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 that students are only beginning to develop an understanding of multiplication. Teaching expanded notation is still very doable, as long as the learning is concrete and there is an awareness that you are teaching 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.
Teacher: Work with your partner to build the number 234 with your base-10 blocks
**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: 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 overhead, she calls on Jayden)
Jayden: We see 2 times 100 and 3 times 10 and 4 times 1.
Teacher: Can you explain that to the other mathematicians?
Jayden: (holds up the two flats) Well, 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: 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.
**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 called expanded notation. (write it on the board). Hmmm, does that remind you of anything?
Angie: 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 in expanded notation. I want you to talk to your partner and see if you can figure out what expanded notation is and how it’s different from expanded form. (Teacher writes 234 = (2 x 100) + (3 x 10) + (4 x 1) on the board and gives the students time to talk) So? Who think they can explain what expanded notation is?
Jamie: It’s writing the value of each digit using multiplication. It’s kind like expanded form because 2 x 100 is 200 and that’s what we wrote in 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.