Pattern Blocks+

Pattern Blocks+
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Description Examples Sample Files
Features of the Tool Keyboard Shortcuts


The Pattern Blocks+ Tool is a virtual manipulative similar to the physical ones commonly available in classrooms. Eighteen different shapes can be dragged into the workspace from the scrollable selection panel at the left. Once in the workspace they can be moved, copied, reflected, or rotated, individually or in groups. Blocks to represent tenths are included. The colour of each type of block can be changed. Students can make designs and develop numerous mathematical concepts. The Pattern Blocks+ workspace shows an isometric grid of small equilateral triangles, which is unique when so many area activities use a square grid (see the discussion of length and area units in the Measurement and Geometry section below).

Access a wide variety of Annotation tools Annotation button to communicate thinking.
Insert picturesImport Picture button into the tool.
Work created in a mathies tool can be savedSave File button and openedOpen Button. A saved file can be shared with peers or submitted to a teacher. The file will contain all solution steps from start to finish.

Take a screenshot to use as part of a portfolio, presentation, web page, etc.


Mathematical Concepts

Pattern Blocks can be used to develop understanding of:


Connecting Fundamental Math Concepts with Pattern Blocks+

Fundamental Concepts and Skills (Link #1) Pattern Blocks+ Connections
Working with numbers:
Understanding and using numbers (e.g., being able to read, represent, count, order, estimate, compare, compose, decompose, and recompose numbers).
Pattern Blocks+ can be used to:
  • represent fractions and decimals using area and set models
  • identify different shapes as the whole and relate other shapes to that whole (e.g., what fraction of the white pentagon is the double hexagon?)
  • practise counting fractional units
  • order fractions by considering the area of a block representation
  • compose and decompose shapes and relate to numerical measurements and operations (e.g., one-sixth of a hexagon plus one-third of a hexagon is equal to one-half of a hexagon)
  • demonstrate and explain the concept of equivalent fractions
  • relate fractions, decimals, percents and ratios
Recognizing and applying understanding of number properties:
Understanding how numbers behave in operations and drawing on that understanding to master math facts and perform calculations.
Pattern Blocks+ can be used to:
  • relate addition to combining
  • relate subtraction to taking away
  • relate multiplication to repeated addition
  • relate multiplication to a fraction "of" a number
  • recognize that the commutative and associative properties apply to the addition and multiplication of fractions
Mastering math facts:
Understanding and recalling math facts, using a variety of strategies.
Pattern Blocks+ can be used to:
  • create arrays representing a product of two whole numbers
  • apply knowledge of whole number facts to assist in recognizing patterns when operating with fractions
  • practise number facts while solving problems involving Patterning and Algebra
Developing mental math skills:
Doing calculations in the mind, with little or no use of paper and pencil or calculator.
Using visual tools when learning to perform mathematical operations allows students to draw on these mental models and visualizations to perform mental calculations.

Students will develop their mental math skills with Pattern Blocks+ as they:
  • determine the area and perimeter of blocks and composite shapes
  • solve problems involving the addition and subtraction of fractions and decimals
  • multiply and divide by 10 and 0.1
Developing proficiency with operations:
Performing calculations with ease, precision, and consistency and with a general understanding of number and operations, number properties, and their appropriate application in problem solving.
Pattern Blocks+ can be used to:
  • recognize the inverse relationship between addition and subtraction (e.g., since 1/6 + 1/3 = 1/2, then 1/2 - 1/6 = 1/3)
  • recognize the inverse relationship between multiplication and division (e.g., since 1/2 x 6 = 3, then 3 ÷ 1/2 = 6).
  • explain why dividing positive numbers by 0.1 gives a larger result
  • describe multiplicative relationships between quantities by using simple fractions and decimals
  • demonstrate an understanding of proportional reasoning using ratios and unit rates

Connecting Fundamental Math Concepts with mathies.ca (Draft)




Students develop spatial sense by creating designs and copying or modifying existing designs.

Some possible activities include:


Number Sense

Many of the WINS counting activities can be used with Pattern Blocks+ (click the Activities link to expand the list).

Count the number of various types of blocks in a design.
Dinosaur design
Composing and Decomposing Numbers
See the WINS activities for Composing and Decomposing Numbers to 5, Numbers to 10 and Numbers to 20.
Activity from WINS

Comparing and Ordering Numbers
See the WINS activities for Comparing and Ordering Numbers to 10 and Numbers to 20.
Activity from WINS


Geometric Properties

Sort a collection of shapes by their geometric properties.
(e.g., number of sides, side lengths, number of interior angles, number of right angles, area, perimeter, symmetry) Arrangement of the rhombus shapes available
Students could be asked how the sort above was organized.

Venn diagram sort of the pattern blocks according to quadrilateral and having parallel sides
The sort above, using a Venn Diagram, raises some interesting questions:
  1. Is it possible to create a quadrilateral with no parallel sides composed of pattern blocks?

    Reasoning using the interior angles of the pattern blocks might help.
    Does the answer depend on whether blocks are allowed to overlap? snap to the grid?
  2. Design a new pattern block that is a quadrilateral with no parallel sides.

    Use triangular isometric grid paper (see NCTM's Dynamic Paper), dynamic geometry software or 2D design software, if available.

    Possible Solution: The Geometer's Sketchpad was used to create a plum coloured quadrilateral block.
    Quadrilateral Tile
    The design below was then created by:
    1. Inserting an image of the plum coloured quadrilateral block into the Pattern Blocks+ tool.
    2. Dragging a red trapezoid block into the workspace; rotating and resizing the plum block image to match one side length of the red trapezoid.
    3. Copying the plum block image and positioning it beside the original image.
    4. Dragging out and positioning the other blocks.
    Notice: the two orange square blocks and a green triangle block are placed on top of the plum block image. Turn off snap to grid Magnet icon to place these blocks in the corners of the plum block. These corner blocks help us to see the 90°, 60°, 90° angles in three of the corners of the plum coloured block. What is the measure of the fourth angle?
    Quadrilateral Block designed in Sketchpad

Create and identify congruent shapes.
In Pattern Blocks+, congruent shapes are created by copying, translating, reflecting and rotating shapes.

Students could also be asked to: Shape with six copies of a pattern core arranged around a hexagonTwo designs created by transforming a block pattern
Note: The first shape above can be accessed as a sample file. Undo and redo through the sequence to see one way that the core can be identified and the transformations applied.

Guess my Shape.
Students work in partners. Each student creates a shape using pattern blocks, hidden from the view of their partner. Students then take turns providing clues that describe their shape using proper mathematical terminology. Based on the clue, the partner tries to replicate the shape. Students continue exchanging clues until each partner is able to replicate the other's shape.
A ring made of 6 hexagons
Some sample clues for the above shape:
What is the perimeter of this shape? (This question might spark some lively mathematical discourse!)


Identify, extend, and create repeating patterns.
Four repeating patterns
Students demonstrate an understanding that patterns result from using a transformation (e.g., slide, flip, turn), or making some other repeated change to an attribute (e.g., shape, size, colour, orientation, number).
Note: In Pattern Blocks, all blocks with a given shape have the same colour, limiting the number of attributes that can change. To create patterns with more attributes, consider using the Set Tool.

Students extend and create repeating patterns that result from reflections and/or rotations.

Create a pattern block train by alternating one green triangle with one red trapezoid. Predict which block will be in the 30th place.
Note: This is a sample problem from the Grade 4 Patterning and Algebra - Patterns and Relationships expectation: make predictions related to repeating geometric and numeric patterns

Identify, extend and create growing patterns.
linear growing pattern
The image shows the first 3 terms of a linear growing pattern.
What will the 10th term look like?
How many blocks will be in the 10th term?
How many blocks will be in the 100th term?
Does a term exist which contains 45 blocks? 300 blocks?

Students create visual representations of linear growing patterns then pose and answer questions about each pattern.


Measurement and Geometry

The Ontario Curriculum (Link #2) suggests using physical pattern blocks as non-standard units of length, area and volume.

Covering designs (see above) also addresses a variety of Geometry expectations found in the Ontario Curriculum.

Students take measurements of composite shapes and relate them to side lengths, areas and angles of basic blocks.

Length and Area Units used with Pattern Blocks

The Pattern Blocks+ tool uses a triangular isometric grid.
A typical length unit is the distance between adjacent dots, which is a side length of the first six standard pattern blocks.
A typical area unit is the area of the small green triangle.

When a 1cm by 1cm square grid is used, lengths would typically be measured in centimetres and areas in square centimetres.

On the Pattern Blocks+ grid below, the small green triangle's area is not the square of its side length. It is the orange block that has area equal to the square of its side length.
Green Triangle on Orange Square
The square's area is more than twice the area of the triangle. Hint: Draw a vertical line through the middle of the shape.
More precisely, the area of the small green triangle is about 43% of the area of the orange square.

Area of the Yellow Hexagon - Two Approaches
When we say that the yellow hexagon's area is 6, we mean 6 little green triangle areas.
Hexagon area compared to six triangles
However, the yellow hexagon's area, measured in orange square areas, is less than 4, as illustrated by the image below.
Hexagon area compared to four squares
The yellow hexagon's area is actually less than 3 orange square areas.
Hexagon area compared to three squares
Hint: Compare the area of the hexagon that is not over squares to the area of the squares that is not under the hexagon.

When measuring, it is very important to be clear about the units being used. The yellow hexagon measures 6, when the area of the small green triangle is the unit. However, it measures approximately 2.6, when a square grid unit is used.


Example 1 - Greatest Perimeter for a fixed Area

Create a composite shape using six yellow hexagon blocks with the greatest possible perimeter. Six hexagons arranged in two different ways to calculate perimeter
Encourage students to use efficient computational strategies to calculate the perimeters.
Calculation of perimeter

Example 2 - Relating the Area of a Triangle, Parallelogram and Rectangle

Relate the area of a triangle to a parallelogram. Triangle reflected to make a parallelogram
Reason that the parallelogram (in this case a rhombus) has twice the area of the large green triangular block by:
Relate the area of a triangle to a rectangle. Triangle compared to surrounding rectangle

Relate the area of a parallelogram to a rectangle.
Parallelogram compared to surrounding rectangle

Like many activities involving Pattern Blocks, this activity is meant to give students an introductory, visceral experience of the area relationships above, which can then be extended to more general cases.

Example 3 - Angle Measures

Determine the interior and exterior angles of all the pattern blocks by relating them to the equilateral triangle's interior angles of 60°.
e.g., Deduce that the angles in the tan rhombus are 30° and 150°.
Three tan rhombii related to a green triangle block

Verify properties including:

Example 4 - Similar Figures

Relate the areas and perimeters of similar shapes.
For example:

Example 5 - Height of an Equilateral Triangle

Note: When one side of a triangle is identified as the base,
Base of triangle identified
then the segment perpendicular to the base containing the opposite vertex is called the height or altitude of the triangle.
Height of triangle identified
In Grade 7, students will learn that no matter which side is treated as the base, the area is one-half the length of the base multiplied by the height from that base.

Relate the height of the green triangle block to its side length.

Sequence of triangles inside squares
Click the image above to see it at full size, without distortion.

From each numbered term above, reason that the height is:
  1. less than its side length
  2. more than one-half its side length
  3. more than two-thirds its side length
  4. more than three-fourths its side length
  5. more than four-fifths its side length
  6. more than five-sixths its side length
  7. really close to six-sevenths its side length
  8. less than seven-eighths its side length

Use the Pythagorean theorem to relate the height of the green triangle block to its side length.
The difference between the estimate of six-sevenths and the actual value is less than 1%.

Create various number patterns from the sequence above. Solve related problems (e.g., what is the next term, the 20th term, the general term).

Fun facts about triangles
(that have precious little to do with Pattern Blocks)
The three altitudes of a triangle always intersect in a single point called the orthocentre of the triangle. This point is the centre of a circle that can be constructed by creating a triangle with sides parallel to the original triangle through its vertices (see below).
A construction of the orthocircle of a triangle
Other "centres" of a triangle include the circumcentre, incentre and centroid, although for an equilateral triangle these are all the same.
A construction of the orthocircle of a triangle
A yellow equilateral triangle with the
red incircle, purple circumcircle and green orthocircle constructed


Fractions and Decimals

What fraction of the area of this figure is green?
Dinosaur made out of triangles and derivative shapes
Students use an area model to determine the fraction.

Students might:
Students could also be asked:
Note: This shape was created by using the Dinosaur sample file and replacing the tan rhombus with a red trapezoid so that the area can be expressed in terms of the green triangle.

Naming Fractions

Naming Fractions example Naming Fractions answer key
On desktop, click on the image above to open this file. Answer Key

Much as students might use Relational Rods to compare various lengths, they can use Pattern Blocks to compare the areas of various shapes, using fractions.

This example can be changed by:
Students could challenge each other, "I have two shapes. The area of one shape is one-eighth and the area of the other shape is three-fourths. What might my two shapes be?"
Or, for a real challenge, use one-fifth and three-fourths.
Naming Fractions challenge example

See the Changing Wholes with Pattern Blocks task from the Fractions Learning Pathways.

Pattern Blocks+ can be used to represent decimals.

Represent 1.8 in more than one way
Representations of 1.8

Represent and calculate 0.5 ÷ 0.1
Boat compared to pink trapezoid



Create expressions and solve equations.
For a collection of triangles, the total number of sides is equal to three times the number of triangles or s = 3n.
Note: This is an example from the Grade 8 Patterning and Algebra - Variables, Expressions, and Equations expectation: translate statements describing mathematical relationships into algebraic expressions and equations

Solve problems related to pattern rules.
PB Industries manufactures boat-shaped tables. Write an expression for the number of seats, s, for t tables. How many tables are needed to seat 100 people?
Boat blocks with circular chairs around it
Assign values to various blocks and come up with expressions, substitute values into those expressions and solve equations involving those expressions.
PB Industries is creating pattern blocks from scrap material.
It determines that its cost to produce a pattern block is $0.05 per cut.
Arrange the blocks from least expensive to most expensive.

The ordering of the first fourteen pattern blocks shown below assumes that a cut is made for each side of the block.
Cost question
A contractor decides to tile a floor using PB Industries' blocks.
If the contractor only cares about coverage for the cheapest price, which blocks should be used the most?

A student might reason that the block of greatest area for each group would be the most cost-efficient and then sort those blocks from most cost-efficient to least cost-efficient.
Cost table
To compare the cost-efficiency of these six blocks, a rate calculation is made. What does the rate of 0.05 mean for the hexagon?


Create an Outline Puzzle

Create an outline puzzle, or template, similar to the grey boat example above, and then challenge others to cover it with pattern blocks (see Grade 1 Curriculum connections [Link #2]).

Creating an Outline Puzzle using mathies Pattern Blocks+ (PDF) describes how to turn

a block design like Dinosaur design into an outline puzzle template like Dinosaur template.


Sample Files

To access a sample file:
See the File Operations page for more details.


Click to open in the tool - desktop only


Type or copy into Open WWW text box, or
save locally by right-clicking or hard-pressing
Design example
Naming Fractions
Naming Fractions example

Naming Fractions - Continued
Naming Fractions example
Naming Fractions - Combined (for larger screens)
Puts the two tables above together in one file.
Grey Boat
Grey Boat example
Dinosaur design
Dinosaur Template
Dinosaur Template
Repeated Transformation
Pattern with core
Symmetric Design 1
Tiling Pattern 1
Inspired by a tweet from @Trianglemancsd and created by @davidpetro314.
Symmetric Design 2
Tiling Pattern 2
With thanks to David Petro.

Note: This file will only work in version 1.0.3 or later, since it includes the three new rhombus shapes added in that version.
Note: These files were designed on a desktop computer and may not open exactly as shown on other devices.


Features of the Tool

Button Description
Block Size buttons

Block Size

Decrease / increase the block size.
Magnet icon


Pattern Blocks snap to each other and to the isometric grid unless this setting is toggled off. When this setting is toggled on, shapes rotate to multiples of 15°.
English buttonFrench button

English / French

Switch between English and French.
Annotation button

Annotation Tool

Hide / Show a wide variety of Annotation Tools which can be used to communicate thinking.

Import Picture button

Insert Image Button

Insert images into the tool. More details.

Undo/Redo button

Undo / Redo

Step backward or forward through the actions taken with the tool.
This feature is not only useful for backtracking when a misstep is made, it enables a student to demonstrate their work from the start to the finish. The student can press Undo until they are at the start of their solution and then press Redo repeatedly, explaining each step.

Note: Undo / Redo is not available for annotation objects.
Reset button


Delete all work and return the tool to its starting state.
Information button


Shows a dialog with a link to this support page, a feedback form as well as copyright details and version number.

Pattern Blocks Information Dialog
Settings button


Shows the settings dialog.

Settings dialog

The Auto Size Annotation setting is selected by default. This means that if the block size is changed, any annotation in the workspace will be scaled to match.

The Apply Template Settings button is used to turn composite shapes into one grey shape with no visible outlines. See Creating Outline Puzzles for more details.

The Use Standard Colours button is used to restore the default block colours. Use the Restore Defaults button to restore all the settings, including block colour.

Open, Import and Save files (see File Operations for more details)
Recycle button


(in the workspace)
Click to clear selected blocks. If nothing is selected, the entire workspace will be cleared.
Alternatively, drag items to the recycle bin to remove them.
Count button

Block Count

Show / Hide counts next to each cloner block. Scroll to see all the blocks and their counts.

Selection Panel showing counts next to each block
Multiplier button

Number of Copies to Drag

Set the number of copies to drag from the selection panel.
Copy Button


Make a copy of the selected objects.

Other Functionality

Multiple Selection

Multiple Select

To select blocks draw a marquee rectangle around them.
Hold down the SHIFT key when drawing a marquee rectangle to add to the previous selection.

Click a block to add or remove it from the selection.

Selected blocks can be moved, copied, rotated, reflected vertically or horizontally, or recycled as a group.

Change Cloner Colour

Configure the Selection Panel

Click on a cloner block in the selection panel to change the colour of all blocks of that type.

Keyboard Shortcuts

On the desktop version of this tool, the standard Keyboard Shortcuts have been implemented.

PDF supports

  1. Focusing on the Fundamentals of Math - A Teacher's Guide
  2. Pattern Blocks Curriculum Links
  3. Connecting Fundamental Math Concepts with mathies.ca (Draft)
  4. Creating an Outline Puzzle using mathies Pattern Blocks+
  5. mathies Tools and Games used in WINS
  6. Pattern Block Template
  7. Pattern Blocks Manipulative Sheet
    (Source - EduGAINS | Ministry Developed Resources | Mathematics | Lessons & Supports | Manipulatives)
    • What are Pattern Blocks?
    • How do Pattern Blocks help students?
    • Sample Activities
    • Recommended Websites (dated)

    Visit EduGAINS for additional Manipulative Sheets.


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