Lesson 1

• Groups of 2
• Display the image.
• “What do you notice? What do you wonder?”
• 1 minute: quiet think time

• “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses.

• “If the large square represents 1, what fraction of it is shaded? What fraction is not shaded? How do you know?” (\(\frac{6}{100}\) are shaded. There are 100 small squares, and 6 of them shaded and 94 are not.)

In this activity, students use a square grid of 100 to revisit the meaning of tenths and hundredths and to make sense of the decimal notation for these fractions. They begin to make connections between the familiar representations of a fraction—using a diagram, fraction notation, and words—and the newly introduced decimal notation, and to notice similarities in their structure (MP2, MP7). It is important for students to consistently hear numbers read as decimals, for example 1.7 as one and 7 tenths so that they can connect decimal notation to visual representations and fraction notation. Later, in the lesson synthesis, the connections between the decimal notation and numbers in base-ten will begin to be made explicit.

Students may begin to notice that there are different ways to write decimals that represent the same fraction. It is not essential to discuss this in depth, as students will look at equivalent decimals more closely in upcoming activities.

• Groups of 2
• Give students access to colored pencils.

• “Work on the first problems independently.”
• 3–4 minutes: independent work time
• Pause for a brief whole-class discussion. Display the diagrams in the first problem.
• “What notation can we write to show each fraction? How do we say the fraction in words?”
• Record students’ responses in both notation and words.
• Display the diagrams of one hundredth and one tenth (in the next problem). Explain that the first fraction, \(\frac{1}{100}\), can also be expressed as 0.01 and is still read “1 hundredth”. This form of writing is called decimal notation.
• “What about \(\frac{9}{100}\)?” (nine hundredths, 0.09) “\(\frac{10}{100}\)?” (ten hundredths, 0.10)
• “We know that \(\frac{10}{100}\) can also be expressed as \(\frac{1}{10}\). How do we say it in words and write it in decimal notation?” (one tenth, 0.1)
• “In the numbers written like 0.1 and 0.01, there are decimal points. The digit to the left of the decimal point is in the ones place. The 0 means that there are no ones.”
• “Now try writing the fractions from the first problem as decimals. Then, complete the rest of the activity.”
• 5–6 minutes: independent or partner work time
• Monitor for the ways students write the fractions in the last problem, which may inform how they write corresponding decimals. For instance, they may write:
  • mixed numbers (\(1\frac{20}{100}\))
  • sums (\(\frac{100}{100} + \frac{20}{100}\) or \(1 + \frac{20}{100}\))
  • fractions with no whole numbers (\(\frac{120}{100}\))

• Invite students to share the decimals for the diagrams in the first problem. Record their responses for all to see.
• “How are the diagrams in the last problem different from those in the first problem?” (They represent numbers greater than 1.)
• “How did you figure out how to write each fraction as a decimal?”
• If not mentioned in students’ responses, point out that we can think of \(1\frac{20}{100}\) as \(1 + \frac{20}{100}\). The 1 whole goes in the ones place, to the left of the decimal point, and the 20 hundredths goes on the right of the decimal point.
• “The decimal 1.20 can be read ‘one and 20 hundredths’. The decimal 1.33 can be read ‘one and 33 hundredths’.”

In this activity, students practice representing and writing decimals given another representation (fraction notation or a diagram). The idea that two decimals can be equivalent, just like two fractions can be equivalent, is made explicit here. When students make connections between quantities in word form, decimal form, and fraction form, they reason abstractly and quantitatively (MP2).

• Groups of 2
• Give students access to colored pencils.
• “Let’s practice representing amounts using diagrams, fractions, and decimals.”

• “Take a few quiet minutes to work on the activity. Then, share your responses with your partner.”
• 7–8 minutes: independent work time
• 3–4 minutes: partner discussion
• Monitor for students who use the idea of equivalent fractions to explain why 0.6 and 0.60 refer to the same amount.

• Ask students to use words and decimal notation to express each amount in the first problem.
• Clarify that 0.02 can be named “two hundredths” and 0.22 can be named “twenty-two hundredths” and so on.
• Select students to share their representations for \(\frac{107}{100}\) and 1.6 and how they reasoned about them.
• Display 1.07 and 1.6. “How would you say these decimals in words?” (One and seven hundredths or one point zero seven for 1.07, and one and sixth tenths or one point six for 1.6.)