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Expand That Number!: Composing and Decomposing Numbers Using Standard and Expanded Form

  • Lesson
Pre-K-2,3-5
2
Number and Operations
Lisa Willman
Hillman, MN

In this lesson, students learn to compose and decompose numbers into the hundreds place as they move from standard form to expanded form and back again. Students explore composing and decomposing numbers using base ten blocks and place value cards.

Before class, copy the Place Value Playing Cards Activity Sheet and Spinner Activity Sheet on cardstock. Students will work in pairs, so make a copy for each pair of students.

pdficon Place Value Cards Activity Sheet 

pdficon Spinner Activity Sheet 

To introduce standard form, write the words "three hundred forty-five" on the board. Write just the words, not the numeral. Ask a volunteer to come to the board and write the words using digits. State that this number, 345, is written in standard form. Explain that standard form is simply the numerical form of a number. To get students to think about what this number means, have them write everything they know about the number in standard form. To focus students' thoughts, encourage them to make observations about each digit and what that digit represents in the number. Discuss their answers.

Students may make these observations about 345:

  • It has three digits.
  • It includes ones, tens, hundreds.
  • It has a 3 in the hundreds place.
  • It has a 4 in the tens place.
  • It has a 5 in the ones place.
  • It is a number written in standard form.
  • It is greater than 340 but less than 350.

Organize students into pairs. Hand out the base ten blocks. Ask groups to model the number 345 using the fewest base ten blocks possible. Remind students that a flat represents 100, a rod represents 10, and a unit represents 1. Groups should end up with 3 flats, 4 rods, and 5 units.

With the entire class, count the value of the blocks. Ask, "How many flats do you have?" [3.] "What is the total value?" [300.] Explain that 3 hundreds is worth 300 while you count the flats aloud, "100, 200, 300." Write "300" on the board. Do the same with the rods and the units, writing "+ 40" and "+ 5" on the same line as 300. That is, when you've finished counting, the following should appear on the board:

300 + 40 + 5

Tell students that what you have written on the board is the expanded form of 345. Provide a few more examples of numbers in standard form, and have students work together to create the numbers using base ten blocks before they write down the expanded form.

Group students into pairs and pass out the sets of place value cards, one set to each pair. Explain that pairs will play a game. The first student uses the place value cards to compose a number in expanded form, and shows it to the second student. The second student writes the standard form of the number on a slip of paper and shows it to the first student so he or she can confirm or reject. Have students reverse roles.

In the following period, collect the place value cards and redistribute the number cards (excluding the "+" symbols), passing out the cards as evenly as possible so that each student has only 1 or 2 cards in his or her hands. For example, if you have 28 students, each student would get 1 card. If you have fewer students, some students may get 2 cards. If you have more than 28 students, pair students so that all may participate. Have a volunteer use their paper clip, pencil, and spinner to generate a 3-digit number—for example, 6, 3, and 5. Ask students to use each of the numbers exactly once to create the greatest number possible in their notebooks.

Ask, "If we put each number in the hundreds place, what would be the value of each number?" For the example roll, students would hold up the 600, 300, and 500 cards, and walk to the front of the room. Discuss which of the numbers is the greatest. [600] Students with the 300 and 500 cards may then sit down. Do the same with the 30 and 50 cards to pick the tens. [50] That will leave 3 in the ones place.

Take the three cards (600, 50, 3) and stack them together (so the 5 overlaps the 0 in the tens place of 600, and the 3 overlaps the 0 in the ones place of 50) to show that standard form is 653. Ask, "How do we know this is the largest number?" Try to elicit that the digit with the greatest value, 6, is in the hundreds place. 

ExpandThatNumber IMAGE NumbersOverlapping 

As time allows, continue this activity with other numbers. Have students overlap the cards to solidify the composition of numbers. Because math is not a spectator sport, it would be good if each student's numbers are used at least once during this activity. To keep kids interested, you can alternatively ask for the least number that could be formed, or the largest odd number, or the least even number, or other types of numbers.

Assessment Options  

  1. As students work with the base ten blocks, observe if they are correctly identifying the value of the blocks. For example, if they have 3 tens rods, are they counting by tens to identify the value as 30?
  2. Give each student an index card and a spinner. Have each student spin a number 3 times and write all 3 digits on the top of his or her card. Have students write the greatest number they can make with those 3 digits in both standard and expanded form. Ask them to write an explanation on the index card of how they know this is the biggest number they can make with those digits.

Extensions  

  1. Extend the lesson to working with numbers in the thousands, ten thousands, and hundred thousands places. With index cards, have students create their own place value cards for the thousands, ten thousands, and hundred thousands places. Provide examples of the numbers in expanded form.
  2. Give each student a bag of the place value cards. Ask students riddles, and have them use the cards to try to figure out the number you are talking about. For example, "I'm thinking of a number that has a 4 in the ones place, and the digit in the hundreds place is twice as big as the number in the ones place. The digit in the tens place is two less than the digit in the hundreds place. What number am I thinking of?" [864] You may also wish to ask students to come up with riddles of their own.

Questions for Students  

1. What is the difference between standard form and expanded form?  

[The standard form of a number is the number written using digits. The expanded form of a number displays the value of each digit and represents the number as the sum of the parts. For example, 345 can be written as 300 + 40 + 5. Both forms are correct; they are just different ways to represent a number.] 

2. If you had the numbers 856 and 578, to which place would you look to find the greater number?

[You need to look at the hundreds place. Eight hundred (or 8) is greater than 500 (or 5), so you know that 856 is the greater number, regardless of the digits in the tens and ones place.]

3. In the numbers 435 and 487, both numbers have a four in the hundreds place. How would you figure out which one is the greater number?  

[Since both numbers have the same digit in the hundreds place, you need to look to the next place over. In this case, you look at the tens. There are 3 tens, which gives you a value of 30, and 8 tens have a value of 80. Since 80 is greater than 30, 487 is the greater number.]

4. Which number is greater, 235 or 84?

[The number 235 has a 2 in the hundreds place. Although the number 84 has a first digit of 8, it occurs in the tens place; therefore, the value of the digit in the hundreds place is 0. Since 2 is greater than 0, then 235 is greater than 84.]

Teacher Reflection  

  • Were you able to challenge all learners in your class?
  • What evidence did you observe that students were effectively using the manipulatives for the lesson objectives and actively engaging in learning?
  • What worked well with classroom management while using the manipulatives? How would you adjust for future lessons that use manipulatives?

Learning Objectives

Students will:

  • Build numbers up to the thousands place with base ten blocks.
  • Determine the value of each digit in a three-digit number.
  • Compose and decompose numbers using standard and expanded form.
  • Use place value to compare the values of numbers.

NCTM Standards and Expectations

  • Develop a sense of whole numbers and represent and use them in flexible ways, including relating, composing, and decomposing numbers.
  • Use multiple models to develop initial understandings of place value and the base-ten number system
  • Understand the place-value structure of the base-ten number system and be able to represent and compare whole numbers and decimals.
  • Recognize equivalent representations for the same number and generate them by decomposing and composing numbers.

Common Core State Standards – Mathematics

-Kindergarten, Counting & Cardinality

  • CCSS.Math.Content.K.CC.C.7
    Compare two numbers between 1 and 10 presented as written numerals.

Grade 1, Number & Operations

  • CCSS.Math.Content.1.NBT.B.3
    Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.

Grade 2, Number & Operations

  • CCSS.Math.Content.2.NBT.A.3
    Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.

Grade 2, Number & Operations

  • CCSS.Math.Content.2.NBT.A.4
    Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

Grade 4, Num & Ops Base Ten

  • CCSS.Math.Content.4.NBT.A.1
    Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division.

Grade 4, Num & Ops Base Ten

  • CCSS.Math.Content.4.NBT.A.2
    Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and <. symbols to record the results of comparisons.

Grade 5, Num & Ops Base Ten

  • CCSS.Math.Content.5.NBT.A.1
    Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.