In this activity, students shade diagrams to represent fractions and decimals to the thousandths place. The first problem reviews IM Grade 4 work in which students filled in the same diagrams to show decimal fractions and decimals to hundredths. After this review, the problems all involve thousandths. First, students interpret how much of a square is shaded and then they shade a part of a square to represent a three-digit decimal. Because the thousandths are so small, students may struggle to count the shaded thousandths and may disagree about how many thousandths are shaded in the diagrams.    

Monitor for students who interpret and draw diagrams of decimals by thinking about each individual digit in a number. For example, in order to show 0.327 in a diagram, students can think of this as:

• 3 tenths
• 2 hundredths
• 7 thousandths

When shading the thousandths and naming them, students must be precise and pay close attention to what they decide to shade (MP6).

• Display the diagram that shows 0.625.
• “What number does this diagram represent?” (six hundred twenty-five thousandths)
• As students respond to each of the following questions, highlight on the diagram the tenths, hundredths, and thousandths.
• “Where do you see 0.6 in the diagram?” (There are 6 rows shaded and each row is 0.1 or a tenth.)
• “Where do you see 0.62 in the diagram?” (There are 62 small squares shaded and each one is 0.01 or a hundredth so that's 0.62 or 62 hundredths.)
• “Where do you see 0.625 in the diagram?” (If we divide each small square into ten tiny rectangles there will be 625 of them and they are each 0.001.)

In this activity, students consider different ways to name a decimal shown on a hundredths grid. The meaning of a decimal such as 0.150 is 1 tenth, 5 hundredths, and 0 thousandths. In words, however, it is usually read as one hundred fifty thousandths. Students see, using a diagram, that 1 tenth and 5 hundredths is equivalent to 150 thousandths. When students interpret the different descriptions of the shaded region, they construct viable arguments and critically analyze claims (MP3).

• Invite students to share their answers and reasoning for each statement.
• As students share, represent their language with fractions and decimals and show their reasoning on the hundredths grid.