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.

## 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.

## 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.

## 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:

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!

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.

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.

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.

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.

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.

## Save it for later

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