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?

Comments
(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?


REMINDER!
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!

Updates:
(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.

Update Week of 8-24-09: Contest Info

REMINDER!
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!

Updates:
(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 will receive a registration form.

An aside...
I've been asking my kids questions every day to sharpen their minds for school which starts next week. I asked my son how he would spell, arachnophobia, the fear of spiders. He was confident he knew the first four letters: iraq....

Challenge Their Minds Day 1 - A 'Means to an End'

With the school year starting for some and soon for others, here are a couple of ideas to set the tone in our math classes early on. Do not assume these are intended only for your advanced youngsters!


Middle School

1) (No calculator!) What is the average of ninety-nine 1's and one 2?

2) (No calculator!) Find 5 different sets of 5 numbers each of which has a mean of 5.
Note: The wording will be problematic here since students often associate the adjective different with the numbers themselves. Basic grammar, cough, cough...


High School (or advanced middle schoolers)


(No calculator!)
Set S consists of 100 different numbers each of which is between 0 and 1.
Which of the following could be the mean of these 100 numbers?

I. 0.01
II. 0.5
III. 0.98

(A) I only (B) II only (C) I and II (D) I and III (E) I, II, and III

[Yes, there will always be some discussion of "between!"]

A few comments...
(1) These problems are intended to be a springboard for your own creativity. You can do better!!

(2) Each of you probably has your own favorite resources of problems so that you don't have to reinvent the wheel. However, finding high-quality Problems of the Day which are matched to your curriculum is not always easy despite the abundant ancillaries supplied by the publisher and resources on the web.

(3) From the previous comment you can guess that I feel strongly about giving more challenging warm-ups to our students - all of our students (adjusted for backgrounds, abilities, skills). Don't worry that discussion of these will destroy your lesson. Students can work together for 5 minutes while you're taking attendance, checking homework, etc. I usually invited students who solved some or all of these to go to the board and explain their methods. To encourage students to look these over, tell them you will include a variation of one of these questions on the next quiz or test. Start by having it as an Extra Credit problem, then worth a couple of points, gradually increasing their value.

(4) Imagine if our students were exposed to these higher-order types of questions about 180 times a year from middle school on. By the time they take their college-entrance exams or other state assessments (or tests like the ADP End of Course Exams), they will have a much higher degree of comfort and should perform better, although we know that there are so many other factors that go into performance on high-stakes tests.

(5) Yes, the above high school problem is in SAT format. Why do you think I included these kinds on my daily warm-ups? By the way, I'm not promoting ETS but middle and high school teachers may well want to invest in (or ask their supervisor to order) the College Board's book of
10 Real SATs. There is no better source for these kinds of problems and many questions are appropriate for middle schoolers.

Another 'Average' Problem for Standardized Tests and Conceptual Understanding

After 4 tests, Barry's average score was 5 points higher than Michelle's. After the 5th test, Michelle's overall average was 5 points higher than Barry's. Michelle's score on the 5th test was how many points higher than Barry's?

Can you find at least three methods for solving this?
Algebraic, "plug-in", conceptual, etc...

As teachers we need to have a deep understanding of these kinds of problems and familiarity with several approaches. Of course, our students will show us a variety of methods, both right and wrong, when we open up the dialog!


Comments
Students from middle school on see many problems relating to means. However, they need to see a variety of problems of increasing difficulty. This question is certainly not a highly challenging math contest problem but I believe it demonstrates some important principles of averages and can be used to review different problem-solving strategies. Middle schoolers would struggle with the algebraic approach (a system of two equations), however they should be thoroughly comfortable with the underlying ideas.

Since the focus is on concept and method, I will give the answer: 45

Using "SAT-Type" Problems to Develop Understanding of Quadratic Functions in Algebra

f(x) = t-2(x+4)² where t is a constant.
If f(-8.3) = f(a) and a > 0, what is the value of a?


This type of question is of the Grid-in type (or short constructed response) that now appears on standardized testing like the SAT-I and ADP Algebra 2.

I administered it to a group of strong SAT students recently and the students who completed Alg II struggled with it. As our president might say, this was a "teachable moment!"

A few thoughts...
Should textbooks include more questions of this type both as examples and regular homework exercises? As you might guess, I'm very much opposed to having questions labeled as Standardized Test Practice in texts or appear in a separate section of the text or in ancillaries.

By the way, by including the label "SAT-type problems" in the title of this post I'm trying to engender both positive and negative response. Those of you who have followed this blog for 2- 1/2 years know that what I'm really referring to are "conceptually-based questions." Some of you react adversely to the idea that standardized test questions should influence our curriculum or how we teach. N'est-ce pas?

Your comments...

Updates, ODDS AND EVENS and some Geometry Packing Problems

Enjoying your summer hiatus or as busy as ever? I know that feeling!

1. 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 for now will be on Geometry, Algebra II and Precalculus. Several other ideas are running through my head but I need the time to bring them to fruition. If any public, charter, prep, parochial or homeschool (including international school) is interested, send me an email ASAP to receive registration materials: "dmarain 'at' geeeemail dot com."

2. CNNMoney.com Article - Something to tell your students in September!
Here is the link. The 2nd paragraph says it all:

The top 15 highest-earning college degrees all have one thing in common -- math skills.

3. Silly Instruments for Math Teachers to Play
I always told my students that I'm predominantly left-brained -- analytical, organized, detailed, process-oriented, algebraic -- as opposed to most of my children and my wife who are creative, spatial, mechanical, who see the forest more than the trees. One of my sons is a musician and another is a dancer so we are not always on the same wavelength! So I mentioned to my SAT students that I wish I had a more creative side and perhaps be able to play an instrument, but, in fact, the only thing I can "play" is my iPod! One of my students in the front row immediately responded, "I know an instrument you can play, Mr. M -- the triangle! I congratulated her for the cleverness and told her that maybe I will learn how to play the "cymbals." (the class actually applauded that lame attempt at word play!). In fact, I've read that many famous mathematicians were also musicians, so let us know: Do you play an instrument or are passionate about music or do you have a silly instrument for a mathematician to play?

4. Circle Packing Problems
Even though I am dominantly left-brained, I still enjoy challenging spatial geometry problems. I find these questions have improved my creativity and my spatial sense and they often involve multi-faceted thinking. Here are a couple of famous 'packing' problems which are accessible to geometry students. More important than solving these is to give our students a sense of the importance of packing problems and the ongoing research in this area. There are still unsolved problems here!

Although you can easily research packing problems on MathWorld and Wikipedia, the diagrams below come from an exceptional website I discovered. The author, Peter Szabo (missing accents), provides diagrams for packing 2-100 circles with accompanying data (radii, density, etc).


PROBLEM I


The two congruent circles at the left are actually enclosed in a unit square which is not shown.The circles are tangent to each other and to the sides of the square. If these circles have the maximum radius possible, determine the radius.
Note: The indicated square (assume it is a square) is helpful in solving the problem. Trig is not necessary here.

Answer (Yes, I'm providing this since the objective is to discuss the method):
[The following is the diameter, not the radius, of each circle. Thanks to watchmath for correcting this error].









PROBLEM II



Again, imagine that the three congruent circles at the left are enclosed in a unit square and are tangent to each other and to the sides of the square. If the circles have the maximum radius possible, determine this radius.
Notes: The indicated square again may be helpful to solve this problem. Trig can be used but clever use of special right triangles is preferred.

Answer:
[The diameter is given below, not the radius. Thanks to watchmatch for correcting this]

A Morning Warmup for Middle and High Schoolers - No Calculators Please!

How many integers from -1001 ro 1001 inclusive are not equal to the cube of an integer?

Hint: This could be a real 'Thriller'!

Click Read more for comments...


Comments
1) Do you think daily exposure to these kinds of problems as early as 7th grade will improve student thinking, careful attention to details (reading!) and ultimately performance on assessments? I think you can guess my answer!

2) I've published many similar questions on my blog but I couldn't resist this tribute to MJ.

3) I strongly believe we must occasionally remove the calculator to force their thinking. The stronger student recognizes immediately that 1000 and -1000 are perfect cubes and that one does not need to count the cubes but rather the integers which are being cubed (aka, their cube roots). The student with less number sense and weaker basics will feel lost at first but eventually their minds will develop as well if challenged regularly.

4) I added some complications to this fairly common 'counting' problem, similar to many SAT problems. This type of question is also typical of 8th grade math contests. Where do you think the common errors would occur assuming the student has some idea of how to approach this? Is understanding the language the primary barrier or not?

5) Let me know if you use this in September to set the tone for the year!

...Read more

Taking Middle Schoolers Beyond Procedures To The Next Level...

Typical Classroom Scenario?
We're introducing the idea of least common multiple of two positive integers and after defining the terminology and illustrating several examples most students are catching on to some procedural method of which there are many:
Listing common multiples of each
Prime Factorization
The "upside down division method" you saw at a conference...

Yes, we are all very good at demonstrating step by step procedures and having students practice repetitively until they catch on and can reproduce this with some speed and accuracy. We feel this is a worthwhile skill (they'll need it for common denominators, clearing denominators in rational equations, useful for solving certain types of word problems, etc), it's in the curriculum and the standards, it will be tested in various places and the lesson plays out. Some students pick up the method(s) quickly, while others struggle, particularly those who haven't learned their basic facts.

BUT how can we raise the bar to stretch their minds? Can the above scenario be restructured to enable students to gain a deeper understanding of the concepts of lcm and gcf? Perhaps we can start the class off with a more open-ended type of question and ask them to work in small groups to solve it. Perhaps, we can ask a different type of question after teaching some standard procedure. A nonroutine, higher-order question that is not in the text...

What resources are available for more open-ended or nonroutine questions to enable our students to delve beneath the surface and actually think about what they are doing? Well, I can't answer all these questions but here are a few thoughts...


1) Write two examples for which the lcm of two numbers is their product.
2) Write two examples for which the lcm of two numbers is not their product. The numbers in each example must be distinct (different).

3) The lcm of 12 and N is 24.
a) What is the greatest possible integer value of N?
b) What is the least positive integer value of N?


These are just a few samples to start you off. You could probably come up with better ones or you've read some excellent ideas in some publication. Please share...

To see a more challenging version of the examples above, click Read more...



You might want to give the following for homework or an extra practice problem in class. Do you think students will require a calculator? How about telling them they cannot use it!

The lcm of 100 and N is 500. What is the least positive integer value of N?

...Read more