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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

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:

Chickens | Cows | Number of chicken feet | Number of cow feet | Total number of feet |

20 | 5 | 40 | 20 | 60 |

21 | 4 | 42 | 16 | 58 |

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

19 | 6 | 38 | 24 | 62 |

Better! The total number of feet are increasing!

15 | 10 | 30 | 40 | 70 |

12 | 13 | 24 | 52 | 76 |

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:

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

Looking back: How would you find the nth term?

Find the 10 th term of the above sequence.

Let L = the tenth term

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

## Strategy: Make a Table

## Practice this math problem solving strategy, Make a Table to Solve a Problem, with the help of these free printable problems.

Download this make a table to solve a problem set of word problems for your 1st, 2nd and 3rd grade math students.

These worksheets will be a helpful addition to your problem solving collection.

This is another free resource for teachers and homeschool families from The Curriculum Corner.

Looking to help your students learn to make a table to solve a problem?

This math problem solving strategy can be practiced with this set of resources.

## Math Problem Solving Strategies

This is one in a series of resources to help you focus on specific problem solving strategies in the classroom.

Within this download, we are offering you a range of word problems for practice.

Each page provided contains a single problem solving word problem.

Below each story problem you will find a set of four steps for students to follow when finding the answer.

This set will focus on the make a table strategy for math problem solving.

## What are the 4 problem solving steps?

After carefully reading the problem, students will:

- Step 1: Circle the math words.
- Step 2: Ask yourself: Do I understand the problem?
- Step 3: Solve the problem using words and pictures below.
- Step 4: Share the answer along with explaining why the answer makes sense.

## Draw a Table to Solve a Problem Word Work Questions

The problems within this post help children to see how they can make a table when working on problem solving.

These problems are for first and second grade students.

Within this collection you will find nine different problems.

You will easily be able to create additional problems using the wording below as a base.

With these word problems students are encouraged to draw pictures, but then to take it a step further by putting the information into a table to help answer the questions.

This is a great start to showing students how to organize information as a necessary step in problem solving.

## The problems include the following selections:

- Fixing Bikes
- Flower Petals
- Lovely Ladybugs
- Spider Legs
- Feet and Inches
- Counting Nickels
- Counting Dimes
- Counting Quarters
- Quarters in a Dollar

You can download this set of Make a Table to Solve a Problem pages here:

Problem Solving

You might also be interested in the following free resources:

- Draw a Picture to Solve a Problem
- Write a Number Sentence to Solve a Problem
- Addition & Subtraction Word Problem Strategies
- Fall Problem Solving
- Winter Problem Solving
- Spring Problem Solving
- Summer Problem Solving

As with all of our resources, The Curriculum Corner creates these for free classroom use. Our products may not be sold. You may print and copy for your personal classroom use. These are also great for home school families!

You may not modify and resell in any form. Please let us know if you have any questions.

chona obregon

Monday 28th of December 2020

Nice worksheets. Thank you for sharing it to us.

Jill & Cathy

Monday 1st of February 2021

You're welcome!

Tammy Nicholson

Friday 19th of July 2013

Love your worksheets! Thanks so much!

Thursday 11th of July 2013

Just wanted to let you know that I really appreciate your website and the wealth of activities, checklists, games, center ideas, etc. that are contained in your website. I also appreciate you sharing these things without charging. Thank you for helping educators make a difference in the lives of the students we teach.

Wednesday 10th of July 2013

I love the simplicity of these for my class. I plan to add them to my learning centers. Thank you for sharing them.

Saturday 29th of June 2013

These are great and will be very useful to me! Thank you.

Question: You save $3 on Monday. Each day after that you save twice as much as you saved the day before. If this pattern continues, how much would you save on Friday? Strategy: 1) UNDERSTAND: You need to know that you save $3 on Monday. Then you need to know that you always save twice as much as you find the day before. 2) PLAN: How can you solve the problem? You can make a table like the one below. List the amount of money you save each day. Remember to double the number each day.

You save $48 on Friday

- PRINT TO PLAY
- DIGITAL GAMES

## Problem-Solving Strategies

October 16, 2019

There are many different ways to solve a math problem, and equipping students with problem-solving strategies is just as important as teaching computation and algorithms. Problem-solving strategies help students visualize the problem or present the given information in a way that can lead them to the solution. Solving word problems using strategies works great as a number talks activity and helps to revise many skills.

## Problem-solving strategies

1. create a diagram/picture, 2. guess and check., 3. make a table or a list., 4. logical reasoning., 5. find a pattern, 6. work backward, 1. create a diagram/draw a picture.

Creating a diagram helps students visualize the problem and reach the solution. A diagram can be a picture with labels, or a representation of the problem with objects that can be manipulated. Role-playing and acting out the problem like a story can help get to the solution.

Alice spent 3/4 of her babysitting money on comic books. She is left with $6. How much money did she make from babysitting?

## 2. Guess and check

Teach students the same strategy research mathematicians use.

With this strategy, students solve problems by making a reasonable guess depending on the information given. Then they check to see if the answer is correct and they improve it accordingly. By repeating this process, a student can arrive at a correct answer that has been checked. It is recommended that the students keep a record of their guesses by making a chart, a table or a list. This is a flexible strategy that works for many types of problems. When students are stuck, guessing and checking helps them start and explore the problem. However, there is a trap. Exactly because it is such a simple strategy to use, some students find it difficult to consider other strategies. As problems get more complicated, other strategies become more important and more effective.

Find two numbers that have sum 11 and product 24.

Try/guess 5 and 6 the product is 30 too high

adjust to 4 and 7 with product 28 still high

adjust again 3 and 8 product 24

## 3. Make a table or a list

Carefully organize the information on a table or list according to the problem information. It might be a table of numbers, a table with ticks and crosses to solve a logic problem or a list of possible answers. Seeing the given information sorted out on a table or a list will help find patterns and lead to the correct solution.

To make sure you are listing all the information correctly read the problem carefully.

Find the common factors of 24, 30 and 18

Logical reasoning is the process of using logical, systemic steps to arrive at a conclusion based on given facts and mathematic principles. Read and understand the problem. Then find the information that helps you start solving the problem. Continue with each piece of information and write possible answers.

Thomas, Helen, Bill, and Mary have cats that are black, brown, white, or gray. The cats’ names are Buddy, Lucky, Fifi, and Moo. Buddy is brown. Thoma’s cat, Lucky, is not gray. Helen’s cat is white but is not named Moo. The gray cat belongs to Bill. Which cat belongs to each student, and what is its color?

A table or list is useful in solving logic problems.

Thomas | Lucky | Not gray, the cat is black |

Helen | Not Moo, not Buddy, not Lucky so Fifi | White |

Bill | Moo | Gray |

Mary | Buddy | Brown |

Since Lucky is not gray it can be black or brown. However, Buddy is brown so Lucky has to be black.

Buddy is brown so it cannot be Helen’s cat. Helen’s cat cannot be Moo, Buddy or Lucky, so it is Fifi.

Therefore, Moo is Bill’s cat and Buddy is Mary’s cat.

## 5. Find a pattern.

Finding a pattern is a strategy in which students look for patterns in the given information in order to solve the problem. When the problem consists of data like numbers or events that are repeated then it can be solved using the “find a pattern” problem-solving strategy. Data can be organized in a table or a list to reveal the pattern and help discover the “rule” of the pattern.

The “rule” can then be used to find the answer to the question and complete the table/list.

Shannon’s Pizzeria made 5 pizzas on Sunday, 10 pizzas on Monday, 20 pizzas on Tuesday, and 40 pizzas on Wednesday. If this pattern continues, how many pizzas will the pizzeria make on Saturday?

Sunday | 5 |

Monday | 10 |

Tuesday | 20 |

Wednesday | 40 |

Thursday | |

Friday | |

Saturday |

## 6. Working backward

Problems that can be solved with this strategy are the ones that list a series of events or a sequence of steps .

In this strategy, the students must start with the solution and work back to the beginning. Each operation must be reversed to get back to the beginning. So if working forwards requires addition, when students work backward they will need to subtract. And if they multiply working forwards, they must divide when working backward.

Mom bought a box of candy. Mary took 5 of them, Nick took 4 of them and 31 were given out on Halloween night. The next morning they found 8 pieces of candy in the box. How many candy pieces were in the box when mom bought it.

For this problem, we know that the final number of candy was 8, so if we work backward to “put back” the candy that was taken from the box we can reach the number of candy pieces that were in the box, to begin with.

The candy was taken away so we will normally subtract them. However, to get back to the original number of candy we need to work backward and do the opposite, which is to add them.

8 candy pieces were left + the 31 given out + plus the ones Mary took + the ones Nick took

8+31+5+4= 48 Answer: The box came with 48 pieces of candy.

Selecting the best strategy for a problem comes with practice and often problems will require the use of more than one strategies.

## Print and digital activities

I have created a collection of print and digital activity cards and worksheets with word problems (print and google slides) to solve using the strategies above. The collection includes 70 problems (5 challenge ones) and their solution s and explanations.

sample below

How to use the activity cards

Allow the students to use manipulatives to solve the problems. (counters, shapes, lego blocks, Cuisenaire blocks, base 10 blocks, clocks) They can use manipulatives to create a picture and visualize the problem. They can use counters for the guess and check strategy. Discuss which strategy/strategies are better for solving each problem. Discuss the different ways. Use the activities as warm-ups, number talks, initiate discussions, group work, challenge, escape rooms, and more.

Ask your students to write their own problems using the problems in this resource, and more, as examples. Start with a simple type. Students learn a lot when trying to compose a problem. They can share the problem with their partner or the whole class. Make a collection of problems to share with another class.

For the google slides the students can use text boxes to explain their thinking with words, add shapes and lines to create diagrams, and add (insert) tables and diagrams.

Many of the problems can be solved faster by using algebraic expressions. However, since I created this resource for grades 4 and up I chose to show simple conceptual ways of solving the problems using the strategies above. You can suggest different ways of solving the problems based on the grade level.

Find the free and premium versions of the resource below. The premium version includes 70 problems (challenge problems included) and their solutions

There are 2 versions of the resource

70 google slides with explanations + 70 printable task cards

70 google slides with explanations + 11 worksheets

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Use this teaching video from iMaths to teach the Problem-Solving Strategy ‘Make a table or chart’. This is one episode from a series of 10 Problem-Solving Strategy videos available at iMaths Online.

## Fun teaching resources & tips to help you teach math with confidence

## Math Strategies: Problem Solving by Finding a Pattern

One important math concept that children begin to learn and apply in elementary school is reading and using a table. This is essential knowledge, because we encounter tables of data all the time in our everyday lives! But it’s not just important that kids can read and answer questions based on information in a table, it’s also important that they know how to create their own table and then use it to solve problems, find patterns, graph equations, and so on. And while some may think of these as two different things, I think problem solving by making a table and finding a pattern go hand in hand!

–>Pssst! Do your kids need help making sense of and solving word problems? You might like this set of editable word problem solving templates ! Use these with any grade level, for any type of word problem :

## Finding Patterns in Math Problems:

So when should kids use problem solving by finding a pattern ? Well, when the problem gives a set of data, or a pattern that is continuing and can be arranged in a table, it’s good to consider looking for the pattern and determining the “rule” of the pattern.

As I mentioned when I discussed problem solving by making a list , finding a pattern can be immensely helpful and save a lot of time when working on a word problem. Sometimes, however, a student may not recognize the pattern right away, or may get bogged down with all the details of the question.

Setting up a table and filling in the information given in the question is a great way to organize things and provide a visual so that the “rule” of the pattern can be determined. The “rule” can then be used to find the answer to the question. This removes the tedious work of completing a table, which is especially nice if a lot of computation is involved.

But a table is also great for kids who struggle with math, because it gives them a way to get to the solution even if they have a hard time finding the pattern, or aren’t confident that they are using the “rule” correctly.

Because even though using a known pattern can save you time, and eliminate the need to fill out the entire table, it’s not necessary. A student who is unsure could simply continue filling out their table until they reach the solution they’re looking for.

Helping students learn how to set up a table is also helpful because they can use it to organize information (much like making a list) even if there isn’t a pattern to be found, because it can be done in a systematic way, ensuring that nothing is left out.

If your students are just learning how to read and create tables, I would suggest having them circle their answer in the table to show that they understood the question and knew where in the table to find the answer.

If you have older students, encourage them to find a pattern in the table and explain it in words , and then also with mathematical symbols and/or an equation. This will help them form connections and increase number sense. It will also help them see how to use their “rule” or equation to solve the given question as well as make predictions about the data.

It’s also important for students to consider whether or not their pattern will continue predictably . In some instances, the pattern may look one way for the first few entries, then change, so this is important to consider as the problems get more challenging.

There are tons of examples of problems where creating a table and finding a pattern is a useful strategy, but here’s just one example for you:

Ben decides to prepare for a marathon by running ten minutes a day, six days a week. Each week, he increases his time running by two minutes per day. How many minutes will he run in week 8?

Included in the table is the week number (we’re looking at weeks 1-8), as well as the number of minutes per day and the total minutes for the week. The first step is to fill in the first couple of weeks by calculating the total time.

Once you’ve found weeks 1-3, you may see a pattern and be able to calculate the total minutes for week 8. For example, in this case, the total number of minutes increases by 12 each week, meaning in week 8 he will run for 144 minutes.

If not, however, simply continue with the table until you get to week 8, and then you will have your answer.

I think it is especially important to make it clear to students that it is perfectly acceptable to complete the entire table (or continue a given table) if they don’t see or don’t know how to use the pattern to solve the problem.

I was working with a student once and she was given a table, but was then asked a question about information not included in that table . She was able to tell me the pattern she saw, but wasn’t able to correctly use the “rule” to find the answer. I insisted that she simply extend the table until she found what she needed. Then I showed her how to use the “rule” of the pattern to get the same answer.

I hope you find this helpful! Looking for and finding patterns is such an essential part of mathematics education! If you’re looking for more ideas for exploring patterns with younger kids, check out this post for making patterns with Skittles candy .

## And of course, don’t miss the other posts in this Math Problem Solving Series:

- Problem Solving by Solving an Easier Problem
- Problem Solving by Drawing a Picture
- Problem Solving by Working Backwards
- Problem Solving by Making a List

## One Comment

I had so much trouble spotting patterns when I was in school. Fortunately for her, my daughter rocks at it! This technique will be helpful for her when she’s a bit older! #ThoughtfulSpot

Comments are closed.

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## Math Problem Solving Strategies

Happy Wednesday! I recently read Chapter 4 of the book, What’s your Math Problem? and this was my favorite chapter so far. This chapter is all about the students using math problem solving strategies to get themselves organized before or while they are solving a problem.

The math problem solving strategies discussed in this chapter are:

- Look for a Pattern
- Create a Table
- Create an Organized List
- Guess and Check (My Favorite! This one rarely let me down in college math..hehe!)

It is important to note that these strategies do not have to be used in isolation. In fact, many students will use the strategies together or with another strategy. For example, a student may organize their guesses from Guess and Check into a table (Create a Table strategy).

Here is a break down of each strategy:

Look for a Pattern : This strategy starts in Kindergarten and has the students looking for numerical relationships through identifying, extending, or generalizing patterns.

Create a Table : This strategy will helps the student organize the information presented in the problem so that they can use another strategy to reach the solution.

Create an Organized Lis t: this strategy is very similar to Create a Table in that it helps the students organize their information.

Guess and Check: This strategy, though often used by many students, could be developed into a more efficient problem solving strategy.

I loved how the book was all about modeling these strategies with the students. If modeled correctly and practiced enough in the younger grades, these students will be able to use these strategies fluently to solve more complex problems in grades 3-5.

The chapter provided different problems for each strategy to try out. The purpose was to have the reader reflect on how they were using the strategy and whether it was effective or not. Yikes! Math problems during the summer?! But, this inspired me to write similar word problems to use to model these strategies in the first weeks of school. Here they are with a chalkboard format. This format would be good for projecting them on a smartboard or promethean board because who has all that ink. 😀

## Share the Knowledge!

Reader interactions, 15 comments.

July 3, 2013 at 1:42 pm

I just LOVE this! The patterning and guess and check strategies are excellent skills to have. I think having these up around the classroom to remind students they are perfectly find 'options' or ways to solve questions! So many students skip questions because they don't know how to solve it and I know I've taught guess and check methods which have really helped my year 9s! 🙂

Thanks for the very useful information Jennifer! 🙂

Liz – BaysideMathTeacher

July 3, 2013 at 2:14 pm

I love guess and check! It has saved me many times on a math exam in college! Thanks for stopping by!

July 3, 2013 at 2:04 pm

I spend some time on these strategies while launching math workshop–it would be so beneficial if they were introduced in the younger grades. I have to do a lot of front loading just to get them familiar with the terms and when to use them…appropriately. (The THINK framework helps with having students choose a strategy and why they chose it as a discussion first, then from the list of discussed strategies they can choose one that they want to use to solve the problem.)

I have to be more vigilant about this in the beginning of the year and spend more time on each strategy (the students had a lot of trouble answering the short and extended response questions)–I especially love Guess, Check, and Revise. I struggle with teaching students the difference between using a table and creating a list–at times, I feel like they are interchangeable.

Thank you for the posters–they are definitely going up in my room 🙂

July 3, 2013 at 2:15 pm

I would love to hear more about this THINK framework you mention. And I LOVE how you call it Guess, Check, and Revise!

Glad you like the posters! Jennifer

July 3, 2013 at 2:48 pm

One strategy I found to work for my kids last year (especially since we were starting a new math series that was WAY different than what they were used to) was draw a model. I used the models you mentioned above as well, but a lot of my kiddos needed to draw it out or use manipulatives to finally understand what was happening in the problem. These posters are great…can't wait to use them!

July 3, 2013 at 7:27 pm

Laura, I love using models! I am always telling my kids, "If you just draw a model, you will figure out the problem." We have an acronym in our room that we use that requires them to draw a model or other visual. We call it LOVE. Labels, Our Thinking, Visuals, Equations.

Thanks for stopping by!

July 3, 2013 at 6:11 pm

Thank you so much for the math posters. These will become my 3rd grade anchor charts! I LOVED this chapter because it is where the rubber meets the road, so to speak. My favorite strategy is/and always has been guess and check. It is my "go to" strategy and has served me well. Having said that, I feel the four strategies present a more complete approach to problem solving and it is my goal to expose my students to these strategies and give them the opportunities to practice and learn each approach. I believe the most difficult strategy for me is making a table. What to put where has always be a bit fuzzy to me. I want to work on this and be more confident as I use it in my classroom. I agree with Jen's post…students need to be familiar with these strategies before they enter the middle grades. It is my goal for my 3rd graders to know these strategies, which is more appropriate when, and if they need to use more than one, as well as, to give them confidence as they grow mathematically.

July 3, 2013 at 7:29 pm

Hi Pamela, I, too, am least comfortable with the Create a Table approach. And, yes can you imagine how great it would be if our kids knew these strategies before they came to us? It would make our lives so much easier!

Thanks for stopping by! Jennifer

July 5, 2013 at 5:35 am

In teaching 6th grade for 3 years now, looking and analyzing tables has been difficult for me, too! I see now how looking for patterns and then creating the table work together. I was analyzing the create a table strategy in isolation from make a pattern strategy. As I read on, I am seeing the pattern and creating the table and loving it. I'm even preferring.

July 3, 2013 at 6:29 pm

Great chapter…I am, and always have been, partial to the Guess and Check method. However, I've noticed when working with students that they are hesitant to use this method while problem solving. I never knew why until I read the following on p.100: "Sometimes we are uncomfortable with making an estimate or guess because we have been taught that in mathematics, we should follow a particular rule or algorithm."

Makes sense to me!

Great posters, Jennifer! Thank you for sharing!! 🙂

July 3, 2013 at 7:31 pm

Mandy, thanks for sharing that quote from the book. It is interesting how much we shape the kids; minds to believe one thing or another about math, intentionally or not. I don't know about you, but I have really changed the way I view math instruction over the last couple of years! There is a wealth of research out there that is opening my eyes!

July 5, 2013 at 7:20 am

Out of these four strategies, I think that Making an Organized List and Looking for a Pattern are my "go-to" strategies when solving math problems. I like a systematic approach as it is not only logical to me but also allows me to interpret misconceptions and be able to see where a mistake was made.

In my class I am supportive of all strategies as long as they can be proven to work in more than just an isolated case.

Love the posters and thing they would work great as a reminder for students!

Jennifer Smith-Sloane 4mulaFun

July 5, 2013 at 2:33 pm

I love this chapter. Teaching how to organize in tables is not my strength. I love guess and check. I worked the problems and had to focus on the tables. I know that with the implementation of STAAR, these patterns have gotten much more complex. I have been working on a good way to teach this! I love seeing my kids use strategies that they have and they share with others. My rule is it has to be mathematically sound (will it work to solve other similar problems). I saw some issues this year with students not answering what the question is asking so reading this book will make a huge difference in my math teaching this year.

July 5, 2013 at 3:33 pm

Great summary of the chapter. Using tables is not one of my students' favorite tools, but I encourage them to try all methods and to become familiar with them. Not all problems can be solved with the same strategies so if they are having difficulties I suggest they try a different method. Many times I think students want an immediate answer and give up if their favorite method doesn't work. I need to work on how to encourage them to continue and finish what they have started.

Thanks for sharing the posters. Ana

August 14, 2013 at 6:58 pm

Some students lack good testing strategies. My son struggled with this particularly in math. He experienced anxiety every time he took his exams. He attends a decent California public school but he just wasn’t doing so well. This school year, I decided to get him a math tutor in San Jose . The tutor really helped him improve his grades by getting him more organized and utilizing the strategies mentioned above.

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## Problem solving strategies

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## What are problem solving strategies?

Strategies are things that Pólya would have us choose in his second stage of problem solving and use in his third stage ( What is Problem Solving? ). In actual fact he called them heuristics . They are a collection of general approaches that might work for a number of problems.

There are a number of common strategies that students of primary age can use to help them solve problems. We discuss below several that will be of value for problems on this website and in books on problem solving.

Common Problem Solving Strategies

- Guess (includes guess and check, guess and improve)
- Act It Out (act it out and use equipment)
- Draw (this includes drawing pictures and diagrams)
- Make a List (includes making a table)
- Think (includes using skills you know already)

We have provided a copymaster for these strategies so that you can make posters and display them in your classroom. It consists of a page per strategy with space provided to insert the name of any problem that you come across that uses that particular strategy (Act it out, Draw, Guess, Make a List). This kind of poster provides good revision for students.

## An in-depth look at strategies

We now look at each of the following strategies and discuss them in some depth. You will see that each strategy we have in our list includes two or more subcategories.

- Guess and check is one of the simplest strategies. Anyone can guess an answer. If they can also check that the guess fits the conditions of the problem, then they have mastered guess and check. This is a strategy that would certainly work on the Farmyard problem described below but it could take a lot of time and a lot of computation. Because it is so simple, you may have difficulty weaning some students away from guess and check. As problems get more difficult, other strategies become more important and more effective. However, sometimes when students are completely stuck, guessing and checking will provide a useful way to start to explore a problem. Hopefully that exploration will lead to a more efficient strategy and then to a solution.
- Guess and improve is slightly more sophisticated than guess and check. The idea is that you use your first incorrect guess to make an improved next guess. You can see it in action in the Farmyard problem. In relatively straightforward problems like that, it is often fairly easy to see how to improve the last guess. In some problems though, where there are more variables, it may not be clear at first which way to change the guessing.
- Young students especially, enjoy using Act it Out . Students themselves take the role of things in the problem. In the Farmyard problem, the students might take the role of the animals though it is unlikely that you would have 87 students in your class! But if there are not enough students you might be able to include a teddy or two. This is an effective strategy for demonstration purposes in front of the whole class. On the other hand, it can also be cumbersome when used by groups, especially if a largish number of students is involved. Sometimes the students acting out the problem may get less out of the exercise than the students watching. This is because the participants are so engrossed in the mechanics of what they are doing that they don’t see the underlying mathematics.
- Use Equipment is a strategy related to Act it Out. Generally speaking, any object that can be used in some way to represent the situation the students are trying to solve, is equipment. One of the difficulties with using equipment is keeping track of the solution. The students need to be encouraged to keep track of their working as they manipulate the equipment. Some students need to be encouraged and helped to use equipment. Many students seem to prefer to draw. This may be because it gives them a better representation of the problem in hand. Since there are problems where using equipment is a better strategy than drawing, you should encourage students' use of equipment by modelling its use yourself from time to time.
- It is fairly clear that a picture has to be used in the strategy Draw a Picture . But the picture need not be too elaborate. It should only contain enough detail to help solve the problem. Hence a rough circle with two marks is quite sufficient for chickens and a blob plus four marks will do a pig. All students should be encouraged to use this strategy at some point because it helps them ‘see’ the problem and it can develop into quite a sophisticated strategy later.
- It’s hard to know where Drawing a Picture ends and Drawing a Diagram begins. You might think of a diagram as anything that you can draw which isn’t a picture. But where do you draw the line between a picture and a diagram? As you can see with the chickens and pigs, discussed above, regular picture drawing develops into drawing a diagram. Venn diagrams and tree diagrams are particular types of diagrams that we use so often they have been given names in their own right.
- There are a number of ways of using Make a Table . These range from tables of numbers to help solve problems like the Farmyard, to the sort of tables with ticks and crosses that are often used in logic problems. Tables can also be an efficient way of finding number patterns.
- When an Organised List is being used, it should be arranged in such a way that there is some natural order implicit in its construction. For example, shopping lists are generally not organised. They usually grow haphazardly as you think of each item. A little thought might make them organised. Putting all the meat together, all the vegetables together, and all the drinks together, could do this for you. Even more organisation could be forced by putting all the meat items in alphabetical order, and so on. Someone we know lists the items on her list in the order that they appear on her route through the supermarket.
- Being systematic may mean making a table or an organised list but it can also mean keeping your working in some order so that it is easy to follow when you have to go back over it. It means that you should work logically as you go along and make sure you don’t miss any steps in an argument. And it also means following an idea for a while to see where it leads, rather than jumping about all over the place chasing lots of possible ideas.
- It is very important to keep track of your work. We have seen several groups of students acting out a problem and having trouble at the end simply because they had not kept track of what they were doing. So keeping track is particularly important with Act it Out and Using Equipment. But it is important in many other situations too. Students have to know where they have been and where they are going or they will get hopelessly muddled. This begins to be more significant as the problems get more difficult and involve more and more steps.
- In many ways looking for patterns is what mathematics is all about. We want to know how things are connected and how things work and this is made easier if we can find patterns. Patterns make things easier because they tell us how a group of objects acts in the same way. Once we see a pattern we have much more control over what we are doing.
- Using symmetry helps us to reduce the difficulty level of a problem. Playing Noughts and crosses, for instance, you will have realised that there are three and not nine ways to put the first symbol down. This immediately reduces the number of possibilities for the game and makes it easier to analyse. This sort of argument comes up all the time and should be grabbed with glee when you see it.
- Finally working backwards is a standard strategy that only seems to have restricted use. However, it’s a powerful tool when it can be used. In the kind of problems we will be using in this web-site, it will be most often of value when we are looking at games. It frequently turns out to be worth looking at what happens at the end of a game and then work backward to the beginning, in order to see what moves are best.
- Then we come to use known skills . This isn't usually listed in most lists of problem solving strategies but as we have gone through the problems in this web site, we have found it to be quite common. The trick here is to see which skills that you know can be applied to the problem in hand. One example of this type is Fertiliser (Measurement, level 4). In this problem, the problem solver has to know the formula for the area of a rectangle to be able to use the data of the problem. This strategy is related to the first step of problem solving when the problem solver thinks 'have I seen a problem like this before?' Being able to relate a word problem to some previously acquired skill is not easy but it is extremely important.

## Uses of strategies

Different strategies have different uses. We’ll illustrate this by means of a problem.

The Farmyard Problem : In the farmyard there are some pigs and some chickens. In fact there are 87 animals and 266 legs. How many pigs are there in the farmyard?

Some strategies help you to understand a problem. Let’s kick off with one of those. Guess and check . Let’s guess that there are 80 pigs. If there are they will account for 320 legs. Clearly we’ve over-guessed the number of pigs. So maybe there are only 60 pigs. Now 60 pigs would have 240 legs. That would leave us with 16 legs to be found from the chickens. It takes 8 chickens to produce 16 legs. But 60 pigs plus 8 chickens is only 68 animals so we have landed nearly 20 animals short.

Obviously we haven’t solved the problem yet but we have now come to grips with some of the important aspects of the problem. We know that there are 87 animals and so the number of pigs plus the number of chickens must add up to 87. We also know that we have to use the fact that pigs have four legs and chickens two, and that there have to be 266 legs altogether.

Some strategies are methods of solution in themselves. For instance, take Guess and improve . Supposed we guessed 60 pigs for a total of 240 legs. Now 60 pigs imply 27 chickens, and that gives another 54 legs. Altogether then we’d have 294 legs at this point.

Unfortunately we know that there are only 266 legs. So we’ve guessed too high. As pigs have more legs than hens, we need to reduce the guess of 60 pigs. How about reducing the number of pigs to 50? That means 37 chickens and so 200 + 74 = 274 legs.

We’re still too high. Now 40 pigs and 47 hens gives 160 + 94 = 254 legs. We’ve now got too few legs so we need to guess more pigs.

You should be able to see now how to oscillate backwards and forwards until you hit on the right number of pigs. So guess and improve is a method of solution that you can use on a number of problems.

Some strategies can give you an idea of how you might tackle a problem. Making a table illustrates this point. We’ll put a few values in and see what happens.

pigs | chickens | pigs legs | chickens’ legs | total | difference |

60 | 27 | 240 | 54 | 294 | 28 |

50 | 37 | 200 | 74 | 274 | 8 |

40 | 47 | 160 | 94 | 254 | -12 |

41 | 46 | 164 | 92 | 256 | -10 |

From the table we can see that every time we change the number of pigs by one, we change the number of legs by two. This means that in our last guess in the table, we are five pigs away from the right answer. Then there have to be 46 pigs.

Some strategies help us to see general patterns so that we can make conjectures. Some strategies help us to see how to justify conjectures. And some strategies do other jobs. We’ll develop these ideas on the uses of strategies as this web-site grows.

What strategies can be used at what levels?

In the work we have done over the last few years, it seems that students are able to tackle and use more strategies as they continue with problem solving. They are also able to use them to a deeper level. We have observed the following strategies being used in the stated Levels.

Levels 1 and 2

- Draw a picture
- Use equipment
- Guess and check

Levels 3 and 4

- Draw a diagram
- Guess and improve
- Make a table
- Make an organised list

It is important to say here that the research has not been exhaustive. Possibly younger students can effectively use other strategies. However, we feel confident that most students at a given Curriculum Level can use the strategies listed at that Level above. As problem solving becomes more common in primary schools, we would expect some of the more difficult strategies to come into use at lower Levels.

Strategies can develop in at least two ways. First students' ability to use strategies develops with experience and practice. We mentioned that above. Second, strategies themselves can become more abstract and complex. It’s this development that we want to discuss here with a few examples.

Not all students may follow this development precisely. Some students may skip various stages. Further, when a completely novel problem presents itself, students may revert to an earlier stage of a strategy during the solution of the problem.

Draw: Earlier on we talked about drawing a picture and drawing a diagram. Students often start out by giving a very precise representation of the problem in hand. As they see that it is not necessary to add all the detail or colour, their pictures become more symbolic and only the essential features are retained. Hence we get a blob for a pig’s body and four short lines for its legs. Then students seem to realise that relationships between objects can be demonstrated by line drawings. The objects may be reduced to dots or letters. More precise diagrams may be required in geometrical problems but diagrams are useful in a great many problems with no geometrical content.

The simple "draw a picture" eventually develops into a wide variety of drawings that enable students, and adults, to solve a vast array of problems.

Guess: Moving from guess and check to guess and improve, is an obvious development of a simple strategy. Guess and check may work well in some problems but guess and improve is a simple development of guess and check.

But guess and check can develop into a sophisticated procedure that 5-year-old students couldn’t begin to recognise. At a higher level, but still in the primary school, students are able to guess patterns from data they have been given or they produce themselves. If they are to be sure that their guess is correct, then they have to justify the pattern in some way. This is just another way of checking.

All research mathematicians use guess and check. Their guesses are called "conjectures". Their checks are "proofs". A checked guess becomes a "theorem". Problem solving is very close to mathematical research. The way that research mathematicians work is precisely the Pólya four stage method ( What is Problem Solving? ). The only difference between problem solving and research is that in school, someone (the teacher) knows the solution to the problem. In research no one knows the solution, so checking solutions becomes more important.

So you see that a very simple strategy like guess and check can develop to a very deep level.

## Problem Solving: Eliminating Possibilities

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## What Is It?

Eliminating Possibilities is a strategy in which students remove possible answers until the correct answer remains. Here's an example of a problem that can be solved by Eliminating Possibilities:

The product of an unknown number multiplied by four is less than 35. The unknown number is divisible by four. What two numbers could the unknown number be?

The answer can be found by eliminating the numbers that do not meet the rules and choosing the numbers (four and eight) that remain.

## Why Is It Important?

This is a problem-solving strategy that can be used in basic math problems or to help solve logic problems. Eliminating possibilities helps students organize information and evaluate which pieces of information they will use, eliminating the information that does not fit. It encourages students to consider all options and narrow the possibilities to reasonable choices.

## How Can You Make It Happen?

Introduce a problem to students that will require them to eliminate possibilities in order to solve the problem. For example:

In the game of football, a team can score either a touchdown for six points or field goal for three points. If a team only scores touchdowns or field goals but does not get any extra points (no points for an extra kick) what scores cannot be achieved if the team scored under 30 points?

Understand the Problem

Demonstrate that the first step is understanding the problem. This involves identifying the key pieces of information needed to find the answer. This may require students to read the problem several times or put the problem into their own words.

In this problem, students understand that there is a finite set of possible answers. Students will have to find all of the possible answers and then narrow down the list according to the criteria in the problem.

The score can be 1 through 29. The score must be a multiple of 3 or 6.

Choose a Strategy

The strategy of eliminating possibilities can be used in situations where there is a set of possible answers and a set of criteria the answer must meet.

Solve the Problem

First, list the numbers 1 through 29, because the problem states that the score was less than 30.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

Next, eliminate answers that are not possible solutions. Work through each criteria to find the solution.

Any multiple of six would be a possible score of the game. If the team only scored touchdowns, they could score 6, 12, 18, 24 and so on. Therefore, all multiples of six should be eliminated.

1 2 3 4 5 * 7 8 9 10 11 * 13 14 15 16 17 * 19 20 21 22 23 * 25 26 27 28 29

Any multiple of three would be a possible score of the game. If a team scored only field goals, they could score 3, 6, 9, and so on. Therefore, all multiples of three should be eliminated.

1 2 * 4 5 * 7 8 * 10 11 * 13 14 * 16 17 * 19 20 * 22 23 * 25 26 * 28 29

The answer to the problem is that the following scores could not be the score of the game:

1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 28, 29

Read the problem again to be sure the question was answered.

The scores that would not be possible in this game are listed.

Check the math to be sure it is correct.

Look through the answers you have eliminated and those that are remaining to make sure you have applied all the criteria in the problem.

Determine if the best strategy was chosen for this problem or if there was another way to solve the problem.

Eliminating possibilities was a good strategy to use for this problem.

The last step is explaining how you found the answer. Because this strategy involves logic, it is important for students to talk or write about their thinking. Demonstrate how to write a paragraph describing the steps you took and how you made decisions throughout the process.

First, I listed the possible scores. Then I started to eliminate scores that were not possible. I found the multiples of six and crossed them out. Then I found the multiples of three and crossed them out. I was left with all of the possible scores.

Guided Practice

Have students try solving the following problem using the strategy of eliminating possibilities.

Find the numbers between 10 and 30 that are divisible by 4.

Have students work in pairs, in groups, or individually to solve this problem. They should be able to tell or write about how they found the answer and justify their reasoning.

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## Problem-Solving Strategy: Make a Table - Reteach

In this math tables worksheet, learners read the word problem to understand the information. Students then picture the information in the table. Learners use the table to help them solve the problem. Students then make a table to help them solve the last two problems.

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- What’s new in GTO Wizard

## Bounty Models Explained: Solving Knockout Tournaments

We’ve just rolled out a revolutionary update! You can now find optimal postflop ICM strategies tailored for any tournament format, including classic freezeouts, satellites, progressive knockouts (PKO), regular knockouts (KO), and mystery bounties. Learn more about the update here !

To achieve this, we developed new methods for accurately modeling the value of bounties. This involved creating algorithms to calculate the expected value (EV) of knocking out players both in the current hand and in future hands. In this article, we’ll unveil the engineering solutions that make solving these bounty formats possible!

## Table of Contents

The scope of bounty tournaments, how gto wizard ai models bounties, future bounty ev, icm value of your stack, putting it all together.

Let’s start with a brief summary of what bounty tournaments are before we dive into the details.

In knockout tournaments, a bounty is a prize you win for each player you eliminate . There are primarily three types of knockout tournaments:

- Standard Knockout (KO) – When you eliminate a player in a KO, you claim their entire bounty. This is immediately added to your bankroll. The buy-in split is typically 50% toward the prize pool and 50% toward your bounty.
- Progressive Knockout (PKO) – When you eliminate a player in a PKO, you cash out half their bounty. The other half of their bounty is added to your own head, to be awarded to whoever knocks you out. The only exception is if you win the tournament, then you unlock your own bounty. The buy-in split is typically 50% towards prizes, 25% to the capturable bounty, and 25% added to your own head.
- Mystery Bounty – Essentially, it is just a KO tournament where the value of each bounty is random and unknown during the event. This can be modeled as a standard knockout tournament using the average bounty value. The average bounty is typically just half the total prize pool divided by the number of entries.

You can read more about knockout formats and how to strategically approach each of them in this article .

The latest update allows you to plug in any stack sizes, assign bounties, set arbitrary payout structures. One of the major engineering challenges we faced was finding a way to model the value of a bounty in knockout tournaments .

At first, this seems like a simple problem. If one player stacks another, they get some portion of that player’s bounty. The solution looks straightforward. Just add the value of a bounty to a player’s EV whenever they knock someone out.

Not so fast! There are two hurdles:

- Knockout probabilities – How do you estimate how likely one player is to knock out another without knowing their strategies on future streets?
- Future bounty EV – If you have a big stack, you’re more likely to collect bounties in the future. How do we model the value of winning bounties in the future?

Let’s search for the puzzle pieces we need in order to lay this postflop ICM+bounty puzzle.

To calculate the expected value (EV) of anything, you need two pieces of information:

- The value of outcomes
- The probability of those outcomes
- In this case, the outcomes = knockouts

Traditional CFR Counterfactual Regret Minimization (CFR) Counterfactual Regret Minimization is an algorithm that is commonly used to solve large imperfect-information games. solvers calculate all the way to the river. That means players’ strategies on every runout and betting line are known and well-defined. This information makes it easy to calculate how often one player will knock out another, and therefore the EV of knockouts is trivial to determine (for a computer). However, GTO Wizard AI only solves one street at a time .

However, the work involved for traditional solvers can take minutes or hours to complete for one flop; it’s not feasible to do this on a web application like GTO Wizard. We use a shortcut to solve spots thousands of times faster than traditional solvers. Instead of calculating the entire strategy to the river, GTO Wizard AI calculates:

- The current street
- Then estimates the value of hands on future streets using AI

Hours worth of solving compressed down to seconds!

While this speedup is great, it also means that if you solve a turn spot, it doesn’t calculate the river strategy. Future streets are abstracted, so there’s no way to directly calculate the knockout probabilities. We needed a new method!

We leveraged the power of artificial intelligence to solve this problem. We generated hand histories by allowing our traditional (CFR) solver to play against itself for hundreds of millions of hands over a wide variety of spots. Then, we trained a neural network on this data to estimate the EV of knockouts directly. In other words, we use AI to predict the expected value of winning bounties . Instead of solving and then iterating through the entire game tree to find the EV of bounties, we just plug the data into a neural network. Simple, fast, and effective!

Keep in mind that this AI estimate is only used on earlier streets. By the river, we can directly calculate how often one player will stack the other.

Knockout probabilities only deal with the likelihood of immediately stacking someone in a hand. But as any MTT pro knows, the size of your stack has a huge influence on your ability to accumulate bounties in the future. If you cover many players, then you have more potential opportunities to snag a bounty. If you’re short-stacked, then you’re at risk of busting and are less likely to collect bounties.

So, how do we account for the value of acquiring future bounties? We use the Proportional Bounty Model .

The Proportional Bounty Model argues that a player’s share of the future bounties is proportional to their share of the chips in a tournament.

## Future Bounty EV = Chip Share × Remaining Bounty Prizes

- Chip Share = (your chips / all chips in the tournament)
- Remaining Bounty Prizes = the amount of money locked up in bounty prizes remaining in the tournament

For example, if you increase your chip share by 1% and the remaining bounties are $10,000, that would mean you can expect to win another $100 in bounties.

The astute reader will realize that this is essentially just a Chip EV calculation . The proportional bounty model claims that your share of future bounties is proportional to your share of the chips, which is precisely what a Chip EV model asserts. Research shows that the proportional bounty method is an extremely close approximation to your true bounty EV.

Note that the Proportional Bounty Model isn’t new. This method has been around since at least 2018 (possibly earlier). Prior to this, PKO players had been using Bounty Power —a Chip EV approximation of bounty values—since well before 2018.

So far we know the value of bounties.

The final piece of the puzzle is estimating the value of stacks using the Independent Chip Model (ICM) —a mathematical formula that converts your tournament stack into a monetary value.

Calculating ICM values for large fields was a long and tedious process in the past. ICM can be calculated directly by using the basic method outlined in the article that’s been linked above, or it can be calculated using Monte Carlo simulation. The problem is that, for large tournament fields, the basic method is too computationally complex, and the Monte Carlo method takes too long to converge.

☑ Value of bounties ☐ Value of stacks

That’s why we developed a new mathematical method to calculate (not estimate) ICM values. This new technique allows us to solve large fields 100x to 1000x faster than traditional methods! This means that you can find optimal strategies for even the largest MTTs. We can’t give out all the secret sauce, but you can read more about our ICM breakthrough here .

Let’s recap. You’re simulating a PKO postflop spot. We use AI to predict the EV of bounties gained this hand. We use a proportional model to figure out how many bounties you’ll get in the future. And we use ICM to figure out how much prize money your stack is worth. With these three pieces of information, we can very accurately predict optimal strategies for any postflop MTT spot.

Let’s walk through a simple example to demonstrate how all this works.

It’s the first hand of a PKO. Some maniac player open-shoves.

- $200 (rake-free) buy-in ($100 to the prize pool, $50 claimable bounty, $50 goes to the head of whoever knocks someone out)
- 100 players
- 100bb starting stacks
- Prizes paid to top 10% of field

You call the shove with AA and stack them. Congrats on winning their bounty! But how much value did you gain? Can we quantify this? Let’s compare your value before and after stacking them to make it simple.

Before the hand began, you invested $200 to play this PKO, and have an equal share of the chips to every other player. Ignoring any skill edge, you’d expect to win $100 in prizes and $100 in bounties. In other words, before the first hand, your stack was just worth the $200 you paid to enter the MTT .

After stacking the maniac, you immediately win $50 from their bounty .

You’ve doubled your stack, so you have 200bb out of 10,000bb or 2% of all the chips. There’s $9950 remaining in bounty prizes, and you expect to win 2% of those, so your future bounty EV is $199 . This is a straightforward Chip EV calculation.

Lastly, we calculate the ICM value of your stack. With a Chip EV approximation, you’d expect to win 2% of the untouched $10,000 prize pool. However, the independent chip model doesn’t value chips linearly. Doubling your stack won’t quite double its value. After all, knocking out that maniac didn’t just benefit you; it also slightly benefits everyone else who remains since they are one step closer to the money. Therefore, the ICM value of your stack is closer to $195 . (This varies by a few percent depending on the payout structure).

All in all, you’ve gained about $244 total!

Modeling the value of bounties is crucial for solving optimal tournament strategy in today’s games. To recap, we break the problem down into three parts:

- Bounty EV – The expected value of winning your opponent’s bounty this hand. Bounty EV is predicted using a neural network on flops and turns (when players are not all-in) and calculated precisely on the river.
- Future Bounty EV – The expected value of winning bounties in future hands. This is calculated using the proportional bounty model, which predicts that your share of bounties is proportional to your share of the chips.
- The ICM value of your stack – As predicted by the Independent Chip Model, powered by GTO Wizard ICM .

To be precise, our neural network predicts current and future bounty EV in one step.

With these three pieces of the puzzle, we have a complete picture that accurately predicts chip utility in tournaments, allowing MTT players to solve optimal postflop strategies for all formats and take their game to the next level.

The BB’s incentives for calling an SB preflop raise differ from those of a cold-caller facing a raise from, say, LJ or CO. And the SB’s incentives for raising differ from those of an open-raiser in any other position.

Tom is a long time poker theory enthusiast, GTO Wizard coach and YouTuber, and author of the Daily Dose of GTO.

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## The facts about Kamala Harris' role on immigration in the Biden administration

By Camilo Montoya-Galvez

Updated on: July 23, 2024 / 11:48 AM EDT / CBS News

Following President Biden's decision to abandon his reelection campaign and endorse Vice President Kamala Harris to be the Democratic nominee for president, Harris' role on immigration has come under scrutiny.

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Harris is all but certain to face even more criticism over the Biden administration's record on immigration, one of American voters' top concerns ahead of the election. And Harris does have an immigration-related role in the Biden White House, but her responsibilities on the issue are often mischaracterized.

## What exactly is Harris' immigration role?

In March 2021, when the Biden administration faced the early stages of an influx in illegal crossings at the U.S. southern border, Mr. Biden tasked Harris with leading the administration's diplomatic campaign to address the "root causes" of migration from Guatemala, Honduras and El Salvador, including poverty, corruption and violence.

The region, known as Central America's Northern Triangle, has been one of the main sources of migration to the U.S.-Mexico border over the past decade.

Harris was not asked to be the administration's "border czar" or to oversee immigration policy and enforcement at the U.S.-Mexico border. That has mainly been the responsibility of Homeland Security Secretary Alejandro Mayorkas and his department, which oversees the country's main three immigration agencies, including Customs and Border Protection.

In reality, the only role close to that of a "border czar" under the Biden administration was held for only a few months by Roberta Jacobson, a longtime diplomat who served as coordinator for the Southwest border until April 2021.

In her immigration role, Harris' main line of work has focused on convincing companies to invest in Central America and promoting democracy and development there through diplomacy. In March of this year, the White House announced Harris had secured a commitment from the private sector to invest over $5 billion to promote economic opportunities and reduce violence in the region.

Efforts to reduce migration by improving conditions in migrants' home countries have always been viewed as a long-term strategy by U.S. officials. In its "root causes" framework , the Biden administration conceded the "systemic change" it envisions for Central America "will take time to achieve."

## Questions about her work on immigration

There are some legitimate questions about Harris' work on immigration.

Before the COVID-19 pandemic, most non-Mexican migration to the U.S. southern border originated from the Northern Triangle. In 2021, it made sense for the administration to focus on the root cases of migration in those countries. But migration flows have changed dramatically in recent years. Record numbers of migrants have been coming from places outside of Central America, including from countries like Cuba, Colombia, China, Ecuador and Venezuela.

In fiscal year 2023, for example, Border Patrol apprehensions of migrants from Guatemala, Honduras and El Salvador made up 22% of all crossings during that time period, down from 41% in fiscal year 2021, government statistics show. On the flip side, however, the administration could point to the fact that illegal crossings along the U.S. southern border by migrants from Guatemala, Honduras and El Salvador have decreased significantly every year since 2021.

While most of her critics have been Republicans, Harris' work on immigration has also garnered some criticism from the left. During a visit to Guatemala in June 2021, Harris told those intending to migrate, "Do not come," a statement that drew ire from some progressives and advocates for migrants.

As the second-highest ranking member of the Biden administration, Harris will also likely face questions over the all-time levels of unlawful border crossings reported in 2021, 2022 and 2023. Those crossings, however, have plunged this year, reaching a three-year low in June , after Mr. Biden issued an executive order banning most migrants from asylum.

Camilo Montoya-Galvez is the immigration reporter at CBS News. Based in Washington, he covers immigration policy and politics.

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Step 3: Solve the Problem. Make a table to organize the data. For this example, create a row for the slower car, a row for the faster car, and a column for each hour. Find the distance traveled during each hour by looking at the distances listed in each column. The distance of the faster car was more than the distance of the slower car in hour ...

In this video, we explore one of eight problem-solving strategies for the primary math student. Students are introduced to the Make Table strategy and then s...

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. ... Problem Solving Strategy 3 (Using a variable to find the sum of a sequence.) Gauss's strategy for sequences. last ...

MAKE AN ORGANIZED LIST OR A TABLE Making a list or a table is a way to organize data presented in a problem. This problem solving strategy allows students to discover relationships and patterns among data. This strategy helps students to bring a logical and systematic development to their mathematics. Example 1:

Math Problem Solving Strategies. This is one in a series of resources to help you focus on specific problem solving strategies in the classroom. Within this download, we are offering you a range of word problems for practice. Each page provided contains a single problem solving word problem. Below each story problem you will find a set of four ...

After you finish reading the rest of the section, you can finish solving this problem for homework. Develop and Use the Strategy: Make a Table. The method "Make a Table" is helpful when solving problems involving numerical relationships. When data is organized in a table, it is easier to recognize patterns and relationships between numbers.

We use the strategy "make a table" to help us solve several word problems that involve multiplication. A graphic organizer is a tool, such as a table, that w...

Practice this math problem solving strategy, Make a Table to Solve a Problem, with the help of these free printable problems. Download this make a table to solve a problem set of word problems for your 1st, 2nd and 3rd grade math students. These worksheets will be a helpful addition to your problem solving collection.

Problem-Solving Strategy: Make a Table. Emphasis: Ratio, Proportion, Per Cent. A table helps the student's mind organize data and understand problem situations. Cognitive psychology supports the proposition that the mind attempts to organize perceptual input, that memory is aided by perceptual and conceptual schemas, and that achievement is ...

Copy the link below in your browser and watch the introduction video about Problem-Solving Strategy: Make a Tablehttps://www.youtube.com/watch?v=lKZjBxsudhM...

Problem Solving Strategies Make a Table . ... Strategy: 1) UNDERSTAND: You need to know that you save $3 on Monday. Then you need to know that you always save twice as much as you find the day before. 2) PLAN: How can you solve the problem? You can make a table like the one below. List the amount of money you save each day.

1. Create a Diagram/draw a picture. Creating a diagram helps students visualize the problem and reach the solution. A diagram can be a picture with labels, or a representation of the problem with objects that can be manipulated. Role-playing and acting out the problem like a story can help get to the solution. Example.

Problem-Solving Strategy Video: Make a Table or Chart. Use this teaching video from iMaths to teach the Problem-Solving Strategy 'Make a table or chart'. This is one episode from a series of 10 Problem-Solving Strategy videos available at iMaths Online.

One of the most useful strategies in math is problem solving by finding a pattern. Teach kids to make a table, find a pattern, and solve the problem! ... find patterns, graph equations, and so on. And while some may think of these as two different things, I think problem solving by making a table and finding a pattern go hand in hand! ->Pssst!

PROBLEM-SOLVING STRATEGIES Guess and check. M ke a table. XP ain when to use the make a table strategy to solve a problem. Explore: We need to find how many students downloaded at least 10 songs in one month. Plan: Make a frequency table of the data. Solve: Draw a table with three columns. In the first column, list the number of songs less than ...

The math problem solving strategies discussed in this chapter are: Look for a Pattern. Create a Table. Create an Organized List. Guess and Check (My Favorite! This one rarely let me down in college math..hehe!) It is important to note that these strategies do not have to be used in isolation. In fact, many students will use the strategies ...

Make a Table Strategy is another very useful strategy for solving Word Problems. It is a systematic way of approaching the solution in a step-by-step way. We...

Common Problem Solving Strategies. Guess (includes guess and check, guess and improve) Act It Out (act it out and use equipment) Draw (this includes drawing pictures and diagrams) Make a List (includes making a table) Think (includes using skills you know already)

The score must be a multiple of 3 or 6. Choose a Strategy. The strategy of eliminating possibilities can be used in situations where there is a set of possible answers and a set of criteria the answer must meet. Solve the Problem. First, list the numbers 1 through 29, because the problem states that the score was less than 30.

Curated and Reviewed by Lesson Planet. In this math tables worksheet, students read the word problem to understand the information. Students then picture the information in the table. Students use the table to help them solve the problem. Students then make a table to help them solve the last two problems. 12 Views 6 Downloads.

Traditional CFR Counterfactual Regret Minimization (CFR) Counterfactual Regret Minimization is an algorithm that is commonly used to solve large imperfect-information games. solvers calculate all the way to the river. That means players' strategies on every runout and betting line are known and well-defined. This information makes it easy to calculate how often one player will knock out ...

This Power Point contains 10 engaging problems using the Make A Table strategy. Teachers can use the problems for whole group demonstrations or cooperative group work on the different problem solving strategies in mathematics. Best practices include problem solving every day. This Power Point wil...

MAT 152 LESSON 5 VERSION 2

On day 11 of illness, fevers and symptoms recurred, including new headaches. The patient returned to the same ED. Her examination was notable for bilateral tonsillitis with overlying exudates, and ...

Chapter 1 Lesson 12

In her immigration role, Vice President Kamala Harris has mainly focused on convincing companies to invest in Central America and promoting democracy and development.