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 Grade 1: Patterns and Algebra
In the area of patterns and algebra, the learning story for Grade 1 is very similar to and builds on children’s experiences in Kindergarten.
In Kindergarten, children have 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 7 to 4. In Grade 1, students continue to explore changes in quantity, bridging over 10 to include quantities to 20 and being able to do so both with concrete materials and verbally. It is helpful to connect this concept to stories and children’s daily lives and could involve the use of story mats and materials. Splat! Is an important routine for children to develop their algebraic thinking around change in quantity. In Grade 2, children will be expected to be more fluent with symbolic representation of these concepts.
As symbolic equations for addition and subtraction are introduced in grade 1, the = sign is introduced and the symbol for inequality can also be introduced. Children need to understand that the = sign is a symbol of equivalence, and that it means both sides of the equation are equal. This is an important connection to computational fluency learning standards in Grade 1 and it is important to play with different formats of equations to emphasize this idea of equality such as writing equations like 9=__ + 4 and 5+6=10+__. In Grade 2, children will be expected to be more fluent with symbolic representation of these concepts.
In Kindergarten, children learn about repeating patterns using two or three elements and in Grade 1 the repeating patterns extend to using multiple elements or attributes such as colours, shapes, and sizes. Children learn to look for what is “regular” and identify the part of the pattern that repeats over and over, which we will call the “core”. Children can “read” patterns and translate between different modes such as reading the colours of the pattern to labeling them with AB notation or using clapping and snapping to read the pattern.Children can predict what comes next or before or between in a repeating pattern, using what they have generalized about the pattern. Children can create and extend patterns using a range of concrete and visual representations. In Grade 2, children will continue their exploration of patterns using more complex repeating patterns and introducing increasing patterns.
Key Concepts
Change, Equality & Inequality
Children will be able to describe a change in quantity to 20 (ie how to change 8 to 12) verbally and concretely. Children will be able to verbally explain how some quantities are equal and others are not equal using concrete materials as well as addition and subtraction equations.
Repeating patterns
Repeating patterns include a core of elements that repeat over and over again. Children create, describe, label and extend repeating patterns that include multiple elements such as using four different colours of tiles.
Key Patterns and Algebra Concept 1: Change, Equality & Inequality
Overview
In Grade 1, students explore changes in quantity, bridging over 10 to include quantities to 20 and being able to do so both with concrete materials and verbally. 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 plus 1, minus 1, plus 2, minus 2 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. Using a ten frame as an example of a tool to support visualization of change, a student may have 8 on the ten frame and then roll a 6 and say, “I know that I need 2 more to change 8 to 10 and then there is 4 left so 10 and 4 is 14.”
Students will be able to verbally explain how some quantities are equal and others are not equal using concrete materials as well as addition and subtraction equations. The symbols of equality and inequality are introduced and can be used with materials and numbers and other symbols such as addition and subtraction symbols. Students need to understand that the = sign is a symbol of equivalence, and that it means both sides of the equation are equal (and not that you put the answer after the = symbol). Students need to be flexible in their thinking about equations and what they mean so it is important to present equations to students in different formats such as 5+__ = 11 and 14 = __ + 9. This is where we can also play with the idea of keeping an equation balanced but changing each side in the same way, such as 6= 4+2 to 6+3 = 4+2+3. A tool like a number balance creates a physical and visual experience with balancing numbers on two sides and can be connected to written symbolic equations.
Patterns and Algebra Foundations:
The following concepts and competencies are foundational in supporting understanding of change, equality, and inequality in Grade 1 and are developed in Kindergarten:
- Change in quantity to 10 using concrete materials
- Composing and decomposing quantities to 10
- Equality as a balance or being the same
- Inequality as an imbalance or being different
Progression:
- Compare magnitude of quantities (concrete, visual, symbolic) as more or less
- Compare quantities (concrete, visual, symbolic) 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 (one or two more or less than) and then using small quantities such as three, four or five
- Use ten frames to support visualization of change in quantities to make 10 or bridge over 10
- Build and change quantities with concrete materials, visualizing and verbally explaining what needs to happen for the change (changing 12 to 8 or 5 to 9)
- Connecting change in quantities to symbolic equations using numbers and addition and subtraction symbols
- Introduce symbols of equality and inequality and symbols of equivalence, that they have meaning to explain the balance or equivalence two sides of an equations
- Create and prove equations are true using symbols of equality and inequality using numbers, materials, words and symbols
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 20 through counting, composing and decomposing as well as being introduced to the concepts of addition and subtraction. This week of lessons focuses on algebraic thinking and the concepts of change, equality and inequality, experienced through different models, contexts, tools, and materials.
Read the story Splash! by Ann Jonas, stopping to have students count the number of animals in the pond from page to page. Ask students at one point, What has changed from this page to this page? (showing illustrations). How many more/less animals are in the pond now?
Model a “change in quantity” story with a story mat (felt or paper) and materials. For example, “Raccoon noticed five fish in a pond and some swam away when they saw the raccoon. Now the raccoon could only see two fish. How many fish swam away?” Invite students to visualize the change and share their thinking verbally. Ask students to create their own stories that involve a change in quantity, including the posing of a problem to solve. This is an opportunity for students to create stories connected to their daily lives, community, or culture.
Sharing Stories: Invite students to share their story with a partner and have the partner solve the change problem and then reverse roles. Gather the students together and ask them to share and reflect on how stories helped them think about the math they were doing.
Open with Splat! using projected slides or paper splat and dot magnets on whiteboard. Focus on a single splat and quantities from 5-20. Have students share the different strategies they use to solve for the missing part/unknown. Model how to record as an equation such as 12= 8+___.
Introduce or review the meaning of the equality and inequality symbols.
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
-equality and inequality symbol cards: using numbers and materials, have students create true equations or representations using symbol cards
Small Group Instruction: In groups of two or three, show students a quantity of 7 tiles or blocks. Ask them to visualize what they need to do to make the quantity 10, then to make the quantity 8 and then 12, asking different students to verbally explain their thinking after each change.
Sharing Circle: Invite students to share what they practiced today or something new that they learned.
Set out tubs of Cuisenaire rods for students to explore. Ask students to share what they found out about the rods and record their findings on a chart or a whiteboard, such as “Each rod is one longer than another” or “There are 10 different rods.”
Create some task cards for students to explore the concepts of change, equality and inequality with Cuisenaire rods. Some examples include:
-What different ways can you make ten?
-Choose a five rod (yellow). What different rods can you add to it to change the value/quantity to 12?
-What different combinations of rods could make this equation true?
8 + _____ = 17
-Use printed equality and inequality symbols to create equations using Cuisenaire rods.
Provide tubs of rods at tables for students to investigate the prompts with. Provide math notebooks or small whiteboards for students to record their findings or equations. Students could also use the digital Cuisenaire rods on Mathigon or NRICH to represent the mathematics they are exploring.
Closing Discussion: Choose one of the tasks and invite students to share the different strategies they used to solve the question and think about change, equality or inequality. Ask students to consider new strategies they might be hearing that they can use next time when they are using Cuisenaire rods.
Do Puzzles 1-4 on SolveMe Mobiles together as a class, having students turn and talk in partners or small groups before sharing their solutions. Introduce the physical number balance to the students, showing them how to balance or create equivalence on each side by adding weights in different ways. Show how once balanced – for example, 3 and 4 on one side and 6 and 1 on the other side that if you add 5 to one side, you can add 2 and 3 to the other side to keep it balanced. Play with the idea of keeping the balance in balance in different ways.
Math Workshop (students choose tasks)
-number balance: students create balanced equations on the number balance and record them using symbols and numbers in their math notebook or on small whiteboards
-equality and inequality symbol cards: using numbers and materials, have students create true equations or representations using symbol cards
-if available, you could set up a small set of iPads or tablets to SolveMe Mobiles and have student solve those puzzles together
-Cuisenaire rods: create balanced equations using Cuisenaire rods and recording the equations in math notebooks or small whiteboards
Small Group Instruction: continue build and change assessment tasks from Tuesday
Closing Discussion: Ask students to turn and talk to a partner or small group about what materials and tools are supporting their thinking about change, equality and inequality. Invite a few students to share their reflections to the whole group.
True or False: Share a few different equations (using equality and inequality symbols) on the whiteboard or chart and have students discuss whether they are true or false, explaining and justifying their responses using mathematical reasoning.
Ask students to record up to 10 true equations using the equality and inequality symbols (five of each) in their math notebooks or small whiteboards. Have students exchange their list with a partner to “check” and confirm that they are true. (Teacher note: notice the flexibility and fluency with numbers that students demonstrate when creating equations.) Students may then choose a practice task from the week such as creating change in quantity stories, splats, using the number balance, change to 10 with ten frames, etc.
Sharing Circle: Invite students to reflect on this week’s learning and share something they feel they learned more about and share a personal goal for this area of mathematics.
You may focus the following week or two of lessons on addition and subtraction fluency and connecting some of the ideas, materials and tools from this week for practicing addition and subtraction in new ways.
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 and “prove” whether an equation is balanced or equal. Provide counters, ten frames, number balance, paper and pencil or small whiteboards and markers for students to record and share their thinking.
By the end of grade 1, 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).
- Verbally explain what they need to do to change 7 to 10 or 12 to 10.
- Using materials and symbols in equations, demonstrate they understand the difference between equality and inequality.
Suggested Links and Resources
Key Patterns and Algebra Concept 2: Repeating Patterns
Overview
In Grade 1, students become more fluent and flexible with their understanding of repeating patterns. They think about “what make a pattern a pattern?” and are able to compare and analyze different types of patterns. Students work with patterns that have three or more elements such as colour, shape, texture, quantity, and size or combinations of these. Students identify the core of the pattern and use that knowledge to label and extend patterns. Students are able to fluently move between or translate, different representations of a pattern using different materials,
Patterns and Algebra Foundations:
The following concepts and competencies are foundational in supporting understanding of patterns in Grade 1 and are part of the learning expectations for Kindergarten:
- Repeating patterns with two or three elements in concrete form
- “Reading” patterns by colour/shape/size
- Identifying the “core” of a repeating pattern
- Labeling patterns using AB notation (orally)
- Creating repeating patterns
- Extending repeating patterns by predicting what comes next
Progression:
- Read and describe AB, ABB, AAB, ABC type patterns that are presented concretely or visually/pictorially (patterns with two or three elements)
- Notice and describe repeating patterns in daily life and in the environment
- Identify and label the core of repeating patterns that have multiple elements (more than three) or attributes introducing attributes beyond colour such as shape, texture, type, and size
- Translate and record (written, pictures) patterns in different forms using AB notation (might now include up to ABCDE notation) or other coding systems or movements
- Predict what comes next, what comes before or what comes between (a missing part) in a repeating pattern
- Discuss and compare patterns, sharing how two patterns are alike and how they are different, using mathematical vocabulary and language
Sample Week at a Glance
Prior to this week of lessons, students would have had a week of soft start explorations with different materials, being invited to create repeating patterns and doing gallery walks to view, read, label 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 identify the core of each. This week of lessons will focus on students developing flexibility and fluency in the ways they discuss and compare repeating patterns with more than three elements and connect patterns to other areas of math concepts, other areas of learning and personal interests.
Read Pitter Pattern by Joyce Hesselberth (or similar picture book with illustrations that include patterns) stopping to have students read and describe some of the patterns. Invite students to make connections between the book and patterns they encounter in their own lives.
Invite students to think about a day in their life and where and when they might encounter patterns such as in the examples in the book – their clothing, in their home or neighbourhood, their music or sports practice, times in the day, routines, etc. Using pictures, numbers, and words, invite students to draw themselves and the patterns in their lives. This could be in a comic strip format or as a booklet or on a large piece of paper like a story map. Encourage students to add labels (AB notation) to their patterns.
Share and Compare: Invite students to share their personal patterns with a partner and compare the different patterns they identified.
In a pocket chart, on a whiteboard or chart, share a pictorial/visual pattern with four or five elements such as an ABBCD pattern. Ask students to choral read the pattern describing the elements such as long, short, short, medium, sideways (lines) and then choral label using AB notation. Invite students to describe what they notice about this pattern – what makes it a pattern?
Math Workshop (students choose an area to practice):
-Provide a collection of materials such as pattern blocks, Unifix cubes, glass gems and loose parts or technology loaded with an app such as Mathigon Polypad, alongside cards with the following pattern labels on them: ABBC, ABAC, ABBCC, ABCD, ABCDE, ABBCCD etc. Invite students to create and label these patterns with the materials provided.
-On paper strips, have some repeating patterns drawn out with some parts “missing” (after the core) for students to solve and use similar materials as the drawings to place in the missing spots.
-Have a collection of letter cards or wooden letters (A-E) and invite students to create patterns with materials and then label them with letters.
Small Group Instruction: In small groups, provide two cards with these cores: ABBC and ABAC and ask students to create a repeating pattern for each of them (provide loose parts or pattern blocks) and then as a group compare and describe how the patterns are the same and how they are different. This is an opportunity for formative assessment.
Math Notebook: In their math notebooks or journal, invite students to record one of the patterns they created today using pictures, numbers and words. In their table groups or with a partner, invite students to compare their patterns using mathematical language and vocabulary.
Share some photographs from a book like I See a Pattern Here by Bruce Goldstone or Patterns in Nature by Philip Ball. Invite students to notice and describe patterns they see and to make connections.
Go on a pattern walk outdoors to a park or neighbourhood. Invite students to look for repeating patterns in nature, on buildings, etc. Stop and have students describe and compare the patterns they are finding. Take photographs of the patterns the students find. Students may bring clipboards with them and draw and label the different patterns they find. Next, invite students to collect some “natural loose parts” that are available on the ground such as fallen leaves, twigs, stones, shells, or cones. Invite students to create repeating patterns with these natural materials and to create them in lines or different shapes. Invite students to do a gallery walk, take photographs of their patterns and discuss and compare how their patterns are the same and how they are different.
Closing DIscussion: Either outdoors or back in the school, share two of the photographs taken of patterns seen and created outdoors and invite students to turn and talk in small groups about what connections they are making and where they might see other patterns like this in their daily lives or in the world around them.
Introduce a local cultural practice that has patterning embedded in the process and item such as Coast Salish cedar weaving. Share a video or images of the process and some of the items created. Invite students to make connections to something they have done or created that involves patterns.
Math Workshop (students choose an area of interest to investigate):
-Photographs of weaving and textiles from many cultures with clipboards with paper for students to record their observations and record the pattern
-Provide construction paper and scissors for students to do paper weaving. Consider providing a QR code to scan to a short video showing how to do paper weaving.
-Provide a collection of rhythm instruments for students to create musical patterns with. Ask them to create a pattern notation of their music on small whiteboards or clipboards.
-Provide templates or paper and crayons/pencils crayons for students to design their own clothing that includes repeating patterns.
Closing Circle: Invite students to bring something they created or thought about to the carpet. Ask them to consider how they used patterning in their investigations. What connections are you making?
Choral Counting: 1-30 by 1’s starting at 1, teacher records the count on whiteboard or chart as students are counting together orally in unison. Invite students to turn and talk with a partner or small group and look for patterns they notice in the record of the choral count. Have students share what patterns they found and annotate the record of the count with the patterns they describe. Note that this routine supports students making connections between the development of their counting and beginning understanding of place value.
Provide students with different choices of how numbers are organized in our world such as calendar pages, a number line/measuring tape or clock. Invite students to notice and name the different repeating patterns they notice in how the numbers are organized. Invite students to do a number pattern search in the classroom or school or do their own choral counts in small groups, recording the number patterns they find on small whiteboards.
Reflection on Learning: Ask students to share what they learned about patterns this week, what they are still curious about and what learning goal they have for themselves regarding patterning.
Many students will likely demonstrate proficiency with repeating patterns after this week of lessons. If you think your students need more practice with repeating patterns, you may repeat similar lessons for the next week by changing the materials or picture books used or find other math connections in a series of lessons such as connecting patterns and shapes or patterns and science/local plants and trees.
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 repeating patterns with more than 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 grade one, students will be able to:
- Describe, label, extend and create repeating patterns with multiple elements (more than three) with concrete materials.
- Record patterns with drawings and label patterns with letters such as ABAC, ABBCD.
Suggested Links and Resources
BC Reggio-Inspired Mathematics Project: Investigating Patterns
Pitter Pattern by Joyce Hesselberth
Mathigon Polypad (web and iOS app) HERE
I See a Pattern Here by Bruce Goldstone
Patterns in Nature by Philip Ball
Coast Salish Weaving video by Jessica Silvey HERE
Paper Weaving Video HERE
Choral Counting HERE
Choral Counting TEDD HERE