.  I feel really inspired by the work we did and want to tell you about it.  It’s my hope that there are other 5th, 6th, and 7th grade teachers out there that might find this lesson useful to use in class.  Please feel free to plagiarize and make it your own as you see fit.  I’d love to hear from you if you use it so we can get better together.  (And as always, special thanks to Andrew Stadel for providing such an array of amazing visual hooks!)

The teachers and I spent two days together doing a lesson inquiry.  On the first day, we spent about 7 hours talking about our passions as math teachers and our vision for a successful math classroom.  Then we examined ways in which their adopted curriculum limits and restricts learning for students and bores students and teachers in equal measure.  (This is another example of a conflict between purpose and practice that I wrote about in another post that you can find here.)

So we sought some ways to inject some energy, curiosity, and inspiration in to the lesson.  The group decided to try out one of Andrew Stadel’s Estimation180 activities that we expanded into a full 3-Act math lesson.  We chose this activity because it aligned with the 6.NS.3 standard (decimal operations) they were working on next week, and we wanted to see what skills the students already had.  The next day, we took turns teaching the lesson (three times total) and observed student learning, examined teacher choices, and revised our lesson plan as we went forward.  We also engaged in several cycles of reflection conversations where we gave each other feedback on missed opportunities in either the lesson plan design or the instructional choices of the teacher.

I’m very grateful that two of teachers let me co-teach the lesson with them.  I miss my classroom days quite a bit.  All in all, the lesson inquiry process led to some meaningful professional growth for the teachers (and me!) and hopefully leads to some long-term instructional shifts for the teachers as they strive to align their vision (an engaged classroom filled with empowered learners) and their practice (working with a cumbersome curriculum that stifles learning).

Our objectives:  

• Students are engaged in thinking about math and talking with and learning from each other.
• Students make estimations and share their reasoning with each other.
• Students show what they know about decimal operations (for teacher to consider for future instruction).

We communicated these objectives to students by embedding these ideas using student friendly language in the Into portion of the lesson.  (Note:  This lesson took about 55-65 minutes, but wasn’t rushed.  It can be made shorter by eliminating the “line-up-in-order-by-estimate” activity at the end of the Into.)

Into:

In less than one minute, we framed the objectives saying something akin to:

Hello everyone!  Today, we’re going to ask you to learn from each other and practice sharing your “math thinking” with your classmates.  We’re going to do this by geeking out and making some estimates.  

We then asked them to do a 30 second pair-share about their understanding of the word “estimate” and think of a time when they had to make an estimate.  After taking a few student responses, students were clear on the goals of the lesson and were grounded in the right mindset for our work.  We wanted this step to be quick, clear, concise, and take less than 2 minutes.

Then a short story to frame student thinking:

So I have this bowl at my house by the front door and everyday I throw my loose change in the bowl.  Do you have a similar bowl, jar, or piggy bank filled with change?  (Heads nod; hands go up; they seem to get what we’re talking about.)

When the bowl gets really full, I like to take it to the coin machine at the grocery store and turn it in to paper money.  But you know what I like to do first?  I like to estimate how much money it is and see how close I got.  It’s a fun math nerdy thing I like to do, and I’m inviting you to do the same.

So I took a picture of it and here it is.  

How much money do you think that is?

We had students confer with their table mates and then write their individual estimates on Post-It notes in large, dark print.  (Note:  Clarify to students that it’s the same bowl, but taken from different angles.)  Then we asked them to take their Post-It notes and line up around the edge of the room forming a human number line. Lots of conversation happening during that moment and it forced students to work together and talk to one another.  Once they were in order, they were standing next to classmates that had similar estimates.  We had them discuss their reasoning briefly in small groups standing around the room.

As a whole class, we had a short discussion about the range of answers and where the middle estimate was.

Lastly, we asked them to ponder the following:

What additional information would you like to know if you wanted to make your estimate more accurate?

Students wanted to know:

1. How many coins in the bowl?
2. How big was the bowl?
3. What types of coins were in the bowl?
4. How many of each type?

Questions 3 and 4 are the most important pieces of information.  We made sure students elaborated their thinking about why this was important and asked students to paraphrase each other:

So you want to know how much of each coin.  What will you do with that information?  How will it help you?

Once we felt that almost all students were clear about the essential question (how much money?) and the sub questions (how many of each coin type?), we had them high-five their neighbors and return to their seats.

Through:

We showed the CoinStar receipt from Andrews website.

So here’s the amount of coins.  Use this information to find the value of the bowl of coins.  I’m curious to see who got the closest estimate.

Just a quick reminder that these 6th graders hadn’t yet studied 6.NS.3 so they were using only the skills they had learned previously.  As a result, we were expecting a diversity of ability levels.  We were ready to identify students who appeared unengaged and quickly diagnosis if it was a lack of motivation issue, a lack of clarity issue, or a lack of basic math skills/confidence issue.  (More analysis about this moment at the end of this post.)

Students got to work.  We gathered some data about what they were doing and floated around the room to ask probing questions to individuals and small groups.

We copied some examples of student work on the board and stopped the students after about 10 minutes.

Two things to anticipate:

1. Some students will add the coin amounts.
2. Some students won’t quite understand how to convert the number of coins into a monetary value.

So one of the examples we posted was showing the sum of all the coins.

Great effort everybody.  I’m really proud of your willingness to dive right in and get curious about this problem.  

I saw several students do this addition problem and get 3053.  What question did these students answer?

We tried to reframe this common reasoning mistake by framing it as “good math but not useful for our current question.”

Then we shared two examples of student work using the quarters:

1. 519 ÷ 4
2. 519 x 25

(Note:  Almost all students who did #2 left out the decimal point and so I did as well.  More on this choice in the analysis below.)

We asked students to explain their reasoning.  We concluded this short conversation by clarifying that these are at least two ways to convert the number of quarters into monetary value.

Armed with this new knowledge, let’s see how much further we can get through this problem.

Students worked for about 10-15 more minutes.  We encouraged really struggling students to think about the value of 917 pennies first, and then try to find the value of 898 dimes.  Some students had calculated the answer all the way through.  We had them pair up and compare their methods and reasoning.  If they finished that, we had them put away their work and help other classmates with their thinking.  We stressed that it was about helping classmates understand and not about just telling them the answer.

Beyond:

Once we felt like students had picked all the meat off the bone that they could, we had some students share their answers.  We compared these answers with their original estimate.  (Note #1:  Almost every student underestimated the amount of money in the bowl.  Note #2:  At the end, there were still about 20-40% of students who still hadn’t arrived at a complete answer, but that we had been able to witness what skills they had and that more time on the problem was only going to lead to unproductive struggle for some students.)

Then we showed the real answer.  (Spoiler alert.)

Who got the closest estimate?

We applauded them for their curiosity and effort.  Then concluded with a short reflection activity to close.

Sometimes the best teacher for you in the classroom is a classmate, and not one of us.  Take a moment and think of a time during the when you learned something from someone else.

They had a pair-share moment after about a minute of silent reflection and then we asked for volunteers to share with the whole class.

Then we ended with an open-ended exit ticket asking students to think of a combination of coins that could total $1.74 and show their reasoning in numbers or pictures.  We wanted to see what conceptual sense making skills they had about numbers and decimal values.  Some of their work is below.

What we noticed and wondered:

• We knew that there was a pivotal moment near the beginning of the Through section when we were asking students to dive in a try some math.  We were worried that they wouldn’t be hooked, but they were.  In reflection, we identified our energy, banter, and story telling during the Into as being one possible causal factor.  Also, the task isn’t terribly intimidating.  They’ve all probably counted change before, just not $264.67.
• There are many many opportunities in this lesson to focus on Math Practice 2:  Reason abstractly and quantitatively.  We were able to see if students really could make contextual sense out of the numbers.  For example, students said that 519 ÷ 4 is $129 R3.  When asked what the R3 meant, they had to grapple with thinking contextually about the meaning of the numbers they were working with.
• Another point about MP2 (that may warrant a future blog post) was how many students jumped in to following algorithms faithfully without thinking.  To find the value of the pennies, most students found the quotient of 917 ÷ 100 using the algorithm.  Which is great if we wanted to see if they knew 6.NS.2 (long division), but not great if we want to see how well they embody the spirit of MP2.  Students should be able to pause and think that 917 pennies is $9.17 without much mental effort.  Students also showed similar blind faith in algorithms when it came to 898 dimes.
• This activity is great for assessing what students know about decimals and place value.  Many students never showed a decimal in their work until the final answer.  Most put the decimal two places to the right of the last digit and knew this intuitively.  “Because cents only has two place values,” said one student.  (I wanted to press him that gas prices sneak on an extra nine-tenths of a hundreth of a dollar, but I let it slide.)  The “two place values” reasoning falls apart when we we examine 898 dimes however.  This is a great opportunity to use this problem to create authentic disequilibrium that causes students to attend to precision (Math Practice 6) and re-evaluate their own thinking.  Some students had minimal awareness about the importance and meaning of the decimal point.  Some students thought that there was $12,975 worth of quarters.  Upon reflection, they realized that that was a lot of money and that something was wrong, but couldn’t quite tell.
• One of the reasons we did this lesson is because we can keep coming back to this problem and their work for teachable moments about why the decimal algorithms make conceptual sense (and cents!).  As students formalize their thinking next week, our hope that this experience creates a lasting memory that helps students see the algorithms aren’t magic and that mathematical rules make logical sense.
• There’s an authentic and fun opportunity during the “line-up-in-order-around-the-room” to weave in some language, vocabulary, and conceptual development around the 6th grade Statistics and Probability standards.  We didn’t take that road, but this estimation activity is certainly primed for it.
• There’s also an authentic opportunity to explore the conceptual overlap and mathematical equivalence of expressions involving rational numbers.
  • For example:  519 ÷ 4 = 519 x (1/4) = 519 x .25 = 519 x (25/100).  All the numbers have concrete representations when thinking about a big pile of quarters.  Written as a decimal, a quarter is $0.25.  Also a dollar is a group of 4 quarters so we need to divide by 4 to find the number of groups.  The number of dollars is 1/4 of the number of quarters because a quarter is a 1/4 of a dollar.  (Why don’t the call dimes “tenths”????)
  • We didn’t have time to dive all the way in to this during our lesson, but the opportunity is there if we wanted to return to it.
• A quick word about differentiation:  Most students were solidly engaged in at least 10-15 minutes of actual math doing.  We encouraged students who were mathematically struggling to focus first on the pennies and dimes and see what they could come up with.  If that didn’t work, we gave them smaller numbers to work with.  “How much would 23 dimes be worth?  How did you get that?”  For stronger students, we were not prepared for an extension within the lesson so we encouraged them to be responsible peer tutors after they had shared their different approaches with each other.

An invitation for feedback and for you to plagiarize:

So that was our experience.  How could we have made it better?  What missed opportunities did we make?  What would you change if you did this in your class?  What would you keep the same?

I’d love to hear about your thoughts and your experiences.  Please share in the comment section of the post!

Until next time math geeks.