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Module 1: Problem Solving Strategies

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Unlike exercises, there is never a simple recipe for solving a problem. You can get better and better at solving problems, both by building up your background knowledge and by simply practicing. As you solve more problems (and learn how other people solved them), you learn strategies and techniques that can be useful. But no single strategy works every time.

Pólya’s How to Solve It

George Pólya was a great champion in the field of teaching effective problem solving skills. He was born in Hungary in 1887, received his Ph.D. at the University of Budapest, and was a professor at Stanford University (among other universities). He wrote many mathematical papers along with three books, most famously, “How to Solve it.” Pólya died at the age 98 in 1985.1

1. Image of Pólya by Thane Plambeck from Palo Alto, California (Flickr) [CC BY

Screen Shot 2018-08-30 at 4.43.05 PM.png

In 1945, Pólya published the short book How to Solve It , which gave a four-step method for solving mathematical problems:

First, you have to understand the problem.

After understanding, then make a plan.

Carry out the plan.

Look back on your work. How could it be better?

This is all well and good, but how do you actually do these steps?!?! Steps 1. and 2. are particularly mysterious! How do you “make a plan?” That is where you need some tools in your toolbox, and some experience to draw upon.

Much has been written since 1945 to explain these steps in more detail, but the truth is that they are more art than science. This is where math becomes a creative endeavor (and where it becomes so much fun). We will articulate some useful problem solving strategies, but no such list will ever be complete. This is really just a start to help you on your way. The best way to become a skilled problem solver is to learn the background material well, and then to solve a lot of problems!

Problem Solving Strategy 1 (Guess and Test)

Make a guess and test to see if it satisfies the demands of the problem. If it doesn't, alter the guess appropriately and check again. Keep doing this until you find a solution.

Mr. Jones has a total of 25 chickens and cows on his farm. How many of each does he have if all together there are 76 feet?

Step 1: Understanding the problem

We are given in the problem that there are 25 chickens and cows.

All together there are 76 feet.

Chickens have 2 feet and cows have 4 feet.

We are trying to determine how many cows and how many chickens Mr. Jones has on his farm.

Step 2: Devise a plan

Going to use Guess and test along with making a tab

Many times the strategy below is used with guess and test.

Make a table and look for a pattern:

Procedure: Make a table reflecting the data in the problem. If done in an orderly way, such a table will often reveal patterns and relationships that suggest how the problem can be solved.

Step 3: Carry out the plan:

Notice we are going in the wrong direction! The total number of feet is decreasing!

Better! The total number of feet are increasing!

Step 4: Looking back:

Check: 12 + 13 = 25 heads

24 + 52 = 76 feet.

We have found the solution to this problem. I could use this strategy when there are a limited number of possible answers and when two items are the same but they have one characteristic that is different.

Videos to watch:

1. Click on this link to see an example of “Guess and Test”

http://www.mathstories.com/strategies.htm

2. Click on this link to see another example of Guess and Test.

http://www.mathinaction.org/problem-solving-strategies.html

Check in question 1:

clipboard_e6298bbd7c7f66d9eb9affcd33892ef0d.png

Place the digits 8, 10, 11, 12, and 13 in the circles to make the sums across and vertically equal 31. (5 points)

Check in question 2:

Old McDonald has 250 chickens and goats in the barnyard. Altogether there are 760 feet . How many of each animal does he have? Make sure you use Polya’s 4 problem solving steps. (12 points)

Problem Solving Strategy 2 (Draw a Picture). Some problems are obviously about a geometric situation, and it is clear you want to draw a picture and mark down all of the given information before you try to solve it. But even for a problem that is not geometric thinking visually can help!

Videos to watch demonstrating how to use "Draw a Picture".

1. Click on this link to see an example of “Draw a Picture”

2. Click on this link to see another example of Draw a Picture.

Problem Solving Strategy 3 ( Using a variable to find the sum of a sequence.)

Gauss's strategy for sequences.

last term = fixed number ( n -1) + first term

The fix number is the the amount each term is increasing or decreasing by. "n" is the number of terms you have. You can use this formula to find the last term in the sequence or the number of terms you have in a sequence.

Ex: 2, 5, 8, ... Find the 200th term.

Last term = 3(200-1) +2

Last term is 599.

To find the sum of a sequence: sum = [(first term + last term) (number of terms)]/ 2

Sum = (2 + 599) (200) then divide by 2

Sum = 60,100

Check in question 3: (10 points)

Find the 320 th term of 7, 10, 13, 16 …

Then find the sum of the first 320 terms.

Problem Solving Strategy 4 (Working Backwards)

This is considered a strategy in many schools. If you are given an answer, and the steps that were taken to arrive at that answer, you should be able to determine the starting point.

Videos to watch demonstrating of “Working Backwards”

https://www.youtube.com/watch?v=5FFWTsMEeJw

Karen is thinking of a number. If you double it, and subtract 7, you obtain 11. What is Karen’s number?

1. We start with 11 and work backwards.

2. The opposite of subtraction is addition. We will add 7 to 11. We are now at 18.

3. The opposite of doubling something is dividing by 2. 18/2 = 9

4. This should be our answer. Looking back:

9 x 2 = 18 -7 = 11

5. We have the right answer.

Check in question 4:

Christina is thinking of a number.

If you multiply her number by 93, add 6, and divide by 3, you obtain 436. What is her number? Solve this problem by working backwards. (5 points)

Problem Solving Strategy 5 (Looking for a Pattern)

Definition: A sequence is a pattern involving an ordered arrangement of numbers.

We first need to find a pattern.

Ask yourself as you search for a pattern – are the numbers growing steadily larger? Steadily smaller? How is each number related?

Example 1: 1, 4, 7, 10, 13…

Find the next 2 numbers. The pattern is each number is increasing by 3. The next two numbers would be 16 and 19.

Example 2: 1, 4, 9, 16 … find the next 2 numbers. It looks like each successive number is increase by the next odd number. 1 + 3 = 4.

So the next number would be

25 + 11 = 36

Example 3: 10, 7, 4, 1, -2… find the next 2 numbers.

In this sequence, the numbers are decreasing by 3. So the next 2 numbers would be -2 -3 = -5

-5 – 3 = -8

Example 4: 1, 2, 4, 8 … find the next two numbers.

This example is a little bit harder. The numbers are increasing but not by a constant. Maybe a factor?

So each number is being multiplied by 2.

16 x 2 = 32

1. Click on this link to see an example of “Looking for a Pattern”

2. Click on this link to see another example of Looking for a Pattern.

Problem Solving Strategy 6 (Make a List)

Example 1 : Can perfect squares end in a 2 or a 3?

List all the squares of the numbers 1 to 20.

1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400.

Now look at the number in the ones digits. Notice they are 0, 1, 4, 5, 6, or 9. Notice none of the perfect squares end in 2, 3, 7, or 8. This list suggests that perfect squares cannot end in a 2, 3, 7 or 8.

How many different amounts of money can you have in your pocket if you have only three coins including only dimes and quarters?

Quarter’s dimes

0 3 30 cents

1 2 45 cents

2 1 60 cents

3 0 75 cents

Videos demonstrating "Make a List"

Check in question 5:

How many ways can you make change for 23 cents using only pennies, nickels, and dimes? (10 points)

Problem Solving Strategy 7 (Solve a Simpler Problem)

Geometric Sequences:

How would we find the nth term?

Solve a simpler problem:

1, 3, 9, 27.

1. To get from 1 to 3 what did we do?

2. To get from 3 to 9 what did we do?

Let’s set up a table:

Term Number what did we do

problem solving lesson 1 9

Looking back: How would you find the nth term?

problem solving lesson 1 9

Find the 10 th term of the above sequence.

Let L = the tenth term

problem solving lesson 1 9

Problem Solving Strategy 8 (Process of Elimination)

This strategy can be used when there is only one possible solution.

I’m thinking of a number.

The number is odd.

It is more than 1 but less than 100.

It is greater than 20.

It is less than 5 times 7.

The sum of the digits is 7.

It is evenly divisible by 5.

a. We know it is an odd number between 1 and 100.

b. It is greater than 20 but less than 35

21, 23, 25, 27, 29, 31, 33, 35. These are the possibilities.

c. The sum of the digits is 7

21 (2+1=3) No 23 (2+3 = 5) No 25 (2 + 5= 7) Yes Using the same process we see there are no other numbers that meet this criteria. Also we notice 25 is divisible by 5. By using the strategy elimination, we have found our answer.

Check in question 6: (8 points)

Jose is thinking of a number.

The number is not odd.

The sum of the digits is divisible by 2.

The number is a multiple of 11.

It is greater than 5 times 4.

It is a multiple of 6

It is less than 7 times 8 +23

What is the number?

Click on this link for a quick review of the problem solving strategies.

https://garyhall.org.uk/maths-problem-solving-strategies.html

Go Math Interactive Mimio Lesson 1.9 Problem Solving Multiplication and Division

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Problem solving

Problem solving lesson plan

problem solving lesson 1 9

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Problem Solving: Lesson plan

Problem solving: Presentation slides

Demonstrating your skills quick fire activity

Problem solving in practice: Interactive worksheet

Our problem solving content focuses on one of these skills and develops understanding of the six stages of problem solving, as well as identifying different types of situations in which young people might already be using these skills. Furthermore, it encourages them to use an adaptive approach, explaining that different types of problems can be approached in different ways.

The activities on this page support your teaching of these skills through an independent activity, quick activities or a full length, curriculum-linked lesson plan. 

Teaching resources:

  • Problem solving: Lesson plan and presentation slides – full lesson plan including icebreaker for use with a group of students in the classroom
  • Demonstrating your skills: Quick-fire activity  – 10 minute activity for a group of students in the classroom, can be used as an icebreaker for the lesson plan
  • Problem solving in practice: Interactive worksheet – activity for independent learning whether remote or in class

Lesson plan

(60 -75 minutes)

This lesson is designed to equip young people with an adaptable approach to solving problems, large or small. It includes a short film and scenarios that encourage development of practical problem solving skills which can be useful for learning, day to day life, and when in employment.

By the end of the lesson, students will be able to:

  • Identify problems of different scales and what is needed to solve them
  • Illustrate the use of an adaptable approach to solving problems
  • Understand that problem solving is a core transferable skill and identify its usefulness in a work setting
  • Work on a problem solving activity in a team

The lesson aims to reinforce students’ understanding of the potential future applications of this skill as they move into the world of work, particularly in an activity differentiated for an older or more able group on creating new opportunities.

Quick-fire activity

(5 - 10 minutes)

The demonstrating your skills quick-fire activity focuses on helping young people understand the key skills that are needed in the workplace, including the importance of problem solving.

Students will be asked to name the skills being demonstrated in a variety of scenarios, and identify ways they’re already using those skills in this short activity.

You might find it useful as a starter or icebreaker activity to begin a lesson, or at the end to allow students to put what they have just learnt in the Problem solving lesson into practice.

Interactive worksheet

(20 - 25 minutes)

Please note that students below the age of 14 cannot sign up for their own LifeSkills account. Any independent tasks must be printed or downloaded and provided digitally for them to complete as they are currently hosted on educator pages.

The Problem solving in practice interactive worksheet introduces some of the themes from the full lesson plan and gives students some practical strategies for problem solving, including introducing the six stages of problem solving. The worksheet can be printed or completed digitally, so can be used flexibly to give students practise putting their problem solving skills into action. You might choose to assign it:

  • As homework following the Problem solving lesson
  • For independent study
  • For remote learning

Looking for more ways to boost self confidence with LifeSkills?

Other lessons that may prove useful for students to build on these activities include the  Adaptability  and  Innovation and idea generation  lessons. Alternatively, consider encouraging them to apply their skills through  Steps to starting a business  or the  Social action toolkit .

Why not build problem solving in as a focus in your students’ wider curriculum? Refer to our  Content guide to find out how this resources can be used as part of your teaching.

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In this lesson, students will learn how to identify when and where to work on solving a problem, and who should be included. Lesson Plan (PDF) Lesson Presentation

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problem solving lesson 1 9

In this activity from the Growth Mindset & Goal Setting unit for Kindergarten–Grade 1, kids learn three strategies that can help them work through challenges and then use those strategies to build a tower from 10 random items.

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McGraw Hill My Math Grade 5 Chapter 1 Lesson 9 Answer Key Problem-Solving Investigation: Use the Four-Step Plan

All the solutions provided in McGraw Hill Math Grade 5 Answer Key PDF Chapter 1 Lesson 9 Problem-Solving Investigation: Use the Four-Step Plan  will give you a clear idea of the concepts.

McGraw-Hill My Math Grade 5 Answer Key Chapter 1 Lesson 9 Problem-Solving Investigation: Use the Four-Step Plan

1. Understand

What facts do you know?

McGraw Hill My Math Grade 5 Chapter 1 Lesson 9 Answer Key Problem-Solving Investigation Use the Four-Step Plan 9

2. Plan To solve this problem, I can work backwards.

3. Solve The smallest size is ___ centimeter. Victor spent 2 × ___ or ____. Subtract to find the remaining amount spent: $61 – ___ = ____ The largest size is 0.011 centimeter. Each package costs $7. Victor bought $21 ÷ $7, or ___ packages of the largest grain sandpaper.

4. Check Does your answer make sense? Explain.

What facts do you know? ________________ ________________

What do you need to find? ________________

2. Plan ________________ ________________

4. Check Is your answer reasonable ? Explain. ________________

Apply the Strategy

Solve each problem using the four-step plan.

McGraw Hill My Math Grade 5 Chapter 1 Lesson 9 Answer Key Problem-Solving Investigation Use the Four-Step Plan 10

Question 3. Mathematical PRACTICE Keep Trying You divide a number by 3, add 6, and then subtract 7. The result ¡s 4. What is the number? Answer: Let the number be ‘n’ Now divide the number(n) by 3. n/3 + 6 – 7 = 4 Now solve the equation. n/3 – 1 = 4 n – 3/3 = 4 n – 3 = 12 n = 12 + 3 n = 15 Now substitute n value in the equation. 15 – 3 = 12 12 = 12 LHS = RHS

Question 4. Mr Toshio lent out 11 rulers at the beginning of class, collected 4 rulers in the middle of class, and gave out 7 at the end of class. He had 18 at the end of the day. How many rulers did he start with? Answer: The number of rulers Mr Toshio lent out at the beginning of the class is 11 The number of rulers Mr Toshio collected in the middle of class = 4 The number of rulers he gave out at the end of the class = 7 The number of rulers he had at the end of the day was 18 The number of rulers he had at the beginning = X X = (11 – 4) + 18 + 7 X = 7 + 18 + 7 X = 32 Therefore, he started with 32 rulers.

McGraw Hill My Math Grade 5 Chapter 1 Lesson 9 Answer Key Problem-Solving Investigation Use the Four-Step Plan 12

Review the Strategies

Use any strategy to solve each problem.

  • Use the four-step plan.
  • Make a table.
  • Act it out.

Question 6. Mathematical PRACTICE Make a Plan Ming-Li spent $15 at the movies. She then earned $30 babysitting. She spent $12 at the bookstore. She now has $18 left. How much money did Ming-Li have to begin with? Answer: The amount she spent on movies = $15 The amount she earned on babysitting = $30 The amount she spent at the bookstore = $12 The amount she has left = $18 The amount she has to begin with = X X – 15 + 30 – 12 = 18 X – 15 + 18 = 18 After solving the equation, we get 18 gets cancelled. X – 15 = 0 X = 15 Therefore, the money she begin with $15.

Question 7. Mr. Jenkins bought payers for some landscaping projects. He used 120 for a small patio, 86 for the border of a flower bed, and 70 for a wall. He has 24 payers left. How many payers did Mr. Jenkins buy? Answer: The number of payers he used for the small patio = 120 The number of payers he used for the flower bed = 86 The number of payers he used for the wall = 70 The number of payers left = 24 The number of payers he buys = X X = 120 + 86 + 70 + 24 X = 300 Therefore, he bought 300 payers.

McGraw Hill My Math Grade 5 Chapter 1 Lesson 9 Answer Key Problem-Solving Investigation Use the Four-Step Plan 13

Question 9. You multiply a number by 3, subtract 6 and then add 2. The result is 20. What is the number? Answer: Let the number be n 3n – 6 + 2 = 20 Solve the equation. 3n – 4 = 20 3n = 20 + 4 3n = 24 n = 24/3 n = 8

McGraw Hill My Math Grade 5 Chapter 1 Lesson 9 Answer Key Problem-Solving Investigation Use the Four-Step Plan 14

McGraw Hill My Math Grade 5 Chapter 1 Lesson 9 My Homework Answer Key

Problem Solving

McGraw Hill My Math Grade 5 Chapter 1 Lesson 9 Answer Key Problem-Solving Investigation Use the Four-Step Plan 15

Question 2. The volleyball team sold 16 items on the first day of a bake sale, 28 items on the second day, and 12 items on the last day. There were 4 items left that had not been purchased. How many total items were for sale at the bake sale? Answer: The number of items sold by the Volleyball team on the first day = 16 The number of items sold by the Volleyball team on the second day = 28 The number of items sold by the Volleyball team on the third day = 12 The number of items left that are not purchased = 4 The number of total items = x x = add all the items to get the total x = 16 + 28 + 12 + 4 x = 60 Therefore, the total items are 60.

McGraw Hill My Math Grade 5 Chapter 1 Lesson 9 Answer Key Problem-Solving Investigation Use the Four-Step Plan 16

Question 4. The distance between Cincinnati, Ohio, and Charlotte, North Carolina, is about 336 miles. The distance between Cincinnati and Chicago, Illinois, is about 247 miles. If Perry drove from Charlotte to Chicago by way of Cincinnati, find the distance he drove. Answer: Cincinnati, Ohio, and Charlotte, North Carolina all are considered as x1 Cincinnati and Chicago, Illinois are considered as x2 The distance between the cities(x1) = 336 The distance between the cities(x2) = 247 The distance we need to find out for Charlotte to Chicago by way of Cincinnati = D D = 336 – 247 D = 89 Therefore, the distance he dove is 89 miles.

Question 5. Mathematical PRACTICE Plan Your Solution Adison earned $25 mowing her neighbor’s lawn. Then she loaned her friend $18, and got $50 from her grandmother for her birthday. She now has $86. How much money did Adison have to begin with? Answer: Let x be the beginning amount of money The amount earned by Addison by mowing her neighbour’s lawn = $25 The amount she loaned from her friend = $18 The amount she got a birthday gift from her grandmother = $50 the amount she has present = $86 the expression to solve this question is x + 25 – 18 + 50 = 86 x + 75 – 18 = 86 x + 57 = 86 x = 86 – 57 x = 29 Therefore, the beginning amount of money is $29

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