*“Our mission is to give every student the edge to be successful in school as well as in life.”*

### Too often, students regurgitate answers to solve problems instead of applying problem-solving skills. At MathEdge, we value the approach much more than the answer.

MathEdge was created in 2001 to teach students to apply problem-

Classes are taught in a communicative environment. Students are encouraged to verbalize how they have reached the answers to problems so as to emphasize the importance of the reasoning process. Additionally, instructors are better able to pinpoint faulty reasoning and students can better understand the missteps in their logic. This teaching process is one of the main components of MathEdge that differentiates it from other learning centers where self-

## Teaching Method

At MathEdge, classes are interactive and dynamic. Effective problem solving strategies are introduced, fun learning is emphasized, and challenges for the students are presented.

With problem solving, there is no conclusive method for reaching an answer. However, certain approaches are far more efficient and effective than others. When a complicated problem is presented, students are taught how to dissect and reinterpret for optimal understanding and analysis.

No matter how complex the problem appears, our students are trained to learn how to approach it, whether by asking questions or looking for information. As a result, our students are able to tackle difficult problems without quitting.

We encourage students to learn by methodically working through problems, articulating their thought processes, and developing better reasoning skills. By employing the approach and techniques taught at MathEdge, students gain critical skills and knowledge that stays with them for a lifetime.

# Why Math Edge?

MathEdge is an educational program focusing on the problem-solving skills that lead students to success in both academics and the real world. It taps into intellectual curiosity and helps foster the mathematic potential of your child.

More importantly, though, MathEdge gives students confidence and the skills to tackle any type of problem, academic or otherwise. Problem solving is what math is all about. As one discovers how to solve problems one previously thought were impossible, he/she would apply the same calm thought process to all of life’s problems!

# What is problem solving mathematics?

Unlike computational mathematics, problem solving mathematics deals with solving word problems.

### What are word problems?

Word problems describe challenges whose solutions are found by applying mathematical techniques. They are solved by creatively applying math in different ways.

### What do word problems look like?

They state the facts and relationships in plain English. Then ask one or more questions in the end.

### Why are they important?

They are examples of how to use math to solve real life problems.

### What is special about them?

Each is a little different from the rest, and requires creative use of strategies to solve. Simple rote math manipulation is not sufficient.

# How to solve them?

Students are required to read and comprehend the given statements and understand what are being asked. They must answer exactly to the questions but not the interim solutions.

**There are techniques and the following states the steps: **

- Underline or highlight the important facts in the problem, or rewrite them.
- Circle the question(s) or rewrite what are needed to find.
- Use appropriate problem solving strategy (or a combination of them) to solve the problem. Most commonly used strategies are drawing diagrams or pictures to help visualizing the problem, or guess and check.
- Write down each step to help explaining the reasoning.
- Show all calculations
- Draw a rectangular box around the final answer
- Showing unit in the answer is optional. If decides to show unit, it must be correct. Wrong unit is considered wrong answer even the numeral portion of the answer is correct. These days, most of math contests don’t require entering unit(s).

# A system for problem solving (The UPIC method)

**U**nderstand the problem

- Read the math problem once quickly to obtain an idea of its general nature.
- Next read the problem carefully. Underline the relevant information or key words. Understand what facts and/or relationships do they have?
- Ignore the irrelevant info. Organize or rephrase the important information.
- Circle the question(s). It’s very important to accurately understand what are being asked. Restate the question(s) if necessary. What do you know that is not stated in the problem?

**P**ick a strategy or method to solve the problem

- Think whether solved similar problems in the past.
- Decide what strategy or a combination of strategies to use. Most commonly used strategies are “draw pictures”, “list/table”, “work backward”, or guess and check.
- Try a strategy that seems as if it will work.
- If it doesn’t, it may lead to another one that may.

**I**mplement and Solve it

- Using the strategy selected, work the steps to solve the problem.
- It may involve in deciding what operations must be performed and in what order, and then carry out the computation.
- Box the answer.

**C**heck and verify the answer(s)

- Reread the question(s).
- Ask whether answered all question(s).
- If answering with unit, make sure it’s correct unit. (In most math contests, unit is not required. However, answer with wrong unit is considered to be wrong answer.)
- Does your answer seem reasonable?
- Verify the answer by using either approximation or puck the answer back in the question to check whether it makes sense.

### While learning, write down the steps and show work:

It is very important that the student be very methodical to record the steps taken to solve the problem. Why? It helps keep the mind organized when working the problem. It helps avoid making careless mistakes. It allows the teacher to correct the thinking or concepts. It helps check to make sure the answer is correct. In the future, one can refer back to solve similar problems.

# Example of A system for problem solving (UPIC method)

**U**nderstand the problem

**Read the math problem once quickly to obtain an idea of its general nature.****Example:**Auntie Stella from Australia sends money to her niece, Alice, in USA on her birthday. She gives her $10 on her first birthday and promises to sends her $20 more than the birthday before on each birthday thereafter. How old will Alice be when she will receive a total of $1,000 from Auntie Stella since she was born?

**Next read the problem carefully. Underline the relevant information or key words. Understand what facts and/or relationships do they have?****Example:**Auntie Stella from Australia sends money to her niece, Alice, in USA on her birthday. She gives her $10 on her first birthday and promises to sends her $20 more than the birthday before on each birthday thereafter. How old will Alice be when she will receive a total of $1,000 from Auntie Stella since she was born?

**Ignore the irrelevant info. Organize or rephrase the important information.****Example:**She gives her $10 on her first birthday and promises to sends her $20 more than the birthday before on each birthday thereafter. How old will Alice be when she will receive a total of $1,000 from Auntie Stella since she was born?**What does it mean that Alice will get $20 more than on the birthday before:**It means that the amount increases by $20 every year. For birthday year 1, she gets $10. On birthday year 2, she gets $10+$20=$30. On birthday year 3, she gets $30+$20=$50, and so on.

**Circle the question(s). It’s very important to accurately understand what are being asked. Restate the question(s) if necessary.**What do you know that is not stated in the problem?**Example:**She gives her $10 on her first birthday and promises to sends her $20 more than the birthday before on each birthday thereafter.**How old**will Alice be when she will receive a**total of $1,000**from Auntie Stella since she was born?

**P**ick a strategy or method to solve the problem

**Example:** To solve this problem, beginner students would use “Table” strategy while advanced students might use “sum of odds = n2 ” formula.

**I**mplement and Solve it

**Method 1:** Using “Table” strategy: Make a table to keep track of the amount given for each birthday and a running total for all the money Aunt Isabella sent. Continue the chart until the total amount you reach is $1,000.

1

2

3

4

5

6

7

8

9

10

$10

$30

$50

$70

$90

$110

$130

$150

$170

$190

$10

$40

$90

$160

$250

$360

$490

$640

$810

$1,000

**Answer: When Natalie turns 10 years old, she will have received a total of $1,000.**

**Method 2:** Using “sum of odds = n2 ”

Formula:

10 + 30 + 50 + 70 + … = 1000

10 (1+3+5+7+…) = 1000

1+3+5+7+… = 100

n2 = 100 n = 10

**Ans: 10 years old**

**C**heck and verify the answer(s)

Does your answer seem reasonable? Getting a total of $1000 in 10 years means an average of $100 per year seems reasonable.

# Possible problem solving strategies are:

- Draw a picture or diagram
- Make an organized list, chart, or table
- Work backward
- Use logical reasoning
- Guess and check or Trial and Error
- Find a pattern
- Deduction and elimination
- Write equation(s)
- Solve a simpler problem
- Compute or Simplify (arithmetic)
- Use formula
- Act it out or experiment

# What will students learn?

The course covers the system for problem solving, its possible problem solving strategies, and the supporting mathematical concepts.

Getting the right answer is only of secondary importance. The primary goal is to understand and apply the methods. Therefore, writing down or articulating on how to get the answer is more important than getting the correct answer.

Working on problem solving math problems can be like solving puzzle games. They can be fun and challenging at the same time. Being able to solve them can be a very fulfilling experience.

# Examples of Different Ways Solving Problems

### Problem example 1:

It’s October time and Farmer Frank is getting ready for his annual pumpkin sale. To attract attention, he is going to arrange his pumpkins in a triangular display. He put one pumpkin in the first row, 3 pumpkins in the second row, 5 pumpkins in the third row, and so on for 20 rows. How many total pumpkins does he need?

### SOLUTIONS

After interpreting the problem (following the steps in “A System for Problem Solving”), most students would know that it is a problem of finding the answer for this sum: “1+3+5+7+…+39”. However, different level of problem solvers would have different approach in solving this problem.

**Beginner’s method: **

Simply adding: 1+3+5+7+…+39 = 400

**Intermediate’s method: **

There are total of 10 pairs (20 numbers ÷ 2) of 40.

Thus, the result is 40 x 10 = 400.

**Advance’s method: **

Using the “sum of odds = n2 ” formula ∑ 1+3+5+…+39 = 202 = 400

### Problem example 2:

Owl A hoots every 3 hours, Owl B hoots every 8 hours, and Owl C hoots every 12 hours. If they all hoot together at the start, how many times during the next 60 hours will just two owls hoot together?

### SOLUTIONS

**Beginner’s method: **Draw picture

Answer: We can see from the graph that there are 3 times that only 2 owls hoot together.

**Intermediate’s method:** Use a “table”

Answer: From the above table, there are only 3 times that two owls hoot together.

**Advance’s method:** using LCM

Finding the hours that all three owls hoot together is LCM (3, 8, 12) = 24. The problem only wants the answer for just two owls hoot together and not three. So we have to find the frequency of the pair(s) of owls that hoot together excluding those that all three are hooting.

First, picking any two owls and find out when they hoot together:

LCM (3,8) = 24, we can’t use this pair at all because it is the same frequency as when 3 hoot together.

LCM (8,12) = 24, we can’t use this pair at all because it is the same frequency as when 3 hoot together.

LCM (3, 12) = 12, we can use this pair and the only pair.

Second, find the frequency in the 80 hours time span by finding the multiples of 12: 12, 24, 36, 48, 60, 72.

Third, exclude those of multiples of 24 (when 3 hoot together) which left with: 12, 36, 60.

ANSWER: 3 times.