More Problems for Fourth and Fifth Grade Students

Triangles, triangles.

Students in Thunder and Lightning have been given two more problems to ponder this week.  The first introduced them to the idea of triangular numbers.  This is a simple idea, and helps students grow in the area of pattern recognition.  It helps them see a nice connection between arithmetic and geometry.  It also sets them up nicely for the next problem!

Pascal's triangle is named after the French mathematician Blaise Pascal.  The triangle has been tracked to Persian and Chinese origins as early as the 11th century, well before Pascal's work in the 17th.  However, Pascal used it extensively and identified many of its properties and applications.  

Mustard Seed School students will be looking for patterns and interesting number sequences that emerge from the triangle.  Among other things, they will find triangular numbers, powers of 2, powers of 11, and the Fibonacci sequence.  See what you can find.

A Classic and Tricky Problem

Right now, middle school students are working on a classic and tricky problem.  This is one that PhD's in mathematics have debated publicly, to the embarrassment of those on the wrong side.

What makes it an appropriate challenge for middle school students is this:

•  It encourages students to begin with a guess
•  Then it encourages them to act out the problem
•  The evidence that emerges will improve their mathematical intuition
•  Then they are encouraged to answer the question "Why?"

If you have a middle school student, you can help by playing the game with them (you can do it with three playing cards, representing the three doors).   Keep track of your results.  What do you observe?  How do you explain this result?  

Thunder and Lightning Join the Fun

Fourth and Fifth Graders at Mustard Seed School (known as Thunder and Lightning) received their first I Love Math problems for 2015 today.  Like the middle school students who began their problem-solving a couple of weeks ago, they will be thinking about non-routine problems that demand a high level of creativity and persistence.

If you are new to the I Love Math season, you might want to look at the last three posts.  They show the kinds of problems that students are solving and the spirit in which they are working right now.  The tone is playful and fun.  Yet our goals for the children are as serious as they could possibly be.

We share the problems here so that parents and students at all grade levels can work on them.  Pick any problem that looks interesting to you.  If you have a child in Thunder and Lightning, then we encourage you to ask your child about these problems.  For the matchstick problem, get out some paper and pencil and begin drawing.  Better yet, get some straws or matches and start making triangles.

The Mind Reader problem begs to be acted out.  It takes 5 minutes, and it is lots of fun.  Play along with your student, and prepare to be amazed.

So if you have tried the Mind Reader a few times, will it always work?  Why?  With so many random choices, why do you keep getting an "o" and an "a"?

While these problems are lots of fun, don't miss the significant math content held inside:

divisibility by 3

divisibility by 9

pattern recognition

uniqueness

randomness

concept of proof (when have you proven something?  how do you build an argument?)

Enjoy!

The Second Problem of 2015

Students have turned in some astonishing work on the Bathtub problem.  Almost all teams came up with correct answers, and provided compelling arguments explaining why the answer would be 20 minutes.  A few submissions had unexpectedly creative ways of envisioning the problem.  The problem can become quite simple if we just "see" it correctly.

Time for the second problem:

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The Sums

of Odd Numbers

Here is a pattern that seems to emerge from observing square numbers.  The sums of odd numbers produce square numbers:

1 = 1  (a square number)

1 + 3 = 4  (another one)

1 + 3 + 5 = 9 (yup!)

1 + 3 + 5 + 7 = 16  (hmmm…….  interesting…….)

1 + 3 + 5 + 7 + 9 = ___________  (well?)

Check a few more.  Will this always be true?

If you don’t think this will always work, come up with a counterexample (in other words, find a case where it does not work).

If you think it will always be true, find a way to show that it will always work (in other words, find a way to convince a skeptic.)

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This problem comes from Proofs Without Words: Exercises in Visual Thinking by Roger B. Nelsen.  We hope that students will get a visual understanding of what square numbers are, where they come from, and how they grow.  

While students are having fun with this problem, they will hardly notice the serious math content they are learning:

square numbers

square roots

exponents

quadratic growth

the distributive law

binomial expansion

for any given square (81 for example), where does the square root come from?

--There are at least three distinct ways to think about this.  Can you think of three different ways that the square root 9 shows up?  is there a fourth?  (I'll give you one interesting answer:  the number of terms in the addition problem.  The sum of the first 9 odd numbers is 81!  Interesting!  Is this an accident?)

rich connections between arithmetic, algebra and geometry

the importance of seeing numbers and relationships visually

the concept of "proof"

pattern recognition

One hint:  Mathematicians draw pictures.  Think of different ways to draw 1 + 3 + 5.  Is there a way to organize your pictures so that you begin to see squares?  If you are stuck, ask your child to help you think!

Enjoy the problem!

The First Problem for 2015

Middle School students received the first I Love Math problem today.  There is much to say about the problem-solving activities we do each year.  In the coming weeks, we will say a great deal!

For today, however, I want to simply (1) share the problem and let you enjoy it with the students, and (2) let Steven Strogatz explain why he loves this problem so much.  Steven Strogatz is a professor of applied mathematics at Cornell, and the author of "The Elements of Math" series which appeared in the New York Times online edition in 2010.

First, the problem:

Could the problem be any simpler?  Three simple sentences.  Forty-nine simple words.  But plenty to think about.

Steven Strogatz was given this problem by his Uncle Irv when he was a boy.  As an adult, he finds that his thoughts return to it again and again.  In his book The Joy of X, a whole chapter is prompted by his memories of the bathtub problem.  Strogatz says, "There are broader lessons to be learned here -- lessons about how to solve problems approximately when you can't solve them exactly, and how to solve them intuitively, for the pleasure of the Aha! moment."

Dr. Strogatz goes on, "Word problems give us practice in thinking not just about numbers, but about relationships between numbers -- how the flow rates of the faucets affect the time required to fill the tub, for example.  And that is the essential next step in anyone's math education.  Understandably, a lot of us have trouble with it;  relationships are much more abstract than numbers.  But they're also much more powerful.  They express the inner logic of the world around us.  Cause and effect, supply and demand, input and output, dose and response -- all involve pairs of numbers and the relationships between them.  Word problems initiate us into this way of thinking."

We have many hopes, dreams and prayers for Mustard Seed's students in these days of problem solving.  May they grow in this next step of abstraction, and see more of the inner logic of the world around them!

Welcome to I Love Math 2015!

I Love Math Day is coming up on February 12, 2015.

I Love Math is a season in the school year when students, teachers, parents, and the whole community celebrate an everlasting love for math.  It is also a time when we work on some of our most profound hopes for Mustard Seed School students for their growth as mathematicians.  This is a season for:

• helping students learn more of the breadth and beauty of mathematics

• expanding children's understanding of what math is

• playful problem solving

• building persistence

• developing mathematical creativity

Already, middle school students have been considering the question:  "What do Mathematicians Do?"  Each grade level generates its own list.  At the beginning of the year, most classes create a list of 25 or 30 things.  We add to the list as we go through the year, and it often grows to 40 to 60 items.  Who knew?

A list like this serves at least three wonderful purposes.  

• It expands student thinking about what mathematics is.

• It helps children realize that they are mathematicians.  As the list grows, we ask, "Do you do these things?  When you do them, you are doing the things mathematicians do.  You must be a mathematician!"

• It helps students develop a broad array of problem solving strategies.  When a student is stuck, or needs a new strategy, we always ask the question, "What do mathematicians do?"  The list usually contains several helpful strategies for any particular problem.

Please look over the list below to see the kinds of things middle school students think mathematicians do.

Please also return to this blog frequently over the next several weeks.  You will find more postings about the activities of I Love Math season.  Importantly, you will find some of our favorite problems that we can all work on together.  You will also get the news about a distinguished visitor or two who love math.  Stay tuned.

What Do Mathematicians Do?

2014-2015

The list is long, gathered from students and adults at Mustard Seed School.  If sometimes a word choice sounds like it came from a child, it probably did.  They own this list!

1.     Solve problems

2.     Create problems

3.     Graph

4.     Predict answers

5.     Make grids

6.     Use coordinates

7.     Discover new things 

8.     Create order

9.     Create disorder

10.  Arrange numbers

11.  Rearrange numbers

12.  Support and build technology

13.  Analyze stocks

14.  Find formulas

15.  Explain formulas

16.  Use what they know about small problems to solve big problems

17.  Cook

18.  Ask questions

19.  Experiment with numbers

20.  Count

21.  Estimate

22.  Approximate

23.  Make boundaries

24.  Think, think, think

25.  Prove things

26.  Disprove things

27.  Calculate

28.  Arithmetic

29.  Draw pictures

30.  Make art

31.  Observe patterns

32.  Simplify problems

33.  Make assumptions

34.  Make mistakes

35.  Consider the infinite

36.  Consider the very small

37.  Act out problems

38.  Factor

39.  Speak the language of creation

40.  Speak the language of the universe

41.  Speak the language of God

42.  Collaborate

43.  Get help

44.  Use resources