Continuing Thoughts on Math Facts

Some of the biggest concerns I get from teachers revolve around math facts.  Oftentimes, when they hear my response, they think that I am saying that math fact fluency is not important.  That is not true.  I do believe math fact fluency is important; I just don't believe that most of our class time should be spent on rote math fact practice.

This is a topic of much discussion among math educators, and books continue to be written addressing the ways to build true math fluency.  Graham Fletcher and Tracy Zager are piloting a math fluency kit that should be available soon, and I can't wait to see it in action!

I have been recently reading No More Math Fact Frenzy by Davenport, Henry, Clements, and Sarama.    It reinforces the ideas that I continue to communicate to teachers.  Rote memorization is not a method for students to best learn their math facts.  It does not lead to a true understanding or flexibility of number that defines fluency.

Some ideas to consider:

Count.  Choral counting is a great way to build number fluency.  Counting forwards, backwards, by different multiples....This not only allows students to think about the strings of numbers, but it helps to build a mental number line for students which is so helpful for future success with mathematics.  Learn more about choral counting here.

Make your fact practice visual. Create structure that students are familiar with, and have them see the fact rather than just memorize the fact.  For example:

Seeing 7 x 6 this way helps students to visualize how knowing 7 x 5 can help them get the answer for 7 x 6.  Using subitizing is still important as students get older!

Rekenrek and ten frame visuals are also great for addition and subtraction facts.  There is nothing wrong with students using fingers for a while either.  They are a built-in tool. We hope that they eventually gain the confidence to know the sums and differences without their fingers, but if using their fingers helps it to make sense, let them!

Use number talks.  Asking students to use dot cards to make sense of math facts is important because it allows them to decompose numbers in different ways and to hear different people's strategies.  The above image could be an example of a dot card number strategy for older students, but more simple dot cards for primary will also allow for students to see the fact.  Traditional number talks with numbers written horizontally also build fact fluency through exposure to multiple strategies.  For example:

When students discuss the way that they solved a problem like this, they gain a better understanding of number.  Maybe one student added 7 + 7 and then added one more.  Another decomposed 7 into 2 and 5 so they could make 10 + 5.  Maybe another started at 7 and counted up to 8.  Number talks give students opportunities to make sense of problems in ways that make sense to them, but they also give you the opportunity to make connections between the strategy they used and those that their classmates used.  The goal is for students to not only think more flexibly but also to look for a more efficient method.

Make connections:  Help students to see the connection between operations.  How does addition help us to do subtraction?  How is multiplication related to addition?  How are subtraction and division related?  Not only asking these questions, but having students explore with manipulatives and discover these relationships will help students to have a better conceptual understanding of the relationship between operations.

Play games:  Games that practice math facts are always good not only because students get the opportunity to use their facts, but also because they get to practice important social skills like taking turns, good sportsmanship, and taking care of materials.  The best games include students having to use strategy besides knowledge of facts.  Encourage parents to play games at home with their children.  This is one of the best ways for parents to help their children become better mathematical thinkers.

Make it real:  Find as many examples as you can in the real world to help students see how the operations are used.  Adding the chairs at one table to the chairs at the other table, finding an array in a display of student work, and talking through the math of your lunch count are all real world ways for the students to make more sense of their math.

Trying some or all of these ideas should help students to gain more flexibility and fluency with their math facts.  I believe it is better use of time than pages of math facts, repeated rote practice on a computer program, or timed tests.  Knowing your math facts fluently does free up some brain space as you work through more complicated mathematical concepts, but conceptual understanding is much more important than rapid fire!

Reminder of math fact progessions:
Kindergarten:  Fluency of +/- facts within 5
1st grade: Fluency of +/- facts within 10
2nd grade: Fluency of +/- facts within 20
3rd grade: Fluency of x and / facts within 100

Playing with Numbers

After reading the book Math Recess by Sunil Singh and Dr. Christopher Brownell, I realized the importance of giving students time to explore numbers by playing with them.  This post shares a couple of ideas for these explorations.

This book is a great read and will make you rethink your instructional practices!

Abundant Numbers:  Have students search for ABUNDANT NUMBERS.  A number is considered abundant if the sum of its divisors is greater than the number.  For example, twelve is abundant because its divisors (1,2,3,4, and 6) is greater than 12.How many can your students find?

Circular Primes:  A circular prime is one that remains prime with the relocation of the first digit to the end.    So for example, 113 is a circular prime:  113 is prime.  When I move the 1 to the end of the number, my new number is 131, which is also prime.  When I again  move the first digit to the end, I get the number 311.  It is also prime; so it means that all 3 of those numbers are CIRCULAR PRIMES.

Happy Numbers: 19 is a HAPPY NUMBER.  How do I know?  To find a happy number, square each digit and find the sum. Continue doing until you find the final number.  If it is 1, then the number is happy.  

Click here to see how I know 19 is happy!

Make it a goal to give your students some time with these ideas.  Can they find more of any type of number?  How many can your class find this week?  before winter break?  this school year?  Can they prove that the numbers they found fit the definition provided?

Let your students spend time playing and thinking about numbers.  

Prove it!

I have really been working with my Firsties groups to teach them how to "prove" their answer.  They need to prove it to themselves first through either manipulatives or a representation, and then be prepared to share their steps so that they can prove it to someone else.

Today's task was a pretty basic one, but I find that those are important for them to do as they grow their ability to explain to each other what they did and how they know.  Today I asked them to find the sum of 6 + 8.

The variety and levels of their strategies was, as always, quite interesting...

This little one counted out 6 green and 8 orange.  She needed to think about what to do with them, but she then realized she needed to put them together.  She then proceeded to count all to get to the sum of 14.

This guy counted out 6 and 8.  He organized them into a line and then counted all.  His counting was a bit easier because of the line and the differentiation between the 6 and 8, but he still counted all of the cubes to get 14.

A similar approach with the counters...this little guy organized his 6 yellow and his 8 red.  He began with the 6 at the top and then counted on the next 8 to get 14.

Here was one who grouped his 8 and 6. He then started with the 8 and counted on the next 6 to get 14.  He decided to build the plus sign while he was waiting for the others.

There was some organization in this guy's work.  He grouped his 8 and 6 and then counted on from 8 to get his 14.  While he was proving it to the group, one of the others,  who had used a much more mottled strategy, blurted out, "It's a ten frame!" We took the time to explore how it was like a ten frame, but not quite, so we moved two yellow to the end of the black blocks to make our ten.  They immediately saw the 10 +4 and called out, "14!"

Now it was my turn.  I started with a stack of 6 and a stack of 8.  This time, however, we broke two off of the 8 to identify that each of the stacks was now worth 6, and the students called out, "6 + 6 =12." They then knew they needed to add the two back to get the 14.

This little activity doesn't take us long, but it shows me so much about the students' thinking and levels.  They have all been taught in the classroom to "start big and add on."  However, when working independently, they don't all do that.  When we look at each other's strategies, we identify how it works for each person.  We identify what looks the same and what looks different.  I model and discuss the way to organize our blocks so that they are easier to total.  In the end, however, we have a discussion about the fact that there are many ways to get to the same place.  We all need to use what makes sense to us.  

I was glad to see when I was in one of these student's classrooms today leading a 3 act task that one of the students that I see regularly was organized in her representation of the problem.   We were able to share her representation with the class and discuss how the organization that she used helped to make finding the sum a little easier.  

They are 6 and working to make sense of it all.  😃

EQUALITY is SO IMPORTANT

This online tool is a great option if you don't have access to a concrete tool in your classroom

Research has shown that a strong understanding of the equal sign in elementary school has a positive effect on student math success as they progress through middle and high school.  The standard sits in first grade (1.OA.7), but it is up to all grade levels to provide practice with the importance of the equal sign so that students can see the equal sign not as operational but as relational.

This is an important concept that I find myself working to build in our 1st and 2nd grade students quite often.

We typically begin by using the scale to think about missing addend problems.  The fulcrum of the scale represents the equals sign in the number problem. Because this concept is still difficult for most of them, we use cubes to look at the relationship of the two sides.  We always tell a story to go along with the problem. What do I need to add to 3 to make it be equal to 9?  I also draw it so that they see a representational model.

When we add the 6 weight, the students confirm that their answer was correct. 

After they become pretty proficient at solving the problems within 10, we stretch to equalities between different sums.

We again use cubes to help us make sense of the numbers and write number sentences to match our work.  We explore these problems always with context, and we make sure to write our number sentences with the missing addend in a variety of places. When the missing addend is in the spot directly after the equal sign is when the students tend to struggle the most. (4 + 5 = ___ + 1)

Giving students much time to explore and think about the numbers with the scale and other manipulatives is important.  It will also help them to better understand addition and the facts that go with it.

Here is an Equality dice game you can play, too.

The Math Coach's Corner has a good game for practicing equality with two-digit numbers on this post.

Illustrative Mathematics has some other good tasks that you can do with your students to help build their understanding of the equal sign.

What do you do to help your students truly understand the equal sign?

Student Accountability for Fluency

A discussion we have all of the time--with each other, with parents, with students--is about learning multiplication facts. "Why don't they know them like we did?" 

It is important for them to understand multiplication and how it works.  It is important for them to have efficient strategies to help them determine the product of two numbers. It does make future, more complicated math less daunting. We do want them to learn those facts--it is a 3rd grade standard: 3.OA.7:  

Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.

However, many of our students do not know them. They do not remember them from year to year. They continue to struggle with recall of these facts throughout their schooling.

We need to remember that fluency is not just measured by automaticity.  Fluency also includes efficiency, accuracy, and probably most importantly, flexibility.  For the purpose of this post, I am mainly focusing on the element of automaticity.

As many of you have heard me say, I do think some of this is the changing of the times.  Students do not have a need to use their memory as we did when we were their age--they have so many tools that will remember things for them.  When I was a child, I knew all of my friends' phone numbers.  People today have no need to memorize their friends' phone numbers.  I can't tell you very many of my friends' cell phone numbers today, but I still remember Gena Corbin's phone number from 1978.  Our students don't memorize things very often anymore.  This is not a skill that is well-developed in them, as it was for us.  And--it makes sense. Why memorize things when you can access them in the click of a button?

I think this question also leads into the next reason I believe they don't learn them as we wish they should.  They know they don't need to--they'll be able to access them easily as they get older.  In their minds, learning these facts has little value.

While we can't change these factors, we can help our students to retain more of these facts.  The best way is through use.  Games and other activities that ask them to know their facts will help them to remember many of them.  Daily usage of facts and discussion/teaching of strategies/Number talks in the classroom will help many of them to make sense of the facts and the patterns found within them.

I had a middle-school teaching friend who shared this with me.  I think it puts the accountability of the facts onto the students.  First, give each student a multiplication chart.

As they practice facts and find that they know some from memory, they can begin to cross them out.

This will allow them to only see the ones they don't remember from memory, and they won't continue to look back at the ones for which they have demonstrated mastery. It would be a good thing for the class to discuss how the facts that you know can help you to learn facts you are still struggling with.  This could be the subject of multiple number talks.

They should be able to better self-assess their knowledge of these facts and will be able to see their growth as their chart becomes darker and darker.

3.OA.7 is a standard that will help our students in the long run.  We want them to be successful in them.  However, too many of us are putting too much pressure on ourselves to get our students to learn them.  Maybe, using this method will put the focus onto our students for mastery.  

We make it our goal for them to "master" these facts by the end of their year with us, but in recent trainings, 4th, 5th, and 6th grade teachers all said that students come to them not knowing these facts, so they had to work on teaching them all over again.  Each year, teachers send students on thinking that they have mastered these facts...

Only a handful of our students are mastering these facts. Because really--students can access this information in an instant. They don't understand why we think it is important.  It is our job to offer students many opportunities to use them, to practice them, to self-assess their knowledge and understanding of them, and to have strategies in place that they can use when they don't remember the product quickly.  The stress we are putting on ourselves now to help students gain mastery isn't working (for them or for us).  

We need to put the ball in their court.  We need to show them how valuable knowing these facts is as the math becomes more difficult.  We need to make them the owners of this learning. 

Distributive Property Interactive Tool = Visual Learning

This interactive tool makes a great visual representation of the distributive property/partial products for our students!  It is from the Math Learning Center, and it will help you to visually show how the distributive property works.  You put in the array size in the top boxes (max of 30 x 30), and then you can decompose either side by sliding the little widgets on the side.

Valentines Fact Game

I saw something like this online, and I decided to create a similar one.  It is a Powerpoint game that you can use to help your students practice their math facts. I have created a version for students to play independently and another one for them to play as teams.

Download the Powerpoint and the Bingo Cards.  The Powerpoint is set to advance to the next slide after 4 seconds.  The team version uses the same Bingo Cards but the slides are in a different order and there are slides inserted to tell the students when to change players.

I think it will be a fun way for them to practice their facts.  Students are expected to be fluent (efficient, flexible, accurate, automatic) in addition and subtraction facts up to 10 by the end of 1st grade. 3rd grade students should be fluent in multiplication and division facts within 100 by the end of the year.

VALENTINE BINGO CARDS Addition 1-10

VALENTINE BINGO POWERPOINT  Addition 1-10

VALENTINE TEAM BINGO POWERPOINT  Addition 1-10

VALENTINE BINGO CARDS  Subtraction 1-10

VALENTINE BINGO POWERPOINT Subtraction 1-10

VALENTINE TEAM BINGO POWERPOINT Subtraction 1-10

VALENTINE BINGO CARDS Multiplication within 100

VALENTINE MULTIPLICATION BINGO POWERPOINT  Multiplication within 100

VALENTINE TEAM MULTIPLICATION BINGO POWERPOINT Multiplication within 100

VALENTINE BINGO CARDS   Division within 100

VALENTINE BINGO POWERPOINT  Division within 100

VALENTINE TEAM BINGO POWERPOINT Division within 100

Please let me know if you have any problems accessing or using these Powerpoints.

Only 8 Fridays 'til Christmas!

And only 7 of those are we in school!

I created these Christmas tree puzzles for practice on math fact fluency.  They could be used in lots of ways in the classroom.

Download the ones you want.  The directions are included.

Addition Facts Under 10

Addition Facts 11-20

Subtraction Facts within 10

Subtractions Facts from 11-20

Multiplication Facts

Division Facts

Merry Christmas!  :)

Have you checked out Greg Tang's website?

You might know Greg Tang from his great math books like The Best of Times,  The Grapes of Math, or Math Potatoes.   Many of his books are included in our Math Reads sets that we received last year.  His books are great mentor texts for reading, writing, or math workshop!  He includes strategies to help the reader make sense of the numbers.

Whether you are familiar with his books or not, his website is a great resource for all grades!
On this site, he has activities that correlate with his books. (One of these could make a great minilesson.) He has a word problem generator which allows you to differentiate your problems depending on student needs.  In addition to those great features, the site also has some pretty awesome games.  These games are leveled and many can be used from K-5.  The games allow for practice on subitizing, coins, integers on a number line, and more.

I think this site deserves a look-see.  Let me know what you think!

Make Ten Resources

Making ten is  an important addition strategy for primary grades.  Here are some resources to help your students comprehend the concept as well as develop some automaticity of facts.  When working on facts, always try to have your students first involved concretely, then pictorially, then numerically/abstractly.  Fact fluency involves good understanding of the concept--not just rote memory. I have labeled each activity with a C, P,or an A.  

Most of these would make great math stations! (Many of the examples have links to PDFs, SMART Notebooks or other things you might need)

This picture shows a life-size idea for ten frames.  You might also consider having your students stand in the spaces of the ten frame for a more kinesthetic experience.  Dr. Nicki recommends using a shower curtain and duct tape.  I have found that plastic tablecloths hold up pretty well to kiddos standing/walking on them, too.   C

Number Bowling:  This game comes from What the Teacher Wants.  Students bowl the tennis ball and then figure out the combination that made 10.  I think you could play it with an addition or subtraction number sheet. Here is a recording sheet to help with subitizing and recording their game. C

Make Ten Bracelets:  This idea comes from Kindergarten Doodles.  You could use pipe cleaners as bracelets or beaded number lines of ten... C

Ten Frames Puzzles:  These ten frame puzzles are from FirstGradeTeacherLady.  P

Visualizing/Modeling Template:  This template has kids show the fact in a variety of ways.  Maybe you make a book of them for Make Ten facts...P

Make Ten games:  This doc contains three different "games" for making ten.  Each game requires a 10 sided die.  They can be played as an individual or with a partner.  A (but could be Concrete if you provided manipulatives)

Number Bonds Game: I have shared the number bonds link before, but this one focuses on making ten.  A great online challenge.  A

This video uses pictures, numbers, and music to teach.  P  

SMARTBoard Games: This doc has two different pages that you could use as a whole class to practice Make Ten Facts.  A

Once you think that a fact strategy has been conceptualized, try it out with story problems. This page is a start to some Make Ten story problems for your children to solve.  Consider including manipulatives and encourage drawings or number lines to show their thinking! P/A

In the end, our goal is to help kids see the relationship between a number and ten.  If you are interested in using subitizing in your classroom in a more systematic way, consider registering for these 3 free videos for some great professional development.  You can do them at a time that works for you. They are presented by Christina Tondevold, the Recovering Traditionalist.

If you have a colleague you think would like this post, please share. Do you have other ways you work to teach Make Ten facts?  Share in the comments.  :)

Building Fact Fluency

Fact fluency is usually an important piece in math success.  It is recommended that you have 5-10 minutes per day involved with individual math fact practice.  I know this is hard to make work some days.  If you can't get that math fact practice in daily, maybe you can just try to get practice in more often?

I have created flashcards to help you with this. Children should use flash cards once they have the understanding of the strategy and can represent it concretely and pictorially.  Flashcards can help to lead to automaticity. These are self-checking flashcards.  Print them out and then run them back to back.  Use a hole punch to make holes where the three answer choices are.

When the students go to use them, they choose their answer and stick their pencil through the hole that they think is the answer.  They look at the back of the card, and if they are correct, the hole will have a ring around it.

These might work well at a center, or you might have students keep a set in their desk to practice when they have finished other work.  You decide how they might work best for your classroom.

There are different versions so that you are able to differentiate as necessary.  Download whichever ones you think might help you.

Addition/Subtraction Facts Under Five

Addition/Subtraction Facts--Doubles

Addition Facts--Doubles + 1

Addition/Subtraction Facts--Make Ten

Addition/Subtraction Facts--Under 20

Addition/Subtraction Facts--Extension

Fact Fluency

Fact Fluency is a big part of the CCSS.  However, we sometimes get caught up in thinking that fact fluency is just automaticity of facts.  Automaticity of facts is one piece of fact fluency, but there are other parts that are important, too!

Fact fluency includes accuracy--Does she get the right answer consistently? Can she prove to you how she knows it is the accurate answer?

Fact fluency includes efficiency--What strategies does the student use to help her know or remember the fact?  e.g. Does she recognize doubles + 1 facts?  Does she understand how place value influences 60 x 70?  Can she explain it?

Fact fluency includes flexibility--Can she use fact families to help her solve an unknown fact?  Can she relate one fact to another easily?  Can she use a variety of strategies to explain the problem? when solving facts? Does she understand how the commutative, distributive, and associative property work Can she move easily from one operation to the next?  Does she know her facts easily in written or mental form?

In reading, we know that it is important not only to decode, but also to comprehend.  There are many pieces to being a good reader, and decoding and comprehension are only two.  The same is true for math.  In order to strengthen our students' fact fluency, we should find ways in our classroom to encourage all of these elements of fact fluency.  Not only will it increase our students' fluency, but it will also lead them to better number sense.

Dr. Nicki recommends that we have a 5-10 minute block daily in our classrooms where students work on improving their fact fluency.  I know that our days are full and finding 5-10 minutes is not always feasible, so I suggest we just be more cognizant of trying to fit facts in more often--and not just instant recall--think of other ways to present your facts to kids.  I will try to share more ideas and resources for this as the year progresses.

To help boost your students' fact fluency, I have created two math path puzzle SMART documents. One covers addition and subtraction, and the other multiplication and division. Each screen is a different puzzle with each group of puzzles becoming progressively more difficult.

There are many ways you can use these in your room.  You could print out a page or two and use them as a station during workshop.  You could use a page every now and then as a warmup activity for the whole class. You could use them as part of a Number Talk.  The choice is up to you!

Parts of these files could be used in every grade K-5.  Click on the links to download the Notebook file or files to your computer.

Addition and Subtraction MathPath Puzzles

Multipication and Division MathPath Puzzles

TIP:  When I use pages from a file like this, I put a blank page in the doc as a bookmark.  That way, when I come to grab another puzzle or two, I remember which ones I have used.