problem solving look for a pattern practice 1 6

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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Problem-Solving Strategies: Finding Patterns

In this lesson, we will learn different problem solving strategies to find patterns.

The following are some examples of problem solving strategies.

Explore it/Act it/Try it (EAT) method (Basic) Explore it/Act it/Try it (EAT) method (Intermediate ) Explore it/Act it/Try it (EAT) method (Advanced) Finding a Pattern (Basic) Finding a Pattern (Intermediate) Finding a Pattern (Advanced)

Find A Pattern (Advanced)

Here we will look at some advanced examples of “Find a Pattern” method of problem solving strategy.

Example: Each hexagon below is surrounded by 12 dots. a) Find the number of dots for a pattern with 6 hexagons in the first column. b) Find the pattern of hexagons with 229 dots.

a) The number of dots for a pattern with 6 hexagons in the first column is 142.

b) If there are 229 dots then the pattern has 8 hexagons in the first column.

Example: Each member of a club shook hands with every other member who came for a meeting. There were a total of 45 handshakes. How many members were present at the meeting?

Solution: Total = 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 45 handshakes

There were 10 members

Example: In the figure, a pinball is released at A.

How many paths are there for it to drop from A to E?

Solution: from A to B: 2 B to C: 6 A to C: 2 × 6 = 12 C to D: 70 A to D: 12 × 70 = 840 D to E: 2 A to E: 2 × 840 = 1680

There are 1680 paths from A to E

Example: A group of businessmen were at a networking meeting. Each businessman exchanged his business card with every other businessman who was present.

a) If there were 16 businessmen, how many business cards were exchanged?

b) If there was a total of 380 business cards exchanged, how many businessmen were at the meeting?

Solution: a) 15 + 14 + 13 + 12 + 11 + 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 120 exchanges

120 × 2 = 240 business cards

If there were 16 businessmen, 240 business cards were exchanged.

b) 380 ÷ 2 = 190

190 = (19 × 20) ÷ 2 = 19 + 18 + 17 + … + 3 + 2 + 1

If there was a total of 380 business cards exchanged, there were 20 businessmen at the meeting.

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Pattern Problem Solving: Teach Students to Find a Pattern in Math Problems

Pattern analysis is a critical 21st century skill.

Need more tips and tricks for teaching math? You can find them in our math resources center .

Pattern problem solving

In this article, we’ll delve into the concept of pattern problem solving, a fundamental mathematical strategy that involves the identification of repeated sequences or elements to solve complex problems.

This method is instrumental in enhancing logical thinking and mathematical comprehension among students. Let's explore how this works, why it's important, and how to teach this critical skill effectively.

What is pattern problem solving? 

Pattern problem solving is a mathematical strategy in which students look for patterns in data to solve a problem. To find a pattern, students search for repeated items, numbers, or series of events.

The following problem can be solved by finding the pattern:

There are 1000 lockers in a high school with 1000 students. The first student opens all 1000 lockers; next, the second student closes lockers 2, 4, 6, 8, 10, and so on up to locker 1000; the third student changes the state (opens lockers that are closed, closes lockers that are open) of lockers 3, 6, 9, 12, 15, and so on; the fourth student changes the state of lockers 4, 8, 12, 16, and so on. This continues until every student has had a turn. How many lockers will be open at the end?

For the answer, visit  The Locker Problem from the Math Doctors

Why is pattern problem solving important? 

Pattern problem solving is an important strategy for students as it encourages them to observe and understand patterns in data, which is a critical aspect of mathematical and logical thinking.

This strategy allows students to predict future data points or behaviors based on existing patterns. It helps students understand the inherent structure of data sets and mathematical problems, making them easier to solve.

Pattern recognition also aids in the understanding of multiplication facts, for example, recognizing that 4 x 7 is the same as 7 x 4. Overall, pattern problem solving fosters analytical thinking, problem-solving skills, and a deeper understanding of mathematics.

How to teach students to find the pattern in a math problem (using an example)

In the upcoming section, we will break down the steps on how to find a pattern in a math problem effectively. We will use a practical example to illustrate each step and provide helpful teaching tips throughout the process.

The goal is to offer a clear and comprehensible guide for educators teaching students about pattern problem solving in math.

Sample question:  If you build a four-sided pyramid using basketballs and don't count the bottom as a side, how many balls will there be in a pyramid that has six layers?

Helpful teaching tip:  Use cooperative learning groups to find solutions to the above problem. Cooperative learning groups help students verbalize their thinking, brainstorm ideas, discuss options, and justify their positions. After finding a solution, each group can present it to the class, explaining how they reached their solution and why they think it is correct. Or, students can explain their solutions in writing, and the teacher can display the solutions. Then students can circulate around the room to read each group's solution.

1. Ensure students understand the problem 

Demonstrate that the first step to solving a problem is  understanding  it. 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.

Sometimes you can solve a problem simply through pattern recognition, but more often you must extend the pattern to find the solution. Making a number table will help you see the pattern more clearly.

In this problem, students understand:

The top layer will have one basketball. I need to find how many balls there will be in each layer of a pyramid, from the first to the sixth. I need to find how many basketballs will be in the entire pyramid.

2. Choose a pattern problem solving strategy 

To successfully find a pattern, you need to be sure that the pattern will continue.

Have students give reasons why they think the pattern is predictable and not based on probability. Problems that are solved most easily by finding a pattern include those that ask students to extend a sequence of numbers or to make a prediction based on data.

In this problem, students may also choose to make a table or draw a picture to organize and represent their thinking.

3. Solve the problem

Start with the top layer of the pyramid, one basketball. Determine how many balls must be under that ball to make the next layer or a pyramid. Let students use manipulatives if needed— they can use manipulatives of any kind, from coins to cubes to golf balls. Let students also draw pictures to help solve the problem, if needed.

If your students are in groups, you may want to have each group use a different manipulative and then compare their solutions. This will help you understand if different manipulatives affect the solution.

Helpful teaching tip:  If students are younger, solve this problem with only three layers.

If it helps to visualize the pyramid, use manipulatives to create the third layer. Record the number and look for a pattern. The second layer adds 3 basketballs and the next adds 5 basketballs. Each time you add a new layer, the number of basketballs needed to create that layer increases by 2.

1 1 + 3 = 4 4 + 5 = 9

Continue until six layers are recorded. Once a pattern is found, students might not need to use manipulatives. 9 + 7 = 16

16 + 9 = 25 25 + 11 = 36

Then add the basketballs used to make all six layers.

The answer is 91 balls .

Look at the list to see if there is another pattern. The number of balls used in each level is the square of the layer number.  So the 10th layer would have 10 x 10 = 100 balls .

4. Check your students' answers

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

Yes, I found the total number of basketballs in the six-layer pyramid.

Also check the math to be sure it’s correct:

1 + 4 + 9 + 16 + 25 +36 = 91

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

Finding a pattern was a good way to solve this problem because the pattern was predictable.

5. Explain 

Students should be able to explain the process they went through to find their answers. Students must be able to talk or write about their thinking. Demonstrate how to write a paragraph describing the steps they took and the decisions they made throughout.

I started with the first layer. I used blocks to make the pyramid and made a list of the number of blocks I used. Then I created a table to record the number of balls in each layer. I made four layers, then saw a pattern. I saw that or each layer, the number of balls used was the number of the layer multiplied by itself. I finished the pattern without the blocks, by multiplying the number of balls that would be in layers 5 and 6. Then I added up each layer. 1 + 4 + 9 + 16 + 25 +36 = 91. I got a total of 91 basketballs

How can you stretch this pattern problem solving strategy? 

Math problems can be simple, with few criteria needed to solve them, or they can be multidimensional, requiring charts or tables to organize students' thinking and to record patterns.

In using patterns, it is important for students to find out if the pattern will continue predictably. Have students determine if there is a reason for the pattern to continue, and be sure students use logic when finding patterns to solve problems.

• For example, if it rains on Sunday, snows on Monday, rains on Tuesday, and snows on Wednesday, will it rain on Thursday?
• Another example: If Lauren won the first and third game of chess, and Walter won the second and fourth game, who will win the fifth game?
• Another example: If a plant grew 13 centimeters in the first week and 10 centimeters in the second week, how many centimeters will it grow in the third week?

Because these are questions of probability or nature, be sure students understand why patterns can't be used to find these answers.

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Patterns Questions

Patterns questions and answers are given here to help students understand how to solve patterns questions using simple techniques. Solving different pattern questions will be beneficial for a quick understanding of the logic in patterns. In this article, you will learn how to solve patterns in maths with detailed explanations.

What are Patterns in Mathematics?

In mathematics, a pattern is a sequence of numbers that are formed in a particular way. Every pattern contains a specific rule. For example, the sequence of even numbers is a pattern since each number is obtained by adding 2 to the previous number.

i.e., 2, 4, 6, 8, 10, 12, 14,….

Here, 2 + 2 = 4

8 + 2 = 10 and so on.

Also, check:

• Number patterns in Whole numbers
• Algebra as pattern

A pattern can be of numbers or figures, which means we can also observe patterns in a sequence of similar figures. Let’s have a look at the solved problems on the various number and figure patterns.

Patterns Questions and Answers

1. Identify the pattern for the following sequence and find the next number.

2, 3, 5, 8, 12, 17, 23, ____.

2, 3, 5, 8, 12, 17, 23, ____

The pattern involved in the given sequence is:

12 + 5 = 17

17 + 6 = 23

23 + 7 = 30

Therefore, the next number of the given sequence is 30.

2. Observe the below figure and identify the missing part.

Consider the question figure, where the design in each part will be obtained by rotating the previous design by 90 degrees in the clockwise direction.

So, the missing part will be option (b).

Hence, the complete figure is:

3. Write the next three numbers of the following sequence.

173, 155, 137, 119, 101

The pattern in the given sequence is:

173 – 18 = 155

155 – 18 = 137

137 – 18 = 119

119 – 18 = 101

So, the next three numbers can be written as:

101 – 18 = 83

83 – 18 = 65

65 – 18 = 47

Thus, the sequence is 173, 155, 137, 119, 101, 83, 65, 47.

4. Observe the following figure and choose the correct option.

Each part of the square box contains a triangle inscribed in the circle in the given figure. Also, the triangle in the next circle is the vertical image of the previous one.

Similarly, in the second row, the triangle in the missing part will be an image of the previous one.

Thus, the missing part is option (a).

5. What is the formula for the pattern for this sequence?

11, 21, 31, 41, 51, 61, 71

The numbers in this sequence are written as:

11 + 10 = 21, 21 + 10 = 31, 31 + 10 = 41, and so on.

This can also be expressed as:

11 = 10 + 1 = 10 × 1 + 1

21 = 20 + 1 = 10 × 2 + 1

31 = 30 + 1 = 10 × 3 + 1

41 = 40 + 1 = 10 × 4 + 1 and so on.

From this, we can write the formula for the above pattern as: 10n + 1, where n = 1, 2, 3, etc.

6. Find the pattern in the sequence and write the next two numbers.

10, 17, 36, 73, 134,…

Given sequence is:

10 = 1 3 + 9

17 = 2 3 + 9

36 = 3 3 + 9

73 = 4 3 + 9

134 = 5 3 + 9

So, the next number = 6 3 + 9 = 216 + 9 = 225

Again, the next number = 7 3 + 9 = 343 + 9 = 352

Therefore, the sequence is:

10, 17, 36, 73, 134, 225, 352.

7. Observe the pattern given below. Find the missing number.

In the given figure, we can observe that the sum of the four numbers is equal to the number written in the middle of the shape.

That means,

11 + 22 + 33 + 44 = 110

16 + 24 + 32 + 40 = 112

? + 23 + 34 + 12 = 114

? = 114 – 23 – 34 – 12 = 45

Therefore, the missing number is 45.

8. What is the next number of the following sequence?

20, 18, 21, 16, 23, 12, 25, 8, 27, 4, 33, ?

Given sequence:

20, 18, 21, 16, 23, 12, 25, 8, 27, 4, 33

Let us find the difference between two consecutive numbers of the sequence to identify the pattern.

18 – 20 = -2

21 – 18 = 3

16 – 21 = -5

23 – 16 = 7

12 – 23 = -11

25 – 12 = 13

8 – 25 = -17

27 – 8 = 19

4 – 27 = -23

33 – 4 = 29

Here, we can see that the differences are the prime numbers.

So, the number number = 33 – 31 = 2

9. Estimate the next number of the following sequence.

1, 2, 6, 15, 31, ?

Let’s write the difference between consecutive numbers.

2 – 1 = 1

6 – 2 = 4

15 – 6 = 9

31 – 15 = 16

Here, 1 = 1 2 , 4 = 2 2 , 9 = 3 2 , 16 = 4 2 .

Thus, the next number of the sequence will be obtained by adding 5 2 , i.e. 25, to the previous number.

Therefore, 31 + 5 2 = 31 + 25 = 56.

10. What will be the next number of the given sequence?

1, 5, 12, 22, 35, ?

Let’s write the difference between consecutive numbers.

5 – 1 = 4

12 – 5 = 7

22 – 12 = 10

35 – 22 = 13

The difference between these numbers follows a pattern that 3 is odd to the previous difference.

So, the next number will be obtained by adding 13 + 3, i.e. 16 to 35.

Hence, the next number = 35 + 16 = 51.

Practice Problems on Patterns

• Find the correct number to complete the pattern given below.

20, 21, 23, 26, 30, 35, 41, ___.

• Find the next number in the sequence, 12, 21, 23, 32, 34, 43.
• Write the missing numbers in the following.

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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:

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

Better! The total number of feet are increasing!

Step 4: Looking back:

Check: 12 + 13 = 25 heads

24 + 52 = 76 feet.

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

Videos to watch:

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

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

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

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

Check in question 1:

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

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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

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Video transcript

How to Solve Pattern Questions

How to solve pattern questions: patterns your child must know for psle math.

P6 pattern questions often appear in the last few questions in Paper 2, and they are usually worth five marks.

So, it’s important to understand how to solve Pattern questions better. Here are 3 simple steps which you can guide your child to solve any Pattern Questions. 1) Identify the Pattern There are 3 common types of patterns which your child must know for PSLE Math.

• Common Difference: 1, 3, 5, 7, 9…
• Square Numbers: 1×1, 2×2, 3×3, 4×4, 5×5…
• Increasing Difference: 1, 3, 6, 10, 15, 21…

2) Think of the strategy Many students only know 1 strategy to solve Pattern questions: Listing. Listing method is too time consuming! In this tutorial, we are going to explore some strategies which your child can use easily.

3) Apply the strategy In this tutorial, we’re solving two questions from last year’s prelim papers. After this tutorial, you’ll realize that pattern questions can actually be solved very easily.

Before you read on, you might want to download this entire revision notes in PDF format to print it out for your child, or to read it later.

This will be delivered to your email inbox.

This question was taken from the Henry Park Primary School 2022 Prelim Paper 2.

Question : Jimena uses circles and triangles to form figures that follow a pattern as shown below.

(a) The table shows the number of triangles and circles for the first 4 figures. Complete the table for Figure 5.

Step 1: Look at the range of numbers on the table to see if there is a pattern.

Skimming through the numbers, we see that the number of circles follows a pattern of a constant difference. A constant difference means a difference that is the same throughout.

The constant difference here is +4. Therefore, the number of circles in Figure 5 would be 18 + 4, which is 22.

Step 2: Look at how the number of triangles is increasing.

There doesn’t seem to be any obvious pattern. But what if we try to relate the figure number with the number of triangles?

Let’s start with Figure 4, looking for a 4 somewhere in the triangle.

We separate the first column of 4 from the main group of triangles. Then, you can see that the other side has a group of triangles that forms a 4 × 4 square.

So, Figure 4 is made up of a column of 4 + a group of triangles that forms a square (4 × 4).

Step 3: Try the same formula for all figures.

If we partition the first column, we get 3 triangles on the left. On the right, we have 3 rows of 3 triangles. This 3 × 3 is a squared number, just like 4 × 4.

This also works in Figure 2.

There are 2 triangles on the left and 2 × 2 triangles on the right.

This formula works for Figure 1 as well, where the left is a 1 and the right is a simple 1 × 1.

Figure 5 also has to follow the same formula. Therefore, 5 + 5 × 5 = 30. From here, we can find the total number of triangles and circles, which is 52.

(b) A figure in the pattern has 240 triangles. What is the Figure Number?

The answer to this question follows the formula where the figure number (n) + the squared number of n = the number of triangles. So, n + n × n = 240.

Here, we can do a simple guess and check. When n = 15. 15 + 15 × 15 = 240. So, the figure number is 15.

(c) What is the total number of triangles and circles in Figure 100?

We can find the number of triangles in Figure 100 using the formula we found in question (a) . So, this will be 100 + 100 × 100 = 10100.

Now, let’s try to find the number of circles in Figure 100.

We know that the number of circles follows a constant difference of +4. So, we find the number of intervals from Figure 1 to Figure 100.

So, we simply subtract Figure 1 from Figure 100. We find that there are 99 intervals.

From Figure 1, all the way to Figure 100, there are 99 additions of 4. That means from Figure 1, we will be adding 99 +4s.

So, 99 × 4   will tell us the total increase from Figure 1’s number of circles.

Figure 1’s number of circles is going to increase by 396. So, we take 6 + 396, which gives us 402 circles.

Therefore, 402 circles + 10100 triangles will give us the total number.

This question was taken from the Catholic High School 2022 Prelim Paper 2.

Question : Raju used white and grey squares to form the following patterns as shown.

The table below shows the number of white and grey squares in each figure.

(a) Fill in the table for Figure 5.

Again, we skim through for an easy pattern. We see an easy pattern of a constant +3 for the grey squares.

So, Figure 5 will have 13 grey squares because it’s 10 + 3. Now we can look at how the white squares are arranged.

We can easily draw the number of white triangles for Fig 5 as follow.

So, the total is 6, and 6 + 6 = 12 white squares in Figure 5.

Figure 5 will also have another 3 grey squares. So, it will look as follows:

The question only gives us 1 mark, meaning it’s perhaps a simple question that may only require one equation.

So, let’s look at the total number of grey squares first to see if there’s a pattern:

These are all squared numbers. Figure 1 is 1 × 1, Figure 2 is 2 × 2, and Figure 3 is 3 × 3 total. Therefore, the total number of squares in Figure 40 has to be 40 × 40 = 1600.

(c) How many more white squares than grey squares are used in Figure 40?

The pattern for white is a bit strange and not as straightforward as grey.

So, we can use the constant difference to find the intervals and then find the number of grey squares in Figure 40.

Once we find the number of grey squares, we can find the number of white because we already found the total number of squares in the previous question.

The number of intervals from Figure 1 to Figure 40 is 40 – 1 = 39 intervals. We know that each interval carries a constant difference of +3.

Therefore, 39 × 3 = 117.

So, we increase the number of grey squares in Figure 1 by 117. Therefore, 1 + 117 = 118 is the number of grey squares in Figure 40.

With this total, we can find the number of white squares because we know that the total number of squares is 1600. So, the total – grey squares = white squares, meaning that 1600 – 118 = 1482.

The question is asking for the difference between the white squares and the grey squares. Therefore, we just subtract the grey, which gives us 1364.

In this question, we used our knowledge of squared numbers, the number of intervals, and the constant difference.

Pattern questions like this are not that challenging if you can spot the constant difference and can immediately use the interval concept or spot the square numbers.

I hope this tutorial was easy to understand and helpful for your child. If you have any questions or suggestions, please feel free to leave a comment below.

You can also watch the full video tutorial here:

Before you go, you might want to download this entire revision notes in PDF format to print it out for your child, or to read it later.

To Your Child’s Success,

Ms Nelly Ke Math Specialist Jimmy Maths and Grade Solution Learning Centre

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