## Lesson: Comparing and Ordering Fractions, Mixed Numbers, and Decimals Developing the Concept

Students should be able to move quickly from the concrete to the symbolic because they already have a good understanding of comparing and ordering whole numbers and comparing and ordering fractions and mixed numbers. Since decimals follow the rules for whole numbers, introduce the place-value chart. If you haven't already done so, spend some time helping students to see the benchmark decimals that match their benchmark fractions: 0, , and 1. (0.25) and (0.75) are also great benchmarks for decimals. Use a number line if necessary.

Materials: place-value charts (Learning Tool 36 in the Learning Tools Folder), a large copy of Learning Tool 26 (visible to all students)

Preparation: Make a sheet containing about a dozen place-value charts like the one shown below for each child, with plenty in reserve for students who need more practice.

 hundreds tens ones tenths hundreths
• Ask: Can someone draw a place-value chart on the board?
Help a volunteer draw a whole-number place-value chart with places for ones, tens, and hundreds.
• Say: A decimal place-value chart starts out just like a whole-number place-value chart, but there is a decimal point just to the right of the ones place.
Add a place for the decimal point to the chart on the board.
• Say: Look at the place-value chart.
Write 1.11 in the place-value chart.  hundreds tens ones tenths hundreths 1 1
• Ask: What must the places to the right of the decimal point represent?
Students should understand that when a number is between two whole numbers, it has a fractional part. The places to the right of the decimal point represent that part.
• Ask: Why, do you think, is the tenths place to the right of the ones place and to the left of the hundredths place? Look at the decimal grid for help.
Help students to explain that since the digit in any place in a place-value chart has a value 10 times the value of the digit one place to the right, the order must be ones, tenths, hundredths.

Concrete explanation: One tenth of the grid is one column. Ten columns fill up one whole, so ten tenths equals one.

Computational explanation: The one in the ones place has the same value as ten times the one in the tenths place: 10 x = = 1.

Concrete explanation: One hundredth of the grid is one unit. Ten units fill up one column, which is one tenth, so ten hundredths equals one tenth.

Computational explanation: The one in the tenths place has the same value as ten times the one in the hundredths place: 10 x = = .

• Ask: Who can write the number three and four tenths in our place-value chart?
Use the place-value chart on the board.  hundreds tens ones tenths hundreths 1 1 1 3 4
• Ask: Which number in our chart is greater? How do you know?
3.4 is greater. Compare the ones place first. 3 > 1, so 3.4 > 1.11.
• Ask: If I write two more numbers in the chart, can you put all the numbers in order from greatest to least?  hundreds tens ones tenths hundreths 1 1 1 3 4 2 3 1 1
• If it's easier for students to use zeros as placeholders, show them how to place them. Students should first compare the ones. 3.4 is the greatest number in the group, 2.3 is second greatest, but 1.11 and 1.1 have the same number in the ones place. Look at the tenths place. Same number! Look at the hundredths place. 1.11 > 1.10, so the order is 3.4, 2.3, 1.11, 1.1.
• Ask: What if I want to compare 2 to 2.2? How can I do that?
Discuss this until students come up with at least two ways to do it.

1) Write 2.2 as a fraction, then find equivalent fractions with like denominators.

 2 ? 2.2 2 ? 2 2 > 2

2) Find an equivalent fraction for 2 that has a denominator of 10, 100, or 1,000, then compare by place-value. Use the place-value chart if it helps.

2 = 2 = 2.25

2.25 > 2.2

• Continue decimals, and mixed numbers until students are comfortable with a variety of techniques. For strings of mixed fractions and decimals, encourage students to change the fractions to decimals, using a place-value chart or a number line.

Wrap-Up and Assessment Hints
After students have had plenty of time to practice mixed numbers, and decimals, ask them to write and illustrate an explanation for how they ordered this list from least to greatest.

1.45, 1, , 2

(The correct order is , 1.45, 1, 2.)