Weaving and Coding– Mathematical Connections

Weaving is a skill and an art that has been used throughout history and in many cultures. It continues to be developed and used throughout the world today. Like all forms of design, Weaving has deep roots in Mathematics and great potential for cross curricular explorations that include coding, mathematics and art.

Where I live, on the unceded territories of the Coast Salish peoples, the Katzie and the Kwantlen, what is now called Surrey, British Columbia, by most people today, weaving was and continues to be an important and evolving art form. I am grateful to my colleague Nadine McSpadden, for the opportunity to participate in one of her projects that helped me learn more about the protocols and importance of Coast Salish weaving, both historically and today. Through this experience, I have learned that weaving was and is a skill that takes a Master Weaver a lifetime to learn. The blankets, originally made from the fur of the now extinct woolly dog and later the wool of the mountain goat, were complex and highly valued. They were gifted and worn for ceremonies. Today, Coast Salish weavers continue to develop the techniques, designs and materials of this living artform. Blankets are still gifted and worn for ceremonial purposes. Weavers are artists and Coast Salish weavings are also created as wall hangings and other art pieces today.

My mother was a weaver in the European tradition and I grew up on a sheep farm, where I came to learn about the process that wool goes through from sheep to blanket. The opportunity to reconnect with weaving has allowed me to reconnect with my childhood roots and to revisit the power of mathematics viewed through the lens of tradition different from my own. It has reminded me, as I am so often reminded, that mathematics runs deep in all cultures of the world and that Math and Art are sisters in the connection between the conceptual and the abstract.

It is important that, as educators, we avoid the tokanism of parachute crafting projects in the classroom. I encourage you to explore for yourself and with your students the cultural roots and significance of weaving in your place and begin your journey there. Involve indigenous community members, elders and weavers in the most authentic ways that you can. Ask before sharing, as some knowledge, stories and weavings are sacred and/or belong only to a particular person or group. It is possible to do the activities below using only the pictures of the weavings I created, but, in doing so, much of the richness will be lost. If we want to connect our children to the richness and power that comes from connecting mathematics and art, then we must value the time it takes to do the learning and respect the protocols of the cultures from which those teachings come.

My sincere wish is that the materials on this page assist you on a journey of discovery. A journey that, whether you are just starting out or already well on your way, connects us to the past, leads us to the future and connects us to each other.

General Routines: 

Notice/Wonder 

What do you notice?  What do you wonder? These two questions are the launchpad for a variety of rich math activities.  They can be used as a warm up, to get students curious, to launch an exploration or task or to encourage students to really focus in on an image, problem, equation, etc.  We can mathematize almost any experience by asking students to focus in on their mathematical noticings and wonderings.  For example, students could be given one or both of the images below.  Very young students might benefit from the weavings without the stickers added (see resources collection below). 

Students might notice:

  • Different colours 
  • Both have thin, thick and medium stripes 
  • One has fringes on all sides and one only has them on three sides 
  • The stripes have 6, 4, 3 or 2 columns of stitches 
  • Every loop in the top row is created by 2 columns, so, if you count the top row and multiply by 2, you know how many columns in the piece. 

The more experience students have with notice/wonder connected to their Numeracy experiences, the richer their mathematical noticings will become.  Students may have separate wonderings, as well as wonderings that stem from what they notice: 

  • Are they knitted? (Students wonderings can reveal their understandings and misconceptions related to the background knowledge they already have) 
  • How big is the weaving in real life? 
  • Why did the weaver choose those colours? 
  • Why are there 2 knots on top of the left one, but none on the right one? 

Educators may choose to use wonderings to practice curiosity, to launch a lesson, to construct an inquiry or to check in with what knowledge students are applying. 

How Many? (Unit Chats) 

Although How Many seems like a straight forward question, it is actually quite complex.  Asking students to discuss How Many encourages counting, identification of what is being counted and the realization that there are many different ways to count.  By following up “How Many?” with “How do you see them?, “we are asking students to share their mathematical lens.  How Many warm-up can lead to lessons about counting strategies, unitizing and the connection between the visual representation and the symbolic numbers or equations that represent what is being described. 

Responses to this image might include: 
1 mat 
7 stripes: 4 white stripes, 2 grey stripes, 1 blue stripe (4 + 2 + 1 = 7) 
14 tassels: 7 stripes x 2 ends is 14 (7 x 2 = 14) 
13 rows of stitches/warp 
13 rows across the top 
169 stitches: I know that area is l x w and there are 13 stitches across and 13 in the warp (13 x 13 = 169) 
16/26 white rows  
60% (white) 
One third is grey:  There are 26 rows and 8 of them are grey.  8×3=24, so 8/26 is about a third of the blanket.

This connection between the concrete work, the visual pictures, the symbolic numbers and equations and the language used in explanations is valuable to building conceptual connections and sense-making. 

Which One Doesn’t Belong (WODB) 

This is an open routine that allows anyone to participate in sharing their thinking. All answers are acceptable. As students become more familiar with the routine, they can be encouraged to make progressively deeper connections to mathematics in the images. For example: A kindergarten student might say that the top right weaving does not belong because it is the only one with tassels at the top. A grade 6 student might choose the same reasoning, but add the deeper connection that the reason it has tassels at the top is because it is woven using multiples of 3, rather than multiples of 2, which effects where the loose ends of the weaving need to be tied off. Warming up student thinking with a WODB can launch a larger exploration or discussion related to the Mathematics. WODB is also an opportunity for teachers to get a sense of what math concepts the students are fluent in and where they may need more experience. WOBD is a great low risk way of surfacing mathematical misconceptions or launching debate. Students can be encouraged to code their own WODBs or to create them using their own weavings or combining theirs with groups of other students. 

Sample Plan – Multiplication/Factors/Multiples: 

Sample weaving pictures available in the resources section below.

Curriculum Overview (B.C., Canada)

GradeBig IdeasCurricular
Competencie
s
ContentQuestions to Ask 
(Questions are progressive
and cumulative) 
*3Development of computational 
fluency in addition, subtraction, multiplication, and division of whole numbers requires flexible decomposing and composing. 
 
Regular increases and decreases in patterns can be identified and used to make generalizations.
Use technology to explore mathematics 
 
 
Model math in contextualized experiences 
 
 
Develop, demonstrate, and apply mathematical understanding through play, inquiry, and problem solving 
 
 
Engage in problem-solving experiences that are connected to place, story, cultural practices, and perspectives relevant to local First Peoples communities 
 
 
Represent mathematical ideas in concrete, pictorial, and symbolic forms 
 
 
Reflect on mathematical thinking 
addition and subtraction to 1000 
 
addition and subtraction facts to 20 (emerging computational fluency
 
multiplication and division concepts 
 
increasing and decreasing patterns 
 
What computation stories/patterns do you see in this design? 
 
Can I create a blanket design using a multiplication pattern? 
 
How can I represent my design using numbers? 
 
How does writing the code help us understand relationships and patterns in addition and multiplication, 
Commutative Property, Odd and even numbers?

How does my design help me understand patterns in computation? 
 
How does understanding patterns and computation effect the decisions I make when planning and coding my blanket? 
4 
Development of computational 
fluency and multiplicative thinking requires analysis of patterns and relations in multiplication and division. 
 
Regular changes in patterns can be identified and represented using tools and tables. 
Same as gr.3addition and subtraction facts to 20 (developing computational fluency
 
multiplication and division facts to 100 (introductory computational strategies) 
 
increasing and decreasing patterns, using tables and charts 
 
algebraic relationships among quantities 
 
one-step equations with an unknown number, using all operations 
 
How does writing the code help us understand relationships and patterns in addition and subtraction, multiplication and division, weaving design, equations and code?

If I want a blanket that is 36 rows long, what additive/multiplicative patterns could I use?  
 
Are there any numbers that will never appear in my pattern? 
 
How can I record my pattern using a table?  How does my table help me expand my pattern? 
5 
Computational 
fluency and flexibility with numbers extend to operations with larger (multi-digit) numbers. 
 
Identified regularities in number patterns can be expressed in tables. 
Same as gr.3addition and subtraction facts to 20 (extending computational fluency) 
 
multiplication and division facts to 100 (emerging computational fluency) 
 
rules for increasing and decreasing patterns with words, numbers, symbols, and variables 
 
one-step equations with variables 
 
How can I record the patterns in my design using words, equations and variables? 
 
What is the relationship between my blanket, my rules and equations and my code procedures? 
*6Computational 
fluency and flexibility with numbers extend to operations with whole numbers and decimals. 
Use tools or technology to explore and create patterns and relationships, and test conjectures 
 
Model math in contextualized experiences 
 
Develop, demonstrate, and apply mathematical understanding through play, inquiry, and problem solving 
 
Engage in problem-solving experiences that are connected to place, story, cultural practices, and perspectives relevant to local First Peoples communities 
 
Represent mathematical ideas in concrete, pictorial, and symbolic forms 
 
Reflect on mathematical thinking 
multiplication and division facts to 100 (developing computational fluency) 
 
order of operations with whole numbers 
 
factors and multiples — greatest common factor and least common multiple 
If I want all my knots/fringe to be on the same side, which factors are possible? 

If I want all my stripes to be multiples, which factor is the best choice?
 
In what ways do computational patterns effect the blanket design? 
7Computational 
fluency and flexibility with numbers extend to operations with integers and decimals. 
Same as gr.6multiplication and division facts to 100 (extending computational fluency) 
 
operations with integers (addition, subtraction, multiplication, division, and order of operations) 
 
Grade 7s can continue to explore the previous questions. 
*Grade levels have NEW content specifically related to multiplication and factors/multiples. Other grade levels benefit from continuing these explorations for depth or the addition of other prompting questions (examples are included below).

possible Activities: 

Students can engage in exploration and inquiry over time.  Exploration should be interspersed with regular opportunities for consolidation of learning about key mathematical ideas, instruction and introduction of new mathematical knowledge as necessary and regular low risk formative assessment to determine growth, challenges and next steps. 

  1. Explore connections to patterns and equations in woven blankets (notice/wonder). 
  1. Draw/weave/code blankets based on a specific computational rule (add/subtract 1, x2 or x3, common factors, greatest common multiple, etc.). This can be done in small groups or individually. 
  1. Discuss and reflect on the variations of blanket patterns possible within the confines of the computational rule.  Vary the prompting questions and examples selected based on the curricular goals for the grade level(s).  Building connections between areas of computation and computational patterns will support the development of computational fluency (Number Talks).  There are many framing mechanisms for these discussions including: consolidation of a mathematical understanding, learning from mistakes and challenges, mathematical strategies, variations and constraints within a pattern rule and representations and connections. 
  1. Code procedures to create blankets using a particular computational rule.  Students can experiment with different designs using their procedures. 
  1. Discuss and reflect on coding decisions.  How do different ways of writing the code illuminate connections (addition/multiplication) and mathematical generalizations (odd vs even numbers)  

Sample Sessions (Grade 6):

While the inclusion of GCF/LCM makes this a grade 6 lesson plan.  It is easily adaptable to a plan for multiplication (gr. 3-4).  Discussions of number and computational patterns are appropriate at any grade level, although what students notice and the numbers/equations they use will vary by experience.  Grades 5 and 7 continue to benefit from deepening of their understandings of multiplication and patterns in different contexts.  It is also easy to adapt this plan to put the emphasis on geometry, proportional reasoning or algebraic thinking by adapting the explore and consolidate questions. 

All of the sessions in this plan may take one or multiple class blocks.  This plan is designed as a template from which to plan for your class needs.  It is appropriate to vary the exploration prompts to reflect other areas of the Math curriculum (eg: fractions) or grade levels.  It is expected that there will be interludes for teaching of needed mathematical ideas.  These needs should be illuminated by the explorations and discussions and taught on an ‘as needed’ basis to small groups, individuals or whole class.  They should not be front loaded at the beginning of the unit or every lesson. 

Warm up Routines (10 min. every period): Warm ups foster vocabulary and strategy development, while helping students connect to what they already know.  They also allow the teacher to see what strategies the class is using and misconceptions that may require clarification.  The weavings in the example images are all woven using multiples of 2 and/or 3 (LCM 6).  These images can be used for How Many and Notice/Wonder, as well as Which One Doesn’t Belong.  Using these routines to warm up daily gives students an opportunity to connect and broaden their understandings of how computation is connected to pattern.  Number Talks based on multiplication are also important to include in your warm-up routine repertoire. 

Session 1 

Exploration How does multiplication live in these weavings? 

  • Use the images attached or other weaving samples where the patterning reflects clear use of multiplication 
  • Encourage students to record their ideas as words, drawings and equations 

Consolidation: 

  • Mathematical idea: The design process requires Computational (multiplicative) Thinking. 
  • Choose and sequence students/groups to share examples that illustrate different ways that multiplication, factors and multiples appear in the weavings or effected the design process. 
  • Possible things to look/listen for may include:  
  • Repeated addition (build to a related multiplication) 
  • Skip counting 
  • equations that include multiplication 
  • Comparison of the widths of the stripes using factors and multiples 
  • Area calculations/explanations 
  • Arrays and area models 
  • Positioning of the fringes related to x2 vs x3 or computation of even vs odd numbers 
  • REMEMBER:  There will be lots of great ideas as you observe students engaging in the exploration.  Be selective in building the consolidation story towards the key mathematical idea. 

Session 2 

Exploration: Create some designs using 2 as the primary factor.  Create some designs using 3 as the primary factor.  What do you notice?  What do you wonder?   

  • Students can use an Excel spreadsheet, graph paper or manipulatives to explore the design possibilities 
  • Encourage students to note how the 2 factors are the same and how they are different.  Nudge connections to mathematical ideas (odd/even, factors/multiples, GCF/LCM, design restrictions) 
  • This exploration can be extended by asking students to write equations reflecting their designs, exploring composite factors or by playing with multiples rather than factors. 

Consolidation: 

  • Mathematical idea: There are mathematical rules in computation that effect the parameters of design OR Understanding the relationships between factors and multiples helps me understand the design process. 
  • Choose and sequence students/groups that build towards a deeper understanding of the relationships between factors and multiples. 
  • Possible things to look for include: 
  • x2 & x3 designs can both use 6 (LCM) 
  • X2 (and other even factors) always end up on the same side, whereas x3 (and other odd factors) don’t. This reflects multiplicative ‘rules’ about odd vs even factors and products. 
  • Multiplication equations that reflect the design or area of the weaving 
  • REMEMBER: It is important to keep track of who is selected to share each day.  While it is important to build the discussion to consolidation of the mathematical understandings, it is equally important that all students get a chance to see their thinking valued.  It is also important that consolidation discussions are not always structured from ‘least’ to ‘best’ or from ‘least sophisticated’ to ‘most sophisticated.’ If consolidation is consistently structured this way, students will soon adopt the idea that early sharings are not “as good” as later ones. 

Session 3  

Exploration: Using Linx, write procedures for x2 (the Turtle will draw one row up and one back) and x3 (the Turtle draws 3 rows).  What blanket designs are possible using only x2 procedures? X3 procedures? Both?  What is the same?  What is different? 

OR 

Parallel Exploration: Here is some Linx code for procedures that draw factors 2 and 3. What do you notice? What do you wonder? (This Notice/Wonder could be used as today’s whole class warm up.) Use these procedures to explore blanket designs.  Why are there 4 procedures instead of only 2?  What designs can you create using only the x2 procedures? Only the x3? Both? What is the same? What is different? 

Turtle will ‘weave’ 2 rowsTurtle will ‘weave’ 3 rows Turtle will ‘weave’ 2 rows (use if turtle is at the top of the blanket)Turtle will ‘weave’ 3 rows (use if turtle is at the top of the blanket)
to x2 
fd 130 wait 5 
rt 90 fd 4
rt 90 wait 5 
fd 130 wait 5 
lt 90 fd 4 
lt 90 wait 5 
end 
to x3 
fd 130 wait 5
rt 90 fd 4
rt 90 wait 5 
fd 130 wait 5 
lt 90 fd 4 
lt 90 wait 5 
fd 130 wait 5 
rt 90 fd 4
rt 90 wait 5 
end 
to x2top 
fd 130 wait 5 
lt 90 fd 4 
lt 90 wait 5 
fd 130 wait 5 
rt 90 fd 4
rt 90 wait 5 
end  
to x3top 
fd 130 wait 5 lt 90 fd 4 
lt 90 wait 5 
fd 130 wait 5
 rt 90 fd 4
rt 90 wait 5 
fd 130 wait 5 
lt 90 fd 4 
lt 90 wait 5 
end 
Like students can write different equations to express the same value, code can be written in different ways. There is great value in exploring the writing of the code and it’s connection to equations used to express the designs. Computational Thinking is the basis of coding and develops from a strong sense of how numbers work, which we call Number Sense in Mathematics.

Consolidation

  • Mathematical idea: Numbers and operations have different properties, which create predictable patterns in computation. 
  • Choose and sequence students who have noticed mathematical properties and patterns that create the “rules” of computation and how those properties effect their coding.   
  • Possible things to look for include: 
  • Odd vs even numbers 
  • Relationships of factors and multiples 
  • Relationships between different representations (weavings, code, equations, images) 
  • REMEMBER: Consolidation does not always have to take place around correct answers or the most efficient strategy.  Mistakes and be powerful sources of learning and tools for learning about strategy and persistence.  Normalizing mistakes as part of learning creates a safer, richer learning environment for all students.  It is appropriate to strategically include the sharing of mistakes in consolidation discussions.  Routines like My Favourite No are a great way of encouraging a positive mindset about mistakes as learning. 

Session 4 

Exploration: Create a procedure for drawing a blanket pattern that uses only your factor 2 procedure(s) from last session.  Create another blanket design that is as similar as possible to your x2 design, but using only your factor 3 procedures.  What do you notice?  Are there more efficient ways of coding the same designs if you are not limited to a factor?  What if you use both the factor 2 and factor 3 procedures to code each blanket? 

OR 

Parallel Exploration: Here are Lynx procedures for a factor 2 blanket and a factor 3 blanket using the procedures from the last session. What do you notice?  What do you wonder?  Use these procedures to draw the 2 blankets in Lynx.  Why are the codes different lengths?  How do the different factors effect the blanket designs?  Are there ways to create the same designs with different code? NOTE: Lynx does not like the way quotation marks are formatted by the website. You will get an error message if you copy/paste the code. To fix the error, just delete and retype the quotation marks in the code. You will also need to input the block procedures from the previous session for this code to work.

To weave a sample blanket based on the factor 2 (using the x2 procedures above)To weave a sample blanket based on the factor 3 (using the x3 procedures above)
to x2blanket 
setpensize 5  pd 
setcolour ‘grey’ 
repeat 3[x2] 
setcolour ‘blue’ 
x2 
setcolour ‘grey’ 
x2 
setcolour ‘blue’ 
repeat 3[x2] 
setcolour ‘grey’ 
x2 
setcolour ‘blue’ 
x2 setcolour ‘grey’ 
repeat 3[x2] 
end 
to x3blanket
setpensize 5 pd 
setcolour ‘grey’ 
x3 
x3top 
setcolour ‘blue’ 
x3 
setcolour ‘grey’ 
x3top 
setcolour ‘blue’ 
x3 
x3top 
setcolor ‘grey’ 
x3 
setcolour ‘blue’ 
x3top 
setcolour ‘grey’ 
x3 
x3top 
end 
Both of these blankets contain the factors 2 and 3 and have a LCM of 6. There is great value in exploring how the blankets are similar and different, as well comparison of the codes. There are other ways to code the same blanket designs. Exploring the coding and design process connects learners to a deeper understanding of Computational Thinking and the properties of multiplication.

 Consolidation

  • Mathematical idea: When we understand computational patterns(rules), we can make thoughtful choices about the strategies we use. 
  • Choose and sequence students to share ideas that relates code writing to established patterns in multiplication.   
  • Possible things to look for include: 
  • Relationships between equations related to the design and decisions made in writing the code. 
  • The commutative property and it’s effect on the efficiency of code writing 
  • Odd vs even factors 
  • Restricting to a must use factor vs open computation when writing/designing 
  • The multiplicative connection/effect of using procedures to code procedures 
  • The connection of the command “repeat” to multiplication (factors and multiples) 

Additions and Extensions:

  • Weave the designs using cardboard looms 
  • Expand the coding explorations to allow students to select their own factors or create designs based on a GCF or LCM. 
  • Allow time for students to explore their own wonderings about the designs, weavings or code 
  • Extend explorations and discussions to include other patterns and equations, area, order of operations or fractions/decimals/ratio/percent.  Some examples of exploration prompts in a variety of Mathematical strands are included below. 

Computational Thinking: 

  • How did you use computation when planning your design and writing your code? 
  • How many? More/less? 
  • What equation(s) are represented in your design/code? Where do you see (+-/x)?  What does this (particular number or operation in code or equation) mean? 
  • Which strategy is more efficient when weaving or coding? 

Measurement (Example photos in Resources below): 

  • What is a reasonable non-standard measurement for cutting my yarn? 
  • Why are standard units oof measure helpful? 
  • How much yarn will I need to weave the area of my loom? 
  • What is the relationship between the diameter of the yarn and the length needed/area woven? 
  • Area 
    • What will the area of my design be once it is woven? 
    • What will the area of the image be when it is coded? 
    • What variables effect what the area will be? 
  • What length of warp thread to ____ size of loom? 
  • What is the relationship between the dimensions/area of the loom (rectangle) and the length of yarn needed to warp the loom? 
  • Triangles/rectangles relationships 

Fractions/Decimals/Ratios/Percent: Mathematical Routine – Fraction Talks 

  • What Fractions/Ratios/Percentages are apparent in your weaving? 
  • What weaving designs are possible if the blanket is: 
  • ½ grey, ¼ white and ¼ blue? 
  • ½ grey, ¼ white and 1/8 blue? 
  • How would you represent the proportions of colour in the design if you used 4 of your small blankets to create one large blanket? (Students may choose to identify that the proportions stay the same, as the large blanket is now the whole.  It is also possible that students will continue to identify the smaller blanket as one whole, leading to the use of improper fractions, mixed numbers or decimals/percentages greater than one whole (100%).  The key is to note logical connections between the identified whole and the proportions.) 

Patterning and Equations: Mathematical Routine – Visual Patterns 

  • Identifying/Creating repeating/increasing/decreasing patterns 
  • Is this a pattern? 
  • What is the core of your pattern? 
  • How do you see the pattern growing? 
  • How do mistakes when you are weaving help me think about pattern? 
  • In what ways can you weave patterns using only one colour? 
  • Representations 
  • Can you write expressions to reflect the increasing/decreasing patterns in your design? 
  • What is the relationship between the Lynx code for your design and the expressions? 
  • What is the relationship between the design (visual pattern), the expression(s) and the graph(s) of those expressions? 
  • What representations would help you extend the pattern?   

Cross Strand Inquiry: 

After Students have made their own small weaving sample: 

  • How long would it take for you to weave a blanket for your bed?  How much yarn would you need?  How many sheep would it take to produce the wool? How much should that blanket be worth? 

Resources

Angles, Shape relationships and properties, Pattern, line, slope Weaving:

Multiplication/Factor/Multiple Weavings: 

Measurement Examples:

Coding Examples:  

multiplication blanket procedures (see code for procedures in the sample sessions above)
CODING TRANSFORMATIONS (THE TURTLE RESPONDS IMMEDIATELY TO EVERY COMMAND ENTERED IN THE LOWER RIGHT SECTION. THIS FUNCTION FACILITATES CONCEPTUAL CONNECTIONS AS STUDENTS EXPLORE THE MOVEMENT OF THE TURTLE. STUDENTS COULD USE THE MULTIPLICATION PROCEDURES OR CREATE THEIR OWN TO PLAY WITH.)

Books and Websites:

Lynx Coding Club

CanCode to Learn

Salish Blankets: Robes of Protection and Transformation, Symbols of Wealth by Leslie H Tepper, Janice George and Willard Joseph