Polya's Approach to Problem Solving

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Questions and Answers

What is Josie's total jogging time in week six?

96 minutes

132 minutes

108 minutes

120 minutes (correct)

How many minutes does Josie increase her jogging time each week?

6 minutes

10 minutes

8 minutes

12 minutes (correct)

If Josie were to continue her routine for another week, what would her total minutes jogged in week seven be?

144 minutes

128 minutes

140 minutes

132 minutes (correct)

In the triangular arrangement of tennis balls, how many balls are in a triangle with 3 rows?

6 balls Signup and view all the answers

What method can be used to find the number of balls in a triangle with 'n' rows?

Adding each row's balls sequentially Signup and view all the answers

If each row has increasing numbers of balls with a pattern, how many balls would a triangle with 5 rows have?

15 balls Signup and view all the answers

What is the total formula for Josie's weekly jogging total if she maintains her schedule for five weeks?

$60 + 12n$ for n weeks Signup and view all the answers

What patterns can be observed in Josie's weekly jogging schedule?

Linear increase in jog time Signup and view all the answers

What is the first step when using the strategy of solving a similar but simpler problem?

Break apart or change the problem into simpler ones. Signup and view all the answers

Which of the following methods is NOT considered a strategy for problem-solving?

Use a calculator for advanced mathematics. Signup and view all the answers

When creating a table to look for a pattern, which of these is the most direct benefit?

It helps visualize relationships between data points. Signup and view all the answers

How many unit squares are found in a 3 unit by 3 unit square picture?

9 Signup and view all the answers

What is the probability of rolling a sum of 7 with two six-sided dice?

1/6 Signup and view all the answers

Which statement best describes the process of 'working backwards' in problem-solving?

Starting from the problem solution and figuring out how to reach the original problem. Signup and view all the answers

In solving a problem, what is the purpose of guessing and checking?

To test multiple scenarios until the answer is found. Signup and view all the answers

When are diagrams particularly useful in problem-solving?

When visualizing and understanding relationships is essential to the problem. Signup and view all the answers

What is the total number of balls in a triangle arrangement with 5 rows?

15 Signup and view all the answers

How do you determine the number of balls in a new triangle based on the old triangle?

By adding the number of rows to the number of balls in the previous triangle Signup and view all the answers

Which of the following correctly describes the pattern for calculating the number of balls?

The number of balls increases by adding 1 for each new row Signup and view all the answers

If there are 28 balls in a triangle with 7 rows, how many balls will there be in a triangle with 8 rows?

36 Signup and view all the answers

What mathematical operation is used to find the total number of balls when adding a new row to the triangle?

Addition Signup and view all the answers

In the context of the problem involving Paul and his piggy bank, what is the main strategy used to find the amount he started with?

Calculating the total from the end and working backwards Signup and view all the answers

When creating a table to represent the number of balls in the triangle, what should be included?

The number of rows alongside the cumulative total of balls Signup and view all the answers

What is the mathematical result of adding the numbers in the first 8 rows together?

36 Signup and view all the answers

Study Notes

Mindfulness Moment

A moment of focused reflection and awareness.

Prayer for Typhoon Relief

Acknowledges the challenges of a typhoon.
Expresses thanks for technology connecting the group remotely.
Provides comfort and support for those affected by the storm.
Recognizes the efforts of responders and caregivers.
Emphasizes resilience, gratitude, hope and determination.

Problem-Solving: Polya's Approach

Focused on heuristic strategies for problem-solving.
Highlights the importance of a practical approach often involving educated guesses, trial and error.

Intended Learning Outcomes

Students will recognize different heuristic strategies (e.g., working backward, guess and check).
Students will understand the heuristic approach and its importance in problem-solving.
Students will apply Polya's four-step process in mathematical and real-life problems (Understand the problem, Devise a plan, Carry out the plan, and Look back).

What is Heuristics

Heuristics are experience-based techniques for problem-solving, learning, and discovery.
Solutions are not guaranteed to be optimal.

Polya's Steps in Problem Solving

Understand the problem
Devise a plan
Carry out the plan
Look back

Polya's Steps in Problem Solving Elaboration

Understand all the words
Restate in own words
Find what you need to find
Diagram
Is enough information provided?

Strategies: Heuristic Approach

Guess and check
Solve similar but simpler problems
Make an organized list
Make a table and look for a pattern
Work backwards
Draw a diagram

Strategies: Devising a Plan

Solve a similar but simpler problem
Make a table and look for a pattern
Work backwards
Guess and check

Solve a Similar but Simpler Problem

Break the problem into parts.
Simplify the problem.
Solve the simplified problem.
Use answers to solve original problem.

Example 1: Squares

To determine the number of squares.
Count squares with sides of 1 unit
Count squares with sides of 2 units
Count squares with sides of 3 unit
Add the results.

Example 2: Probability of Rolling a Sum of 7

To find the probability of rolling a sum of 7 when two six-sided dice are rolled.
Simplify the problem by first trying for a sum of 4.
List all possible outcomes (pairs) that sum to 4.
Divide the number of favorable outcomes (sum to 4) by the total number of outcomes for two dice (6*6 = 36).
Apply the simplified probability to the original problem.

Example 3: Solving an Exponential Equation

To solve for x in 3^2x = 81
Simplify by writing 81 as a power of 3.
Set exponents equal and solve for x.

Make a Table and Look for a Pattern

This method involves systematically organizing data in columns and rows to spot patterns.

Example 1: Josie's Jogging Time

Given Josie's increasing jogging time each week.
The total jogging time in week six is calculated using tabulated week-by-week increase in time and multiplied by six days that Josie jogs.

Example 2: Tennis Balls in Triangular Arrangement

To solve for the number of balls in a triangle with 8 rows.
Create a simple table to track number of rows.
Determine the pattern.
Add the numbers in each row to find the total number of balls.

Work Backwards

A problem-solving strategy.
Solving a problem by beginning at the end point.
Then doing the reversal steps from the end to get to the beginning

Example: Paul's Piggy Bank

Paul's original amount of money in his piggy bank is calculated by reversing the transactions:
Calculate the final total minus the amount for birthday and calculate the remaining money.
Next calculate the remaining amount minus the amount spent on ice-cream.
The last calculation is the original amount saved.

Example 2: Solving for a Variable

A problem involving a variable is simplified into a simpler equation.
The simpler equation is solved for the variable.
The variable is calculated back using the solution to the simplified equation.

Guess and Check

A trial-and-error strategy for problems.
Begin by making an educated guess.
Check the result against the given problem condition.
If result is not correct, adjust your next guess based on your previous answer

Example 1: I am a 2-digit Number

Using the clues, make educated guesses to determine the number.
Narrow the possibilities until the correct answer is found.

Example 2: Nadia's Ribbon

A problem about dividing a ribbon into pieces using numbers.
Devise a guess-and-check strategy by guessing different values for the lengths of the pieces.
Continue to adjust the guesses until correct answer is found.
Using the information on lengths to get the appropriate sizes.

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Description

This quiz focuses on understanding Polya's problem-solving strategies and their application. Students will explore heuristic methods and the importance of practical approaches to solving mathematical and real-life problems. Participants will learn to recognize various strategies and actively apply them in different contexts.