Patterns and Algebra
Across K-7, students are developing big ideas that connect patters 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 4
Patterns and Algebra
Grade 4 students build on previously developed concepts of patterns and algebra and are introduced to how patterns are represented to charts and tables. They work on solving one-step equations using addition, subtraction, multiplication, and division. Exploring algebraic relationships supports the understanding and representation of patterns and the solving of equations.
In primary years, students have had experience with repeating, increasing, and decreasing patterns. Grade 4 students continue exploring increasing and decreasing patterns and extend their learning to include describing and representing patterns in charts, graphs, and tables (e.g, multiplication chart, T-charts). Patterns can be created and represented in concrete and embodied ways with classroom manipulatives, with natural materials, in movements (dance), with sounds, etc., in pictures and diagrams, and in symbolic form. Students describe the rules for extending the pattern or determining a term further in the sequence using words or diagrams and begin to represent the rules in more formal symbolic notation. The use of symbolic notation of pattern rules will be further developed in grade 5. Connections to place, local Indigenous knowledge, and other subject areas can be made by exploring patterns in content from the grade 4 science curriculum and data representations in social studies.
Students continue to build fluency in solving one-step equations from addition and subtraction in grade 3 to using all four operations in grade 4. One-step equations involve solving for one unknown number and can be a start unknown (e.g., __ + 4 = 16), change unknown (e.g., 3 x ___ = 15), or result unknown (e.g., 24 ÷ 2 = ___). The unknown number may be represented by a line, box (▢), or other symbol (e.g., ▲ or ⭐). Students will represent the unknown number with a variable (e.g, 3 + n = 8) in grade 5 but may be comfortable experimenting with this in grade 4. The understanding that the equal sign represents balance and can be placed in various places in an equation (e.g., 24 = 3 x ?) and there can be various number of terms on either side (e.g., 3 + 4 + ? + 7 = 4 x 5).
It is appropriate to explore increasing and decreasing patterns and one-step equations with smaller numbers. Therefore patterns and algebra can be explored by mid-year after some experience with simple multiplication and division is provided.
Key Concepts
Increasing and decreasing patterns
Students continue exploring increasing and decreasing (growing and shrinking) patterns and are introduced to how patterns are represented in charts and tables. Students use words to describe how they extend patterns and predict terms.
One-step equations
One-step equations involve solving for one unknown number and can be a start unknown (e.g., __ + 4 = 16), change unknown (e.g., 3 x ___ = 15), or result unknown (e.g., 24 ÷ 2 = ). Grade 4 students explore one-step equations using all four operations.
Key Patterns and Algebra Concept 1: Increasing and Decreasing Patterns
Overview
Students will be familiar with repeating patterns (including identifying a core) and with describing increasing (growing) and decreasing (decreasing) patterns represented in concrete materials, pictures, and symbols. Grade 4 students continue creating and identifying increasing and decreasing patterns in a variety of concrete ways (e.g., tiles, pattern blocks, sounds, pictures, nature), in pictures, and in symbolic form with numbers. Patterns may be algebraic ( linear) sequences (growing/shrinking by a fixed amount) (e.g., 3, 8, 13, 18,… [fixed increase by 5]), geometric sequences (each term is a fixed multiple of the proceeding term) (e.g., 2, 4, 8, 16… [double the previous term]), or other number sequences that do not have constant increase or decrease (e.g. 36, 33, 27, 18,… [decrease by 3 more each time]) (this vocabulary is provided for teachers; students do not need to describe sequences using these terms). Some students may rely on creating and identifying repeating patterns and need to be supported in exploring increasing and decreasing patterns, beginning with concrete materials (classroom manipulatives or natural materials). In grade 4, students are introduced to identifying and representing patterns in charts, graphs, and tables, such as the multiplication chart or T-charts. For example, students will learn to record a visual pattern created from classroom or natural materials as a term number and a number representing how many components in the element.
(image from https://visualpatterns.org)
Students will use charts to extend patterns and predict the next element and elements later in the sequence (e.g., the 15th term). They will represent the rule they use to extend patterns in words (e.g., “double the previous term” or “take the term number and triple it”) or diagrams and may play with representing the rule in more formal symbolic notation. Representing pattern rules in symbolic form will continue in grade 5.
Connections to place, local Indigenous knowledge, and other subject areas can be made by exploring patterns in content from the grade 4 science curriculum, such as moon phases, changes in length of day or in plants across seasons, and patterns in tides, patterns in charts and graphs representing information in social studies, and rhyming and syllable patterns in poetry. Examining and creating products through cultural practices from a variety of cultures, such as weaving or beading, can also allow students to explore patterns in a meaningful and contextualized manner.
Patterns and Algebra Foundations:
The following concepts and competencies are foundational in supporting understanding of one-step equations in grade 4:
- Creating and describing repeating patterns, including identifying the core
- Creating and describing increasing and decreasing patterns with concrete materials (classroom and natural), in pictures, and with numbers
- Extending repeating, increasing, and decreasing patterns based on an identified rule (described in words)
- Skip-counting up and down by 2s, 3s, 5s, and 10s starting from any number
- Addition and subtraction facts to 20
- Multiplication and division concepts and experience with simple multiplication and division (e.g., doubling, halving, multiplication by 5 and 10,…)
Progression:
- Exploration of identifying, creating, and representing (in words and pictures) a variety of repeating, increasing, and decreasing patterns with concrete materials (classroom and natural), embodied expression (clapping, dance steps, arm movements, etc.), senses (sounds, colours, etc.), numbers, charts, tables, graphs
- include discussions that ask students to describe how the pattern repeats or changes and to predict what comes next
- Move to a focus on practicing creating and describing increasing and decreasing patterns and extending these patterns
- Formal introduction of representing visual or concrete increasing or decreasing patterns in T-charts and vice versa, describing the pattern rule in words, and using the rule to extend the pattern by several elements
- Patterns can be created by students or be presented from other sources
- Exploration of increasing and decreasing number patterns, including describing a rule for a given number sequence, recording a pattern from a given rule, creating number patterns with an associated rule, and extending the number pattern based on the rule
- Patterns in charts, graphs, and tables within mathematics (e.g., 100 chart, multiplication table) and in other subject areas (e.g., science, social studies)
Sample Week at a Glance:
Students have had an array of experiences with creating patterns with a variety of materials since kindergarten. They have created and described increasing patterns in grade 2 and grade 3 and decreasing patterns in grade 3. In primary grades, students have looked at charts and graphs for patterns (e.g., in a calendar or 100 chart). This week at a glance is a suggested first week of exploring patterns in grade 4. There may have been some discussion of patterns when exploring number concepts, including patterns in the 100 chart and the multiplication chart. Students will have worked on concepts of multiplication and division to support number patterns after this first week of formal instruction on patterns.
Focus: creating and describing patterns with a variety of materials and modalities
Before: Body percussion patterns
Teacher uses body percussion moves (stamp foot, pat thighs, clap hands, snap fingers) in both repeating and increasing patterns. For repeating patterns, students join in when they have figured out the core (e.g., clap-snap-clap-pat [vary the length of the core on different examples]). Pause and discuss how they knew when to join in. For increasing patterns, teacher demonstrates the first 3 or 4 terms (e.g., snap-clap, snap-snap-clap-clap, snap-snap-snap-clap-clap-clap) and students repeat the sequence and extend it. Pause and discuss how the pattern changed. Discuss what ‘pattern’ means and how identifying the pattern and its rule helps continue the pattern.
If time allows, invite individual students to be the leader of the patterns.
During: Math workshop with pattern stations
Students engage with prepared stations to explore creating and identifying patterns. Have enough stations so each station has 4-6 students at a time. Stations could include:
- Creating patterns with pattern blocks, loose parts, or natural materials (with prompts such as “create a repeating pattern with 4 items as its core”, “create a pattern that increases (grows)”, “create a pattern that decreases (shrinks)”) and documenting them on paper in pictures
- Describing and extending visual patterns such as those from Fawn Nguyen’s Visual Patterns website (https://www.visualpatterns.org/); print out images from the website and provide paper and pencils crayons for students to record their descriptions and extension of patterns
- Poems with a variety of rhyming patterns; students represent the patterns with materials (e.g., pattern blocks or loose parts), pictures, and/or symbols
- Body percussion patterns – students work in pairs in the same manner as the ‘before’ part of this lesson; provide prompts to create repeating, increasing, and decreasing patterns
- Number patterns – provide cards with different number patterns (e.g., and paper and pencils for students to record a rule for the pattern and to extend the pattern; provide cards with rules for students to translate to a number pattern (e.g., “start at 3 and add 2 each time”); provide prompts for students to create their own number pattern (e.g., “create a decreasing pattern starting at 65 and describe the rule you used to make the pattern”)
Students might only experience 2 or 3 stations during this lesson. Stations can continue to be available to students throughout the week (and further) to engage with during choice times.
After: Invite students to share the patterns they created or described in the stations. Teacher facilitates class discussion about what students noticed about the patterns they created and ones created by classmates. Incorporate vocabulary such as repeating pattern, increasing pattern, decreasing pattern, term, core, and extend in the discussion.
Formative assessment is important on this first day for the teacher to gauge the students’ incoming understanding of patterns. Look for students who rely on creating repeating patterns and those who have a strong understanding of how to describe and create increasing and decreasing patterns. A checklist with student names and these three types of patterns, ‘create’, ‘describe’, and ‘extend’ on a clipboard may help track what you notice through observations, conversations, and student work (see image below). You may not collect evidence for each section for every student and you may collect evidence over the week if the stations are offered another time. Adjust upcoming lessons and plan for whole class, small group, or individual instruction based on what is observed today.
Focus: increasing and decreasing visual patterns
This lesson is designed to take place outdoors but can be modified to be done indoors with classroom materials.
Before: Provide books, printouts of (in plastic sleeves to protect against weather), or digital links (as technology is available) to the works of sculpturists who work in nature, such as James Brunt or Andrew Goldsworthy. In small groups, students look at the images and describe patterns they see in the artwork. Alternatively this can be done with one image at a time for a whole group discussion.
During:
Play – students work in pairs or groups of 3 to use natural materials from the surrounding area to create sculptures (may be flat) that utilize patterning. They describe the pattern to each other and to the teacher as needed.
Debrief – allow groups to visit the sculptures made by others and describe the patterns they see. Bring the whole class together to discuss the patterns they saw. Specifically draw students’ attention to sculptures that use increasing or decreasing patterns.
Replay – in the same or different groups, students engage in creating new sculptures that specifically use increasing and/or decreasing patterns
After: Have volunteer groups describe the pattern(s) in their sculpture and confirm with the whole class whether the sculpture illustrates an increasing or decreasing pattern. Discuss challenges and successes in using increasing and decreasing patterns in the students’ sculpture art.
Formative assessment – you may wish to continue observing, looking at the products, and having conversations and use the chart suggested in Monday’s lesson to track what you are learning about students’ understanding of patterns.
Focus: describing, representing (pictorial and symbolic), and extending increasing and decreasing patterns
Before: Using Fawn Nguyen’s website, Visual Patterns (https://www.visualpatterns.org/), choose a visual pattern (such as the one pictured below) to project for a class discussion. Discuss how the pattern grows, how students would build the pattern with classroom materials, and how they would extend to create the next two elements..
(image from Visual Patterns website)
During: Present students with the Squares Upon Squares task from Youcubed https://www.youcubed.org/tasks/squares-upon-squares/ as a handout. Students work individually or in pairs to explore how the pattern can be built and how they would describe it. Refer to link for task instructions.
After: Debrief Squares Upon Squares task in a class discussion. On a projected surface, allow students to demonstrate their approach for increasing the pattern. Discuss how many squares the next element would have and predictions for the 10th element.
Focus: representing visual patterns in T-charts
Before: Visual Patterns discussion – choose a visual pattern from Faye Nguyen’s website (https://www.visualpatterns.org/) or other source and project for a class discussion. Ask students to describe how the pattern is changing and what the next two terms would look like. Can they predict what the 10th term will look like?
During: Using the visual pattern from the ‘Before’ part of the lesson, demonstrate how the elements can be translated to a T-chart. Discuss how putting the pattern into a T-chart can support identifying the pattern rule and extending the pattern. Do the same by revisiting the elements from the Squares Upon Squares task from Wednesday’s lesson.
Print a number of Visual Patterns from Faye Nguyen’s site – working in pairs, students represent these in T-charts.
Provide a variety of materials (tiles, pattern blocks, paper and pencil crayons, natural materials, etc.) and have students (individually or in pairs) create and record their own increasing and decreasing patterns and record the terms in a T-chart, describe the rule in words, and continue the T-chart for three terms.
Have students look at another student’s pattern and create the T-chart, including extending it by three terms.
After: Exit ticket – provide students with a new visual pattern and ask them to describe the rule for extending the pattern, to extend the pattern visually for 2 more terms, and to translate the pattern to a T-chart.
Formative Assessment: use the exit tickets, observations, and conversations to gather evidence of understanding of increasing and decreasing patterns, translation of patterns to T-charts, and ability to extend patterns. Plan focus for small group and one-to-one instruction for Friday.
Focus: introduction to increasing and decreasing number patterns, review and exploring pattern stations, small group and 1-1 instruction
Before: Count Around the Circle routine from Number Sense Routines: Building Mathematical Understanding Every Day in Grades 3-5 (Jessica Shumway, Pembroke). Use a variety of increasing and decreasing patterns (e.g., “start at 16 and go up by 3 each time”, “start at 90 and decrease by 5 each time”) and have students predict what the last student in the circle might say.
During: Set up the stations that were introduced on Monday and allow students to continue to explore the stations. While students work with the stations, pull small groups and individual students based on formative assessment data for instructions and review of increasing and decreasing visual patterns and translating these to T-charts.
After: Math games – time for math games to reinforce and extend understanding of basic facts of addition, subtraction, multiplication, division and larger number operations. This may be established as a weekly routine throughout the year.
Following this set of lessons, students will continue to explore increasing and decreasing number patterns. They will also explore how patterns are found and can be described in graphs, tables, and charts (e.g., 100 chart, multiplication chart, calendars, number lines), including looking at graphs, tables and charts related to content in science and social studies.
Suggestions for Assessment
Through student work, observations, and conversations, gather evidence that students can:
- Create increasing and decreasing patterns with a variety of materials and in different modalities (sound, numbers, etc.)
- Describe the pattern rule to a pattern they or others created
- Represent visual and number patterns in charts or tables
- Use charts to support extending patterns and predict subsequent terms
There are opportunities to document student learning of patterns in a portfolio (digital or physical). For a physical portfolio, students may select a favourite pattern they created and extended by recording it on paper with pictures or numbers. For a digital portfolio, students might take a picture of patterns created with concrete materials (classroom or natural) or on paper. Students may be asked to describe why they chose that pattern as an example of their understanding of increasing and decreasing patterns, what questions they have, and what their next steps might be to continue their learning. Exit tickets can be included in the portfolio.
By the end of Grade 4 students will be able to:
- Create and identify increasing and decreasing patterns using a variety of materials and modalities
- Describe and represent pattern rules in words
- Use pattern rules to extend patterns and predict subsequent terms in the sequence
- Represent a visual and number patterns in a table or chart
Suggested Links and Resources
Youcubed https://www.youcubed.org/tasks/
Visual Patterns https://www.visualpatterns.org/
Book – Land Art: Creating Artworks in and with the Landscape by James Brunt (Schiffer Craft)
Book – Andrew Goldsworthy: A Collaboration with Nature by Andrew Goldsworthy (Abrams)
Book – Do They Really Understand? How We Can Make Sure Students Understand the Math We Teach by Marian Small (Rubicon)
Book – Open Questions for Rich Math Lessons: Patterns and Relations/Statistics and Probability by Marian Small (Rubicon)
Book – Making Math Meaningful to Canadian Students K-8 by Marian Small (Nelson)
Book – Number Sense Routines: Building Mathematical Understanding Every Day in Grades 3-5 by Jessica Shumway (Pembroke)
Key Patterns and Algebra Concept 2: One-Step Equations
Overview
Building on students’ previous experiences of working with one-step equations with addition and subtraction, grade 4 students additionally work with one-step equations involving multiplication and division. One-step equations involve solving for one unknown number and can be a start unknown (e.g., ▢ + 4 = 16), change unknown (e.g., 3 x ▢ = 15), or result unknown (e.g., 24 ÷ 2 = ▢). Foundational to solving one-step equations is an understanding that values on each side of the equal side must balance and grade 4 students review and build upon their experience with this concept from primary years. This includes working with balancing equations with the equal sign in different places and various lengths of expressions. Another key understanding that supports solving one-step equations is the relationship between operations. In primary years, students will have had experience with the various meanings of addition and subtraction and how these two operations are related. They will have some understanding of multiplication and division from grade 3, however, prior to solving one-step equations with multiplication and division, grade 4 students need further experience with the meaning of both operations and how they are related to each other. Students also require an understanding of the meaning of the expressions they are asked to solve; this is linked to the operation they are working with. For example, the equation ▢ + 4 = 16 means there is a number and when 4 is added to it (or you travel 4 jumps on a number line) the result is 16. Approaches include using concrete or pictorial representations (e.g., pan-balance, balance scale, number lines, arrays), inspection (using understanding of basic facts), using opposite operations, and guess and check. A full understanding of what one-step equations can represent and having strategies for solving them supports solving multi-step equations in future grades.
Patterns and Algebra Foundations:
The following concepts and competencies are foundational in supporting understanding of one-step equations in Grade 4:
- Composing and decomposing numbers to 20
- Addition and subtraction facts to 20
- Multiplication and division facts to 100 (beginning understanding)
- Understanding the various meanings of operations (e.g., subtraction as taking away/remaining as well as comparing two quantities. See Marian Small’s Making Math Meaningful for Canadian Students K – 8 p. 156 to 177), especially addition and subtraction
- The relationship between addition/subtraction and multiplication/division (inverse operations)
- Understanding the equal sign as balance (i.e., expressions on either side of the equal sign are different representations of the same amounts)
- Experience working with the equal sign in different locations in an equation (e.g., 5 = 2 + 2 + 1)
- Solving one-step equations with addition and subtraction
Progression:
- Reviewing the relationship between operations (addition/subtraction, multiplication/division, multiplication/addition, division/subtraction), including fact families
- Equality as balance
- Solving one-step equations with addition and subtraction (some bullets below may be done simultaneously)
- Reviewing various meanings of addition and subtraction
- Balanced equations using concrete and pictorial representations (e.g., physical or digital balance scales, Cuisenaire rods, number lines)
- Creating and interpreting equations as representations of situations; include all of start unknown (e.g.,▢ + 4 = 11), change unknown (e.g.,18 – ▢ = 15) and result unknown (e.g, 8 + 7 = ▢ ) and various lengths of equations (e.g., 5 + 3 + 2 + ▢ = 9 + 3)
- Using inspection and inverse operations to solve one-step equations on their own and in context
- Solving one-step equations with multiplication and division (some bullets below may be done simultaneously)
- Reviewing various meanings of multiplication and division
- Balanced equations using concrete or pictorial representations (e.g., physical or digital balance scales, Cuisenaire rods, number lines, arrays)
- Creating and interpreting equations as representations of situations; include all of start unknown (e.g.,▢ x 4 = 12), change unknown (e.g.,18 ÷ ▢ = 2) and result unknown (e.g, 8 x 4 = ▢ ) and various lengths of equations (e.g., 5 x 3 x 2 = ▢ x 10)
- Using inspection and inverse operations to solve one-step equations on their own and in context
Sample Week at a Glance
Students will have explored the relationships between operations and understand opposite operations (addition/subtraction, multiplication/division), including work with basic fact families for operations (e.g., 3 x 5 = 15, 5 x 3 = 15, 15 ÷ 5 = 3, 15 ÷ 3 = 5). This may have happened directly before this week or earlier in the school year. This first lesson of this week offers a review and may be scheduled for a shorter time than other lessons, if appropriate for students.
Focus: relationships between operations (inverse or opposite operations)
Before: Class discussion. Pose the question (with the equation projected for students to see), “I have a subtraction equation 15 – 7 = ▢. How can my addition facts help me figure out what goes on the right side of the equal sign?” After relating 7+ 8 = 15 to this equation, pose a question with division/multiplication and discuss (e.g., “I have a division problem 16 ÷ ▢ = 2 . How can my multiplication facts help me solve this equation?”)
During:
Review fact families as a set of four related facts using opposite (inverse) operations e.g., (2 + 5 = 7 , 5 + 2 = 7, 7 – 2 = 5, 7- 5 = 2). Students engage in individual practice with dice, paper, and pencil. Students roll 2 dice (6-sided or 10-sided; 6-sided are enough for multiplication/division practice at this grade level). They use the resulting number to create a fact family, recording it on paper. Students could work in pairs, rolling one set of dice, individually writing the fact family, and buddy-checking their work. Examples:
(images from Mathigon)
After: Come together as a whole group. On board, pose the problem, “What fact family could you create with 12 and 3?” Have students record their answers on paper or individual white boards. Discuss solutions.This extends the previous activity to use the sum or product in the given numbers and shows that a pair of numbers might be used for an addition/subtraction (e.g., 12 – 3 = 9) family AND a multiplication/division (e.g., 3 x 4 = 12)) family. Do more than one example as time permits. Consider collecting one or more samples from students for formative assessment purposes.
Focus: balanced equations
Before: Class discussion: Pose a problem using a balance scale (virtual or concrete) such as this one (choose a number with many factors for a more interesting discussion):
(image created with Polypad from Mathigon)
What number blocks can be used to balance the scale? Encourage discussion of a variety of answers (e.g., 10 and 2, 6 and 4 and 2, 1 and 1 and 1 and 9, etc.). Record equations as either 3 + 3 + 3 + 3 = (student responses) or 4 x 3 = (student answers).
Then ask, “How can we balance using only one kind of number block?” One possible response is:
(image created with Polypad from Mathigon)
4 x 3 = 6 + 6
4 x 3 = 2 x 6
Record all student responses similarly. Note that each side of the equation is a different representation of the same value (here, 12) so the equation is considered balanced.
Lead a similar discussion using Cuisenaire rods (magnetic or virtual such as https://nrich.maths.org/4348), asking students to contribute ideas to make trains of rods that are the same length (using white as value of 1). For example, a train can be made with four rods of 5 length for a total length of 20 and other rods can be combined to the same length. Record equations to show equality, making sure to show some use multiplication. Solutions might look like:
5 + 5 + 5 + 5 = 10 + 10
4 x 5 = 10 + 10
4 x 5 = 2 x 10
4 x 5 = 3 + 6 + 6 + 5
10 + 10 = 5 x 4
(image created with NRich Cuisenaire Environment)
During: Set up virtual or concrete balance scales and Cuisenaire rods and divide the students in two groups to explore one of these materials at a time. Create a set (5-7 for each manipulative) of challenge cards for balancing (by weight or length). For example, “Create a balanced scale that has 3 of one number on one side and more of another number on the other” or “Make a train of four 6s. Make several trains of the same length. Be sure one train is made of more rods, one made of fewer rods, and one is made from the same number of rods as the first train”. Students record equations on paper or individual white boards to represent their solutions. Switch groups half way through the allotted time for this activity.
After: Consolidating discussion – pose one or two challenges from each type of manipulatives and invite students to share solutions. Record equations and discuss. Confirm that students understand that each side of the equal sign can have a different number of terms.
Focus: solving one-step equations with addition and subtraction
Ideas: SPLAT
Making riddles and translating to equation
Before: Do a few Splat! Discussions as a whole group (choose from 1-10 Splats and 1-20 Splats). Demonstrate how a Splat! Image can be translated into an equation with an unknown in the first discussion and have students contribute equations on subsequent discussions. Example:
Image from www.stevewyborney.com
Some equations related to this Splat! are:
15 = splat + 6 or 15 – 6 = splat or 6 + splat = 15 or 15 – splat = 6
During: Word problems with unknowns. Present one or two word problems that involve addition and/or subtraction, such as, “Morgan was putting apples in 3 baskets to fill customer orders. Morgan started with 12 apples and put 5 in one basket and 4 in another basket. How many apples did Morgan put in the third basket?” Discuss what equations could represent this problem and discuss the similarities and differences of the equations. For example, 12 = 5 + 4 + ? or 5 + 4 + ? = 12 or 12 – 5 – 4 = ? Discuss how students would solve to find the unknown.
Individually, students create their own word problems and record an equation that represents the problem. They solve the equation for the unknown. After students have created a few problems, they partner up and share their problems. The partner records an equation and solves for the unknown and the two students compare their work.
As students work, circulate and support students with their work. As needed, encourage students to create problems with different types of unknown (start unknown, change unknown, result unknown).
After: Show What You Know exit ticket. Provide 1-2 word problems on paper for students to solve by recording a corresponding equation and solving for an unknown. Collect for formative assessment purposes. Adjust remaining lessons (e.g., inserting another lesson or making time for small group instruction while other students do math games or stations) as needed.
Focus: solving one-step equations with multiplication and division using number concrete and pictorial representations.
Before: Class discussion in two parts – pose a question such as, “How can a number line help us find the unknown value in ▢ x 2 = 24?” Using a number line drawn or projected onto the board, record students’ contributions. One answer might be, “We can see how many times we skip count by 2 to get to 24.” Additionally, using virtual manipulatives, chips with a document camera, or magnets, pose a question such as, “How can an array help us solve 16 ÷ ▢ = 8? “ and use manipulatives to demonstrate students’ answers.
During: Math workshop (stations)
- Solving with number lines – provide dry-erase (laminated) number lines and cards with equations with unknowns (multiplication and division with samples of all types of unknowns)
- Solving with arrays – provide chips or other manipulatives and cards with equations with unknowns (multiplication and division with samples of all types of unknowns)
- Splat! – provide cards with multiple splats and a total. Students record a multiplication equation (start or change unknown) or division equation (change or result unknown) that matches the picture. Provide felt splats and counters as concrete materials.
- Balance scales – leverage experience from Tuesday’s lesson – provide cards with one “weight” on one side of the scale and one sample weight on the other side (vary which side has the single weight).. Record an equation to match the scenario and use virtual or concrete balance scale to solve. Example:
24 = ▢ x 6 or 24 ÷ 6 = ▢
(images created with Polypad from Mathigon)
- Cuisenaire rods – leverage experience from Tuesday’s lesson – provide cards with one train made of one kind of rod and ask, “How many [factor of train length] rods would make a train of the same length?’ Students record an equation and solve it. Provide concrete or virtual Cuisenaire rods. Example:
▢ x 4 = 24 or ▢ x 4 = 3 x 8 or 24 ÷ 4 = ? or 24 ÷ ? = 4
(image created with NRich Cuisenaire Environment)
After: Class discussion – How is your knowledge of basic facts of multiplication and division helping you find the unknown quantity in equations? (Note: this will be reinforced and extended in the “Before” part of Friday’s lesson)
Focus: solving one-step equations with multiplication and division using fact families and opposite operations
Optional warm-up: students may be ready to solve Splat! puzzles from the Multiple Splat collections by Steve Wyborney
Before: Class discussion – pose the question “How can understanding how multiplication and division are related help us find the unknown in equations? For example, 18 ÷ ▢ = 6 .” Help students connect to fact families and that multiplication is the opposite operation to division as necessary.
During: Math workshop (stations) and small group work – repeat stations from Thursday possibly adding one that involves creating and solving a word problem (with multiplication or division) and/or one or two math game stations (such as Net Zero game by Bay-Williams and Kling – video instructions provided by Love Maths https://www.lovemaths.me/operations-f-2 ) as needed. Students rotate through stations while the teacher pulls small groups to work on solving for unknowns by relating to fact families and opposite operations (use this small group time to collect information for formative assessment to plan additional whole class or small group instruction if necessary).
After: Class discussion – What is your preferred method of solving for unknowns with equations that involve multiplication and/or division? Why is that your preferred way? (Provide an equation to anchor the discussion as necessary.) Students may give an “it depends” answer with explanation.
This sample week may be enough for students to reach proficiency in the learning standards for one-step equations in Grade 4. It is possible that a couple more lessons (or repeats of the outlined stations and small group work) focusing on equations involving multiplication and division may be needed, depending on students’ previous experiences with these concepts. Cognitive science principles encourage revisiting concepts throughout the year. Consider having students play games and play with balance scales over the rest of the school year and pose one-step equations for solving as an opening or closing activity.
Suggestions for Assessment
Through student work, observations, and conversations, gather evidence that students can:
- Connect their understanding of fact families and opposite operations to solving one-step equations using addition, subtraction, multiplication, or division
- Solve one-step equations using a variety of methods including concrete materials (e.g., balance scale, Cuisenaire rods), pictures (number lines, arrays), and using mental math strategies (opposite operations, fact families)
- Represent information presented in concrete materials, pictures, and word problems in equations with an unknown (all of start unknown, change unknown, result unknown) and solve for the unknown, using addition, subtraction, multiplication, or division
There are opportunities to document student learning for one-step equations in a portfolio (digital or physical). For a physical portfolio, students may select a piece of work from a math workshop station (Thursday/Friday). For a digital portfolio, students might take a picture of work completed with concrete materials (classroom or natural) or on paper. Students may be asked to describe what the piece of work or image shows about their learning, what questions they have, and what their next steps might be to continue their learning. Exit tickets can be included in the portfolio.
By the end of Grade 4 students will be able to:
- Create and identify increasing and decreasing patterns using a variety of materials and modalities
- Describe and represent pattern rules in words
- Use pattern rules to extend patterns and predict subsequent terms in the sequence
- Represent a visual and number patterns in a table or chart
Suggested Links and Resources
Book: Math Fact Fluency by Jennifer Bay-Williams and Gina Kling
Math Loves – Operations games (Net Zero game is on this page) https://www.lovemaths.me/operations-f-2
Book – Making Math Meaningful to Canadian Students K-8 by Marian Small (Nelson)
Mathigon Polypad Algebra tools: https://mathigon.org/polypad#algebra (including balance scale)
NRich Cuisenaire Rod environment: https://nrich.maths.org/4348