# Expanded Form vs Expanded Notation

Among all the tsunami-force changes that took place as we moved to our new elementary TEKS (Texas Essential Knowledge and Skills) in 2012, one rather subtle shift might have been overlooked. I’m talking about the change from expanded form toย expanded notation in grades 3-5. Also gone is writing numbers in word form past 2nd grade. Which makes sense, really. When was the last time you wrote a number in the millions or billions in word form?

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So what is the difference between standard form and standard notation? Take a look at these examples:

Expanded form: 234 = 200 + 30 + 4

Expanded notation: 234 = (2 x 100) + (3 x 10) + (4 x 1)

I think it’s easy to see that expanded notation is definitely a mathematical step up from expanded form. Remember that our new standards, whether the TEKS or CCSS, are all about understanding the relationships in numbers, and expanded notation emphasizes the place value understandings that we want our students to develop.

One thing that is tricky about teaching expanded notation in 3rd grade is thatย students are only beginning to develop an understanding of multiplication. Teaching expanded notation isย still very doable, as long as the learning is concrete and there is an awareness that you are teachingย multiplication as well as place value. While the standard in 3rd grade is for numbers up to 100,000, you’ll want to introduce and practice the skill with 3- or 4-digit numbers, so you can use manipulatives, such as base-10 blocks to provide support for the learning. Another great manipulative that will allow you to extend the concept up to larger numbers, while still using hands-on materials, is place value disks. But the manipulatives don’t stop in 3rd grade. Because expanded notation is a relatively new concept for both 4th and 5th grade, and it is incredibly abstract, hands-on materials are still essential for understanding even in the upper grades.

So how can we introduce the concept of expanded notation? As with any other math skill, it’s best if the students can discover it on their own. To me, it would look and sound something like this:

Teacher: Work with your partner to build the numberย 234 with your base-10 blocks

ย **Note: when working with base-10 blocks, I like to use relatively small digits, so it doesn’t take them forever to build the numbers

Teacher: Who can describe what you built?

Andrew: 2 hundreds, 3 tens, and 4 ones

Teacher: Excellent! How could we write that in expanded form? (2nd-grade skill)

Paula: ย ย ย ย  200 + 30 + 4 (teacher writes it on the board)

Teacher: Perfect. Now, look at your materials. Does anybody see multiplication represented by your blocks? Remember, multiplication is equal groups. Talk to your partner. (gives students time to talk and then, based on a conversation she overheard, she calls on Jayden)

Jayden: ย ย  We see 2 times 100 and 3 times 10 and 4 times 1.

Teacher: Can you explain that to the other mathematicians?

Jayden: ย ย  (holds up the two flats) Well,ย there’s two of these and each is 100, so that’s 2 times 100. It’s the same with the tens and the ones.

Teacher: (to the class) Hmmm, what do you guys think about that? Who can restate what Jayden just said?

Haley: ย ย ย ย  Jayden said it’s 2 times 100 because there are two flats and each is worth 100.

Teacher: (to the class) Give me a thumbs up if you agree with Jayden and understand what he said. Hmmm, so I wonder if we could write the value of these materials using multiplication? Jayden said that this (holding up the two flats) is 2 times 100. Work with your partner to write that as a multiplication expression.

ย **Note: some kiddos might write 2 x 100 = 200. That’s okay for now.

Teacher: Okay, I see lots of you wrote 2 x 100. (writes it on the board) Now, can you write expressions for the tens and ones? (Gives students a minute or two) Great! I see 3 x 10 and 4 x 1. Now, mathematicians, I’m going to show you a special way that mathematicians describe numbers. It’s calledย expanded notation. (write it on the board). Hmmm, does that remind you of anything?

Angie: ย ย ย ย  Yes! Expanded form!

Teacher: Right! And expanded form shows us the value of each digit, right? (Points to the expanded form for 234). Alright, now I’m going to write 234 inย expanded notation. I want you to talk to your partner and see if you can figure out whatย expanded notationย is and how it’s different fromย expanded form. (Teacher writes ย 234 = (2 x 100) + (3 x 10) + (4 x 1) on the board and gives the students time to talk) So? Who thinks they can explain whatย expanded notationย is?

Jamie: ย ย ย ย  It’s writing the value of each digit using multiplication. It’s kind of like expanded formย because 2 x 100 is 200 and that’s what we wrote inย expanded form.

Teacher: Great explanation! Thumbs up if you agree and understand.

What comes next? Practice! Lots of concrete practice. Put up your right hand and pledge that you will not skip to the abstract too fast. Manipulatives are not something we need to wean the students off of as quickly as possible. Manipulatives provide a means for deep, conceptual understanding. Will it take longer to work problems using manipulatives? Absolutely, but the understanding is cemented when the learning is concrete.

Once your students have a good grasp of the meaning of expanded notation, here’s a little workstation activity you can use. Within the file, there are three sets of cards–6-, 9-, and 12-digit. numbers. One idea is to copy each set on a different color card stock to easily differentiate. You could also mix and match the cards for greater variety and create a deck with 6-, 9-, and 12-digit numbers. Looking to extend the activity? Have kiddos make their own card decks on index cards.

Grab your freebie here. I hope you’ll leave a comment and let me know how you used this activity in your classroom, or tweet a picture and tag me (@MathCoachCorner).

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1. Becky says:

Thanks for this lesson and the freebie. I start back next week and this will be perfect for my 4th and 5th graders!

1. Donna Boucher says:

Awesome! Always nice to start back with something new in your toolbox!

1. DeeDee says:

I think page 16, second card, has a mistake. Could you check it out?

1. Donna Boucher says:

Thanks for catching that, DeeDee! It’s fixed now. ๐

2. Tracy says:

This is a great explanation of how to cement student learning in a concept and a wonderful lesson to guide teachers. Thank you!

1. Donna Boucher says:

My pleasure, Tracy!

3. Amanda says:

Would love an example like this using the place value disk..

4. Stephanie Sheffield says:

Thanks for the matching cards Donna. This is just what we were looking for, for the expanded notation TEKS. (Now that we know how TEA is going to test it.) Do you have any ideas for TEKS 3.2B, beyond just using a place value chart and having kids write in arrows to show that as you move from right to left each place increases x10 and left to right is 1/10? I’ve used base 10 blocks but they are limited as they only go to thousands, and this is such an abstract concept for 3rd graders, especially this early in the year.

Thanks,
Stephanie

1. Donna Boucher says:

Stephanie, that’s a concept I working with my 4th graders on right now, and it’s a toughie! Although base 10 blocks only go to thousands, if they get the concept with smaller numbers, it should translate to larger ones. I think what’s hard in third grade is that the kids haven’t developed fraction concepts yet and are just really learning multiplication. That TEK might be better revisited later in the year.

5. DeeDee says:

I love this! We a easy transition to multiplication with multiples of ten as well. We used your post as discussion at our math team meeting this week. We move into decimals in another week and will use the expanded notation with them as well. Thanks for the amazing insight!

6. Jan says:

Donna, Can you explain the difference between expanded form and decomposing the number? If 200 + 30 + 4 is the expanded form–is 100 + 100 + 30 + 4 also the expanded form?

1. Donna Boucher says:

Great question, Jan! While numbers can, and should, be decomposed in many ways, I think expanded form shows the actual value of each digit. So the value of the 2 in 234 is 200. Does that make sense?

7. This was so helpful and came for me at the perfect time in my teaching sequence. Thanks for breaking it down for me so that I can break it down better for my students ๐

8. Anna says:

I teach 3rd grade math. I have to say that last year I didn’t even know which was which and that they were called different things. I also never had to represent a number like that my whole career in school. That being said, I don’t disagree with teaching them a deep understanding but, like you said, they can barely do multiplication starting at the beginning of the third grade year. I think tea should have thought this through before placing in the teks. I also have taught 1st grade math and I found it incredibly difficult to teach expanded form when they were not use to dealing with an addition symbol. I think common core is better thought out than the teks. On another note, if you have any suggestions for teaching parallel lines and right angles/ non-right angles without first teaching basic geometry (point, line, ray, line segment, angle, etc.) I could really use those too!

1. Donna Boucher says:

That’s why it definitely needs to be taught at a concrete level, Anna. Another option is to teach place value concepts at the beginning of the year, but bring in the expanded notation piece a little later, when students are studying multiplication. It’s a nice connection!

As for your other question, I’m a big fan of teaching lines and angles kinesthetically. That is using arms and/or fingers to show parallel lines, angles, etc. Using drinking straws is another great option. The ones that bend are great for angles!

1. Anna says:

I agree, it would be best to try it later but according to our scope and sequence they will be assessed on it next week in addition to the tek Stephanie mentioned about understanding the 10 x’s bigger thing. Basically they will be assessed for mastery of all place value concepts next week, and I am feeling the stress. Every lesson has to be laser focused but we still won’t make it to all. I love your blog and activities btw, and thank you for the reply!

9. Jennifer O. says:

Thanks for helping me understand the difference between expanded form and notation. The CCSS only call out standard form specifically for 4th grade, so I am not sure we need to teach expanded notation. Is it your opinion that we should teach it anyway? If so, how do you help students understand the difference?

1. Jennifer O. says:

I want to add that I realize that this lesson is designed to help students learn about both expanded form and notation. However, my question was more about how to have students remember the difference. Sometimes students get confused when mathematical terms are similar.

1. Rose hamdan says:

E X P anded notation

E each digit
X times
p place value.

Written like the above on my word wall and directly below the explanation of each and a visual of expanding the word

10. Sandra Consilio says:

I tweaked some of your questioning to help my 1st grade teachers with expanded form. Made them some expanded form mats and told them they can’t take them away from the kids because the standard in 1st grade says represent with objects and pictures. Why would they take it away then? Thanks for this article to help me provide them with appropriate questioning that builds a foundation for upper grades.

11. Brian says:

Hi Donna,
Thanks for another great post.
In the CCSS standard 5.NBT.3a states “A. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 ร 100 + 4 ร 10 + 7 ร 1 + 3 ร (1/10) + 9 ร (1/100) + 2 ร (1/1000).”
To me this implies that expanded form is written as above.
However, in the glossary they explain expanded form as “643 = 600 + 40 + 3”

Is this example in 5.NBT.3a a mistake by the writers of the CCSS or should they have used “expanded notation”?

1. Donna Boucher says:

Not a mistake, Brian, just an inconsistency I think. In doing further research after this post, it seems that there are not hard and fast definitions of expanded form and expanded notation. Expanded form typically involves adding the values, while expanded notation usually involves multiplying by the place value. But again, different sources say different things.

1. Christine Rowlan says:

Did you ever find more information on this? I am now doing the same research. Thanks!

2. TAMMY M says:

I have used something similar to this activity. I call it find my friends. I give each student a card and they have to find their 2 other friends that have a card that matches theirs in another form.

12. Britt says:

How Awesome! I just learned something new! Thanks