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Problem Solving: Make a Table

What Is It?
Make a Table is a problem-solving strategy that students can use to solve mathematical word problems by writing the information in a more organized format. Here is an example of a problem that can be solved by making a table:
Juanita checked a book out of the library, and it is now 7 days overdue. If a book is 1 day overdue, the fine is 10¢, 2 days overdue, 20¢, 3 days overdue, 30¢, and so on. How much is her fine?
Why Is It Important?
This problem-solving strategy allows students to discover relationships and patterns among data. It encourages students to organize information in a logical way and to look critically at the data to find patterns and develop a solution.
How Can You Make It Happen?
Introduce a problem to students that will require them to make a table to solve the problem. For example:
How many hours will a car traveling at 65 miles per hour take to catch up with a car traveling at 55 miles per hour if the slower car starts one hour before the faster car?
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 need to understand that there is a slower car going 55 miles per hour and a faster car going 65 miles per hour. The slower car starts one hour before the faster car. Students need to find how many hours it will take the faster car to catch up to the slower car.
Choose a Strategy
Because there are three sets of data to organize, you should use the Make a Table strategy. Generally, if there is data associated with a certain category, it can be organized easily by making a table. This strategy also overlaps with the Find a Pattern strategy because it is often easier to find a pattern when the data is organized in a table.
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 seven. The faster car traveled six hours to catch up to the slower car.
Slower Car | 55 | 110 | 165 | 220 | 275 | 330 | 385 |
Faster Car | 0 | 65 | 130 | 195 | 260 | 325 | 390 |
Read the problem again to be sure the question was answered.
Did you find the number of hours it took for the faster car to catch up? Yes, it took 6 hours.
Check the math to be sure it is correct.
55 x 2 = 110, 55 x 3 = 165, 55 x 4 = 220, 55 x 5 = 275, 55 x 6 = 330, 55 x 7 = 385 65 x 2 = 130, 65 x 3 = 195, 65 x 4 = 260, 65 x 5 = 325, 65 x 6 = 390
Determine if the best strategy was chosen for this problem or if there was another way to solve the problem.
Making a table is a good way to solve this problem.
The last step is explaining how you found the answer. Demonstrate how to write a paragraph describing the steps you took and how you made decisions throughout the process.
I set up a table for the miles each car had gone during each hour. I kept adding columns until the faster car caught up to the slower car. At the end of the seventh hour, the faster car had gone 390 miles, which is more than the distance traveled by the slower car, 385 miles. Because the faster car didn't start traveling in the first hour, it traveled for six hours.
Have students try solving the following problem using the strategy Make a Table.
The printer in the media center can print 1 page every 30 seconds. The printer in the office can print 4 pages every 30 seconds. If both printers are printing, how many pages will the office printer have printed by the time the media center printer prints 5 pages?
Have students work in pairs, 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.
How Can You Stretch Students' Thinking?
This strategy can be stretched when combined with other strategies such as looking for patterns or drawing a picture. By combining this strategy with others, students can analyze the data that is given to find more complex relationships.
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Problem Solving: Make a Table

Problem Answer: Make a Table
That is it.
Making a Table is one problem-solving strategy so pupils can application at solve mathematical word problems by writing the information in a more organized format. Here is an example of a concern that can be solved on making a table:
Juanita selected a book out of the archive, press it is now 7 days overdue. If a book is 1 sun overdue, the fine is 10¢, 2 days overdue, 20¢, 3 days past, 30¢, and then on. How much is her nice?
Why Is It Important?
This problem-solving strategy allow students to discover relationships and patterns beneath data. It encourages students until manage information in ampere logical way and until look critique at that data to finding patterns and develop an solution. Math Problem Solving Policy: Making and Using a Table to Solve adenine Problem - free math word problems from The Curriculum Corner
Instructions Canister To Make It Happening?
Introduce a matter to students the will require them to make adenine table to unlock the symptom. For example:
How many hours will a your traveling at 65 miles per total take to arrest up with a motorcar traveling at 55 miles per hour if the slower vehicle starts one hour before which faster car?
Know the Problem
Demonstrate that the first step is understanding the problem. This involves identifying and key pieces of information needed to found the answer. This may require students to get to problem several playing or put the problem into their own words. Concern Solving Create A Round Teaching Research | TPT
In this problem, students need to understand that there is a slower car going 55 miles per hour or a faster car going 65 miles per hours. The flatter auto starts one hour before the faster car. Students needs to find how lot hours it will take one fast car to catch up to the slower car. Results 1 - 24 of 3100+ ... This pack focuses upon the math problem-solving strategies Induce a Table and Find the Pattern. Set for homework or use such ampere warm-up ...
Choose adenine Approach
Because here are three sets of data to organize, you should use the Make a Table strategy. Generally, if there is data associated equal a determined category, it can must organized easily by manufacturing a table. This strategy also overlaps with which Detect a Pattern tactics because it is often easier to find a pattern whenever the data is organized in a table.
Solve and 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 post for jede hour. Find the distance traveled during each hour by looking with the spacing listed in each column. The distance on the quicker car was more when to remote of the slower passenger in total seven. That faster automotive traveled six hours to catch up for the slowlier car. Procedure: Make a table reflecting the file in the problem. If done in an orderly way, such a table will often reveal patterns and relationships ...
Slower Passenger | 55 | 110 | 165 | 220 | 275 | 330 | 385 |
Faster Car | 0 | 65 | 130 | 195 | 260 | 325 | 390 |
Read the problem further to be safely the get was answered.
Performed you find the number of hour it took for the faster car to catch up? Yes, to took 6 hours.
Check the mathematic to be sure it is correct.
55 x 2 = 110, 55 scratch 3 = 165, 55 x 4 = 220, 55 x 5 = 275, 55 x 6 = 330, 55 x 7 = 385 65 x 2 = 130, 65 x 3 = 195, 65 scratch 4 = 260, 65 expunge 5 = 325, 65 x 6 = 390
Find if the best strategy was chosen for this problem or if there was further way to solve the problem.
Making an table is a great way to solve this problem.
The last select is explaining how you found the answering. Prove how to record a edit describing the steps you took both how you made decisions throughout to process. Problem Solving Strategies
I set upwards a table for the miles each car had dead during each hour. I stopped totaling support until the faster car catching raise to the slowed automotive. At the end of the seventh time, the quick car has gone 390 miles, which is more than the distance traveled by the slower car, 385 distance. Because the faster car didn't start traveling in the first hour, it moved for six hours.
Have students try solving the following problem using the approach Make a Table.
The printer in the media center can pressure 1 page every 30 seconds. The printed with the office cannot print 4 pages every 30 alternates. If both printer are imprint, how many pages bequeath the office printer possess printed by and start the media center printer prints 5 pages?
Have students work in pairs, groups, either individually to resolve this problem. They should be able to tell or write about how they found the answer both justify their arguments.
How Can You Stretch Students' Thinking?
Aforementioned strategy can be stretched when combinated by other core such as looking for patterns or drawing a picture. Of combining this strategy with others, students can analyze the data that is given till find more complex links.
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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.
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Problem Solving Strategies
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.
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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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- Pre-Algebra
More Problem-Solving Strategies: Look for a Pattern
- January 24, 2014
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One of the problem-solving strategies that is often used in math is to look for a pattern. Often when exploring problems, the student can notice a relationship between numbers. This relationship can help to solve the problem by shortening the number of steps it takes to get to a solution.
Step 1: Make a Table
The first step to look for a pattern is to make a table showing the relationships that are there. For example, suppose the problem were to find the next three numbers in the series 2, 4, 6, 8. They are related in a simple way, such that 2 +2 is 4 +2 is 6 +2 is 8. In order to extend the pattern, 8 +2 is 10, +2 is 12 +2 is 14. Therefore, the next three numbers are 10, 12, and 14.
Step 2: Find the Relationship Between Numbers
In the previous example, the relationship was very simple, as the pattern was the set of the first 7 even numbers. Each number in the series was separated by the same distance. Suppose the relationship weren’t quite as simple. This time, the numbers in the sequence are 1, 5, 14, 30, and 55. What are the next 3 numbers in the series? 1 +4 is 5 +9 is 14 +16 is 30 +25 is 55. The numbers aren’t even the same distance from one another.
Step 3: Make a Prediction
What do these numbers have in common? After finding a pattern, it’s possible to predict what the next numbers will be. Those differences 1, 4, 9, 16, and 25, have a pattern in themselves. They are the first 5 squared numbers, as 1 2 is 1, 2 2 is 4, 3 2 is 9, 4 2 is 16, and 5 2 is 25. So the next difference might be 6 2 or 36, and 55 + 36 = 91. The next squared number will be 7 2 , or 49, and 91 +49 = 140. The next squared number will be 8 2 or 64, and 140 + 64 = 204.
Step 4: Check the Answer
The sequence with the next three numbers is 1, 5, 14, 30, 55, 91, 140, and 204. The pattern fits in every step of the sequence. If the numbers were added carefully, it would be easy to check each one, as 1 + 4 is 5, 5 + 9 is 14, 14 + 16 is 30, 30 + 25 is 55, 55 + 36 is 91, 91 + 49 is 140, and 140 + 64 is 204. The next squared number would be 81 (9 2 ), and 204 + 81 is 285.
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Math Review of Equivalent Sentences
More problem-solving strategies: identify subgoals.
The next competitive advantage in talent: Continuous employee listening
How are your employees feeling? Arguably, there’s no question more critical in organizations today. But truly understanding how employees—individually and collectively—feel and what they want has become increasingly difficult, particularly over the past few years.
Companies may have used annual surveys in the past to understand what’s going on with employees, identify and characterize emerging workplace trends, and act quickly to seize opportunities or address any issues. But in reality, annually taking stock is not enough anymore.
The pace and complexity of work has continued to increase in the wake of the COVID-19 pandemic. Working teams have become more agile. Hybrid models and gig and project work have become more prevalent. Employees’ expectations of their employers continue to evolve, and many have expressed the desire for more flexibility, connectivity, and fulfilling and purposeful work alongside traditional compensation and benefits.
At a time when organizations are facing wave after wave of disruption rather than occasional upheavals, the traditional survey approach is no longer sufficient. It should be supplemented by a continuous-listening strategy—a process driven by people analytics and supported by a people analytics team for capturing employee sentiment in both the short and long terms.
How are your employees feeling? Arguably, there’s no question more critical in organizations today.
Through this continuous-listening process, business leaders can both monitor the pulse of the organization at the moment and create an ongoing dialogue with employees. Such a dialogue can engender trust and partnership and can spur long-term improvements to employees’ workplace experiences and performance, as well as companies’ ability to retain top performers.
To understand what a continuous-listening strategy looks like and how to build and staff such a strategy, consider our experience at McKinsey.
In March 2020, at the beginning of the COVID-19 pandemic, McKinsey closed its offices and switched to a remote-work model. Employees’ personal and professional lives had been significantly altered, and we were keen to quantify and address their needs in a way that was timely and effective and that rigorously protected their privacy. Here’s what happened next.
Leveraging the power of people analytics
McKinsey had already established a strong people analytics team that was collecting data through an annual employee survey and using advanced analytics to generate longer-term predictors of employee satisfaction and performance. The team was well positioned to support data-driven decisions about important talent issues. Now faced with crisis, leaders needed to push that capability further to access more immediate employee sentiment in addition to the foundational longitudinal data. In essence, leaders needed to build a continuous-listening strategy that would allow them to quickly identify important changes that were affecting colleagues in the moment and determine how to address them. A key component was a weekly pulse survey.
A weekly pulse check
Leaders began to send a simple but effective survey made up of just two or three questions to all colleagues across the global organization each week. The first question in the pulse survey asked colleagues how they were feeling, with multiple-choice answers; the second prompted a free-text response. When more information was needed about, say, perceptions about professional development or work–life balance or inclusion, a third question, with multiple-choice answers, was added (Exhibit 1).
The people analytics team combined these data with existing employee information, anonymized them, and, using advanced analytics, pinpointed colleagues’ key concerns during the height of the pandemic period. These included financial and childcare concerns and the impact of remote work on mental health, inclusion, and productivity. Through the pulse survey, leaders were better able to understand the relative importance of various issues to different groups of colleagues (depending on role, tenure, geographic location, and parental status, for instance). Using these data, they could target interventions where they would have the most impact while protecting employees’ privacy.
The team continues to send the pulse survey. Results are shared with employees weekly, reiterating how their contributions can help leaders identify and shape changes and reinforcing the shared responsibility of colleagues at all levels of the organization to monitor and improve the culture (Exhibit 2). To date, McKinsey’s pulse survey has recorded more than one million responses from more than 40,000 employees across 140 offices around the world. It has collected feedback from more than 90 percent of the firm’s employees.
But, as McKinsey has learned, the pulse survey doesn’t just happen . Such an approach requires robust data management and IT systems, analytical expertise, and leaders’ time and attention.
Technical and data capabilities
Transforming survey data into insights that leaders can use requires close partnerships among data engineers, data scientists, analysts, translators, and the leaders themselves. Indeed, behind the pulse survey instrument is a diverse set of data sources (behavioral data, digital exhaust, 1 “Digital exhaust” refers to the residue—information or data—that internet users may leave behind through their use of various applications. and demographic data) and a team made up of engineers who can combine these data sets quickly and seamlessly. They use advanced analytics, including natural-language processing, to glean insights from integrated and unstructured data sets.
Critical to this team are analytics translators who work closely with McKinsey leaders to shape pulse survey questions, interpret the resulting data, and provide recommendations. This translator role, while relatively new in many organizations, is important for securing the success of a continuous-listening strategy. Individuals in these roles must be able to navigate complex organizational dynamics and serve as the liaisons between the analytics team and the business at large. The translators must ensure alignment among all relevant stakeholders. They must be able to synthesize the wealth of data generated and develop compelling, data-driven narratives that can highlight for business leaders any clear concerns or opportunities.
In response to demand from managers, the people analytics team built a self-service portal through which pulse-survey-driven reports can be filtered by date, population, and other characteristics. The portal also allows users to compare the reports with longitudinal data to guide real-time decisions (Exhibit 3).
Demonstrating the impact of continuous listening
The measure of success with any continuous-listening strategy is its relative impact on the decisions that leaders and employees need to make. Several core themes relating to employees’ personal and professional lives have emerged from the findings from McKinsey’s pulse survey.
Employee experience
Through the pulse surveys, the people analytics team has gained a clearer view of how different colleagues were experiencing working at McKinsey. While most consultants and employees have reported high satisfaction with their roles, enthusiasm about their work and the organization, and sustainable lifestyles, others have shared their need for support. Managers wanted to address these needs in real-time. Through the self-service tool, they were able to do just that.
Managers can use the portal to track the sentiment of colleagues within their teams and subgroups and quickly address any changes. Further, as analytics translators recognize significant changes in the pulse survey data, they can deliver these insights to senior leaders through weekly reports. With these insights, leaders have been able to explore and introduce new ways of working and guidelines to address the needs of those colleagues with lower levels of satisfaction, enthusiasm, and work–life balance.
Team health
To offer leaders and managers more of a team-level perspective on the employee experience, the people analytics team rolled out a manager alert capability. It combines pulse survey results and other employee data to identify issues that certain teams and groups may be experiencing—for instance, concerns about overly long working hours and a lack of opportunities to connect as a team.
Every two weeks, leaders and managers receive alerts that include an overview of these potential concerns, as well as the specific teams and groups that may be experiencing them. Leaders and managers are using the alert tool to identify emerging challenges before they turn into deeper issues, facilitate more constructive team conversations, and guide teams toward more sustainable norms and practices (Exhibit 4).
Connectivity and remote work
Toward the end of 2020, McKinsey, like other companies, sought to identify an optimal model for returning to the office. Initial data from the pulse survey showed that only a subset of employees—mostly new joiners and more junior employees—reported excitement for returning to a fully in-person model. Many colleagues expressed a preference for a hybrid model that would address their needs for development, connectivity, collaboration, and flexibility, as well as heads-down, focused work time.
The team analyzed colleagues’ contexts, needs, and preferences to develop collaboration archetypes. Through this work, it gained several critical insights—for instance, how much time colleagues should spend in person, which activities might be most impactful while colleagues are together, and how best to motivate colleagues to shift to an optimal model, taking into account the needs of each community.
Growth and learning
In late 2021, the pulse survey showed a decline in experience sentiment among some client-facing employees who had joined within the previous six months. Through an archetype analysis, the team identified a group of new hires who were receiving less coaching in the new hybrid environment. As a result of this analysis, McKinsey leaders established several new initiatives focused on purposeful apprenticeship and increased resources for new learning programs—for instance, a revamped onboarding curriculum and digital-learning support for new joiners.
In other instances, leaders have used pulse survey data to assess the ROI from various training programs, differentiating the programs’ impact according to the skills being developed and whether programs were being delivered online or in person.
One week, the pulse survey asked employees about behaviors that reflect courageous inclusion—that is, engaging with curiosity, sharing openly, and empowering others. Employees’ experiences were varied, according to survey data. To identify potential roadblocks to inclusivity and recommend ways to incorporate more open sharing, the people analytics team used natural-language processing to analyze colleagues’ free-text submissions.
A consistent thread throughout the survey comments was the need for a psychologically safe environment as a precondition for actively pursuing inclusivity and for experimenting and learning from mistakes. McKinsey’s leaders tied this theme to broader people data. In this way, they were able to target recommendations to specific areas of the firm and further reinforce the firm’s culture of inclusion across geographies and populations.
Individual outreach
The weekly pulse survey platform provides a means by which employees can reach out to a leader in times of need in a way that preserves their confidentiality, since pulse-survey-related correspondence is never tied back to an individual. Survey participants need only click on a link within the survey, and the request is forwarded to a relevant contact in their region. The response time is typically within 24 hours from the time of the original request. Over the past three years, the people analytics team has facilitated about 3,000 of these requests.
COVID-19-related lockdowns in 2020 magnified our colleagues’ caregiving challenges. Leaders were concerned about the health of McKinsey colleagues and their families, as colleagues tried to juggle parenting, schooling, and remote work. Insights from the pulse survey data prompted leaders to make a fast and data-informed decision to make resources available to support colleagues’ childcare needs and allow them to work from home more comfortably. More recently, leaders expanded McKinsey’s parental-leave policy in locations that were facing the biggest childcare challenges.
Mental health
A core benefit of the pulse survey is its anonymity. Creating such a safe space for employees to voice their opinions proved invaluable when leaders asked colleagues how the company could better support their mental health. The people analytics team used free-text analytics to structure and quantify colleagues’ written submissions about specific mental-health-related topics that were top of mind for them and then identify potential solutions.
Solutions resulting from the analysis have included events dedicated to acknowledging the stigma associated with mental health and discussing ways to access virtual therapists and other forms of digital support, as well as physical-fitness resources. The firm has also established month-to-month part-time programs that allow employees to flex their working days in situations of special need.
Worldwide events
The pulse survey platform has proved particularly valuable when news—global health emergencies and local crises, for instance—breaks and leaders need to understand quickly which colleagues are affected and how to help. Through pulse surveys, colleagues can share their real-time reactions to external events, and leaders can respond in turn by, say, organizing townhall discussions about the event, providing updated crisis-related information, and offering information about fundraising efforts.
In the current fast-paced talent market, the companies that adopt a continuous-listening strategy stand to build a distinct competitive advantage. They can transform their understanding of employees’ needs and address them in real time, increasing the likelihood of higher performance, higher engagement, and higher retention. Under this approach leaders will, of course, need to continually monitor and collaborate with technical experts and other key stakeholders across the organization to generate insights, consider changes, and make them. But the results are worth it. A continuous-listening strategy can create outsize impact and long-term value, regardless of organization or industry.
Alice Damonte is a data analytics specialist in McKinsey’s Milan office; Elizabeth Ledet is a partner in the Atlanta office; Daniel Morales is a people analytics and strategy leader in the Washington, DC, office; and Sarah Tobey is a people analytics director in the New York office.
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Make a Table is a problem-solving strategy that students can use to solve mathematical word problems by writing the information in a more organized format. Here is an example of a problem that can be solved by making a table: Juanita checked a book out of the library, and it is now 7 days overdue.
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. Making & Using a Table to Solve Problems
Make a Table | Primary Math Problem Solving Strategy Learning With Jeff 460 subscribers 934 views 2 years ago Problem Solving Strategies for the Primary Math Student In this video, we...
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!
Make a Table is a problem-solving strategy that students can getting to solve mathematical word issue by script the information in a more structured format. Skip on main content Profile Button
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:
Problem-Solving Strategy: Make a Table Problem-Solving Strategy: Make a Table Emphasis: Ratio, Proportion, Per Cent A table helps the student's mind organize data and understand problem situations.
Solve: Draw a table with three columns. In the first column, list the number of songs less than 10 and at least 10 that students downloaded. Then complete the table by indicating the frequency or number of songs less than 10 or at least ten that were downloaded. 7 students downloaded at least 10 songs.
Problem-Solving Strategy: Make a Table Part 1 - YouTube Copy the link below in your browser and watch the introduction video about Problem-Solving Strategy: Make a...
Make a list of each relevant task you think of. Then organize each step by listing them in order of when they need to be accomplished. Finally, divide the list by assigning different tasks to individual members of your team. 5. Redefine the problem If the problem seems like it might really be unsolvable, consider redefining the problem.
Make a Table is a problem-solving strategy that students can use to solve numerical word problems by handwriting the information in a more organized format.
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
These resources allows students to practice making a table, or chart, to organize information in multi-step math word problems. Making a table is a great way to show students how
Make a Table For Students 1st - 2nd First and second graders participate in an experiment then create a table of tally marks to identify the totals. They problem solve, tally twenty experiences, and generate an answer to one question. + Worksheet Lesson Planet: Curated OER Math Stars: A Problem-Solving Newsletter Grade 5
This maths problem - solving pack will help your students develop strategies for solving maths word problems. This pack focuses on the math problem - solving strategies Make a Table and Find the Pattern. Set for homework or use as a warm-up activity, these worksheets are so versatile for helping your students learn these essential skills.
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.
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 and more Advertisement 1. Create a Diagram/draw a picture Creating a diagram helps students visualize the problem and reach the solution.
Step 1: Make a Table. The first step to look for a pattern is to make a table showing the relationships that are there. For example, suppose the problem were to find the next three numbers in the series 2, 4, 6, 8. They are related in a simple way, such that 2 +2 is 4 +2 is 6 +2 is 8. In order to extend the pattern, 8 +2 is 10, +2 is 12 +2 is 14.
The following leadership practices can help drive the shift: encouraging and empowering people in small self-regulating entrepreneurial teams. fostering horizontal transparency and collaboration throughout the network and beyond. moving from a hierarchy of individual leaders to networks of leadership teams.
A continuous-listening strategy can create outsize impact and long-term value, regardless of organization or industry. Alice Damonte is a data analytics specialist in McKinsey's Milan office; Elizabeth Ledet is a partner in the Atlanta office; Daniel Morales is a people analytics and strategy leader in the Washington, DC, office; and Sarah ...
But this approach to problem-solving will not protect your heart. On the contrary - this puts you at greater risk." Raised blood pressure: a killer that can be stopped Dr Chikovani is at the centre of the nationwide battle against hypertension, one of the most prominent health risk factors in the country.