4.4: Growing Patterns - Mathematics

Here is a pattern made from square tiles.

Think / Pair / Share

  • Describe how you see this pattern growing. Be as specific as you can. Draw pictures and write an explanation to make your answer clear.
  • Say as much as you can about this growing pattern. Can you draw pictures to extend the pattern?
  • What mathematical questions can you ask about this pattern? Can you answer any of them?

Here are some pictures that students drew to describe how the pattern was growing.

Ali's picture

Michael's picture

Kelli's picture

Think / Pair / Share

Describe in words how each student saw the pattern growing. Use the students’ pictures above (or your own method of seeing the growing pattern) to answer the following questions:

  • How many tiles would you need to build the 5th figure in the pattern?
  • How many tiles would you need to build the 10th figure in the pattern?
  • How can you compute the number of tiles in any figure in the pattern?

Problem 15

Hy saw the pattern in a different way from everyone else in class. Here’s what he drew:

Hy's picture.

  1. Describe in words how Hy saw the pattern grow.
  2. How would Hy calculate the number of tiles needed to build the 10th figure in the pattern?
  3. How would Hy calculate the number of tiles needed to build the 100th figure in the pattern?
  4. How would Hy calculate the number of tiles needed to build any figure in the pattern?

The next few problems present several growing patterns made with tiles. For each problem you work on, do the following:

  1. Describe in words and pictures how you see the pattern growing.
  2. Calculate the number of tiles you would need to build the 10th figure in the pattern. Justify your answer based on how the pattern grows.
  3. Calculate the number of tiles you would need to build the 100th figure in the pattern.
  4. Describe how you can figure out the number of tiles in any figure in the pattern. Be sure to justify your answer based on how the pattern grows.
  5. Could you make one of the figures in the pattern using exactly 25 tiles? If yes, which figure? If no, why not? Justify your answer.
  6. Could you make one of the figures in the pattern using exactly 100 tiles? If yes, which figure? If no, why not? Justify your answer.

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

A fractal is a detailed pattern that looks similar at any scale and repeats itself over time. A fractal's pattern gets more complex as you observe it at larger scales. This example of a fractal shows simple shapes multiplying over time, yet maintaining the same pattern. Examples of fractals in nature are snowflakes, trees branching, lightning, and ferns.

A spiral is a curved pattern that focuses on a center point and a series of circular shapes that revolve around it. Examples of spirals are pine cones, pineapples, hurricanes. The reason for why plants use a spiral form like the leaf picture above is because they are constantly trying to grow but stay secure. A spiral shape causes plants to condense themselves and not take up as much space, causing it to be stronger and more durable against the elements.

Repeating and Growing Pattern

Repeating Pattern: Identifying Next Picture

Circle the picture that comes next in each pattern. These pdf worksheets are most suitable for Pre-K and Kindergarten kids.

Repeating Pattern: Cut-Paste Activity

Cut out the graphics at the bottom of each printable worksheet. Paste the graphic that comes next in each picture pattern.

There are growing sequences in each question. Complete the picture that comes next in the sequence. Few questions have partially drawn pictures to make the drawing easy for you.

Grade 3 and grade 4 children are expected to draw the graphic that goes next in each pattern.

Repeating and Growing: Mixed Pattern

This section contains both repeating and growing pattern in each printable worksheet.

Each pattern contains pictures in different sizes. Choose the size that comes next in the sequence.

There are three pdf worksheets in each level. 2nd grade and 3rd grade children are expected to identify and extend the shape pattern.

Write the rule followed by each picture pattern.

Pattern Making

Children often enjoy spotting patterns and making patterns with lots of different things, including themselves.

Adults could provide a 'pattern of the day' with objects for children to copy, extend and create their own. They could make deliberate mistakes for children to spot. Parents and carers could join in.

Encouraging mathematical thinking and reasoning:

The Mathematical Journey

• copying the pattern by matching cubes one at a time
• copying by complete units e.g. picking up a red and a blue cube together
• continuing the pattern in ones or in complete units
• correcting an error e.g. spotting a missing cube or reversed colours
• identifying the rule of the pattern - "It should go red, blue, blue"
• noticing a growing pattern - "It's like a staircase"

Counting and cardinality:
• counting the number of items in the unit of repeat, or the towers in a staircase pattern

Adding and subtracting:
• generalising about a staircase pattern - "It's one more each time!"

Position and spatial properties:
• using the vocabulary of position - "The red comes next", "It's blue between the red and the yellow"
• reversing units to make reflecting patterns e.g. ABC CBA

Development and Variation
Provide more complex repeating patterns: ABC, ABB, ABCD.
Vary the materials and media, indoors and out.
Make action or sound patterns and record them with invented symbols.
Make growing patterns, e.g. going up in ones.
Make reflecting patterns with peg boards, mirrors and constructions.

Story, rhyme and song links
The Shopping Basket by John Burningham
There Was an Old Lady Who Swallowed a Fly
The Gingerbread Man

Coloured cubes, beads, small toys, leaves and twigs printing and sticking shapes.
Pegboards, mirrors, construction materials.
Long strips of paper for making patterns.

See the Erikson Early Math Collective website for more activities about patterns.

4.4: Growing Patterns - Mathematics

The Staircase Problem / Towers / Fancy Staircases

The Staircase Problem -Towers (&ldquoAlgebraic Strategies&rdquo activities)

The activity actually has three main parts to it. I had students work in pairs on each activity for about 5- 10 minutes and then we discussed each part as a group. The first part is entitled &ldquoGrowing Squares&rdquo and uses table tops made out of square blocks. The first table top has one block, the second table top has four blocks, the third table top has nine blocks, and so on. Students were all able to come up with the pattern (nth table top has n2 blocks) very quickly.
The second part entitled &ldquoThe Staircase Problem&rdquo uses pictures of staircases that have more and more steps. They are again asked to find a pattern. Most of the groups made a table of values similar to the following:

The third part, &ldquoTowers&rdquo, was more challenging. This used three-dimensional shapes. They again made tables like the following:

Creating a formula was challenging for them. Most looked at each tower as a column surrounded by four staircases, when they calculated the number of blocks to be used. They then tried to use the previous formula from the staircases here in this problem as well. Two of the groups concluded the formula for the nth tower as: 2n^2 - n. During the last few minutes of the class period we worked together as a class to see how this formula could be derived.

Towering Numbers

It was a short class, so students had about 20 minutes to work on it. Almost all could figure out the number of bricks in a row when they knew the actual row number. Only about half of them could describe a rule to figure out the number of bricks in a row for any number. No one used variables to describe it (even though we have done a lot of work with variables in this pre-algebra class.)
About ¼ of the students could figure out how to find the total number of bricks in a tower when they knew how many rows there were. And only 2 or 3 of those could describe the rule in words. Again, no one used variables.
I went over the problem the next class day and we talked about using variables. I showed them how to use variables for this particular problem. Hopefully, some will be able to on the next exercise.

Towering Numbers

I decided to try to do one of the &ldquoAlgebraic Strategies&rdquo activities (Sum of Consecutive Numbers) with two of my calculus classes on a Friday afternoon after having taken a chapter test the previous day. I had them work in groups of two in one class and in groups of three students in the other.
The objective of the activity was to find all the possible ways to express each number from 1 to 35 as a sum of two or more consecutive counting numbers. They were given a chart to fill in and then were to answer some questions about patterns they discovered while completing the chart. Using these patterns, they were then asked to make predictions as to whether given numbers greater than 35 could be expressed as a sum of 2, 3, 4, or more consecutive counting numbers.
I told the students that they had 40 minutes to look at the chart and the follow-up questions and then we would get together during the last 10 minutes of class to discuss the activity.
While the students worked on the activity, I tried to walk around the classroom and listen to the discussions that were going on in the individual groups. At first, they had questions about whether they could use the number zero or negative numbers and had to be reminded what a &ldquocounting number&rdquo was.
I was somewhat surprised that a few of the groups started off filling in their charts in a quite disorganized fashion. Some just took a number at random and tried to express it as different sums. It seemed as if it took them a lot longer to complete the chart than I would have expected. Because of the length of time used to fill in the chart, most groups did not have enough time to really do justice to answering the six questions posed in the worksheet. All of the groups eventually came up with a plan that allowed them to get the chart filled in.
When we got together as a class during the last ten minutes to discuss any patterns they discovered, both classes made the comment that they could see patterns but that they had a difficult time putting the patterns down on paper as an algebraic expression of some type. They definitely had a hard time abstracting from the computation. One group did mention that they noticed that if they multiplied the middle number in a sequence by the number of numbers in the sequence that that would give them the sum.
Overall, I was disappointed in the results of the activity. If I were to do this activity again, I would probably either spend a little more time at the beginning giving them more detailed directions or maybe go through a short, similar type of activity with them first. I would probably also give them the entire period to work on it and then have them write something up and maybe spend the first 10-15 minutes of the next day&rsquos class period discussing their results.

I used The Staircase Problem / Towers / Fancy Staircases from the Algebraic Thinking class in my HOTS class. (HOTS stands for Higher Order Thinking Skills and is a non-mandatory mini math class that we offer opposite band where we play with math topics as well as puzzles and thinking games. I have 5 students in the class this semester, which I divided into 2 groups.

One of the groups immediately saw a pattern in the staircases and computed the answers. While they could describe the rule, they could not put it into an algebraic form. The second group, while having less formal math training, actually attempted to create an algebraic formula. It was cumbersome and ugly &ndash but it worked. They were somewhat frustrated with the what their results looked like after working the whole period on it so I sat down and we made it nicer looking together &ndash but pointed out that it was the same thing that they created.

On the towers they developed strategies to compute the 1, 2, 3, 4, and 10th towers. I told them that once they had these done I had a story to tell them that might help them with the 100th (since they haven&rsquot learned about arithmetic sequences yet) and then related the fable of Gauss and his teacher asking him to add all the numbers of 1 to 100 and how he arrived at the added the sum forward and backwards etc&hellip It was a nice extension and eased some of the arithmetic while still concentrating on the patterns of the towers.

The fancy stairs were very difficult to take to an abstract level, but seem to become easier if you break time into &ldquoodd fancies&rdquo and &ldquoeven fancies&rdquo.

Towering numbers. This activity went very good. They did the first one done by using the picture. When we got to the third part to find a rule the faster students had it right away, but were so eager to tell the other students that they didn&rsquot have the chance to think of it on their own. The next time I have them do this activity I will have them work in pairs or in groups of three. I think things went well and I will do towering numbers next year.

Growing and Shrinking Number Patterns (A)

Teacher s can use math worksheets as test s, practice assignment s or teaching tool s (for example in group work , for scaffolding or in a learning center ). Parent s can work with their children to give them extra practice , to help them learn a new math skill or to keep their skills fresh over school breaks . Student s can use math worksheets to master a math skill through practice, in a study group or for peer tutoring .

Use the buttons below to print, open, or download the PDF version of the Growing and Shrinking Number Patterns (A) math worksheet. The size of the PDF file is 11791 bytes . Preview images of the first and second (if there is one) pages are shown. If there are more versions of this worksheet, the other versions will be available below the preview images. For more like this, use the search bar to look for some or all of these keywords: math, mathematics, patterns, patterning .

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Generating Number Patterns

Looking for high-quality Math worksheets aligned to Common Core standards for Grades K-8?

Our premium worksheet bundles contain 10 activities and answer key to challenge your students and help them understand each and every topic within their grade level.

In this lesson you will learn how to describe, extend, and make generalizations about numeric patterns. Working with number patterns is an very useful skill for solving many types of problem.

Identifying a pattern when you look at individual examples helps you to generalize and find a broader solution to a problem.

Work through the examples and explanations in this lesson with your children and then try the worksheet that you will find at the bottom of this page.

Patterns You Know Already

Probably without even knowing it, you have been observing and creating patterns ever since you were a very small child. You probably made repeating patterns with shapes, such the one below with triangles, circles, and squares.

Get your children to explain the pattern they see in the above sequence of shapes.

When you got just a little bit older, you probably learned skip-counting, which is nothing more than applying a pattern to counting.

Skip Counting

Skip-count by 2’s: 2, 4, 6, 8, 10, 12, 14, 16…

Skip-count by 5’s: 5, 10, 15, 20, 25, 30, 35, 40 …

Skip-count by 10’s: 10, 20, 30, 40, 50, 60, 70…

All of these are patterns, or mathematical rules.

Generating & Analyzing Number Patterns

A fun part of math is creating and playing with patterns. Math is organized with rules to follow. If you know what the rule is, you can create a pattern. The rule is often organized on a function table as shown in the examples below.

Function Table Example: x + 5 = y

Function Table Example: 2x = y

Function Table Example: x - 3 = y

Work through the next set of number pattern examples with your children. You can check your answers by clicking the empty boxes to show and hide the Y numbers.

Function Table Example: x + 9 = y

How did you do filling in the missing information on the Y side? Click in the boxes to check your answers . Since the rule is X +9 = Y, you add 9 to each of the X numbers to get the corresponding Y number.

Function Table Example: x - 7 = y

How did you do filling in the missing information on the Y side? Check your answers by clicking the boxes. Since the rule is X -7 = Y, you should have subtracted 7 from each of the X numbers to get the corresponding Y number.

Function Table Example: 5x = y

How did you do filling in the missing information? Since the rule for this function table is "multiply by 5," the first missing number is 15, since 3 x 5 = 15. For the next missing number, you have to think "what number times 5 would give me 25?" You can also think of it as division: "25 divided by 5 = what number?" since division is the inverse, or opposite of multiplication. The final missing number is 5 x 7, or 35.

Function Table Example: What is the Rule?

How did you do? Were you able to figure out the pattern? This table shows "multiply a number by 3." You can use the filled in numbers to figure out the rule by asking, "what is the relationship between the X numbers and their corresponding Y numbers?" Look for patterns. Once you have determined the pattern, it is a simple matter to fill in the missing numbers. Here, 21 (3 x 7) and 33 (3 x 11) were missing. If you were asked to extend the table, you would get the pair: 13, 39.

Generating & Analyzing Number Patterns With Function Tables: Recap

So when it comes to number patterns, remember these things:

  • Look for a relationship between the input “X” side, and the output “Y” side.
  • Check the pattern against every row. It has to be true for the whole table, or it’s not the rule.
  • Use the rule, and the numbers you know, to complete or extend the pattern.

Number Patterns and Function Tables Worksheets

Click the link below and get your children to try the Number Patterns and Function Tables worksheet. This worksheet has 3 pages and includes a recap of the above, guided practice, and independent questions.

Exploring Multiplication Patterns Lesson

Please Note: this material was created for use in a classroom, but can be easily modified for homeschooling use.

This lesson enables students to begin the process of mastering multiplication facts. Students will learn to use patterns and property theories as strategies for recalling those facts.

  • develop computational fluency by exploring patterns in multiplication for products involving one-digit factors.
  • understand and use the zero property for multiplication and the property of one as a factor in multiplication.

This lesson can be divided into two or three smaller lessons, each lasting about 20-25 minutes.

Teacher: Teacher’s Chart of multiplication patterns to be printed as an overhead transparency or copied onto the board.

  1. Introduce key vocabulary: multiple, factor, product, double.
  2. Display the Teacher’s Chart an overhead transparency, or copy it onto the board. Hand out copies of the Hundred Chart.
  3. Have students count by 2s, shading multiples of 2 yellow on their hundred chart. Ask them to examine the numbers carefully. Ask:
  • What patterns do they notice? (The multiples of 2 are even and always end in 0, 2, 4, 6 or 8.)
  1. Have students count by 5s, circling the multiples of 5 with a blue marker on their number charts. Ask:
  • What patterns do they notice? (The multiples of 5 end in 5 or 0).
  1. Model your thinking:
  • When I look at the multiples of both 2 and 5, I see that they all end in zero. It’s like counting by 10s. I notice that 2 x 5 is 10.
  1. Have the students count by 9s on the number chart. Write out the multiplication sentences and answers on the chalkboard (9 x 1 = 9, 9 x 2 = 18, and so on) and ask students to find a pattern and discuss what they find.
  • (The sum of the product’s digits is 9. The tens digit is 1 less than the other factor. Make it clear that they will have to memorize the 9s, but that these patterns may help them remember and can be used to verify the products.)
  1. Have students look at their charts and find:
  • the multiples that 2 and 9 have in common (18, 36, 54, 72, 90, and so on).
  • the multiples that 2 and 5 have in common (10, 20, 30, 40, 50, and so on).
  • the multiples that 5 and 9 have in common (45, 90).
  1. Ask students what would happen if they shaded in all the multiples of 1 on their charts. (They should soon realize that they’d be shading in everything.) Articulate the property of one:
  • The product of a number and 1 is that same number.
  • Every number is a multiple of 1 and itself.
  1. To illustrate, ask several easy questions to the class at large. What is 8 x 1? What is 9 x 1? Get increasingly harder: What is 52 x 1? What is 1 million x 1?)
  2. Ask students to think about multiplying with zero in terms of repeated addition. What is 0 + 0? What is 3 x 0? What is 52 x 0? What is 1 million x 0? Help students to determine the zero property for multiplication:
  • the product of a number and 0 is 0.
  1. Ask students to name the double of 2 (2 x 2 = 4). Share the following problem:
  • For his family reunion Ariel wants to make 2 lemon pies that use 5 lemons each. How many lemons should he buy? (2 x 5 =10). Then he remembers that his Uncle Bob loves lemons and is likely to eat 2 pies all by himself. Ariel better make 4 pies! How many lemons will he need to make 4 pies that require 5 lemons each?
  1. Explain to students that they can arrive at the answer through the idea of the double. Example:
  1. Hand out the Doubles Worksheet that develops the concept of the double and complete it with your students until the pattern is clear. Answer Key
  2. Hand out one or more of the Independent Practice Worksheet for students to practice finding the product. Answer Key
  • Have each student answer the Assessment Questions .
  • Review multiplication facts daily, using patterns and properties for recalling those facts that are not yet automatic.
  • Students should be able to:
  • recognize patterns in multiplication for products involving one-digit factors.
  • understand and use the zero property for multiplication and the property of one as a factor in multiplication.
  • understand and use the technique of the double to solve for more difficult products.
  • know multiplication facts by using the patterns of factors 2, 5, and 9.
  • Checking for automaticity should be ongoing and can be as simple as calling out facts for individuals to give products as quickly as they can. This can be done while standing in the cafeteria line or during other windows that occur in a typical school day. A variety of games can be used as tools to assess students and promote memorization.
  • Have students complete the Multiplication Table through 9 x 9. For extra credit, challenge them to complete the table through 12 x 12. Remind them to use what they have learned about patterns and properties to help. Answer Key
  • Hand out the Extension Worksheet . You may wish to go over the answers as part of a class discussion. Answer Key
  • Hand out the Enrichment Worksheet and have students solve for n. Answer Key

Visit these sites for more Web resources:

Math in literature. Lists books with multiplication themes

Encourage students to begin the process of mastering multiplication facts as you enhance your mathematics lesson by using this detailed outline lesson. Students will learn to use patterns and property theories as strategies for recalling those facts. Guided practice, assessment, and extension printables are included. This lesson can be broken down into two or three smaller ones to use over a span of days.

Additions Since Book Was Published

Growing Patterns: Fibonacci Numbers in Nature by Sarah C. Campbell. Illustrated by Sarah C. Campbell and Richard P. Campbell. (2010, Boyds Mill. ISBN 9781590787526. Order Info.) Nonfiction Picture Book. 32 pages. Gr 3-5.
Eye-catching close-up photographs show the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, 34, 55 . . . ) in flower petals, pine cones, pineapples and seashells. The opening description is a bit confusing but is soon made clear. A great general exposure to patterns for the youngest students and a clear introduction to this crucial pattern for slightly older students.


It's possible even, to create patterns which emulate logic gates (and, not, or, etc.) and counters. Building up from these, it was proved that the Game of Life is Turing Complete, which means that with a suitable initial pattern, one can do any computation that can be done on any computer. Later, Paul Rendell actually constructed a simple Turing Machine as a proof of concept, which can be found here. Although Rendell's Turing Machine is fairly small, it contains all of the ideas necessary to create larger machines that could actually do meaningful calculations. One of the patterns in Jason Summers' collection will compute prime numbers, and another will compute twin primes (two primes that only differ by adding or subtracting 2).

Watch the video: Year 1. Maths. Number Patterns (December 2021).