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5 Ways to Represent Numbers that Our Students Must Know

An important part of having a good number sense foundation is knowing that numbers can be represented in a variety of ways.  But it is also important for students to know that each of those different representations means the same thing.  As students learn to see numbers in different ways, it helps them to expand their understanding of the number and what it means.  It's easy for students to see what a number represents when dealing with small numbers like 5, 13 or even 37.  But it's a lot harder for a student to count out a set of 1,359...and impossible when it comes to numbers like 3.42x1018.   So, instead of relying on the ability to count and see what the number represents, we must give our students other tools to use.  Here are five ways to represent numbers that our students must know.

number representation and number sense

1.  Standard Form

This one might be obvious, but our students need to know how to write and read numbers in standard form.  Standard form is writing numbers using digits, and in math we most often write numbers in standard form.  Here are some examples of standard form: 138 and 3,297.  As we teach our students how to write numbers in standard form, we should also teach them how to read numbers.

Teaching Tip: Did you know that it is incorrect to say "and" when you come to a comma?  Many people would read this number (3,297) as three thousand and two hundred ninety-seven.  But this is not correct, and if we teach this to our students, we are actually setting them up for misunderstandings in the future.  You see, the word "and" should only be used when reading a number with a decimal.  It should sound like this (3,297.5) three thousand, two hundred ninety-seven and five tenths.  

Standard form might be basic, but it's important as this is how most numbers our students interact with will be written.  They should be able to properly write and read numbers in standard form.  A fun way to practice this is with number cards.  Give each student a set of number cards and include a comma if needed.  Then say a number and have the students build it with the card.  After everyone is done, have them say the number back to you.  

building numbers with number tiles to show standard form

2.  Word Form

The second most common way that students will see numbers in their life is word form.  It is important for students to know that 'thirty-five' and 35 mean the same thing.  It is also important for students to know how to correctly write numbers.  Don't forget the important hyphen starting at number 21!

Play Number Show and Tell.  Give each student a white board and dry erase marker.  Say a number or write the standard form of the number on the board.  Students will write the number in word form. When you say "SHOW" students will hold up their white boards so you can see them.  When you say "TELL" students will say the number.  This is a great way to practice both word form and standard form.

student shows number in standard form and word form

3.  Place Value Chart

Another important way to represent numbers is inside a place value chart.  It is important for students to develop the understanding that the digit '4' does not always mean the same thing.  In fact, where that digit is placed inside the number is very important.  A place value chart is a very helpful tool for seeing and understanding this.

For primary age students, or any students that are adding a new column to the place value chart, it is very helpful to practice saying what each digit in a number represents.  Here's an example:

place value chart for representing numbers

Students should learn to identify this number by its standard form "two hundred sixty-eight," but also by place value: "2 hundreds, 6 tens, and 8 ones."  By learning to do both, we are helping students to make connections between the different number forms.

4. Expanded Form

After students are comfortable with the place value chart and what it means, teaching students about numbers in expanded form is the logical next step.  I love introducing expanded form to my students because we get up and get a little active.  I have students stand up and say this with me: "Expanded form is when I stretch out a number using addition."  Then we add some motions.  For the words "expanded form,” students clap their hands and hold them together in front of them.  Then, as we say "is when we stretch out a number" student stretch their arms straight out sideways.  Finally, we bring our arms in front of us and cross them like a plus sign as we say "using addition."  After repeating this a couple of times I stop doing it and simply ask, "What is expanded form?"  The kids love showing me that they know the definition and their movements.  Then for the next few days, and periodically throughout the year I was ask "What is expanded form?" and they know just what to do!

kinesthetic math activity

After we learn what expanded form is, then we learn how to write it with numbers.  I take the place value chart and add some plus signs to the bottom of it.  Then I add a number and we start our normal routine of saying the number that is written in the chart—first in standard form, then according to its place value.

using place value to teach expanded form

This time, as we say the number with its place value I stop the class after each digit.  I might get a little melodramatic and say something like, "What did you say?" or "Did you say that this number has 4 hundreds?"  When they answer again, I model how we would write that at the bottom of the chart in a form.  

This is a very concrete way for students to connect place value to expanded form.  As students get older and have experience with skip counting, I also like to connect expanded form with some skip counting practice.  I might say something like "This number has 4 hundreds.  Let's count by 100 four times." 

We would write it into our expanded form and then do the same with the next digit.

5. Pictorial or Object Representation

The final way that I teach my students to represent numbers is with a picture or objects.  In kindergarten and first grade we do a lot of using real objects and counting them into sets.  But as students get older, this becomes more difficult.  One way to help with this is the use of place value blocks.  These blocks are a great way to connect their knowledge of place value to a physical representation of the number.  

counting blocks or objects to show number meaning

The problem is that you often don't get enough blocks in a set to represent numbers higher than 2 or 3 thousand.  Pictures to the rescue!  I love to teach my students to draw a picture to represent the number.  Not only does this take a lot less space and having blocks and manipulatives on hand is not necessary, but it is a great problem-solving technique too.

I teach my students how to use a small square as a 1, a long rectangle as a 10 rod, a large square as 100, and a cube as 1,000.  They love learning to draw a cube!  They will not only use these pictures to represent numbers but there is a very high rate of carry-over when solving word problems too.  

drawing pictures is a good way to represent numbers

There are more ways to represent numbers than just drawing place value blocks.  Students can see a pictorial representation of numbers through tally marks, ten frames, or even drawing a set of objects.  When students can make a pictorial representation of the number then you know they are well on their way to mastering what the number stands for.

Practice Everyday

While introducing and teaching each of these number representations takes a lesson or two, students need more than this to practice and master these number representations.  That's why I love Number of the Day as a daily math number sense activity.  In just a few minutes a day, students get lots of practice—not only with number representations, but also with a variety of key number sense skills.

Use number of the day to practice representing numbers in different ways

You can see that on one Number of the Day activity page, students are working number representations in multiple ways.  Through the course of a week, they will work on all these important number representations.  

Want to learn more about Number of the Day and how you can use it with your students?  Check out this blog post for the ins and outs of this amazing daily math activity.

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Pin this to your favorite classroom Pinterest board so that you can quickly come back and find these number representations tips and ideas to use in your classroom.

Teaching your students how to represent numbers in different ways is important to developing a good number sense foundation.  This post goes over 5 different ways to represent numbers that your students must know.

Number of the Day: A Great Way to Build Number Sense

One of my absolute favorite times of the day is when I do number talks with my students.  In my classroom, we call this Number of the Day, and it is a daily math routine that my students love too.  Number of the Day is an excellent way to build strong number sense skills and gives my students opportunities to work with numbers in a variety of ways.

Number of the Day is a great way to help students build solid number sense skills.

Number of the Day in my Classroom

  • Every. Single. Day.  We do Number of the Day. It is part of our routine and the students come to expect it and even anticipate it.  If we have a schedule change for an assembly or other out of the ordinary activity they are quick to ask about Number of the Day.  

Number of the Day is our time as a class to dive deep into what a number is, how it can be represented, what we can do with it and more.  But it is so much more than just learning about one number.  We are learning and practicing key thinking skills that can be applied to all numbers and to math. 

students working on breaking down the number 45 as part of number of the day
What started out as a whole class oral activity with a whiteboard and a few manipulatives has turned into year-long printable and digital resources that I have to put together for kindergarten, first grade, second grade, and third grade.  Yep - I love Number of the Day and its effectiveness so much that I wanted an easy way to share it with you.

Now, we still do Number of the Day together talking through each activity, but my students can either follow along with their own Number of the Day page or they later reinforce their learning with a Number of the Day center activity.  

two boys working on a number of the day worksheet during math centers

Every Number of the Day page covers the same number sense skills and concepts, but the activities vary and the rigor obviously looks different based on the grade level and complexity of the number. 

I could write more, but I've decided it would be better to just show you.

Number of the Day in Action

Each day students will work with numbers in a variety of ways.  There will be number identification and oral practice saying the number the correct way.  There will be opportunities to represent the number in many different ways.  These include standard form, word form, expanded form, place value, ten frames, tally marks and by marking or creating groups.  There are also opportunities to work with numbers in basic operations when that is grade-level appropriate.  

a variety of number of the day worksheets to show variety of activities

When all of these activities are repeated with different numbers, we help our students learn to think about numbers in many different ways.  This helps them as they learn to problem-solve and complete more complex math.

Let's take a look at what Number of the Day activities look like.   I have pulled all of the following examples from Month 1, Week 3 of the school year so that you see can the variation and building that happens by grade level.  

sample pages of kindergarten and first grade number of the day activities

sample of second and third grade number of the day pages

Number Identification

On each page, I have marked all of the number identification areas in red.   These areas show all the different places in which students must be able to identify the number in order to complete the activity.  You can see that it progresses from repeated number identification and writing in kindergarten to identifying the number in order to compare it in second and third grade. I love how one simple page can reinforce a skill multiple times. 

Number Representation

On each page, I have marked all of the number representation activities in blue.  It's the variety in these representations that really help kids grasp what a number means.  Learning to see the number as its place value, in a set, with tally marks, equations and words are all part of learning different ways to represent numbers.  It is when a number is explored in multiple ways that students gain an understanding of what the number is, and that each representation ultimately means the same thing.

Working With Numbers

I have marked in green the different places where students are working with the numbers.  These activities help students understand more about how numbers relate to each other and eventually how they relate to real life.  Students must take that understanding of the number as the basis for working with the number as they apply different mathematical processes to it.  

Do you see the progression?  Isn't it exciting?  First, you identify the symbol we call a number and learn its name.  Then you find out what that symbol (AKA number) stands for and practice showing it in many different ways.  Finally, you practice using it to relate with other numbers.  It's an amazing process, and one that can have profound impacts on our students and their future math abilities.

I really love how every page covers every skill and sometimes those thinking skills even overlap.  As you can see, some days one skill might be more focused on than others.  But on other days another skill gets the focus.  It's all part of the progression that leads to mastery.

Try it Out for Free!

How would you like to see and try out three weeks of Number of the Day for free?  You can!  Try the first and second grade freebie by clicking the yellow picture or try the third grade freebie by clicking on the blue and green picture.  Just find the correct grade level and give it a try!

three weeks of number of the day worksheets for free for first grade and second grade

three weeks of number of the day for third grade for free

Ready to Use Number of the Day in Your Classroom?

Grab a full year of Number of the Day and start using it in your classroom today.  It's never too late to add this amazing math routine to your daily schedule.  As an added bonus, the first grade, second grade and third options include both a printable and digital version!  The digital version is great to use whole group on a Smartboard or by projecting.  It is also a great math center using technology or a wonderful daily math activity through remote or distance learning.

full year bundle of kindergarten number of the day activities

full year bundle of first grade number of the day activities

full year bundle of second grade number of the day activities

full year bundle of third grade number of the day activities

Save it for Later

Pin this to your favorite classroom Pinterest board so that you can come back later.

Number of the Day activities are an excellent way for students to build strong number sense skills.  These printable worksheets and digital activities will provide skills based number sense for the entire school year.Number of the Day activities are an excellent way for students to build strong number sense skills.  These printable worksheets and digital activities will provide skills based number sense for the entire school year.

Building a Strong Math Foundation with Number Sense

The day has finally come. The foundation of your new home, your dream home, is being poured.  With excitement, you head over to the property to see the big slab of concrete that will literally support everything you have worked so hard for.   When you arrive, you get a glimpse of the shiny, still wet gray material that glistens in the sun.  Your heart skips a beat as you watch your dream start to take shape.  As you walk around you notice that the concrete is a little uneven.  Some spots are lower than others, and there appears to be a small hill right where your living room is.  You also notice that the foundation doesn't come close to filling to mold that was created for it.  You decide it's okay because what really matters is what has yet to be built.
Building a Strong Math Foundation With Number Sense Tips and Ideas

Building a Strong Math Foundation

If you were going to build a house, would you build it upon a foundation that was only 1/3 the depth of concrete that was required by code or recommended by the engineer?  What about a foundation that was uneven with obvious high spots and low spots?  I sure hope not!  Why?  Because we know how important the foundation is to the stability of the structure being built upon it.  Well, in the math world, number sense is this foundation. 

Everything we do in math rests on this foundation.  I know that this is a strong statement to make, but think about it.  Place value . . . number sense.  Algebra . . . number sense.  Even geometry . . . number sense.  That is why helping out students build a strong number sense foundation is so important.  If we look at math in the early years as just learning to count and identify shapes, then we are completely missing the importance of building a foundation.

So What is Number Sense?

Well, that can actually be a tricky question, since number sense is a broad and abstract concept.  One of the best definitions that I have found came from Janette Bobis, a math educator and researcher in Australia.  She defines number sense as "a well organized conceptual framework of number information that enables a person to understand numbers and number relationships and to solve mathematical problems not bound by traditional algorithms."  It's the ability to see numbers and their relationships in different ways, to work fluidly with numbers, and to be able to easily adapt to number representation without losing meaning. 

Building and Representing Numbers with tally marks is part of number sense

Number sense is what allows students to think about numbers in different ways and solve problems differently, while still reaching the same conclusion.  Number sense is what leads to true mastery and understanding of math.  Check out this short video to see how building a strong number sense in the primary grades can affect a student's math thinking in the future.

What Makes Up Number Sense?

In 1989, the United States National Council of Teachers of Mathematics identified five key components to this abstract notion of number sense:
  1. Number meaning
  2. Number relationships
  3. Number magnitude
  4. Operations involving numbers
  5. Referents for numbers and quantities
It is these skills, when woven together, that create a solid and strong foundation for more complex math concepts.  

Helping Our Students Develop a Solid Foundation

Now that we can better wrap our minds around this nebulous concept of number sense, let's turn to the more important topic of how to help our students get it.  You see, number sense is not something that can really be taught.  It's really something that is learned through experience. And that right there is the key to developing number sense in the classroom - experience.  We need to give our students lots and lots of opportunities to experience numbers.

While we can include some instruction in that equation, without the experience our students will develop that deep foundation that we want them to have.  This is the difference between the "old school" approach of memorizing math and today's (often misunderstood) methods.  Sure, memorization might be an easy way to remember math facts, but it doesn't give our students the underlying problem-solving skills that are developed as number sense strengthens.  

Number Sense Activities for the Classroom

Boy working on building and showing addition problems as part of building a solid number sense

I want to share with you a variety of number sense activities that you can easily add to your classroom.  

1.  Math Centers
Math centers are a great way to give students independent time to experience numbers.  As you prepare and plan your centers and math activities I would encourage you to make sure that you are including variety.  For example, don't just include number identification activities.  Even if identifying numbers to 20 is the standard you are working on, make sure to provide some variety that gets your students experiencing those numbers in different ways.

Building number sense through hands on math centers

2.  Number of the Day
This one is near and dear to my heart because I have used it and seen its effectiveness with students.  Number of the Day is a daily number routine, usually done with the whole class, that gives students with a variety of number sense activities all related to the summer number.  It can be a very powerful tool in helping students develop a strong number sense.  Find out more about how I use Number of the Day in my classroom.

Building Number sense through Number of the Day

3.  Real World Math Connections
Math by nature is abstract.  By connecting these abstract concepts to the real world, we help our students make personal connections.  When they make personal connections, they understand better, and are then able to apply that understanding to future abstract concepts.  You know those word problems that get such a bad reputation (yeah, the ones about two trains heading towards each other)?Well, ty are a great way to connect math to the real world.  Now I will be the first to admit that not all word problems are written well.  But when they are, they can really help students connect to math in a new way.

Another great way to make real-world connections with my students is by sending them out to find math in their world. These days, with technology so readily available, it can be as easy as taking a picture of something that shows 10 and email it to me.  Send them out with a real-world math challenge and I bet they will surprise you and amaze you at what they come back with.

Time to Get Started

So grab your hard hat (I couldn't pass up the opportunity for one more construction connection) and get busy building that foundation.  Our number sense foundation might not be as easy as pouring liquid rock, but over time, we can build it strong by laying it brick by brick.

Don't lose these thoughts and ideas.  Just pin this to your favorite classroom Pinterest board so you will be able to get back here whenever you need to.  And don't forget to check out more number sense ideas.

Building a strong math foundation starts with strong number sense skills.  These tips and ideas will help you incorporate number sense building activities into your classroom.Building a strong math foundation starts with strong number sense skills.  These tips and ideas will help you incorporate number sense building activities into your classroom.

Prevent the Summer Slide!

The Summer Slide - don't let the term fool you.  It's not a playful summer day at the park.  Actually, this term is used for the backward slide of academic skills during the summer.  Yep, it is true and if you have been in the classroom you have seen the effects of it.  But we can and should fight against the summer slide.

the summer slide and a summer math review

Fighting against the summer slide is not hard.  In fact, in just a few minutes a day students can keep their skills fresh and in tip-top shape.  A skills-based review that students can complete independently is the perfect answer for the summer slide.

Summer Math Review

A few years ago I put together a daily math review activity to send home with my first-grade students.  I sent it home with students at the end of the school year.  I helped parents know what to do and assured them it would be easy and just take a few minutes.  The students and parents loved it!  Now your students can too!  Using a fun summer beach theme, students will practice all of the first grade math skills.

review page for math during summer

In addition to basic math problems, this summer math review also provides students practice with word problems.

Each day students complete one page of this spiral math review.  In just a few minutes students will practice a variety of different skills.  The spiral nature of this review means that students don't get bored doing the same thing but instead stay on top of all skills by doing a little practice every day.

summer math review to prevent the summer slide

This summer review includes a total of 40 practice pages (5 a week for 8 weeks).  Now let's just be honest for a sec - I know that not all students are going to work over the summer.  That's a problem that has be tackled from the top down and probably best left for a different post.  But if we could get our students to do just half, they would be in a better place starting the next school year. 

Print or Digital?  How About Both!

This year the look of education has gotten turned on its head.  With the COVID pandemic, distance learning is the new norm.  I have worked quickly to turn this great summer review into a digital resource in order to meet the demands of our current educational state.  

math review for google slides

The Summer Math Review is now available in both print and digital options.  With the print format, teachers or parents simply print out the pages, and students complete one each day.  

summer slide math review sheet

The digital format is just like the paper format except it is completed on a device like a computer or a tablet.  Using Google Slides, students will answer questions in the provided text boxes (no formatting experience necessary).  Students can zoom in on a tablet (or even a phone) and easily see where to answer due to the colors!

summer math review for computer or tablet

The simple student instructions tell them that can type their answer inside any orange area.  Any green elements can be moved into place to answer the question.  Students LOVE the interactive nature of this digital math review.

The digital word problems using Google Jamboard are a hit with students.  They really love the opportunity to create a picture for the problem and show their work.  For more on Jamboard click here.

first grade summer math review

Don't you just love the options for showing work on the digital version!  I think it's my favorite part!

Let's Equip Our Students to Fight the Summer Slide

Grab the Summer Math Review and equip your first-grade students to fight the summer slide.  Choose from the Print Version, the Digital Version, or the Bundle that has both print and digital!  Give students and parents access to the math review (print or digital version).  Then give parents simple instructions and one page a day guideline.  I believe that if parents know upfront what to expect they are more likely to buy into the summer review.  

Not sure sending summer review will work?  I totally get that!  Why not end the year with a review of all the skills you worked on.  This 40-day review is great for the end of the school year too!  Use it as a daily math warm-up, a center activity, or morning work during the last 8-9 weeks of school.  It makes a great informal assessment and you can clearly see what concepts your students have mastered.

You can do a hybrid review too!  Complete the first 20 pages in school during the last month of the year.  Then send the last 20 pages home for students to complete over the summer.  Parents can do just a couple each week, or do them daily in the month leading up to the new school year.  

Try it First!

I know that an important review like this needs to work for you and your students.  That's why I created this FREE sample so you could try it out.  The preview packet includes three sample pages and a detailed preview of the review.

summer slide math review free

Grab the Full Summer Math Review! 

Printable Version

Digital Version

Print and Digital Version

Pin It to Save It! 

If you are not quite ready for the summer review, just pin this to one of your favorite Pinterest boards.  That way when summer arrives you will be able to get back here quickly and help your students stop the summer slide.

Summer Math Practice

It's A First Grade Math Fiesta

The end of the year can be so hard.  Student focus is at an all time low and your brain is whirling with all of the things that have to be done in order to wrap up another school year.  But school isn't over and its a great time to review the concepts from the year in order to solidify learning.  This first grade math review is just what you need.

math review for first grade

It's Fiesta Time!

I'm from Texas, and in these neck of the woods we love a good fiesta.  That's Texan {Spanish really} for a good ol' party!  Why not share some fiesta fun with your first graders.  I think math is a great reason for a party and if you offer your students the choice between a math assignment or a math party I'm pretty sure they will choose the party too!

This resource was designed to be a fun and engaging end of the year activity.  You know that time of the year when students are getting a little antsy and its harder and harder to get them to focus.  Yep - that time of the year when we need to pull out something extra special to keep them going.  Well here is that extra special - it's a First Grade Math Fiesta!  Just the excuse you need for a party!

Balloons, streamers and banners are optional - but boy would they add to the fun atmosphere!  I know I'd have more fun doing math or anything for that matter in a fiesta environment!  If you'd like to set the stage for your math party just check out this great kit with everything you would need to create a classroom Fiesta! {These links are affiliate links which means I might get a small commission from Amazon - but your price doesn't change!}

And to kick off or finish up your math fiesta - these are some super cute photo booth props that I know your kids would LOVE!

Obviously you want to take out the margarita and tequila props.  Maybe save those for the teacher's lounge.  :-) The props and decorations might not be necessary but the Math Fiesta is!

First Grade Math Fiesta

The First Grade Math Fiesta is a year end review of all first grade math standards.  This resource gives you everything you need, and more, to review the skills and give students a chance to practice them again too!  Did I mention that every.single.standard is covered in this resource!  

The First Grade Math Fiesta covers:
  • All Operations and Algebraic Thinking standards
  • All Number and Operations in Base Ten standards
  • All Measurement and Data standards
  • All Geometry standards
Each of those broad categories is a unit in this first grade math review resource.  This makes it super easy for you to find what you need when you need it.  It's also really easy to see related skills together and choose the best review activities for your students.

While designed as a year end review, this resource could easily be used all year long as a supplement to your math curriculum.  Need extra practice on a skill?  No problem!  Need some reteaching activities or intervention resources?  No problem!  The First Grade Math Fiesta is filled with lots of great teaching tools and student practice.

Review Teaching Tools

For each math standard you receive a full color teaching sheet that is perfect for projecting or viewing on the SmartBoard.  You could also use them as guides for small group instruction.  These teaching sheets give practice problems and tips for solving.  They are a great way to remind students of the math steps and strategies they have learned.  It's also a great guided practice activity to do together.  

first grade math review with teaching tools and practice pages

First Grade Math Review includes Base 10 and all standards

Student Practice 

Each standard also includes student practice pages that are great for whole class independent practice, morning work, math centers, small group instruction or even intervention.  These pages don't have all the color and graphics which make them great for printing and copying.  

End of Year Math Review for First Grade

math practice pages for first grade all standards

Easy Differentiation

In addition to having practice for every standard it makes the ability to differentiate so easy.  Many of the math standards build on one another.  So give students review and practice on the highest related skill that they can work at.  This keeps every child challenged without being frustrated.  If a student is ready and able to work on addition with 3 numbers then challenge them there.  But if a student is struggling with addition, challenge them with addition of 2 numbers.  Both students will be working on the same important math skill.

It's Fiesta Time!

Give the First Grade Math Fiesta a try.  Your students will love the change of pace and you'll love knowing that the end of your year is filled with Easy Prep, Standards Based Math.  

End of Year Review for First Grade

Standards based math review for first grade - includes teaching tools and practice for all math standardsStandards based math review for first grade - includes teaching tools and practice for all math standards

I hope you and your students enjoy ending the year with a math FIESTA!  If you aren't ready to party just yet - pin one of the images above to your favorite classroom Pinterest board.  That way when it's Party Time - you will be able to jump back over here to find everything you need!

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