Thursday, August 27, 2009

A Middle School Coin Puzzle To Start The Year

I have an equal number of pennies, nickels and dimes. I also have some quarters which have the same value as the pennies, nickels and dimes combined. If I have no other coins, what is the fewest possible total number of coins I could have? What is the value of all the coins?

(1) An opening day problem?
(2) Would you have students working alone or in small groups?
(3) Would you allow the calculator?
(4) Appropriate for prealgebra students? Students below grade 6?
(5) Is zero a possible answer?
(6) Wording too confusing for most students? Is it ambiguous or clear?
(7) Do you feel there are important underlying concepts and ideas embedded here or is it just a fun puzzle to engage students?
(8) Do students have difficulty in separating number of coins from their value?

MathNotations' Third Online Math Contest is tentatively scheduled for the week of Oct 12-16, a 5-day window to administer the 45-min contest and email the results. As with the previous contest, it will be FREE, up to two teams from a school may register and the focus will be on Geometry, Algebra II and Precalculus. If any public, charter, prep, parochial or homeschool (including international school) is interested, send me an email ASAP to receive registration materials: "dmarain 'at' gmail dot com."
Read Update (4) below!

(1) The first draft of the contest is now complete.
(2) As with the precious two contests there will be one or two questions which require demonstration, that is, the students will have to derive, explain or prove a statement. This is best done freehand and then scanned as a jpeg image which can be emailed as an attachment along with the official answer sheet. In fact, the entire answer sheet can be scanned but there is information on it that I need to have.
(3) Some of the questions are multipart with the last part requiring more generalization.
(4) Even if you have previously indicated that you wish to participate, please send me another email using the title: THIRD MATHNOTATIONS CONTEST. Please copy and paste that into the title. Also, when sending the email pls include your full name and title (advisor, teacher, supervisor, etc.), the name of your school (indicate if HS or Middle School) and the complete school address. I have accumulated a database of most of the schools which have expressed interest or previously participated but searching through thousands of emails is much easier when the title is the same! If you have already sent me an email this summer or previously participated, pls send me one more if interested in participating again.
(5) Finally, pls let your colleagues from other schools in your area know about this. Spread the word! If you have a blog, pls mention the contest. If you're connected to your local or state math teachers association, pls let them know about this and ask them to post this info on their website if possible.
Note: Sending me the email is not a commitment! It simply means you are interested and will receive a registration form.


Totally_clueless said...

Nice one, Dave.

A couple of comments:
(1) Maybe 'amount of money' could be an alternative way to express 'value'
(2) The arithmetic itself becomes more interesting if you leave out the pennies (leading to a non-trivial LCM).


Anonymous said...

I just stumbled across your blog, and I like this problem!

I don't think you can have zero as an option, since you disclose "some quarters". I don't think the pennies take anything away from the LCM connection, since the sum of quarters will always result in either "5" or "0" in the "ones" place, which means that the pennies will all have to be multiples of either 5 or 10. (You can't have, for example, 8 of each coin, p-n-d, because the three extra pennies could never match a multiple of the quarters.) Since nickels are also 5, and dimes are also 10, the pennies will match one or both, logically.

It's a lovely, non-routine problem!

Dave Marain said...

tc and Anonymous--
Thanks for the supportive comments. I hope some middle school teachers will try this out and look for more conceptually based questions of this nature. IMHO, our texts should include more of these types to challenge our children to think more deeply. Imagine if some questions were labeled "Non-Calculator" or "Mental Math!"

Anon, I've always used the term "non-routine" as you did, but I'm wondering if we should drop that label and encourage authors to make them more routine! If we label a problem as more challenging, some students may shy away from them...

tc, I intentionally put the pennies in to help students recognize the lcm intuitively and "see" that one needs 25 of each of the pennies, nickels and dimes and 16 of the quarters. Since some will not "see" this, it becomes a teachable moment.

jd2718 said...

I'd expect kids to quickly catch the need for multiples of 5, and after a go or two, jump to 25. I also like that there is work AFTER they make the realization.


Dave Marain said...

Thanks, Jonathan...

I usually try to create multi-step problems to develop student reasoning. In this problem, one of my many objectives was to highlight the difference between number of coins vs. their value, a common source of confusion for some. I also consider it critical for the instructor to stress the idea that the number of coins can be computed mentally by representing the lcm as 25x16, no calculator needed!

David said...

I really like this opening day scenario. I teach high school and could see this as a great opening and challenge problem to get students thinking ... Thanks!