Lesson 9

• Display one statement.
• “Give me a signal when you know whether the statement is true and can explain how you know.”
• 1 minute: quiet think time

• Share and record answers and strategy.
• Repeat with each statement.

• “Is the statement \(0.909 > 0.91\) true or false? How do you know?” (It is false because 0.909 has 9 tenths and 9 thousandths and 0.91 is 9 tenths and 1 hundredth. 1 hundredth is greater than 9 thousandths.)
• Display: 0.909 and 0.910
• “How does writing the numbers like this help to compare them?” (I can see that 0.910 has 10 thousandths compared to 9 for 0.909.)

The purpose of this activity is for students apply what they have learned about comparing decimals to find numbers that lie between two other decimal numbers. Students may draw number line diagrams, if it helps them, or they may use their understanding of place value.

In each case, there are many different decimal numbers between the two and this will be brought out in the activity synthesis. The last question in this activity is exploratory. Students may say that there is no number between 1.731 and 1.732 or they may say that it looks like there is and they cannot name it yet. The important observation is that the number line suggests that there are numbers in between but we cannot name any of those numbers yet. This question gives students an opportunity to make sense of a problem and some students may propose an answer, using fractions for example (MP1). 

• Groups of 2

• 10 minutes: independent work
• 5 minutes: partner discussion
• Monitor for students who:
  • use a number line
  • count up by hundredths or thousandths between the intervals to find a number in the middle
  • use place value understanding to introduce new places in the decimals when needed

• Display the inequality: \(0.99 < \underline{\hspace{2cm}} < 0.999\)
• “What are some possible numbers that will make this true?” (0.995, 0.991, 0.997)
• “What do you notice about all the possible numbers?” (They all have 9 tenths and 9 hundredths and also some thousandths.)
• “Why does this make sense?” (They need 9 tenths and 9 hundredths to be as big as 0.99 and some thousandths to be bigger. There can be at most 8 thousandths so the number will be less than 0.999.)
• Display number line from the student solution or use a student generated image.
• “Do you think there are numbers between 1.731 and 1.732?” (It looks like there are lots of them but we don’t know what any of those numbers are.)

The purpose of this activity is for students to apply what they have learned about place value and decimals to order several decimals from least to greatest. Students may draw number line diagrams, if it helps them, but will need to think strategically about the endpoints that they choose if they want all 3 numbers to fit. They can also order the numbers by looking carefully at place value to compare pairs of decimals (MP7). 

• Groups of 2

• 8 minutes: independent work time
• 2 minutes: partner discussion
• Monitor for students who:
  • use their understanding of place value to compare the numbers
  • use a number line to visualize how the numbers compare

• Display the numbers: 1.101, 1.02, 1.1
• “How did you decide which of these numbers is the smallest?” (They all have 1 and some more. The second one only has hundredths while the other two have tenths, so 1.02 is the smallest.)
• “How did you decide which one of these numbers is the greatest?” (1.101 has a tenth and a thousandth while 1.1 only has a tenth, so 1.101 is the greatest.)
• Use a student generated number line or display the number line from student solutions and consider asking:
• “How do you know that 99.091 is between 99.09 and 99.99 on the number line?” (It has a thousandth more than 99.09 and has no tenths so is smaller than 99.1 and definitely smaller than 99.99.)
• “Why is it hard to locate 99.091 precisely on the number line?” (It is really close to 99.09, just one thousandth to the right.)