Here you will learn about shape patterns, including how to create and justify a rule and extend a pattern for repeating and growing shape patterns.
Students will first learn about shape patterns as part of the operations and algebraic thinking in 4th grade.
What are shape patterns?
Shape patterns are groupings of different shapes that follow given rules. The shapes can be polygons, 3D shapes, letters or any other type of image.
There are two types of shape patterns – repeating and growing.
Repeating pattern
For example,
This is a simple repeating pattern.
In a repeating pattern, a part of the pattern is repeated over and over again. This part is called the core.
The core of this pattern is:
Growing pattern
For example,
This is a more complex growing pattern.
There are parts that stay the same and other parts that change.
Here, the yellow hexagon stays the same, and 2 green triangles and 1 blue rhombus are added each time.
What are shape patterns?
Common Core State Standards
How does this relate to 4th 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.
How to use shape patterns
In order to identify and extend repeating shape patterns:
Identify the core – the part of the pattern that repeats.
Use the core to find and justify the next part in the pattern.
In order to identify and extend growing shape patterns:
Identify what is changing and what is staying the same.
Create a rule based on Step 1.
Use the rule to find and justify the next part in the pattern.
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Assess math progress for the end of grade 4 and grade 5 or prepare for state assessments with these mixed topic, multiple choice questions and extended response questions!
[FREE] End of Year Math Assessments (Grade 4 & Grade 5)
Assess math progress for the end of grade 4 and grade 5 or prepare for state assessments with these mixed topic, multiple choice questions and extended response questions!
Create a rule for the pattern and find the next shape.
Identify the core – the part of the pattern that repeats.
This pattern goes blue triangle, red heart, yellow circle and then repeats.
2Use the core to find and justify the next part in the pattern.
The next shape is:
because the last shape shown is:
Example 2: letter 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 letters that goes A, B, B, H, A, T.
Notice that A is used twice in the core. Sometimes parts of the pattern are repeated within the core. Always look at all the shapes given to confirm the pattern.
Use the core to find and justify the next part in the pattern.
The next letter is A because the next part starts at the beginning of the core.
Example 3: 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 goes 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.
Use the core to find and justify the next part in the pattern.
The next shape is because the last part of the core shown is
.
Example 4: simple shape pattern, growing
Create a rule for the pattern and find the next shape.
Identify what is changing and what is staying the same.
Changing – 1 purple triangle, then 2 purple triangles, then 3 purple triangles, then 4 purple triangles.
Staying the same – There is a always a red heart between each group of triangles.
Create a rule based on Step 1.
Start with 1 purple triangle, then 1 red heart. Repeat, adding 1 more purple triangle each time.
Use the rule to find and justify the next part in the pattern.
The next part in the pattern is , because after the there is 1 more purple triangle than before.
Example 5: simple shape pattern, growing
Create a rule for the pattern and find the next shape.
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.
Example 6: shape pattern, growing
Create a rule for the pattern and find the next shape.
Identify what is changing and what is staying the same.
Changing – The first shape has 2 squares on the bottom and is 2 squares tall, the second shape has 3 squares on the bottom and is 3 squares tall, the third shape has 4 squares on the bottom and is 4 squares tall.
Staying the same – The top left square is always missing
Create a rule based on Step 1.
Add 1 row and 1 column of squares each time, but no square to the top left corner.
Use the rule to find and justify the next part in the pattern.
The next part in the pattern is , because 1 row is added to the bottom and 1 column is added to the side, but there is not a square in the top left corner.
Teaching tips for shape patterns
It may be useful to review the simple shapes that students learn in 2nd grade before beginning this skill. While it is not imperative that they use the correct terminology, shape patterns are a good way for students to practice shape recognition and use precise mathematical language.
Printable pattern worksheets and other math worksheets can be a great tool to use for this skill, but there are also many opportunities for more engaging pattern activities. Use educational games that involve finding patterns.
Also, give students access to pattern blocks, flashcards, colorful building blocks or any other objects that can be used to create patterns. You may even have a permanent pattern center, where students can create and record their own patterns that can be used when explicitly teaching this skill.
Instead of focusing on just identifying and using the pattern rule, also place emphasis on justifying it. This is not only a good mathematical practice, but the act of justifying can help students write better rules.
Asking why a rule works, naturally draws attention to the general aspects of the pattern – identifying what is changing and what is staying the same, which are the building blocks of a valid mathematical generalization.
Easy mistakes to make
Identifying the wrong pattern core This mistake is particularly common when a shape repeats within the pattern. It is important to look at all the shapes given, before deciding that it is the pattern core.
For example, At first glance it seems is the pattern core. However, when considering all the rhombuses given, it becomes clear that the core is larger.
Not writing a general pattern rule When writing a rule for a growing pattern, the rule should always tell what is changing and what is staying the same. Writing a rule that is too vague may make it hard to extend the pattern. Writing a rule that is too specific does not require students to create generalizations and can set them up for misconceptions in later algebraic topics.
For example, Too vague: Add two blue squares each time. Too specific: Start with 2 red squares, then 2 red squares and 2 diagonal purple squares, then 2 red squares and 4 diagonal purple squares, then 2 red squares and 6 diagonal purple squares. General pattern rule: Start with 2 red squares and add 2 purple squares to the top left and bottom right diagonal each time.
This pattern goes green triangle, purple pentagon, orange cross, red pentagon and then repeats.
The next shape is because the last shape shown is .
2. What is the next shape in the pattern?
3
b
u
B
This pattern goes B, U, G, 3 , b, u, g, 3 , 3 and then repeats.
The next shape is because the last part of the core shown is .
3. What is the next shape in the pattern?
This pattern goes yellow star, red star, purple star, yellow star, green star, red star.
The next shape is because the last shape shown is .
4. What is the next part in the pattern?
Identify what is changing and what is staying the same:
Changing – 1 blue circle, 2 blue circles, 3 blue circles.
Staying the same – There is a always an orange sun and a purple sun between each group of blue circles.
Rule: Start with 1 blue circle, 1 orange sun and 1 purple sun. Repeat, adding 1 more blue circle each time.
The next part in the pattern is , because after the there is 1 more blue circle than before.
5. What is the next part in the pattern?
Identify what is changing and what is staying the same:
Changing – 1 purple square on the left and right, 2 purple squares on the left and right, 3 purple squares on the left and right.
Staying the same – There is always a green circle in the middle, 1 purple square above, and 1 purple square below.
Rule: Start with a green circle in the middle and a purple square above, below, to the left and to the right. Repeat, adding 1 more purple square to the left and right each time.
The next part in the pattern is , because there is a purple square being added to the left and the right each time.
6. What is the next part in the pattern?
Identify what is changing and what is staying the same:
Changing – a row of 1 and 3; a row of 1, 3, and 5; a row of 1, 3, 5, and 7.
Staying the same – There is always a row of 1 and 3.
Rule: Start with a row of 1 orange square and 3 orange squares. Repeat, adding a row with 2 more orange squares each time.
The next part in the pattern is , because there is a row of 2 more orange squares being added each time.
Shape patterns FAQs
How do shape patterns help students’ mathematical understanding?
Shape problems are a great way to encourage students to sharpen their problem solving skills. Creating a rule and extending patterns requires students to analyze and generalize relationships – an important math skill. Justifying growing patterns in particular is a form of algebraic thinking that can help students create and support valid justifications at an elementary level.
When can students start understanding shape patterns?
Students as young as Pre-K have the ability to notice and extend simple patterns. Even though this is a 4th grade skill in the Common Core, it is a good idea to introduce patterns and encourage pattern thinking in younger students.
Can these types of patterns include fractions and decimals?
These types of patterns do not typically include fractions or decimals. However, they can be included if used in a way that is not operational (for example in the repeating pattern \frac{1}{2}, \frac{1}{3}, \frac{1}{2}, \frac{1}{3}, \frac{1}{2}, \frac{1}{3} … ). This pattern can be extended without operating.
Can shape patterns involve subtraction?
Yes, decreasing patterns (which are the opposite of growing patterns) involve less and less shapes each time, which would include the use of subtraction in the rule.
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because the last part of the core shown is
.
, because after the 
there is 1 more purple triangle than before.
, because 1 triangle is added to the top and 1 triangle is added to the side each time.
, because 1 row is added to the bottom and 1 column is added to the side, but there is not a square in the top left corner.
is the pattern core. However, when considering all the rhombuses given, it becomes clear that the core is larger.
because the last shape shown is 
.
because the last part of the core shown is 
.
because the last shape shown is 
.
, because after the 
there is 1 more blue circle than before.
, because there is a purple square being added to the left and the right each time.
, because there is a row of 2 more orange squares being added each time.