Math Fluency

Does Fluency = Timed Test?

What is fluency?

The word fluency surfaces a variety of emotions. Whether it brings tears of frustration or cries of joy for you, it is necessary to relook at fluency. So, let's take a look at what fluency means, activities & lessons to development fluency, and best-practices for assessing fluency.

Wherever the word fluently appears in a content standard, the word means quickly and accurately. It means more or less the same as when someone is said to be fluent in a foreign language. To be fluent is to flow: Fluent isn’t halting, stumbling, or reversing oneself. A key aspect of fluency in this sense is that it is not something that happens all at once in a single grade but requires attention to student understanding along the way. It is important to ensure that sufficient practice and extra support are provided at each grade to allow all students to meet the standards that call explicitly for fluency.

excerpt from PARCC Model Frameworks version 3.0 pg. 9

The Common Core State Standards identify fluency requirements at each grade level. Here is a chart showing the fluency requirements for K-8:

Traditionally, many of these fluencies have been assessed through timed tests. However, when I read the list and saw grade 3 has a fluency requirement for addition and subtraction within 1,000,  I wasn't  sure my definition was complete: fluency = timed tests. Time to do some research and gain a better, more well-rounded perspective.

Common Core has given us the opportunity to rethink, reflect, and retool ourselves with the best practices for our profession.  Through the process of implementing Common Core Math, I have begun my own exploration on fluency to redefine what it means, how to develop fluency, and how to assess fluency. I don't have all the answers, yet (and probably never will) but I am finding some really great resources. Let me share what I have found...

How can we development fluency with math facts?

Principles and Standards for School Mathematics states: 

“Computational fluency refers to having efficient and accurate methods for computing. Students exhibit computational fluency when they demonstrate flexibility in the computational methods they choose, understand and can explain these methods, and produce accurate answers efficiently. The computational methods that a student uses should be based on mathematical ideas that the student understands well, including the structure of the base-ten number system, properties of multiplication and division, and number relationships” (p. 152).

Check out the article from NCTM President Linda Gojak from NCTM Summing It Up November 1, 2012

Using a variety of activities creates multiple experiences for students to become efficient and flexible with their math facts. Although you may use typical fluency activities like flash cards and timed-tests, consider a wider variety of activities that support students' mathematical thinking around numbers and their operations. Games and apps can be a fun way to work on basic math skills. There are a lot of games and apps available - depending on your grade level and skill of focus. Check Edutopia's list of apps to develop fluency here. Another way to address math fluency is to use problem solving activities that require students to think about efficient methods and strategies for solving 

• Incorporate a variety of activities that develop fluency: flash cards, timed tests, games, apps, problem solving activities, conceptual development lessons, math centers, etc. 
• Fluency also requires efficiency & flexibility
• Scaffold learning from concrete to semi-concrete to abstract
• Make connections among strategies 
• Have students explain their thinking
• Model your mathematical thinking 

"The best way to develop fluency with numbers is to develop number sense and to work with numbers in different ways, not to blindly memorize without number sense." Jo Boaler

How can fluency be assessed?

Timed Tests are still a part of fluency. This type of assessment provides us with quick assessment data that we can gather on a regular basis. Timed tests are a clear way to assess a student's speed and accuracy for specific skills.

However, Common Core is bringing mathematics back to a place of balance. Conceptual development works together with drill and practice. Both are necessary. Both support understanding.

Some things to keep in mind:

• Timed tests can cause anxiety for some students
• Consider focusing on growth - students try to improve upon their own score
• Carefully design and plan lessons that support students' sense of a number
• Explicitly make connections within an across the mathematics content of your grade level
• Purposefully plan learning experiences that build to quick recall
• Use problem solving activities to highlight key understandings or common misconceptions
• Consider using a rubric that describes a more complete picture of fluency - a combination of evidence including timed tests, observations, and other assessments. 

Resources for Developing Fluency

Articles: 

Fluency Without Fear:  Research Evidence in the Best Ways to Learn Math Facts and Appendix A
By Jo Boaler, Professor of Mathematics Education, co-founder youcubed
with the help of Cathy Williams, co-founder you cubed & AmandaConfer, Standford University
October 24th, 2014 on youcubed.org

Fluency: Simply Fast and Accurate? I Think Not! By NCTM President Linda M. Gojak. NCTM Summing Up, November 1, 2012

Towards Meaning-Driven Math Fluency  by Dr. Jonathan Thomas, Kentucky Center for Mathematics Faculty Associate. Kentucky Center for Mathematics. 

Resources for Elementary Teachers:

Mathematically Minded & The Recovering Traditionalist
Check out the free downloads for number sense activities, number paths, subitizing, place value cards, and rekenreks activities.

Math Wire: Basic Facts Fluency
Check out the links to resources for a variety of activities to support fluency.

Fact Fluency for Addition & Subtraction, Multiplication

Howard County Public Schools - Fluency Assessment Resources
Includes Learning Targets, rubrics, assessment tasks, and scoring/recording sheets.

Grade 2: Assessing Fluency 2.OA.2

Grade 4: Assessing Fluency: 4.NBT.4

Resources for Primary Teachers:

Math Rack
Math Facts Pro
Math Wire
Web Mad Minute

Math & Literature

Using literature in a math lesson can be a way to draw students in, create interest, make math fun, and bring variety to math time. And who doesn't like a good story? Consider using pictures books as a way to discuss math in a fun and exciting context. Below you will find some background on using pictures, a sample of a book & activities, and finally some resources for where to find some good picture books.

Teaching Math With Picture Books Part 1 

The Three Tiers of Math Picture Books

Picture books fall into three tiers - all three type of math picture books are important and can lead to great math discussion. 

Ø  Tier 1: Fundamental Math Picture Books — These are books in which the math content is the primary purpose of the book. It either dominates the plotline (for fiction books), or is an informational math text. These books are generally read with the specific purpose of learning math content.

Ø  Tier 2: Embedded Math Picture Books — These are books in which the plot has deliberate connections to math, but the story stands on its own as well. These books feel more natural as read-alouds, but may require the teacher to direct the focus onto the content connections.

Ø  Tier 3: Connected Math Picture Books — These books do not have any explicit connections to math, but the teacher can create connections through think-alouds or class discussions. Sometimes, the teacher may challenge students to come up with the connections to math.

adapted from Alicia Zimmerman, Scholastic

http://www.scholastic.com/teachers/top-teaching/2012/11/teaching-math-picture-books-part-1

Ten on a Sled

This fun sing-song book is a great addition to  math time. Like many pieces of literature we incorporate into our classroom, there may be some vocabulary and background knowledge we need to teach students before reading the book. This book has some great connections to language arts lessons - like alliterations, song or poem like structure, etc. When using this book for a math lesson, I encourage you to stick to the mathematics. Don't try to do too much in one lesson. This book could be used over the course of the week and incorporated into math, language arts, reading, poetry, science, etc. Don't be afraid to use the book multiple time for multiple purposes with your students.

Now, let's talk math...

Consider using a Ten-Frame or Rekenrek while reading the book. As each animal falls off the sled, remove one counter. Engage students by asking them questions related to the math of counting, addition, or subtraction. Here is a possible list:

• How many animals are on the sled?
• If one animal falls off, how many will be on the sled?
• How many animals do you think will be on the sled on the next page? How do you know?
• How many animals are on the sled and how many animals have fallen off the sled?
• How many animals have fallen off in total, so far?

Here are a few resources I found for using the book Ten on a Sled:

• Activity - possible for choice time, math centers, etc.
• Ten-On-A-Sled & Number Recognition
• Recovering Traditionalist: Ten on a Sled Ten-Frame
• It's not math, but here is another lesson resource: Ten On a Sled Printable Activities

More Math & Literature

Remember that using literature during math time can be a way to engage students in thinking mathematically. Ask good questions. Dig deeper.

Check out the article by Alicia Zimmerman,  Teaching Math With Picture Books Part 1 for more information about selecting literature for math, various purposes for using literature, and more resources.

Resources

Below is a list of resources for math & literature. Some of these links are lists of books, while others provide lessons and descriptions for using literature during math time.

Let's Read Math
A list of books by math topic

Another Book List
A list of books by math topic or grade

Investigating Number Sense, Addition, and Subtraction read-aloud summaries

Marilyn Burns, 3 Lessons: Using Storybooks to Teach Math, Instructor Magazine April 2005

Marilyn Burns Classroom Library

Math and Literature Series, published by Math Solutions founded by Marilyn Burns

Marilyn Burns' Webinar on Math Re-Alouds: Using Children's Literature to Teach Math Grades K-5

Evens & Odds

Grade 2: Conceptual Understanding of Even & Odd

2.OA.3 Determine whether a group of objects (up to 20) has an even or odd number of members, e.g. by pairing objects or counting by 2's; write an equation to express an even number as a sum of two equal addends.

Students enjoy learning about even and odd numbers. It is a fun topic. Let's take a look at a few activities that can support students' understanding of even and odd numbers.

Counters

Using linker cubes or counters allows students to physically pair the individual pieces. Let's say we have a group of 9 counters and we want to determine if the number is even or odd.

We can start by pairing the counters. There are 4 groups of 2 and 1 counter that does not have a pair. Think about how the visual can support the learning. Some visuals may support components of the topic better than others. Think about your lesson objective, the related standards and the sequence of the learning activities. Also consider possible misunderstandings that might arise. As you make choices about the sequence of activities, questions asked, and models used, keep in mind the ultimate learning goal and where you are in getting your students to that point. It is journey. 

If all the counters are red, paired in 2's and one is not paired, students will be able to see the pairs and that one member does not have a partner. Turing the one counter to yellow highlights the fact that all of the red counters are paired and one counter does not have a pair - it is yellow. Students could flipped the counters over to the opposite color as the pair up the counters, which will leave the one that does not have a partner a different color. This is an instructional choice we will need to make. 

Working in 2's shows that a number is even, or that the number is not even - it is odd. In the picture above, the counters are paired and somewhat scattered. If you are working on the connection between counting by 2's (starting at 0) and even numbers, this representation can be supportive of making the bridge from skip counting to even numbers. Students can write a number sentence 2 + 2 + 2 + 2 + 1 = 9.

However, it is also important that students have an understanding around even numbers as equal addends. Scattered pairs do not emphasize the understanding that 8 can be written as 4 + 4 = 8. So, let's take a look at a different way to arrange and represent even and odd numbers with counters and linker cubes.

If the counters we arranged in 2 columns (or 2 rows), pairs can be made as you create the array. If one counter does not have a partner, then the number is odd. Again, I flipped the one non-partnered member to yellow to emphasize that it was not paired. This can be done as counters are paired. Start with all counters yellow, as you make a pair flip to red and place in the array. An odd number will have one counter left as yellow.

Similarly, you can pair the counters,of different colors using the array representation. Once all counters have been paired - one red with one yellow - either all counters will have a partner (even) or  one counter will not have a partner (you will have one more of one color).

The same idea can be used with Linker Cubes as well.

Linker Cubes

Lesson Idea - Check out the Math Coaching Consortium website, hosted by West Contra Costa Unified School District - www.wccusd.net/math. There are tons of lessons and resources on this website. The lesson for Even and Odd: A Conceptual Understanding can be found under Lessons - Grade 2.

Good teaching always starts with the learning objective. There are several things happening in this lesson plan. You may not want to do all parts in one lesson. This may be something you revisit over a few days or over the course of the year as you see students are ready to add to their understanding around even and odd numbers. However, many students are ready to begin this discussion early in the school year.

Lesson Objective: Students will be able to determine if a number is even or odd by pairing linker cubes. Students will be able to write an equation for an even number using equal addends. 

In the lesson, you start as a whole class and model how students are going to explore numbers by paring counters. After modeling a few numbers, 1, 2, 3, 4, 5, & 6... begin looking at patterns.

What patterns do you notice?

Define even numbers and odd numbers. Begin creating a class anchor chart that displays the information about numbers, the representation, equations, and whether the number is even or odd. Here is a sample of an in-progress anchor chart I need in a recent professional development workshop. 

  

After creating the models, look at each number and determine if it is even or odd. Students should discover that every other number is even; every other number is odd. Let them discuss and discover these patterns. Then help students connect that mathematics to the patterns that they notice. Give them the vocabulary and structures that they need. 

Writing the equations can be a tricky part to this activity. Writing equations for even numbers as equal addends will be easier than writing equations for odd numbers (writing equations for odd numbers is not a part of the second grade standard). This might be a piece of the lesson that is revisited or left out to use as a separate activity. 

The second part of the lesson has students working on a number, pairing up members of the number using either linker cubes or counters. Students will need to tell if their number even or odd and why.

At the end of the lesson, students present their number to the class. Continue adding to the class anchor chart and making connections among the numbers, the equations, and whether the number is even or odd. 

Evens as Equal Addends:

By displaying the linker cubes in two towers, the pairs are next to each other, one cube from the left tower with one cube from the right tower. This allows the equation to match the visual. 

   

For example, if I want to write the equation for 10, we can see the the left tower has 5 cubes and the right tower has 5 cubes, therefore, 5 + 5 = 10.


This works well for all even numbers.

Contrast this with with showing 5 groups of 2's. Although this shows that 10 is even, it leads students to write 2 + 2 + 2 + 2 + 2 = 10.  Again going back to our purpose, if our lesson wants students to write an equation for an even number with two equal addends, then showing 2 columns (or rows) leads students to write the equation 5 + 5 = 10.

Now, your students might be curious about writing equations for odd numbers. There are a variety of ways to write equations for numbers. If I was doing a lesson on breaking a part number and recomposing numbers, then we would explore a variety of equations. But since we are on the topic of even and odd, let's look at how the equal addends extends to odds. 

Continuing from the even number 10, we know that 11 is one more than 10. We can think 11 = 10 + 1. Then using the equal addends for 10, 

we can write it as 5 + 5 + 1= 10. 

Sentence Frames

To support students in their understanding and in their presentations, sentence frames are a great tool. Here is a sentence frame that I used. 

The lesson includes an exit ticket that incorporates the sentence frame as well.

Go check out: www.wccusd.net/math

Go to Lessons, then Grade 2 ... look for the lesson titled Even and Odd: Conceptual Understanding

Tips for using Manipulatives:

• Allow students time to explore with manipulatives before  requiring them to do the math activity. 
• Use a placemat or construction paper - all materials should stay on the mat.
• Be realistic about your procedures & rules
• Model your procedures
• Follow-through with your rules

Math Virtual Manipulatives for K/1 Teachers

Math Resources:

Math Learning Center Apps (Apps for iPad/iPhone & Web-Versions):

The Math Learning Center has free apps and online versions of their apps available for many math manipulatives. Virtual manipulatives can be an easy way to have access to a variety of concrete models for mathematics. The complete list can be found here. Each virtual manipulative has a tutorial video, so check it out and explore!

Number Rack

The Number Rack has rows of moveable, colored beads that encourage learners to think in groups of fives and tens, helping them to explore and discover a variety of addition and subtraction strategies.
This virtual version of the manipulative is an open-ended educational tool, ideal for elementary classrooms. (from Math Learning Center catalog).

The one-minute tutorial video here.
The web version is available here.

Number Frames

The Number Frames app or web-version from Math Learning Center allows you to explore number frames: 5-Frames, 10-Frames, 20-Frames or even 100-Frames. You can use various counters and colors to support the addition and subtraction strategies you are working on.  The calculator allows you to write equations to match your number frame model.

Check out the one-minute tutorial video, download the app, click  here.
To open the web-version or to learn more, click here.


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For more Apps & online math manipulatives from Math Learning Center visit: 
http://catalog.mathlearningcenter.org/apps

Check out: Mathematically Minded

Sometimes I get lost on the internet. I start googling something I need and end up with 20 tabs open, finding great things... great unrelated things, and using precious minutes and even hours on things I never intended to find or need. Sound familiar?

Today I would like to share with you an amazing resources for you to explore. The website is Mathmatically Minded, where  Christina Tondevold writes about her experiences in the classroom, working with teachers, and doing math with her kids. She blogs at The Recovering Traditionalist .

Here are a few of the things I love:

Mathematically Minded

FREE Downloads! There are activities for subitizing, number sense, place value, and rekenreks. You can also find presentations, resources, and blackline masters. Go check it out!

The Recovering Traditionalist 

Since Thanksgiving is just a couple weeks away and winter is approaching, the math and literature connections using the books 5 Silly Turkeys by Salina Yoon and Ten on a Sled by Kim Norman, are great lessons and activities appropriate for this time of the year.
You can click on Math and Literature at the top of the blog or to read her post about 5 Silly Turkeys here, and Ten on a Sled here.

Math Routines: Ten Frames

Using Routines with Math Tools: Ten Frames

Routines in the math classroom are important for a variety of reasons. While some routines are just necessary for the organization and structure of the classroom (and teacher's sanity), other routines can build students' mathematical understanding and reasoning. In this series of blog posts on Using Routines with Math Tools, I will discuss various routines that can enhance the teaching and learning in the classroom. I will reference various articles and books that can support us in implementing these routines in our classrooms. However, many math routines are very simple in nature and can be implemented with very a little prep and resources.

What makes something a routine? In the book High-Yield Routines: Grades K-8, McCoy, Barnett & Combs describe a high-yield routine with the following characteristics:

• a structured activity that helps students gain proficiency with a range of concepts and practices
• offer access to the big ideas of mathematics and allow deep understanding of concepts
• give students opportunities to develop expertise with the eight Standards for Mathematical Practice
• offer opportunities for students to demonstrate their thinking and for teachers to gain insight into the thinking of their students

In my work with Kindergarten and 1st grade teachers, we are looking at a variety of methods, strategies, and tools that support students' learning around their sense of numbers, combinations of numbers, and how numbers relate to each other. So, let's engage in some mathematics by looking at routines with Ten-Frames!

Let's explore some routines with Ten Frames.

What is a Ten Frame?

A Ten-Frame is a 2 by 5 array. Most often you see it this way:

A ten-frame can be used to support the development of land-mark numbers of 5 and 10. Using a double ten-frame can be useful as students are working towards the number 20. Ten-frames can be used as a model and support counting and subitizing. And can be great tools to support students' strategies for addition, and subtraction. The routines I am going to share are not ones I have created and are probably being used in many classrooms. I enjoy the work of Marilyn Burns and find that it resonates with my understanding of teaching and learning. I use the book, It Makes Sense! Using Ten-Frames to Build Number Sense (K-2) as one resource to support my work with Kindergarten and 1st grade teachers. Some of the routines can be found in this book.

Ten Frame Routine: Look, Quick!

Overview:
In this routine, the teacher will show some counters on a ten-frame and students will look for groupings without counting. This routine builds students ability to subitize - to know a number just by looking at it.

Materials:
Ten-Frame, counters

Directions:
*Begin with modeling in the Introduction, Example 1, and Example 2 so that you set the stage for the routine. Students should become familiar with the ten-frame, counters, and the questions you are going to ask before beginning the routine. 

Introduction

• Show students the blank ten-frame. Ask:"How many squares are there?" Rotate the ten-frame and ask again. 

Model: Example 1

• To begin, tell students that you are going to place some counters on the ten-frame and they will need to determine the total amount of counters. You will need to place the counters on the ten-frame without students seeing - either students close their eyes or cover up your work until you are ready to show students the ten-frame with counters. 
• Example 1 - model the routine
• Place 5 counters on the ten-frame, possibly like this: 

• Reveal the Ten-Frame to students. Ask: "How many counters do you see?" 
• Ask students to whisper the number of counters. Then ask "How do you know?"
• Ask students if there is another way to see how the counters are grouped. Ask "How do you see the counters?"

Model: Example 2   (repeat the routine)

• Place 4 counters on the ten-frame - keeping it a secret from the students. 
• Reveal the Ten-Frame to students. Ask: "How many counters do you see?" Encourage students to look for groupings of counters. 
• Have students whisper the total amount on the count of 3. 
• Ask a student to share how he or she knew how many counters were on the ten-frame. 
• Record students explanations as a number sentence. 
• For example, 4 = 2 + 2 or 4 = 1 + 1 + 1 + 1. 

Now that students are comfortable with looking at the ten-frame and counters, introduce the routine, Look Quick! 

Ten-Frame Routine - Look Quick! 

1. Teacher places an amount of counters on the ten-frame (hidden from students). As you repeat the activity, vary the groupings of the counters. 
2. Give student 3 seconds to recognize the amount of counters. Cover/Hide the ten-frame and counters. 
3. Ask: "How many counters do you see?" Then have students whisper, on the count of three, the total number of counters. 
4. Ask students: "How do you know how many counters are on the ten-frame?" or "How did you see the counters?" 
5. As students share their idea, record the number sentence. For example, 4 = 2 + 2 (you can vary the equal sign at the beginning and end so that students don't development a misconception that the equal sign means the answer is coming.)  

I saw this routine done in a vary similar way on the Teaching Channel.
The Teaching Channel is a great place to go to see classroom lessons that support the learning of Common Core.

Check out the video in a Kindergarten classroom:  Quick Images: Visualizing Number Combinations

Ten Frame Routine: Make the Number

Overview:
In this routine, the teacher will quickly show some counters on a ten-frame and students will build what they saw with counters on their own ten-frame. This routine builds on students' ability to subitize and combines it with spatial reasoning.

Materials:
Ten-Frame, counters

Directions:
1. Give students a ten-frame and counters. Ask students to keep their counters off thei ten-frame mat until the routine begins.

2. Model: Example #1 Place 3 counters on the ten-frame. Leaving it in sight, have students build exactly what they see. Ask students to whisper how many counters they used. Then, have students turn and talk to their partner about what they built.

3. Have students clear their Ten-Frames.

4. Ten-Frame Routine: Make the Number! This time the students will only see the Ten_fRame & counters for 3 seconds. Place 3 counters in a different location on the Ten-Frame. Show students the Ten-Frame for 3 seconds, then hide the frame and counters. Ask students to use their counters and Ten-Frames to build what they remember seeing.

5. Lead the class in a discussion by asking questions:

• How many counters did you see?
• What did you build first?
• How did you know where to place the counters?

Students should have an opportunity to talk to a partner before discussing as a class.

6. Show the original Ten-Frame that you built to give students time to check their answer.

Resources & Digging Deeper

Hand Signals

Hand Signals can be very helpful with routines like the ones discussed above. Number Talks use hand signals as a silent gesture to show a students' response to the talk. Here is a poster that shows the hand signal and the meaning of the signal. 

Additional Resources for Routines with Ten-Frames:

• Article on Subitizing
• Illuminations: Ten-Frames 
• K-5 Math Teaching Resources - "Ten Frames and Dot Cards"
• North Carolina Lessons for Learning - Grade K   Grade 1

References: 

Conklin, Melissa. It Makes Sense!: Using Ten-frames to Build Number Sense. Sausalito, CA: Math Solutions, 2010. Print.

McCoy, Ann C., Joann Barnett, and Emily Combs. High-yield Routines for Grades K-8. N.p.: n.p., n.d. Print.

Shumway, Jessics F. Number Sense Routines: Building Numerical Literacy Every Day in Grades K-3. Portland, Me.: Stenhouse, 2011. Print.

The Standards for Mathematical Practice (K-1): Part 2

Recently I have did professional development for teachers in our district that included a segment on the SMPs. The purpose for discussing the SMPs was for teachers to have conversations with grade-alike colleagues around the Mathematical Practice and think about the connections and application to their specific grade level  content standards. Teachers were divided into 8 groups, one group per mathematical practice, and asked to read and discuss the Mathematical Practice. Then, create a poster that describes and shows the SMP connected to content standards at the various grades levels represented (teachers were encouraged to use words, pictures, illustrations, etc). This would allow a snapshot of a possible continuum of how that mathematical practice grows and develops from grade level to grade level. After groups finished their posters, we did a gallery walk and pairs of teachers walked around to look at, discuss, and ask questions around the other SMPs,

Standards for Mathematical Practice: Kindergarten and 1st Grade

The Kindergarten and 1st grade teachers were able to discuss the SMPs and did a great job illustrating the Mathematical Practices. Here are their posters and some descriptors of what each SMP could look like at this level. 

SMP #1 Make sense of problems and persevere in solving them.

Grade K & 1: SMP#1

• make predictions about what the answer and determine a possible strategy that might help them solve the problem.

• choose a method or strategy to solve to a problem. 

• explain mathematical problems in their own words. 

• explain how their picture, model, or equation represents the problem. 

• draw on classroom experience with variety of concrete objects, models, and pictures so that he or she can adjust his or her strategy, as needed, when solving a problem.

• check their answer using a different method or strategy and ask themselves, "Does this make sense?" 

• listen to classmates to understand the approaches of others. 

SMP#2 Reason abstractly and quantitatively.

Grade K & 1: SMP#2

• make sense of quantities by repetitive experiences with a variety of objects, models, and situations. 

• decontextualize a given situation by representing it symbolically, including acting out a situation, modeling it, drawing a picture, using manipulatives, or other representations. 

• contextualize by referencing the real-world mathematical situation to support, plan, and adjust the strategy and solution.  

• think quantitatively by making meaning of numbers, referencing units, and flexibly using different models and representations. 

SMP#3 Construct viable arguments and critique the reasoning of others.

Grade K & 1: SMP#3

• understand and use previous learning when discussing mathematics. 

• explain his or her thinking, and justify his or her answer. 

• communicated explanations and justifications to others. 

• respond to the arguments of others.

• construct arguments using concrete models such as objects, drawings, diagrams, and actions.

• make hypothesis and conjectures that build on a logical progression of statements from prior learning and experiences while connected it to current learning.  

• analyze situations and can recognize and use counterexamples that demonstrate when a mathematical idea is incorrect or a general statement cannot be made.  

• reason about data through making plausible arguments that take into account the context from which the data arose. 

• compare two ideas or arguments and determine when the thinking is correct or when the  reasoning is flawed. 

• explain the flaw in the mathematically thinking or procedure. 

• listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

SMP#4 Model with mathematics.

Grade K & 1: SMP#4

• apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. 

• write simple addition or subtraction equations to describe a situation.

• identify important information required to solve the mathematical real-world problem and use tools as diagrams, tables, graphs, number lines and formulas. 

• interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

SMP#5 Use appropriate tools strategically.

Grade K & 1: SMP#5

• consider the available tools when solving a mathematical problem. These tools might include pencil and paper, base-ten blocks, linker cubes, other objects, ten-frames, counters, a hundreds charts, a number line, a ruler, etc. 

• decide when best to use a tool,  when a tool might be helpful, and recognize when a tool might not be helpful. 

SMP#6 Attend to precision.

• communicate precisely to others by using academic and mathematic vocabulary.

• explain their thinking, their mathematical expressions/sentence and their answers to classmates.

• specify units when measuring and refer to the corresponding quantities in a problem. 

• accurately and efficiently calculate addition and subtraction problems. 

SMP#7 Look for and make use of structure.

• look closely to discern patterns or structures. 

• notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. 

• look at the patterns of making ten, the basic structure of our base-ten system.  

SMP#8 Look for and express regularity in repeated reasoning.

• make connections between various counting experiences to develop a sense of numbers and their sequence, including noticing patterns within a counting sequence.

• recognize what happens to numbers when you add or subtract ten to a number, and developing an understanding of what happens when you add or subtract groups of ten to a number.

• look for and notice patterns when combining numbers.  

• evaluate the reasonableness of their results when solving problems.

Of course, these examples are not an exhaustive list of what each SMP could look like at the Kindergarten and 1st grade levels. 

Standards of Mathematical Practice Resources:

SMP Examples for Elementary Math
SMP Posters for Elementary
Standards for Mathematical Practices Progression through Grade Levels (K-12)