Updated August 12, 2015

Note: the files are available in both dot-doc Word and pdf formats.

Contents (this website has the same material as the file Lesson 0 Overview in the dropbox)

**1. ****Reason for these
lessons **

Given a math story problem where you have no idea where to begin, what do you do? Figuring out what to do is mathematical problem solving. These lessons are designed to help teachers teach mathematical problem solving. Each lesson has a handout and accompanying questions that enable the teacher to guide the student toward thinking like a mathematician. The problems are story problems appropriate for upper elementary grades.

Please modify and use the lessons as best works for you. The lessons are divided into sections and a very rough time estimate is given for each section. These time estimates are provided only as an approximate guide.

This
introduction provides an overview of problem solving and
suggestions for teaching the ideas.

2.
**What
is a problem
**

** **

I say to my classes, “A group of your friends are going to Disneyland and invite you to go. Your parents say fine, but you need to earn the money.” Is this a problem?

If your parents say “here’s the money,” is there a problem? No, but by saying you need to earn the money, we now have a problem. What are some components of this problem – things you need to figure out to solve it?

- How much do you need?
- How can you earn it?
- When are you going?
- How much time do you have to earn it?

A problem is when you are faced with a situation and you don’t know immediately what to do. If how to proceed is clear, then it is not a problem. If I asked “what is 3 times 4” we know the answer is 12, and that is not a problem!

3.
**It
takes TIME to do a math problem
**

I tell my
students that when they get a problem, they need to
“Stop-Look-Think.” My goal is to teach what “think” means in
terms of mathematical problem solving. I tell them that math
problems take time to solve, and in fact, most word problems
take me a few minutes to do and some problems take me much
longer than that, so why should they think they can do it
faster? You cannot just read a problem as if it were a story.
Instead, you must analyze each word and sentence to find out
what information is given and how it all fits
together.

4.
**Steps
in Mathematical Problem Solving
**

Mathematicians think of problem solving in terms of four steps:

Step 1: Unpack the problem

Step 2: Pick a strategy

Step 3: Solve the problem

Step 4: Answer with a few words

**Unpacking
the problem** is
critical to solving it. Use wwwww (who, what, why, when,
where) to guide you. Investigating a math problem is like a
journalist researching a story; you need to ask the questions
to uncover the clues.

**Pick a
strategy** refers to
all the different ways we can approach a problem such as make
a table, draw a picture, work backwards, guess and check,
group terms, use a Venn diagram, and so on. There are many
strategies and we frequently use more than one in solving any
problem. There are many lessons devoted to introducing the
various approaches.

**Solve the
problem** is when
you actually do some arithmetic. It is the third step, with
two very important steps before it.

**Answer with
a few words **to make your answer clear. In more
advanced problem solving approaches this fourth step is
usually “check your work.” For the children at the
elementary level, having them go back and answer the question
posed by the problem using a few words is a first step in the
checking process. ** **

5.
**Focusing
on
unpacking
(understanding) (step 1)
**

Getting my students to independently focus on understanding the problem had always been a challenge. Over the past several years, I have developed an approach that enables students to really “get it.” I have a poster with the questions as reference for them:

**What to do when you don’t know what to do!**

**Answer these questions:**

- What is the question you have to answer?
- Who is involved in the problem?
- What is involved in the problem?
- What are key words in the problem?
- What else do we know from the problem?
- How will you use this information in solving the problem?
- Make up an example of what is described.

When we start a problem I have my students put their pencils away – they are to do the first part orally – NO WRITING. I have them go through the problems with their learning partner and DISCUSS the questions. I then have a few people share what they have learned.

I tell my classes that stopping to really understand the problem is a very hard step. Most of us never really do this, it is new for them and takes lots of practice to have it become “normal.” I reinforce that this is an important and necessary part of the time it takes to solve mathematical problems.

Then I have
the students use their pencils to do the problems by
themselves, then review their answers with their learning
partner, and finally we discuss as a class.

As I said in item 3 above, problem solving takes time!
Accordingly, be sure to limit the number of problems you
expect your students to do at any one time.

6.
**
Math Sprints in FIVE minutes
**

I call my students Mathletes and tell them that Mathletes are

- Arithmetic Champions
- Word Problem Experts

To assist them in becoming Arithmetic Champions, I do mathematical sprints to start each class. A sprint is something you run quickly. Basically I have two sheets of related arithmetic problems – drill problems – for my students to complete; for example, 60 multiplication facts. I give them one minute to do as many as they can, read the answers, lead a short one minute break activity, and then give them the second page with one minute. The goal is to see improvement scores between first and second sprint. The total time after the first couple of days is FIVE minutes, and the kids love this form of drill. For the break I use the counting activities described in item 7 below.

I use Bill
Davidson’s sprint material and sometimes develop my
own. Bill also has material to enable you to learn the
methods he has developed to create and administer sprints.

7.
**Counting
activities
**

At different points when I feel my students need a break or when I’m transitioning from one part of a lesson to another, I use counting activities. I start these very simply and add complexity over the course of the year. When possible, I like them to cross into the hundreds before stopping. Here’s a partial list:

- Count by 10
- Count by 10 starting at 5 (and then start at 7, then 33, etc.).
- Count down by 10 starting at 155 (and then at 273, etc.)
- Count by 5 ( and then starting at 2, then 6, then 77, etc.)
- Count down by 5 etc
- Count by 6 starting at 6 (and then starting at 9, etc.)

And for later
in the year: Count by ˝ and reduce to lowest terms (1/2, 1,
3/2, 2, …), by 1/4, etc.

8.
**Mathematical
Buzz
**

I try to have
a few minutes at the end of class to play “Buzz.” The
traditional game is one where everyone stands up, you go
around the room and each person says the next number in the
series, saying "buzz" instead of the number where appropriate.
If a person makes a mistake, they sit down; the game continues
until there is a winner. I allow people sitting to stand up
again if they correct a mistake of someone standing. In my classes, I call out the numbers
and the students give the response. That way, I can say 12
three times to get the three different factorizations of 12
(1x12, 2x6 and 3x4), or to repeat a number if someone gives an
incorrect answer.

Traditional Buzz: pick a number, say 7, and students say “buzz” (instead of the number) for all multiples of 7 (7, 14, 21, …) and for any number that has the digit 7 in it (17, 27, 37,…) even though they are not multiples of 7.

Factor Buzz: As you go round the room, students have to give a factorization of the numbers rather than the number. For example, as we go around, the students would say…

1 = 1x1

2 = 1x2

3=1x3

4=2x2 or 1x4

and so on.

I tell the
students that they should ONLY use 1x number only if there is
no other factor. So, for 4, I accept 2x2 and not 1x4. For 12,
I accept 2x6 or 3x4. This activity is also a chance to
reinforce prime and composite numbers, which is an upper
elementary standard.

Prime Buzz: Only a slight
change: once they have gotten more experience with Factor
Buzz, they say “buzz” for a prime number (numbers whose only
factors are 1 and itself) and give the factors for composite
numbers. For example, answers would be as follows: 1, buzz
(for 2), buzz (for 3), 4=2x2, buzz (for 5), 6=2x3, buzz (for
7), 8=2x4, 9=3x3, and so on.

Other variations I do later in the school year are:

Remainder Buzz: Take a number, say 5, for which they have to give remainders. For example, if you say 6 they say 1. If you say 28, they say 3, and so on.

Prime Factorization Buzz: if the number is prime, they say “buzz;” if not they have to give the prime factorization. For 12, they would say 2x2x3, etc. I jump around with the numbers to keep it interesting.

**Multiples
of Ten Buzz:** For
example, 70=7x10, 7=35x2, 70=14x5, etc. My rule for this
version is that I will call each number three times; the first time the
student can give me the simple multiple of ten, and the second
time the multiple of 2, and the third time the multiple of 5.

**Fraction
Buzz:** Simplest
form, do only factors of 2:
say 10, response is 5;
say 5, response is 2.5;
say 1, response is ˝;
etc.

MOEMS, Mathematical
Olympiads for Elementary and Middle Schools, is the
organization I have used to guide my teaching over the
past decade. MOEMS
provides five monthly
contests, given from November to March, which I
administer in my classroom.
The problems are stimulating, challenging, and
provide the structure for teaching mathematical problem
solving. I
extensively used the MOEMS books listed below as a
resource. I
strongly recommend every teacher investigate participating
in the MOEMS program as an option for some or all of their
students.

*Crea**tive Problem Solving in School Mathematics (2nd Ed)**Math Olympiad Contest Problems (Volume 1)**Math Olympiad Contest Problems (Volume 2)**MOEMS*® Contest Problems Volume 3

10.
**Note
about
the people who prepared these materials
**

Click
here
to go the lessons and handouts.
Once in the drop box,
double click on the file name to download it to your
computer. Please feel free to modify the lessons and
handouts to meet your needs and to share them with
others. More materials will be added as the year
progresses.

Return to Enrichment Directory Return to Jason Frand's Homepage

Please feel free to email me with
questions and suggestions.

Los Angeles County-wide Math Olympiad Tournament Chairperson

Linwood Howe Elementary School

Culver City Unified School District

Adjunct Assistant Professor, Retired

Assistant Dean and Director, Retired

UCLA Anderson School of Management originally prepared August 3, 2014