### 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 3

#### Patterns and Algebra

Students in grade 3 have an understanding of repeating and increasing patterns as well as identifying the pattern core as these are skills students have worked on from kindergarten to grade 2.

In grade 3, students are continuing to build their understanding, as well as learn new concepts of algebraic thinking and representing patterns. Students will expand their knowledge of increasing patterns and will be introduced to decreasing patterns using manipulatives, diagrams, sounds, and actions. Students will develop their understanding of increasing and decreasing patterns, using materials to create them, and using numbers and words to describe them.

Students in grade 3 will also be introduced to pattern rules, where they will show understanding of where the pattern starts and describe how the pattern continues. Students will explore what is changing in the pattern and what stays the same. Once students are able to identify the pattern rule, ask them to either extend it or describe it to show their understanding. Grade 3 students are encouraged to get out in their environment and explore patterns within nature and their community. Provide students with opportunities for collaborative discussions and hands-on activities to reinforce their learning.

In previous grades students have built their understanding of equality and inequality and naturally progress from numbers and pattern work to interpreting algebraic expressions with solving one-step equations using addition and subtraction. In grade 3 these equations can vary from the start unknown, change unknown, or result unknown. This will help students interpret algebraic expressions.

As students extend their learning in grades 4 and up, they will solve one-step equations with multiplication and division and later start to focus more on variables, equations, and computational rules. In upper grades students will also be introduced to relationship, or function (input and output), as well as explore graphs.

### Key Concepts

#### Increasing and decreasing patterns and pattern rules

Students build their understanding of increasing (growing) and decreasing (shrinking) patterns using concrete, pictorial, and numerical representations. Students are introduced to pattern rules using words and numbers, recognizing where the pattern begins and how it continues.

#### One-step addition and subtraction equations

Students are introduced to one-step addition and subtraction equations with an unknown number (start unknown, change unknown, result unknown).

#### Key Patterns and Algebra Concept 1: Increasing and Decreasing Patterns

##### Overview

In grade 3 students will build on their understanding of repeating and increasing patterns as well as identifying the pattern core. Students will be introduced to decreasing patterns and pattern rules, where they will show understanding of where the pattern starts and describe how to determine the next term, figure or step in the pattern by thinking about where the pattern stays the same and what changes. Students will build understanding by creating patterns in a variety of concrete ways (e.g.,tiles, pattern blocks, sounds, pictures, nature), in pictures, and in symbolic form with numbers to represent increasing and decreasing patterns in multiple ways.

##### Patterns and Algebra Foundations:

The following concepts and competencies are foundational in supporting understanding of patterns in Grade 3:

- Creating and describing repeating patterns, including identifying the core
- Creating and describing increasing patterns with concrete materials (classroom and natural), in pictures, and with numbers
- Extending repeating and increasing patterns based on an identified pattern rule
- Skip-counting up and down by 2s, 5s, and 10s starting from any number
- Adding and subtracting to 100

##### Progression:

- Exploration of identifying, creating, and representing (in words and pictures) a variety of increasing (growing), and decreasing (shrinking) patterns with concrete materials (classroom and natural), embodied expression (clapping, dance steps, etc.), senses (sounds, colours, etc.), numbers (0 to 1000)
- include discussions that ask students to describe how the pattern repeats or changes and to predict what comes next
- Growing means numbers increase in size and shrinking means they decrease in size.
- move to a focus on practicing creating and describing increasing and decreasing patterns, as well as identifying the regularity and extending the patterns

- Use a variety of materials to create patterns and use numbers and words to describe them.
- Exploration of pattern rules (describing patterns), to show understanding of where the pattern starts and describe how the pattern continues. Explore what is changing in the pattern and what stays the same.
- Pattern rule is described as how each and every element of the pattern is described, including the first element.
- Identifying the pattern rule – increasing and decreasing patterns by a fixed amount.
- 3, 5, 7, 9,… adding 2. The pattern rule is to start at 3 and keep adding 2.

##### Sample Week at a Glance:

Grade 3 students will be introduced to patterns. Students will have developed prior understanding to explore and practice repeated patterns in kindergarten to grade 2 and increasing patterns in grade 2. Students in grade 3 continue with exploring and practicing increasing (growing) patterns and are being introduced to decreasing (shrinking) patterns. Students will make connections between increasing and decreasing patterns and be introduced to pattern rules by describing how each and every element of the pattern is described, including the first element.

This sample week would be placed at the beginning of patterning, after students have reviewed the foundations.

**Focus:** Creating and describing increasing patterns using a variety of materials. Introduction to pattern rule.

**Before:**

Provide students with Unifix cubes to play with creating patterns. Discuss and practice creating an increasing (growing) pattern as a class and what is the pattern rule. Use concrete materials, pictures, and numbers

- Pattern rules (describing patterns), to show understanding of where the pattern starts and describe how the pattern continues. Explore what is changing in the pattern and what stays the same.
- 4, 6, 8,… adding by 2
- The pattern rule is: start at 2 and keep adding 2.

- Ex.
- Questions to consider:
- What would come next?
- What would come before? (if applicable)
- What is the pattern rule?

- Questions to consider:

**During:**

Invite students to find different materials (ex. pattern blocks, colored tiles, blocks). Have students explore creating different increasing (growing) patterns. Have “Questions to Consider” visible for students to consider.

Possible student examples:

Start at 3 and keep adding 3. Start at 10 and double it.

**After:**

Closing circle – Have students share their increasing pattern with a partner (If there is enough time, have students share their findings with the class). Go through questions to consider.

**Focus:** Creating and describing decreasing patterns using a variety of materials. Continuing on with the pattern rule.

**Before:**

Provide students with Unifix cubes to play with creating patterns. Discuss and practice creating a decreasing (shrinking) pattern as a class and what is the pattern rule. Use concrete materials, pictures, and numbers

- Pattern rules (describing patterns), to show understanding of where the pattern starts and describe how the pattern continues. Explore what is changing in the pattern and what stays the same.
- 9, 6, 3… decrease by 3
- The pattern rule is: start at 9 and decrease by 3.

- Ex.

**During:**

Invite students to find different materials (ex. pattern blocks, colored tiles, blocks). Have students explore creating different decreasing (shrinking) patterns. Have “Questions to Consider” visible for students to consider.

**After:**

Closing circle – Have students share their decreasing pattern with a partner (If there is enough time, have students share their findings with the class). Go through questions to consider.

**Focus:** Creating and describing increasing and decreasing patterns using visuals. Continuing on with the pattern rule.

** **

**Before:**

Present visual pattern images to your class

- Refer to Fawn Nguyen’s website:
**visualpatterns.org**for samples or create your own. - Make sure visuals are large enough for students to see.
- Questions to consider:
- What would come next?
- What would come before? (if applicable)
- What is the pattern rule?

- Have students share their ideas with partners and then the whole class.

- Questions to consider:

Example:

4, 8, 16…

- What would come next?
**32** - What would come before?
**0** - What is the pattern rule?
**Start at 4 and double it**

**During:**

After providing whole class examples. Invite students to create their own visual patterns that will be posted around the class to share.

**After:**

Invite students to do a gallery walk to share their visual pattern creations.

**Math Workshop:**

- Provide students with different pattern rules and a variety of manipulatives to practice creating patterns.
- Provide students with
**“Pattern Talks” (p.31) from “Investigating Patterns” a K-3 multi-lesson exploration (BC Reggio-Inspired Mathematics Project).**

- Four quadrant math check-in with increasing and decreasing questions.

- Teacher led small group instruction: provide white boards or paper to explore different increasing and decreasing patterns.

Additional suggested technology station: Provide students with iPads. Create a QR code to **Mathigon.org** to create increasing patterns on Polypad

Before:

Share some images of land art involving patterns such as those by James Brunt or Andy Goldworthy. Invite students to describe the patterns they see.

During:

Take students on a math walk to explore patterns within nature and their community. Using natural loose parts found outdoors (twigs, leaves, rocks, cones, shells, etc) invite students to create the first three terms of increasing or decreasing patterns. The class can then visit each other’s patterns and complete the fourth term with materials or by drawing it with sidewalk chalk if the students have created the patterns on or near concrete areas.

After:

To consolidate their understanding about patterns from this week, invite students to share and compare a pattern they created outdoors today with a pattern they created with materials or visuals in-class earlier in the week.

For the following, week continue the development of understanding increasing and decreasing patterns through investigating number patterns. Patterns involving “skip counting” or multiples helps to bridge additive and multiplicative thinking and students can use hundred charts and number lines as tools to support their understanding of number patterns.

##### 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 increasing and decreasing patterns using concrete materials. Students are expected to be able to create the fourth term/step/figure of the pattern and explain what the pattern rule is, such as “add 3”. This could be done during a task-based interview during Math Workshop or during centre or explore time.

By the end of grade three, students will be able to:

- Can create, extend and describe increasing and decreasing patterns with concrete materials or with pictures.
- Can describe the pattern rule (using words and numbers such as “add 3”) of increasing and decreasing patterns that are created with concrete materials
- Can extend number patterns (additive or multiplicative) or solve for the missing part of a number pattern by generalizing the pattern rule (ie. 35, 40, 45, 50, ___, 60, 65)

##### Suggested Links and Resources

BC Reggio-Inspired Mathematics Project: Investigating Patterns

#### Key Number Concept 2: One-Step Equations

##### Overview

In previous grades students have built their understanding of equality and inequality and playing with ideas of building and changing quantities in various ways. In grade 3 this is more formalized with one-step addition and subtraction equations where students solve for the unknown using a range of mental math math strategies. The “unknown” at this grade level is often represented by an empty box or underlined space. It is important that students are able to think flexibly about different ways to solve equations and to explain their thinking. Equation types can vary from the start unknown, change unknown, or result unknown. These equation types and instructional ideas using them draw upon the Cognitively Guided Instruction (CGI) research. As students experience different equation types, it is also important for them to see the equals sign or symbol of equivalence (=) in different positions in equations such as 14 = __ + 9. When beginning to discuss solving one-step equations, you may use numbers within 20 to focus on the concepts and strategies, and with time, this is an opportunity to make connections to thinking with numbers to 1000 in grade 3.

As students extend their learning in grades 4 and up, they will solve one-step equations with multiplication and division and in grade 7 begin to formally solve two-step equations with all operations. .

##### Patterns and Algebra Foundations:

- Understanding of balance, equality and inequality
- Addition and subtraction strategies within 100
- Understanding of relationship between addition and subtraction

##### Progression:

- Addition and subtraction facts within 20, using different mental math strategies and recall
- Place value concepts to 100, then 1000, composing and decomposing numbers and using benchmark numbers
- Addition and subtraction concepts and strategies to 1000
- Relationship between addition and subtraction

##### Sample Week at a Glance

This week of lessons could take place early in the grade 3 school year, as a way to practice and review addition and subtraction concepts and strategies from grade 2 and introduce the idea of solving for an unknown through algebraic thinking. Students will know how to play the card game Salute.

Begin with a number talk such as 8 + __ = 15 and invite students to share the different mental math strategies they use to solve this equation. For the next problem, use 38 + __ = 75. Record the different strategies the students share. Name this equation type as change unknown if students are unfamiliar with that terminology.

Invite students to reflect on the type of thinking they do for change unknown problems as compared to result unknown problems. How are they the same and how are they different? Introduce the term “algebraic thinking” if students are not familiar with this term. On a whiteboard or chart, list several change unknown equations for students to choose from. Include a range of numbers from single digit numbers to three-digit numbers. Ask them to choose 3-5 equations to solve, using materials, pictures, numbers and words to show how they solved the equations. Students can use materials, small whiteboards, etc. *Students will need to reflect on their personal learning in this area and choose numbers/equations that are going to help move their learning forward.

Consolidation: Select a few students to share one of their equations and how they solved it. Invite students to compare the different strategies used.

Begin with a Splat! Involving one splat. Have students share their solutions and how they figured it out.

Math Workshop:

-mini Splats: students can play Splat! with a partner

–**Salute:** card game in groups of three

-CGI change unknown equation cards: students choose equations to solve on mini whiteboards (use all three equation types and include the equals sin in different positions)

-Find Sums app: students can select their practice range – 10, 20, 100

Small Group Instruction: Present change unknown equations for students to solve and ask them to share their strategy for solving and then asking them to consider another way to solve.

Sharing Discussion: Ask students to share what they practiced today during Math Workshop and what they think they need to continue practicing.

Begin with a **numberless word problem** such as: *There were some students in line to get on a bus to go on a field trip. Some of the students got on the bus. Some students were still waiting to get on the bus. *Ask students to think about what numbers might be reasonable for this problem and discuss.

And then reveal one statement at a time including the numbers:

*There were 143 students in line to get on a bus to go on a field trip.*

*95 students got on the bus.*

*Students were still waiting to get on the bus.*

Ask students to think about what needs to be solved. What is the math problem or question?

Ask students to solve and share their strategies for solving and record their ideas. Discuss how the strategies use both addition and subtraction to think about this problem.

(if this is the first time students have done a numberless word problem, this may take a good part of the math lesson)

Math Workshop (choices for practice)

-students create their own numberless word problems

-mini Splats: students can play Splat! with a partner

–**Salute:** card game in groups of three

-CGI change unknown or result unknown equation cards: students choose equations to solve on mini whiteboards (use all three equation types and include the equals sin in different positions)

-Find Sums app: students can select their practice range – 10, 20, 100

Math Journal: Ask students to self-reflect on their learning so far this week and share a math goal related to solving one-step equations.

Begin with a number talk such a __ – 15 = 22 and invite students to share their different strategies for solving. Next use the equation __ – 215 = 422 and record students different strategies for solving. If students are not familiar with the “start unknown” terminology, introduce that here. If students are not using “jumping” or an open number line mentally to solve these questions, this is a strategy you could introduce.

Write the following equations on a whiteboard or chart: __ – 7 = 9, __ – 12 = 32, __ – 25 = 98, __ – 300 = 575, __ – 350 = 82, __ – 599 = 251

Invite students to solve each equation (or you choose which equations for them to solve) and then choose one to pose a problem (create a word problem) using that equation structure and numbers. As students complete their word problems, they can write them on index cards and other students can choose each others’ problems to solve.

Choose one word problems written by a student and read it aloud to the class. Ask students to turn and talk to a partner or in a small group to share how they would solve the problem. Collect and record some of the students’ strategies and ask students to compare the strategies used.

Engage in another numberless word problem with the students, this time focusing on the start unknown structure. Choose a context that is of interest to the students. Have students share their strategies and record their thinking.

Focus on using the open number line for equation practice. If your students are not familiar with using the open number line, begin with a mini-lesson. Provide students with mini whiteboards or laminated open number lines to record their “jumps”. Provide a collection of equations on a whiteboard or chart or printed on equation cards. Students choose equations and solve each in two different ways using the number line to show their thinking. *Provide some equations using single digit numbers for students who are still practicing the different strategies and using single digit numbers will help them focus on the strategies

Class Reflection on Learning: Ask students to reflect on their learning during this week of lessons. Record a list of strategies that students use to solve one-step equations and reflect on how the strategies they use might change depending on the equation type or numbers being used.

Instructional routines, practice tasks and strategies introduced this week can be used throughout the school to practice addition and subtraction computation and to foster a connection between these two operations. As multiplication and division are introduced in grade 3, be mindful of using different equation types as well to foster the connection between multiplication and division and develop algebraic thinking.

##### Suggestions for Assessment

After this week of lessons, assess students to determine their fluency with strategies for solving one-step equations with numbers to 20 and then if they can extend those strategies to numbers to 1000.

By the end of grade three, students will be able to:

- Solve one-step equations involving addition and subtraction using different strategies and different forms of equations
- Represent solutions and strategies for solving one-step equations in concrete, pictorial and symbolic forms
- Make connections between addition and subtraction when solving one-step equations