In this activity, students measure the length, height, and surface area of the boxes they built earlier. This work requires students to apply their knowledge of operations with decimal numbers and to exercise care in measurement (MP6). In measuring the lengths of a three-dimensional object, some imprecision can be expected, for instance:

• The lengths will not be an exact number of millimeters, so students round to the nearest millimeter. In some cases, this may be a guess between two different values.
• The folding process is not exact, so the box is not exactly a rectangular prism with a square base. Measurements of the length, width, and height vary depending on which part of the box is measured.
• When finding the surface area of their box, students will add and multiply their measurements. In performing operations, any errors in the measurements propagate.

Focus the discussion on the variation in the values students recorded, possible factors that contributed to the variation, and appropriate degree of precision in this context.

Display a blank table and invite groups to share their measurements of the length, height, and surface area of the boxes. Record variations in each measurement. Ask questions such as:

• “Why do you think the measurements were not all the same?” (Some likely reasons:
  • Measuring challenges: It was tricky to line up the edges of a paper box with the ruler and to measure to the nearest millimeter.
  • Folding challenges: The sides of the box were not completely flat, the height of the box was different at different places, or the edges weren’t fully creased.
  • Rounding variations: Different students might have rounded differently or to different numbers of decimal places.
  • Calculation variations: Some students might round before multiplying, and others might round afterward. Using slightly different factors would lead to different products.)
• “For the surface area of each box, how precise was the value you recorded? Did you record it to the nearest ten square centimeters, square centimeter, or one-tenth of a square centimeter?”
• “Did anyone record their answer to hundredths of a square centimeter? Why or why not?” (Yes, because the lengths were in tenths of a centimeter, so the products were in hundredths of a square centimeter. No, it was rounded to the nearest tenth of a square centimeter.)

In this activity, students check their predictions about how the measurements of the boxes they built compare. Doing so involves reasoning about quotients of decimals. Students may opt to estimate the quotients, calculate them more accurately, or do both depending on the numbers involved. Each approach is acceptable provided that students can offer an explanation to support their decision. The work here offers students an opportunity to consider the level of precision in communicating their answers (MP6).

Invite students to share their findings on how the measurements of Box 2 and Box 3 compare to those of Box 1, as well as how accurate their earlier predictions were, especially the prediction for how the surface area of Box 3 compares to that of Box 1. Ask questions such as:

• “Were your predictions for how the edge lengths of the boxes would compare correct? Were they close?”
• “Were your predictions for how the surface areas of the boxes would compare correct? Were they close?”
• “What do you notice about the numbers in each row of the second table?” (When the side length of the paper is 1.3 times or 2 times as long as that of another sheet of paper, the edge lengths of the box are also 1.3 times or 2 times as long, respectively, but the surface area is 1.78 times or 4 times greater, respectively.)