activity 3.2 problem solving with patterns

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

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Course: 3rd grade   >   Unit 8

• Finding patterns in numbers
• Recognizing number patterns

Math patterns

• Intro to even and odd numbers
• Patterns with multiplying even and odd numbers
• Patterns with even and odd
• Patterns in hundreds chart
• Patterns in multiplication tables
• Arithmetic patterns and problem solving: FAQ

• (Choice A)   160 ‍   A 160 ‍  
• (Choice B)   180 ‍   B 180 ‍  
• (Choice C)   200 ‍   C 200 ‍  
• (Choice D)   220 ‍   D 220 ‍  

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Number patterns

Here you will learn about number patterns, including how to find and extend rules for sequences, input/output tables and shape patterns.

Students will first learn about number patterns as part of operations and algebraic thinking in 4th and 5th grade. They continue to build on this knowledge in middle school and high school.

What are number patterns?

Number patterns are groups of numbers that follow rules. They can use input/output tables to create sequences .

Two types of sequences are arithmetic and geometric .

• An arithmetic sequence is a list of numbers where the same amount is either being added or subtracted every time.

Each sequence has a starting number, a rule and terms (the numbers that make up the sequence).

For example,

Rule: Subtract 6 each time.

There are also patterns, between the terms of two or more arithmetic sequences.

Both sequences start at 0. If you multiply the left column by 3, you get the terms in the right column.

This is because using the rule +9 is three times more than the rule +3.

Step-by-step guide:   Arithmetic sequence

• A geometric sequence is a number pattern where the rule is multiplication or division.

Rule: Multiply the previous term by 5.

Rule: Divide the previous term by 3.

Step-by-step guide:   Geometric sequence formula

Step-by-step guide:   Sequences

This page will highlight rules that involve whole numbers only.

[FREE] Number Patterns Worksheet (Grade 4 to 5)

Use this quiz to assess your grade 4 to 5 students’ understanding of number patterns. 10+ questions with answers on 4th and 5th grade number pattern topics to identify areas of strength and support!

• Input/output tables are tables that are used to show two sets of numbers that are related by a rule. The rule can be one step or multi-step, but has to work for each relationship shown in the table.

What is the rule for the table below?

To find the rule, look for the relationship between the input and the corresponding output.

Notice, 11 is being added to each input to get the output, so the rule is ‘add 11. ’

Step-by-step guide: Input/output tables

As students learn to work with number patterns, they can learn more about generalizing patterns by working with shape patterns.

• Shape patterns are any set of polygons, 3D shapes, letters or symbols that follow non-operational rules. There are repeating shape patterns and growing shape patterns .

A repeating pattern has a core that repeats over and over again.

The core of the pattern above is:

The core can be used to extend the pattern.

The next shape in the pattern would be:

because the last part of the core shown is:

• A growing pattern has parts that stay the same, but other parts that change.

• Changing – the left column starts with 0 and increases by 2.
• Staying the same – the 1 cube on the top right.

Step-by-step guide: Shape patterns

Common Core State Standards

How does this relate to 4th grade math and 5th grade math?

• Grade 4 – Operations and Algebraic Thinking (4.OA.C.5) Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule “Add 3 ” and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.
• Grade 5 – Operations and Algebraic Thinking (5.OA.B.3) Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule “Add 3 ” and the starting number 0, and given the rule “Add 6 ” and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.

How to use number patterns

There are a lot of ways to use number patterns. For more specific step-by-step guides, check out the pages linked in the “What are number patterns?” section above or read through the examples below.

Number patterns examples

Example 1: geometric sequence.

What are the next three terms in the pattern?

3, 6, 12, 24, 48…

Identify the rule.

Rule: Multiply the previous term by 2.

2 Use the rule to extend the pattern.

Multiply the last term by 2 to extend the pattern to the next three terms.

3 State and explain any patterns within the terms.

Except for the first term, all the terms are even.

The first term is odd, but multiplying it by 2 (even), makes the next term even.

From then on it is always an even term times 2 \rightarrow \text { even } \times \text { even }=\text { even }.

Example 2: compare arithmetic sequences

Compare the following sequences:

0, 2, 4, 6, 8… \, and \, 0, 12, 24, 36, 48…

Identify the rule of each sequence.

Find a pattern between the related terms of each sequence.

Look at the related terms for each sequence. What do you notice?

Both sequences start at 0. If you multiply the left column by 6, you get the terms in the right column.

Use the pattern to write a comparison statement.

The terms in the second sequence are 6 times the terms in the first sequence.

This is because using the rule +12 is six times more than using the rule +2.

Example 3: input/output table – identify the rule

What is the rule for the table?

Look at the relationship between the input and the corresponding output.

Looking from each input to output, see if you notice an obvious relationship. If not, use subtraction to find the difference between each input and the corresponding output.

Decide if the rule is add/subtract or multiply/divide.

Each output is 27 more than the input, so the rule is addition.

Write the rule.

Rule: Add 27 to the input.

Example 4: input/output table – find a missing value

Find the missing value in the table.

From each input to output, the difference is not the same, so the rule is not addition or subtraction.

Try to find a relationship that involves multiplying or dividing. Since the relationship from input to output is decreasing, try division.

Rule: Divide the input by 7.

*Note, the rule ‘Multiply the input by \, \cfrac{1}{7} \, ’ is also correct.

Use the rule to find the missing value(s).

70 \div 7=10

The missing value in the table is 10.

Example 5: 3D shape pattern, repeating

Create a rule for the pattern and find the next shape.

Identify the core – the part of the pattern that repeats.

This is a repeating pattern with cubes that go purple, purple, orange, purple, red, purple.

Notice that purple is repeated multiple times within the core. Always look at all the shapes given to confirm the pattern core.

Use the core to find and justify the next part in the pattern.

The next shape is:

Example 6: shape pattern, growing

Identify what is changing and what is staying the same.

• Changing – 0 triangles on the top and left side, 1 triangle on top and left side, 2 triangles on top and left side, 3 triangles on top and left side.
• Staying the same – The first triangle at the bottom is always there.

Create a rule based on Step 1.

Start with 1 triangle. Add 1 triangle to the top and 1 triangle to the left side each time.

Use the rule to find and justify the next part in the pattern.

The next part in the pattern is:

because 1 triangle is added to the top and 1 triangle is added to the side each time.

Teaching tips for number patterns

• There are many printable number patterns worksheets that can be used, but also give students the opportunity to create their own patterns. Then let their classmates find the rules and extend them. This helps keep students interested and deepens their understanding of patterns by challenging them to create and giving them opportunities to engage in discourse around patterns.
• Support students who are not yet confident in all operations by providing useful tools such as number lines, hundreds boards or counters. Completing calculations should not be a burden when working with patterns.
• Focusing on justifying pattern rules is as important as identifying them. Justification is not only a necessary mathematical practice, but doing it can teach students how to write better rules. Thinking about why a rule works, draws attention to the general parts of a pattern – how they are changing or staying the same, which is the basis of a valid mathematical generalization.

Easy mistakes to make

• Making an operational error Since finding the rule and extending the pattern require a calculation, this leaves room for mistakes to be made. Always double check your work and take time to think about if your answers are reasonable.
• Thinking there is only one way to write a growing shape pattern rule Often there is more than one way to describe what is changing and what is staying the same in a growing shape pattern. Encourage students to look for more than one way to describe their rule, as this is a helpful stepping stone to equivalent expressions.

Practice number patterns questions

1. What is the next number in the pattern?

4, 20, 100, 500, 2,500…

Multiply the last term by 5 to extend the pattern.

2. Starting number: 68

Rule: subtract 6 each time.

Which statement is true about the terms of the sequence described above?

All the terms are even

All the terms are odd

The terms alternate between even and odd

The first term is even and the rest are odd

Extend the sequence based on the starting number and the rule.

Since 6 (even number) is being subtracted each time, and even – even = even, all the terms are even.

3. What is the rule for the table?

Add 20 to the input

Multiply the input by 6

Multiply the input by 8

Add 25 to the input

Looking from each input to output, see if you notice an obvious relationship.

If not, use subtraction to find the difference between each input and the corresponding output.

Try to find a relationship that involves multiplying or dividing.

Since the relationship from input to output is increasing, try multiplication.

Rule: Multiply the input by 6.

4. Find the missing value in the table.

Each output is 12 less than the input, so the rule is subtraction.

Rule: Subtract 12 from the input.

Since the input is missing, thinking about what number subtracted by 12 will equal 83 :

The missing value in the table is 95.

5. What is the next shape in the pattern?

This pattern goes e, c, S, 8, o, e, 8 and then repeats.

6. What is the next part in the pattern?

Identify what is changing and what is staying the same:

• Changing – one purple in the middle and one green on either side, two purple in the middle and two green on either side, three purple in the middle and three green on either side.
• Staying the same – There is always a purple x with a green x beside it and below it at the top.

Rule: Start with a purple x with a green x to the right and below it. Repeat, adding the 1 st term again, but to the left and down one .

because it adds the first term again, but to the left and down one (shown in purple outline).

Number patterns FAQs

No, since students are most familiar with whole number operations, the standards start with these numbers. However, in middle school, sequences grow to include integers and other types of rational numbers. This also continues in later grades to include sequences with complex numbers.

Fibonacci’s sequence (or fibonacci numbers) and triangular numbers (pascal’s triangle) are two sequences that are commonly explored in upper level mathematics. Step-by-step guide : Triangular numbers (coming soon)

While shape patterns do not typically include fractions or decimals, they could be used in a way that does not involve calculations (for example in the repeating pattern \, \cfrac{1}{5} \, , \, \cfrac{1}{8} \, , \, \cfrac{1}{5} \, , \, \cfrac{1}{8} \, , \, \cfrac{1}{5} \, , \, \cfrac{1}{8} \, … ).

Yes, there are sequences that involve square numbers, cube numbers, and other more complex operations.

The next lessons are

• What is a function
• Laws of exponents
• Scientific notation
• Explicit formula
• Recursive formula
• Quadratic sequences

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1.3: Problem Solving Strategies

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• Page ID 82975

• Michelle Manes
• University of Hawaii

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Think back to the first problem in this chapter, the ABC Problem. What did you do to solve it? Even if you did not figure it out completely by yourself, you probably worked towards a solution and figured out some things that did not work.

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.

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]

George Pólya, circa 1973

• Image of Pólya by Thane Plambeck from Palo Alto, California (Flickr) [CC BY 2.0 ( http://creativecommons.org/licenses/by/2.0 )], via Wikimedia Commons ↵

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!

We have already seen one problem solving strategy, which we call “Wishful Thinking.” Do not be afraid to change the problem! Ask yourself “what if” questions:

• What if the picture was different?
• What if the numbers were simpler?
• What if I just made up some numbers?

You need to be sure to go back to the original problem at the end, but wishful thinking can be a powerful strategy for getting started.

This brings us to the most important problem solving strategy of all:

Problem Solving Strategy 2 (Try Something!).

If you are really trying to solve a problem, the whole point is that you do not know what to do right out of the starting gate. You need to just try something! Put pencil to paper (or stylus to screen or chalk to board or whatever!) and try something. This is often an important step in understanding the problem; just mess around with it a bit to understand the situation and figure out what is going on.

And equally important: If what you tried first does not work, try something else! Play around with the problem until you have a feel for what is going on.

Last week, Alex borrowed money from several of his friends. He finally got paid at work, so he brought cash to school to pay back his debts. First he saw Brianna, and he gave her 1/4 of the money he had brought to school. Then Alex saw Chris and gave him 1/3 of what he had left after paying Brianna. Finally, Alex saw David and gave him 1/2 of what he had remaining. Who got the most money from Alex?

Think/Pair/Share

After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What did you figure out about the problem? This problem lends itself to two particular strategies. Did you try either of these as you worked on the problem? If not, read about the strategy and then try it out before watching the solution.

Problem Solving Strategy 3 (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, like this one, thinking visually can help! Can you represent something in the situation by a picture?

Draw a square to represent all of Alex’s money. Then shade 1/4 of the square — that’s what he gave away to Brianna. How can the picture help you finish the problem?

After you have worked on the problem yourself using this strategy (or if you are completely stuck), you can watch someone else’s solution.

Problem Solving Strategy 4 (Make Up Numbers).

Part of what makes this problem difficult is that it is about money, but there are no numbers given. That means the numbers must not be important. So just make them up!

You can work forwards: Assume Alex had some specific amount of money when he showed up at school, say $100. Then figure out how much he gives to each person. Or you can work backwards: suppose he has some specific amount left at the end, like $10. Since he gave Chris half of what he had left, that means he had $20 before running into Chris. Now, work backwards and figure out how much each person got.

Watch the solution only after you tried this strategy for yourself.

If you use the “Make Up Numbers” strategy, it is really important to remember what the original problem was asking! You do not want to answer something like “Everyone got $10.” That is not true in the original problem; that is an artifact of the numbers you made up. So after you work everything out, be sure to re-read the problem and answer what was asked!

(Squares on a Chess Board)

How many squares, of any possible size, are on a 8 × 8 chess board? (The answer is not 64... It’s a lot bigger!)

Remember Pólya’s first step is to understand the problem. If you are not sure what is being asked, or why the answer is not just 64, be sure to ask someone!

Think / Pair / Share

After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What did you figure out about the problem, even if you have not solved it completely?

It is clear that you want to draw a picture for this problem, but even with the picture it can be hard to know if you have found the correct answer. The numbers get big, and it can be hard to keep track of your work. Your goal at the end is to be absolutely positive that you found the right answer. You should never ask the teacher, “Is this right?” Instead, you should declare, “Here’s my answer, and here is why I know it is correct!”

Problem Solving Strategy 5 (Try a Simpler Problem).

Pólya suggested this strategy: “If you can’t solve a problem, then there is an easier problem you can solve: find it.” He also said: “If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?” In this case, an 8 × 8 chess board is pretty big. Can you solve the problem for smaller boards? Like 1 × 1? 2 × 2? 3 × 3?

Of course the ultimate goal is to solve the original problem. But working with smaller boards might give you some insight and help you devise your plan (that is Pólya’s step (2)).

Problem Solving Strategy 6 (Work Systematically).

If you are working on simpler problems, it is useful to keep track of what you have figured out and what changes as the problem gets more complicated.

For example, in this problem you might keep track of how many 1 × 1 squares are on each board, how many 2 × 2 squares on are each board, how many 3 × 3 squares are on each board, and so on. You could keep track of the information in a table:

Problem Solving Strategy 7 (Use Manipulatives to Help You Investigate).

Sometimes even drawing a picture may not be enough to help you investigate a problem. Having actual materials that you move around can sometimes help a lot!

For example, in this problem it can be difficult to keep track of which squares you have already counted. You might want to cut out 1 × 1 squares, 2 × 2 squares, 3 × 3 squares, and so on. You can actually move the smaller squares across the chess board in a systematic way, making sure that you count everything once and do not count anything twice.

Problem Solving Strategy 8 (Look for and Explain Patterns).

Sometimes the numbers in a problem are so big, there is no way you will actually count everything up by hand. For example, if the problem in this section were about a 100 × 100 chess board, you would not want to go through counting all the squares by hand! It would be much more appealing to find a pattern in the smaller boards and then extend that pattern to solve the problem for a 100 × 100 chess board just with a calculation.

If you have not done so already, extend the table above all the way to an 8 × 8 chess board, filling in all the rows and columns. Use your table to find the total number of squares in an 8 × 8 chess board. Then:

• Describe all of the patterns you see in the table.
• Can you explain and justify any of the patterns you see? How can you be sure they will continue?
• What calculation would you do to find the total number of squares on a 100 × 100 chess board?

(We will come back to this question soon. So if you are not sure right now how to explain and justify the patterns you found, that is OK.)

(Broken Clock)

This clock has been broken into three pieces. If you add the numbers in each piece, the sums are consecutive numbers. ( Consecutive numbers are whole numbers that appear one after the other, such as 1, 2, 3, 4 or 13, 14, 15.)

Can you break another clock into a different number of pieces so that the sums are consecutive numbers? Assume that each piece has at least two numbers and that no number is damaged (e.g. 12 isn’t split into two digits 1 and 2.)

Remember that your first step is to understand the problem. Work out what is going on here. What are the sums of the numbers on each piece? Are they consecutive?

After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What progress have you made?

Problem Solving Strategy 9 (Find the Math, Remove the Context).

Sometimes the problem has a lot of details in it that are unimportant, or at least unimportant for getting started. The goal is to find the underlying math problem, then come back to the original question and see if you can solve it using the math.

In this case, worrying about the clock and exactly how the pieces break is less important than worrying about finding consecutive numbers that sum to the correct total. Ask yourself:

• What is the sum of all the numbers on the clock’s face?
• Can I find two consecutive numbers that give the correct sum? Or four consecutive numbers? Or some other amount?
• How do I know when I am done? When should I stop looking?

Of course, solving the question about consecutive numbers is not the same as solving the original problem. You have to go back and see if the clock can actually break apart so that each piece gives you one of those consecutive numbers. Maybe you can solve the math problem, but it does not translate into solving the clock problem.

Problem Solving Strategy 10 (Check Your Assumptions).

When solving problems, it is easy to limit your thinking by adding extra assumptions that are not in the problem. Be sure you ask yourself: Am I constraining my thinking too much?

In the clock problem, because the first solution has the clock broken radially (all three pieces meet at the center, so it looks like slicing a pie), many people assume that is how the clock must break. But the problem does not require the clock to break radially. It might break into pieces like this:

Were you assuming the clock would break in a specific way? Try to solve the problem now, if you have not already.

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