Patterns and Algebra
Across K-7, students are developing big ideas that connect patterns and algebra to other areas of mathematics such as number and operations. Students learn to identify regularities whether in repeating patterns or changes in increasing or decreasing patterns and generalize what is happening mathematically such as being able to predict what comes next. Students learn to look for number relationships when exploring a variety of patterns, including numbers in a hundreds chart, visual patterns, and patterns in art, music and nature. Students develop algebraic thinking across the grades by making generalizations, looking for or creating patterns and seeking number relationships and learn to notate these relationships using symbols that include expressions and graphing. Other big concepts that develop across K-7 include the meanings of equality and inequality, change, and solving for unknowns.
As students explore patterns and mathematical relationships there are many opportunities to connect to students’ lives, community, culture, and place. With these experiences we are honouring the following First Peoples Principle of Learning: Learning is holistic, reflexive, reflective, experiential, and relational (focused on connectedness, on reciprocal relationships, and a sense of place).
As we learn about key concepts in patterns and algebra, we will also be developing many curricular competencies. Three that we have chosen to focus on in our designing of lesson ideas are:
- Represent mathematical ideas in concrete, pictorial and symbolic forms
- Connect mathematical concepts to each other, other areas of learning and personal interests
- Develop, demonstrate, and apply mathematical understanding through play, inquiry, and problem solving
Although these three curricular competencies have been highlighted, there will be many opportunities to develop many curricular competencies during the investigation of patterns and algebra.
Learning Story for Kindergarten: Patterns and Algebra
Patterning is fundamental to the study of mathematics. Students will be looking for patterns throughout their years at school, and those experiences will become increasingly abstract. Kindergarten and primary students need many opportunities over time to develop their understanding of patterns and to make connections across contexts, materials and representations.
In the area of patterns and algebra, Kindergarten children play matching games (what is the same about a group of objects), explored composing, decomposing and comparing quantities to 10 and have used concrete materials to investigate changes in quantity to 10 such as how to change the quantity from 6 to 2.
In Kindergarten, students work with simple repeating patterns that they see and hear in their everyday lives or that they create and extend with simple materials, sounds, and movement. Children are natural pattern-seekers, helping them to make sense of the world, providing order and routine. For kindergarten, students bring their understanding of patterns from their toddler and preschool years and often naturally create patterns when exploring with materials. In kindergarten, students focus on repeating patterns with multiple elements, creating their own and building on and extending patterns created by others. They learn to “read” patterns and name the part of the pattern that repeats over and over. This is often called the “core” of a repeating pattern and these are often labeled with letters such as AB. The mathematical thinking they are doing is looking for regularities and beginning to make a generalization. It is essential that Kindergarten children have many experiences creating and describing patterns through play, movement, stories, songs and with a variety of materials, both in and outdoor (eg. patterns in nature).
It is important to give Kindergarten students time to play with materials and ideas before starting more focused explorations. This is especially important when materials are new and the ideas being explored are new.
Change, Equality & Inequality
Children will be able to show a change in quantity within 10 (ie how to change 5 to 9) using concrete materials.. Children will be able to show how some quantities are equal and others are not equal using concrete materials.
Students are able to instantly recognize and name small quantities of objects or images/dots without counting. Kindergarten students generally can subitize 1-5 and flexible subitizing with different types of dot patterns is developed.
Key Patterns and Algebra Concept 1: Change, Equality & Inequality
In Kindergarten, students explore changes in quantity with quantities within 10 using concrete materials. Visualization and use of tools such as ten frames helps students to see changes in quantity beginning with just comparing magnitude of more or less and then moving more specifically to quantities that are one more, two less, etc. A “build and change” game or assessment task can involve using materials or contexts of interest to the child. For example, place 6 toy cars on a road mat and say, if two cars drove off, how many would there still be on this road? Ask students to visualize and verbally explain their thinking. This understanding and fluency with change supports mental math strategies that will begin to be developed in Grade 1.
Students will be able to verbally explain how some quantities are equal and others are not equal using concrete materials. In Grade 1, students will be introduced to the equality and inequality symbols as they are formally introduced to symbolic notation in equations for addition and subtraction.
Patterns and Algebra Foundations:
The following concepts and competencies are foundational in supporting understanding of change, equality, and inequality in Kindergarten and may be knowledge that children bring with them to the Kindergarten classroom:
- Counting to 5 and 10 with one to one correspondence
- Comparing quantities of more than and less than
- Understanding of one more and one less (what number comes after 2)
- Compare magnitude of quantities (concrete) as more or less
- Compare quantities (concrete) as equal or not equal
- Compare quantities as one or two more than or less than
- Visualize to compare quantities and to predict amount needed to change quantity (1-3 more or less)
- Use ten frames to support visualization of change in quantities to make 10
- Build and change quantities within 10 with concrete materials, visualizing and verbally explaining what needs to happen for the change (changing 12 to 8 or 5 to 9)
Sample Week at a Glance:
This week of lessons could occur part way through the school year, once students have had some experiences exploring numbers to 10 through counting, composing and decomposing. This week of lessons focuses on foundational algebraic thinking and the concepts of change, equality and inequality, experienced through different models, contexts, tools, and materials.
Play Splat! using projected slides or paper splat and dot magnets on the whiteboard. Focus on a single splat and quantities from 5-10. Have students share the different strategies they use to solve for the missing part/unknown.
Math Workshop (students choose tasks)
-ten frames and dice: have students roll a die and place that number of counters on the ten frame and then state how many more counters are needed to change that number to 10
-splat mats; have students work in partners to create splats for each other to solve using felt or paper splats and counters
Small Group Instruction: Using a story context such as bears in a cave (model with materials such as bear counters and a small box), engage in change in quantity stories with small groups of students. Show 5 bears and place a box over them and say that there are five bears in the cave. Lift the box a bit and remove two bears. Have students visualize and explain how many bears are still in the cave and have them share how they know.
Sharing Circle: Invite students to share what they practiced today or something new that they learned.
Before reading “Balancing Act” by Ellen Stoll Walsh, you may need to invite children to share their experiences with being on a “teeter-totter” (or see saw) as some may be unfamiliar with it. Invite children to make predictions as you read the book. “What will happen when the salamander joins the mice? How do you know?” With each additional animal, ask children, “How does the stick become balanced or unbalanced” Talk about balancing. “Do you think one side weighs more than the other side, or do both sides weigh the same? How can you make sure the stick is balanced? The intention is to surface the idea that balancing does not depend on the number of animals on either side of the stick. Instead it depends on the weight on either side.
Math Workshop (students choose tasks)
-Provide children with a variety of materials (blocks and other loose parts) and ask, “How might you use these materials to explore the idea of balance?
-Provide children with either a pan balance scale or create a “balanced” structure using wooden unit blocks (a long wooden block over a wooden cylinder). Working in pairs, ask children to take turns placing objects (loose parts from your counting collections or small animals on either side, exploring the idea of equivalency – “What collections of items did you use to balance your objects?” Partners can take turns adding objects to either side to see what happens.
-share photos of some “balanced” structures that children have created during explorations/centres and invite others to recreate their balanced structures using the same materials..
Ask children to share what they noticed as they tried to balance different objects. “What did you find out about balance? What else did you find out?”
Open exploration with a variety of structured materials. Working in pairs, provide children with either Numicon Shapes, Sumbloks or Cuisenaire rods to explore. Ask students to share what they found out about each material.
Using the same material or exchanging for another, ask students to create different ways to make ten. Take photos (to share at closing circle) and/or go on a gallery walk to see the variety of ways partnerships have created ten with their materials. As you are circulating, prompt students with change questions such as: What could you do to change this composition of 10 to make 8?
Using photographs of students’ “10” compositions, share different ways to make 10. Highlight one or two compositions and pose change questions to discuss together such as: What are the different ways we could change this 10 to make 7?
Read “Balance the Birds” by Susie Ghahremani. Start by choral counting the birds together. As you read, invite children to make predictions, “What will happen when half of the birds leave? How can these birds balance again? Continue inviting predictions until the end of the story
Math Workshop (students choose tasks)
-using small world and story materials such as trees, nests, birds or loose parts, invite students to play with the concepts of balance and change in quantities
-using drawing materials or small whiteboards, have students draw birds or other animals in balanced (equality) and not balanced (inequality) contexts
Small Group Instruction: With small groups of students present two sets of glass gems or other counters (sets of 5 and 3) and ask students is they are equal or not equal. Ask them how they could make them the same or equal. Listen and observe for understanding of equality, inequality and change in quantity.
Sharing circle. Invite students to connect and consolidate their thinking about balance, equal and not equal.
Help to activate and connect students’ learning from the week around key concepts of change, equality, inequality and balance. Show an image of birds on a wire or branch of a tree and ask students to share what math they notice and what math questions they could ask inspired by the image.
Take a math walk outdoors, inviting students to notice and wonder about quantities they notice such as leaves under a tree, birds in the sky, or flowers in a row in a garden. They might also be interested in creating their own “balances” with sticks, logs or rocks they find outside and weigh and balance different items. Students could take clipboards and pencils with them to record their findings. Collect wonders and questions that emerge during the math walk for future inquiry.
Invite students to share and discuss connections they made outdoors to the math they did this week in the classroom.
After this week of lessons, continue to use the concepts and language of balance, change, equality and inequality throughout the school year and as you further develop number concepts with the students.
Suggestions for Assessment
During math workshop, centres, or explore time, do a task-based interview with each student where you can watch and listen for their understanding and reasoning as they solve change in quantity tasks or stories. Provide counters, ten frames, pan balance, paper and pencil or small whiteboards and markers for students to record and share their thinking.
By the end of Kindergarten, students will be able to:
- Use materials such as blocks to show increases and decreases in quantity of a set (two more, one more, two less, one less) with quantities within 10
- With concrete materials, show what they need to do to change 4 to 6 or 8 to 5.
- Using sets of concrete materials demonstrate they understand the difference between equality and inequality.
Suggested Links and Resources
Key Patterns and Algebra Concept 2: Repeating Patterns
Kindergarten students will have likely had both informal and formal experience with repeating patterns in their pre-K contexts at home, in daycare settings or preschool. In Kindergarten, they are formally introduced to pattern notation (AB) and identifying the core of a repeating pattern (the part that repeats over and over again). Kindergarten students read, label and create patterns that have two or three elements such as colour, shape and size. Many Kindergarten students focus on pattern only through colour so it is important to introduce different attributes as well such as shape and size. An example of this is creating patterns with twigs found outside that are first sorted into piles of short and long twigs. The short sticks could be labelled A and the long twigs labelled B and students could then create repeating patterns with them. As students become more proficient with creating and labelling patterns, they are able to predict what comes next in patterns that are not their own.
Patterns and Algebra Foundations
The following concepts and competencies are foundational in supporting understanding of patterns in Kindergarten and may have been developed in home and preK settings before students come to Kindergarten:
- Repeating patterns with two or three elements in concrete form
- “Reading” patterns by colour/shape/size
- Creating repeating patterns
- Noticing patterns in their daily lives (clothes, wallpaper, art, design, nature)
- Read and describe patterns that are presented concretely and have only two or three elements (colour, shape, size)
- Notice and describe repeating patterns in daily life and in the environment
- Identify and label the core of repeating patterns that have two or three elements (what is the part that repeats over and over again)
- Create patterns with two or three elements.
- Introduce how to label the elements of the pattern beginning with A, next different element is B, then C (this is called AB pattern notation and may look like AAB, ABC, ABB, and AB for Kindergarten.
- Translate patterns in different forms using AB notation or other coding systems or movements (clapping and snapping for example).
- Predict what comes next in a repeating pattern
Sample Week at a Glance
Before this week of lessons, Kindergarten children will have had many experiences exploring patterns in their environment with teachers helping them “tune in” to the patterns all around, from recognizing patterns on the railings on the stairs, the tiles in the hallway, patterns on the playground equipment, or those found in a flower petal. They continue to have many opportunities to create repeating patterns with two or three elements, using a variety of materials. They have explored copying, extending, and creating repeating patterns and doing gallery walks to view and discuss each others’ patterns. The teacher will have been checking in with students to see if they are able to make AB, ABB and ABC patterns and continually collects observations (photos, notes) as documentation to revisit with children throughout ongoing math explorations.
This week of lessons will focus on identifying the core of a repeating pattern.
Read the picture book: Beep Beep, Vroom Vroom! by Stuart J. Murphy Provoke students’ mathematical thinking with questions such as: “What do you notice? What makes it a pattern?” They may notice the word-pattern “beep, beep, vroom, vroom, crash, crash,” or “zoom, honked, banged”, or Kevin’s colour pattern of red,blue,yellow, or Molly’s repeating pattern changes.
Pause throughout the book and invite students to choral read the different patterns in which Molly places the cars on the shelf using words to describe the two or three parts of the pattern. (red, blue, yellow or beep,vroom, crash, etc.) Ask, “How could we name the part of Molly’s pattern that repeats?”
Provide a collection of red, yellow and blue objects (they don’t have to be cars). Invite students to work together in partners. Have one partner create a repeating pattern with cars/objects. The other partner will “read” the pattern, naming the part that repeats. The other partner uses the same materials to create a different pattern. Partners continue to take turns “reading” each other’s pattern “core”. Take photos of some of the repeating patterns that partners create to use for Wednesday’’s lesson.
Bring students together to share and reflect on their “core” pattern identifying explorations.
Math Routine: Pattern Talk
Pattern Talks build on the discussion and thinking developed during similar instructional routines such as Number Talks and Which One Doesn’t Belong. Pattern Talks, once established as an instructional routine, can be used as a short 5 minute introduction to a lesson or for small group instruction. Present a Pattern Talk image to students of Indigenous beadwork. Ask them: “What do you notice?” and either have them turn and talk to a partner or share their ideas with the whole group. This introduction should focus on observable attributes – colours, shapes, and quantity. Encourage a variety of different responses asking, “What is a different idea?” If no one mentions the “core” of the pattern, ask students: “What is the part that repeats?”
You can read, “We Can Bead”(Mathology), if you have the book, or model how you make a beaded bracelet with a repeating pattern using different colours or sizes of beads. It may be easier for some children to use wire to link their beads instead of string or fishing line. Provide a selection of beads of different colours and discuss their attributes (colours, sizes). Invite children to create repeating patterns using two or three different colours or sizes of beads. Invite children to go on a “Gallery Walk” to view each others’ beaded patterns.
Invite students to share what they did, what they learned, asking them what they noticed about the similarities and differences of each others’ creations. You could also invite them to share their successes and challenges with making their patterned bead counter and you might ask how they might use it for further mathematical explorations. You can find and share some information about the importance and significance of beading in Indigenous cultures here.
Project a digital photo of a child’s pattern from Monday. Provoke students’ mathematical thinking with questions such as: “What is the core of the pattern? How many parts are in the core? How can we “read” this pattern using letters? Have students move so that the group is seated in a circle and place a collection of loose parts in the middle. Invite students to come to the middle of the circle, choose a label such as ABC and create an ABC pattern using loose parts.
Invite students to work together with partners. Have partners choose a label and a set of loose parts to create their pattern with. Students can trade their label card and/or their set of loose parts to explore creating different kinds of patterns. Pause during explorations and walk around to look at each others’ patterns. Ask, “can you find a pattern with different materials that has the same label as yours?”
Bring students together to share and reflect on their pattern core labeling explorations. Project a digital photo of a student pattern (ABC)
Invite students to read the pattern using words to describe the two or three parts in the “core” of the pattern. (red, blue, yellow).
Invite students to turn and share with a partner another way that you could make an ABC pattern with these materials. (blue, red, yellow)
Read “Pattern Bugs” or “Pattern Fish” by Trudy Harris
Pause throughout the story and ask, “What do you notice?” In “Pattern Bugs” they will likely notice the visual and sound patterns (word patterns, bee’s pattern, and the colour patterns around the borders of each page). Make time for children to guess how each pattern will end (revealed on subsequent page). You could also invite a few students to act as one of the bugs, repeating each bug’s “core” word pattern. Near the end of the story, invite children to stand and recreate the “core” movement of the firefly- “Up,down, around, around.”
Math Workshop (students choose tasks)
-Working in pairs, provide a selection of simple musical instruments and invite children to create repeating sound patterns. (eg.chime, bean shakers, chime, bean shakers).
-Working in pairs, ask children to create movements to demonstrate a pattern. (eg.Jump, squat, jump, squat.)
Invite partnerships to share their repeating sound or movement patterns with the whole group, asking others to identify the “core” of their musical pattern.
Read “Pattern Breakers” by Daniel Finkel
You can continue to use this routine of showing a pattern with an “error” for the children to spot, describe and correct, as a soft start for your ongoing lessons.
Math Workshop (students choose tasks)
Offer children a variety of experiences to “make and break” patterns.
-Working in pairs, children can use pattern blocks (or other loose parts with two or three elements) to create a pattern. Their partner closes their eyes while they make a change to their pattern for their partner to spot (adding an element that “breaks the pattern”). Their partner is invited to point out what the “pattern breaker” element is. Partners continue to switch roles to “make and break” repeating patterns.
-Working in pairs, children are invited to use mark making tools (felts, crayons, etc.) and different surfaces (paper, whiteboard, etc.) to play “make and break” with their partner
-Using premade paper booklets (or children can make their own here), invite
children to create their own Pattern Breakers book. You can find examples on the Math For Love site here
Bring students together to share and reflect on their “making and breaking patterns” experiences. You may decide to highlight a few examples (from your digital photo observations using the projector or show a child’s “Pattern Breakers” book in the hopes that it might inspire others, both during math workshop and writer’s workshop times.
Kindergarten children will likely demonstrate proficiency with repeating patterns after this week of lessons. You may need to continue highlighting the “core” of a pattern as you connect patterns to other areas of mathematics or areas of learning such as visual arts or ADST.
Suggestions for Assessment
After this week of lessons, assess students to determine if they understand how to create, describe, label and predict what comes next for a repeating pattern with two or three elements. This could be done during a task-based interview during Math Workshop or during centre or explore time. If many students are proficient in their understanding of repeating patterns, you may continue to work with those that need more experiences during math workshop or other opportunities for small group instruction.
By the end of Kindergarten, students will be able to
-create repeating patterns with two or three elements using concrete materials
-identify the core of a repeating pattern they have created
-create a pattern following a given label for a pattern core such as AB, ABC or ABB
Suggested Links and Resources
Beep, Beep, Vroom, Vroom! by Stuart J. Murphy
Pattern Bugs by Trudy Harris
Pattern Fish by Trudy Harris
CBC Kids: Do you know what beading is? https://www.cbc.ca/kids/articles/do-you-know-what-beading-is
We Can Bead book(Mathology)
Natural: Simple Land Art through the Seasons by Marc Pouyet
(Note add -Marc Pouyet French titles as well)
BC Reggio-Inspired Mathematics Project: Investigating Patterns