Instructional Routines
Instructional Routines are used across the grades in mathematics to build coherence across the grades in mathematical discourse, developing curricular competencies and content and building a culture of mathematics in classrooms and schools.Once teachers and students understand the structure and expectations of the routine, they can move right into doing math together. Instructional routines are used throughout the school year and some teachers use them as part of their daily and weekly planning for mathematics learning. They are often used at the beginning of a math lesson, to engage in doing mathematics together. Number Talks is one routine that is specifically named in the elaborations of our BC Mathematics Curriculum.
General Instructional Routines Resources
High-Yield Routines: K-8 (NCTM Publication)
Number Sense Routines (K-3) by Jessica F. Shumway
Number Sense Routines (Grades 3-5) by Jessica F. Shumway
The SD38 Numeracy YouTube Channel has a collection of short videos sharing how instructional routines can be used across the grades.
https://www.youtube.com/playlist?list=PLwD5gaAZPrjYCP4AETsSw88PAZ7XXkXsF
The BC Numeracy Network has a page on their website focusing on instructional routines here:
https://sites.google.com/view/bc-numeracy-network/balanced-numeracy-framework/instruction/what-routines-will-support-mathematical-thinking-and-reasoning?authuser=0
Number Talks
The Routine:
- The teacher presents a math question (ie. 49 + 26) usually horizontally or a string of related math questions (9+6, 49+6, 49+26) which is often called number strings.
- The students think about different ways to solve the question. Classes often have a silent communication system (thumbs up or fingers showing how many different ways the student has thought of to solve the question) so that students are not calling out or raising their hands, which interferes with others’ thinking.
- The teacher may ask the students to call out what they think the answer is and record the range of responses. In sharing their strategies, students may then share their “proof” or justification of a solution by saying, “I think it is ____ because _____.”
- Students are invited to share the way the solved the problem. The teacher records teh students’ different strategies (on whiteboard/chalkboard/chart, using displayed technology, etc). After a student shares, the teacher asks the students if someone has a different way of thinking about the question and invites more sharing.
- After a collection of responses is recorded, the teacher might invite students to compare and analyze different strategies or to reflect on a strategy that was new to them that they might try.
Resources:
Making Number Talks Matter by Cathy Humphreys and Ruth Parker
Digging Deeper: Making Number Talks Matter Event More by Ruth Parker and Cathy Humphreys
Number Talks: Helping Children Build Mental Math and Computation Strategies K-5 by Sherry Parrish
Number Talks: Fractions, Decimals, and Percentages by Sherry Parrish
Number Talks in the Primary Classroom by Kathy Richardson and Sue Dolphin
SD38 High-Yield Routines Video on Number Talks
https://www.youtube.com/watch?v=2osLQRH2HbQ&list=PLwD5gaAZPrjYCP4AETsSw88PAZ7XXkXsF&index=7
Mathematics Curricular Connections: number concepts and operations, computational fluency, mental math strategies, communicate mathematical ideas, explain and justify
Number Talk Images
The Routine:
- The teacher projects or shows a dot image or photograph and asks students to think about: How many? How do you see them? Or How do you know?
- Students are invited to share the different ways they see the quantity in the image and the teacher records their thinking using diagrams, drawings, words, symbols and equations.
- Some teachers make multiple copies of the image on one slide for the sharing component so they can record the students’ sharing on the image. For example, The copy the image and create a 2×3 array of images so that when they project on the whiteboard, they can draw over and annotate on an image, one image for each student who shares. You then have a collection that students can look at, compare and analyze.
- Students can be invited to create their own number talk images to use in the classroom.
Resources:
The Number Talk Images website is in English and French and is curated by Pierre Tranchemontagne, an educator iin Ottawa. There are ideas for using Number Talk Images as well as collections of dot images and photographs that are sent in from all around the world.
SD38 Number Talk Images poster (English):
https://blogs.sd38.bc.ca/sd38mathandscience/wp-content/uploads/sites/14/2019/01/NTI-poster.pdf
SD38 Number Talk Images poster (French):
Mathematics Curricular Connections: number concepts and operations, mental math strategies, visualize, communicate mathematical ideas, represent mathematical ideas in different forms (pictorial, symbolic), explain and justify
Counting Collections
The Routine:
- The teacher has collections of materials (quantities suitable for the number range the students are working within) curated in bags, containers or cups for students to choose from.
- Pairs of students choose a collection to count together and any tools they might like to use to support the count, such as ten frames, little cups or bowls.
- The pair of students count the collection in a few different ways (if counting by 1s, maybe moving into a container or putting in a line, or counting by different multiples such as 2s, 5s, and 10s).
- Sometimes the students are invited to record their count and how they counted somehow – on a recording sheet or on a collaborative chart or whiteboard.
- The teacher might pause students and do a gallery walk so they can see and be inspired by the different ways students have counted their collections.
Resources:
Choral Counting and Counting Collections: Transforming the PreK-5 Math Classroom by Angela Chan Turrou, Elham Kazemi, and Meghan L. Franke
SD38 High-Yield Routines Video: Counting Collections
https://www.youtube.com/watch?v=yGd7evZ_cVs&list=PLwD5gaAZPrjYCP4AETsSw88PAZ7XXkXsF&index=6
Some SD38 blogs posts about counting collections:
https://blogs.sd38.bc.ca/sd38mathandscience/2015/11/03/counting-collections/
https://blogs.sd38.bc.ca/sd38mathandscience/2017/01/03/extending-counting-collections/
SD38 Counting Collections poster to print (English):
SD38 Counting Collections poster to print (French):
Mathematics Curricular Connections: number concepts and operations, counting, multiples, mental math strategies, communicate mathematical ideas, explain and justify, connect math concepts to each other
Choral Counting
- The teacher pre-plans a count in response to what students are learning. Students could count by different multiples (2s, 5s, 12s, etc) or using different numbers (frctions, decimals, etc) with different starting and end points. The teacher plans how they will record the count (vertical or horizontal, how many rows and columns).
- The teacher invited the students to all count aloud together as they record the count as planned.
- After the choral count, students are invited to describe what they notice in the record of the count. What number patterns do you notice? How could you describe them?
- The teacher annotates the record of the count with the students’ observations, comments and connections.
Ways to Make a Number
The Routine:
- The teacher invites students to represent a target number as many ways as they can.
- Students may use a variety of tools and/or representations, e.g. ten-frame(s), tally marks, equations, base ten blocks, coins, etc.
- Begin by making it open for any way, then later the teacher can add constraints such as use three addends, include subtraction, etc.
- The focus is about thinking flexibly about numbers, and how to compose/decompose them.
Resources:
Number Sense Routines: Building Numerical Literacy Every Day in Grades K-3 by Jessica Shumway
Mathematics Curricular Connections: number concepts and operations, visualize, represent mathematical ideas in different forms (concrete, pictorial, symbolic)
Fraction Talks
The Routine:
- The teacher presents an image with one part/fraction of an area or set shaded and asks: “What fraction is shaded?
- Students can think to themselves or discuss in small groups.
- Students share their solutions and their reasoning or visualizing that supports their solution.
Resources:
Fraction Talks website created and curated by Nat Banting (includes many possible extensions to the core routine):
SD38 High-Yield Routines Video: Fraction Talks
https://www.youtube.com/watch?v=6whiGQa_R-E&list=PLwD5gaAZPrjYCP4AETsSw88PAZ7XXkXsF&index=17
Mathematics Curricular Connections: number concepts (fractions), measurement, decomposing and composing 2D shapes, visualize, communicate mathematical ideas, represent mathematical ideas in different forms (pictorial, symbolic), explain and justify, spatial reasoning
Estimation 180
The Routine:
- A photograph is presented and students are asked to consider “About how many…About how tall…?” depending on the context.
- To narrow to a reasonable range, the teacher might collect “too low” estimates and “too high” estimates and invite students to share their reasoning behind these.
- Have students think about what a reasonable range for the quantity or measurement would and collect and record their responses on a whiteboard or chart.
- Play part of the “reveal” video which provides students with some information and ask if they would like to refine their estimate.
- Complete the reveal video and compare estimates and the final revealed amount and discuss the reasonableness of the estimates.
Resources:
Estimation 180 website created and curated by Andrew Stadel which include 180+ images to use as well as “reveal” videos and images:
SD38 High-Yield Routines Video: Estimation 180
https://www.youtube.com/watch?v=YVjke6rFqW8&list=PLwD5gaAZPrjYCP4AETsSw88PAZ7XXkXsF&index=11
Mathematics Curricular Connections: number concepts, measurement, use reasoning, estimate reasonably, visualize, communicate mathematical ideas, explain and justify, spatial reasoning
Quick Images
The Routine:
- The teacher presents an image slide or holds up an image card (ie dot card) for 1-2 seconds and asks students to hold the visual image in their mind.
- With the image not visible, the students are invited to describe what they saw – how many and how they saw them.
- The image is shared again and discussion continues about the different ways you can see the quantity.
Resources:
Quick Images are one of the routines included in the following books:
High-Yield Routines: K-8 (NCTM Publication)
Number Sense Routines (K-3) by Jessica F. Shumway
Number Arrangement Cards by Kathy Richardson that can be used for Quick Images can be found here:
https://numbertalkscom.wordpress.com/number-talks-models/
Mathematics Curricular Connections: number concepts and operations, mental math strategies, decomposing quantities, subitizing, visualize, communicate mathematical ideas, explain and justify, spatial reasoning
Clothesline
The Routine:
- The teacher hangs a clothesline (string, twine, rope) and places 1-3 benchmark numbers on either using cards clothes-pegged to the line or tent cards put over the line. For example, if you are ordering number from 1-100, the benchmark numbers you might place are 0, 50 and 100. If you are ordering fractions, the benchmark numbers you might place are 0 and 1.
- Each pair of students is provided with a number card and asked to discuss together where that number would go on the clothesline and decide how they will explain their decision. Different representations of numbers can be included (pictorial representations such as ten frames for example).
- Pairs of students are invited to add their number to the clothesline, explaining their choice. For example, a pair of students might say, “We are putting 18 here as we know that it is quite a bit more than 10 and two less than 20.”
- Sometimes, equivalent numbers or representations are included in the routine. Students can either clip the equivalent under the original number card that was placed or if using tent cards, it can be placed over the original number card.
- Once all the numbers are placed, the teacher can facilitate a discussion about the different strategies students used and which numbers were straightforward to place and which ones were more challenging.
Resources:
The Clothesline Math blog is hosted by Chris Shore, a math consultant in California:
SD38 High-Yield Routines Video: Clothesline
https://www.youtube.com/watch?v=fQYjL3m7Xys&list=PLwD5gaAZPrjYCP4AETsSw88PAZ7XXkXsF&index=9
A blog post about introducing Clothesline to a kindergarten class can be found here:
SD38 Clothesline poster (English):
SD38 Clothesline poster (French);
Mathematics Curricular Connections: number concepts, comparing and ordering numbers (whole numbers, fractions, decimals, percentages), visualize, communicate mathematical ideas, explain and justify, spatial reasoning
Same But Different
The Routine:
- The teacher presents a set of two images of items for students to discuss.
- The students discuss and share how the images are the same and the teachers record their ideas.
- The students discuss and share how the two items are different and the teacher records their ideas.
Resources:
The Same But Different website has a collection of images available across areas of mathematics. Each set of images includes two items (ie numbers) that can be presented for students to discuss and engage in categorical thinking.
https://www.samebutdifferentmath.com/
Mathematics Curricular Connections: number concepts and operations, visualize, communicate mathematical ideas, represent mathematical ideas in different forms (pictorial, symbolic), use mathematical vocabulary and language to contribute to mathematical discussions, explain and justify
Brian Bushart has created a website of images for the Same or Different routine here: https://samedifferentimages.wordpress.com/
Which One Doesn't Belong
The Routine:
- The teacher presents a set of four items such as numbers (two-digit numbers, fractions, percentages or combinations of numbers) in a 2×2 array for students for discuss
- The students discuss and share the numbers belong together.
- The students discuss how each number is unique and the teacher records the students’ ideas.
- In partners or small groups, students discuss if they had to choose one of the numbers to “not belong” or which one is the most unique, which one would it be and why.
- Students share their thinking and the class might try to reach consensus but discussing their reasoning.
Resources:
Which One Doesn’t Belong? A Shapes Book by Christopher Danielson (and accompanying teacher’s resource)
The Which One Doesn’t Belong website has images submitted by educators from around the world. The images are categorized by math topic such as numbers or shapes.
Mathematics Curricular Connections: number concepts and operations, visualize, communicate mathematical ideas, represent mathematical ideas in different forms (pictorial, symbolic), use mathematical vocabulary and language to contribute to mathematical discussions, explain and justify
Splat!
The Routine:
- The teachers shares an image or collection of dots and invites students to estimate how many there are. They then count how many there are and the total is posted.
- A SPLAT! then covers some of the dots.
- Students think or discuss with others how many dots are hidden by the SPLAT! and how they know.
- Students are invited to share the different ways they figured out how many dots are under the SPLAT! and the teacher may record their approaches.
- The SPLAT! is removed to reveal how many dots are under the SPLAT!
Resources:
Steve Wyborney is a math coach in Oregon and creates many online resources for teachers to use. Splat in different forms and in google or powerpoint slides is available to download from Steve’s website:
https://stevewyborney.com/?s=splat
SD38 High-Yield Routines Video: Splat!
https://www.youtube.com/watch?v=Gc-2zbKLkMY&list=PLwD5gaAZPrjYCP4AETsSw88PAZ7XXkXsF&index=4
Mathematics Curricular Connections: number concepts and operations, solving for unknowns, change in quantity, decomposition of numbers, visualize, communicate mathematical ideas, explain and justify
Visual Patterns
The Routine:
- Share the first three steps (also called terms or figures) of a visual pattern, projected on a screen or whiteboard.
- Ask students what they notice, what stays the same and what changes.
- You may invite students to build or extend the pattern using tiles or counters.
- Invite the students to make connections to numbers and number relationships within the pattern, including recording the step number and number of items in a table or chart.
- Have students generalize what the pattern rule is and predict and describe the next step in the pattern.
- Have students determine how many items would be in the 43rd step, explaining their strategies and reasoning. This can be recorded as an algebraic expression such as 2n +1.
Resources:
Fawn Nguyen has created and curated 100s of visual patterns available on this website:
https://www.visualpatterns.org/
All patterns include the first three steps of the pattern and the solution for the 43rd step of the pattern.
Mathematics Curricular Connections: increasing and decreasing patterns, tables and expressions, visualize to explore mathematical concepts, explain and justify
SolveMe Mobiles
The Routine:
- Project a SolveMe mobile puzzle on a screen or whiteboard.
- Have students turn to a partner or think in their head about how to solve the puzzle.
- Invite students to share their solutions and strategies for solving the mobile.
Resources:
The SolveMe Mobiles website can be found here:
https://solveme.edc.org/Mobiles.html
Mathematics Curricular Connections: equality and inequality, one and two-step equations, solving for the unknown, reasoning, explain and justify
I Have, You Need
The Routine:
- Choose a target number such as 100, 25, 4.75 etc
- State: For a total of 25, I HAVE 18, YOU NEED ____
- Provide brief thinking time with students not calling out.
- Collect responses through a choral or popcorn reply.
- Invite students to respond to: How do you know? What did you think about when you were figuring this out?
- Repeat with a related, more complex combination of numbers.
Resources:
Pam Harris includes a video and teacher guide for this routine on her website:
https://www.mathisfigureoutable.com/blog/i-have-you-need
Mathematics Curricular Connections:
addition, subtraction, composing and decomposing numbers, one-step equations, solving for the unknown, reasoning, explain and justify
Slow Reveal Graphs
The Routine:
- Share the first “layer” or part of a graph without titles, labels, words or identifying features about what the graph is about. Ask students what they notice and what they wonder.
- Reveal some more information about the graph such as quantities. Ask students to consider what the graph could be about.
- Slowly reveal more and more information, pausing to have students predict and discuss the graph.
- Finally, reveal the last piece of information. Ask students to reflect on what information they used to make their predictions.
Resources:
Jenna Laib has curated a collection of slow reveal graphs on this website:
Mathematics Curricular Connections: data representation in different forms of graphs (bar, line, circle), interpreting and analyzing graphs, reasoning, visualizing, explain and justify, connecting
Always, Sometimes, Never
The Routine:
- Present students with a statement
- Students categorize the statement as always, sometimes or never true
- Justify your answer
- Search for examples and non-examples
Note: The discussion component here is very important. Students should be encouraged to listen for evidence from others that confirms their answer or causes them to shift their thinking.
Resources:
Where’s Polygon
The Routine:
- Display a blank coordinate plane for students to see.
- Have the coordinates of a specific polygon drawn on another, secret, paper
- Have students offer coordinate pairs
- Colour code their guesses:
Red = outside the shape
Yellow = inside the shape
Blue = the perimeter of the shape
Green = vertices of the shape - Continue until all 4 vertices are found.
Resources: Math = Love – Where’s Polygon math routine