# Lesson 23

Making and Measuring Boxes

## 23.1: Folding Paper Boxes

### Optional activity

In this activity, students work with decimals by building paper boxes, taking measurements of the paper and the boxes, and calculating surface areas. Although the units are specified in the problem, students need to measure very carefully in order to give an estimate to the nearest millimeter. Next, students compare the side lengths of several paper squares and estimate what the measurements of the boxes will be once the squares have been folded.

Closely monitor student progress on the first question to make sure that students are careful in their measurements. On the second question, make sure they think carefully about what level of precision to use in reporting the relationship between two measurements. For examples:

• If one sheet of square paper is very close to twice the length of another, reporting the answer as 2 is reasonable given the possible error in measurement.
• If the relationship is very close to a fraction (e.g. $$\frac{3}{2}$$ for the 6 inch and 9 inch squares) students might report the number as a fraction.
• If students report the quotient as a decimal, three digits (ones, tenths, hundredths) is appropriate because the measurements in the first problem have 3 digits.

Encourage students to make strong creases when folding their paper at the end of the activity. Suggest that they use the side of a thumbnail or a ruler to flatten the crease after making the initial fold. Students will use these boxes in the next activity.

### Launch

Choose at least three different sizes of square paper for students to use. To see the mathematical structure more clearly, the smallest should be 6 inches by 6 inches and the largest should be 12 inches by 12 inches. For the other size squares, common length and width sizes of origami paper include 7 inch, 8 inch, 9 inch, and 9.75 inch. Pre-make boxes of different sizes from square paper with lengths 6 inches, 8 inches, and 12 inches.

Arrange students in groups of 3–4. Provide each group with at least three different sizes of paper. Since the folding process is involved and students are asked to measure the squares they use to make the boxes, make some extra squares available for each group. Students will need metric rulers or tape measures, marked in millimeters.

Give students 10 minutes for the problems. When students are finished, demonstrate how to fold a paper square into a box and then have students fold their paper squares. This can be done with an accompanying video or with instructions such as those included in the blackline master. In either case, consider going through the process with students step by step and practicing beforehand to make sure that it goes smoothly.

Note that these particular instructions make a box with a square base; the following activity, which prompts students to record the length and width of the box’s base, is based on this premise. If a different origami construction is used, the instructions and possibly the task statements will need to be adjusted.

Representing, Conversing: MLR2 Collect and Display. As students work on the problems, use this routine to capture the ways they communicate about the measurements, including the precision at which they were taken. Additionally, capture the ways students discuss the relationships between the dimensions of the sheets of paper and predicted dimensions of the boxes. Listen for terms such as quotient, divide, multiply, and rounding. Notate student language and relevant sketches/diagrams to reference while students are working and then discuss in the synthesis. This will help students communicate about the accuracy of the measurements and the size comparisons between boxes.
Design Principle(s): Support sense-making; Maximize meta-awareness

### Student Facing

Your teacher will demonstrate how to make an open-top box by folding a sheet of paper. Your group will receive 3 or more sheets of square paper. Each person in your group will make 1 box. Before you begin folding:

1. Record the side lengths of your papers, from the smallest to the largest.

• Paper for Box 1: ______________ cm
• Paper for Box 2: _______________ cm
• Paper for Box 3: ________________ cm
2. Compare the side lengths of the square sheets of paper. Be prepared to explain how you know.

1. The side length of the paper for Box 2 is ________ times the side length of the paper for Box 1.
2. The side length of the paper for Box 3 is ________ times the side length of the paper for Box 1.
3. Make some predictions about the measurements of the three boxes your group will make:

• The surface area of Box 3 will be ________ as large as that of Box 1.
• Box 2 will be ________ times as tall as Box 1.
• Box 3 will be ________ times as tall as Box 1.

### Anticipated Misconceptions

Check that students are obtaining accurate measurements to the nearest tenth of a centimeter. Are they placing the ruler at the edge of a side? Are they starting at zero?

### Activity Synthesis

(Note: the discussion questions below assume square paper of side lengths 6 inches, 8 inches, and 12 inches were used).

The goal of this discussion is for students to think critically about the accuracy of their measurements and predictions. Consider asking some of these discussion questions. Sample responses are shown in parentheses, but expect students' answers to vary.

• What did you find for the length and width of the smallest square? (15.2 cm. Also expect some answers of 15.3 cm and possibly a wider range of values.)
• What was challenging about measuring the length of the squares? (The millimeters are so small that it was hard to tell which millimeter it was closest to. It was hard to measure straight across.)
• How confident are you about the accuracy of your measurements (For the first square, very confident about the 15 in 15.2 cm, but not confident about the 0.2.)
• How many times as long as Paper 1 was Paper 3? (About 2, very close to 2, or a decimal number that is close to 2.)
• What are some advantages and disadvantages of reporting the quotient of the side length of Paper 3 and that of Paper 1 as 2? (Advantage: it describes the general relationship clearly, and 2 is an easy number to grasp. Disadvantage: it was not exactly twice as long.)
• What are some advantages and disadvantages of reporting the quotient of the side length of Paper 3 and that of Paper 1 as 2.007 (or another value that is very close to 2 and is proposed by students)? (Advantage: the number is more accurate than 2. Disadvantage: it is too accurate. The measurements were done by hand and were not precise enough to judge the quotient to the nearest ten-thousandth.)
• How much taller do you think Box 3 will be compared to Box 1? (Twice, because the side length of the paper making Box 3 is twice as long and twice as wide as the that of paper making Box 1.)

## 23.2: Sizing Up Paper Boxes

### Optional activity

Using the boxes that they built in the previous activity, students now measure and compare the length, height, and surface area of the boxes. This work requires fluency in operations with decimal numbers and care in measurement. In measuring the dimensions of the box, there are multiple layers of imprecision that can be expected.

• The length, width, and height will not be an exact number of millimeters and so students round to the nearest millimeter. In some cases, this may essentially be a guess between two different values.
• The box is made by folding paper and this process is not exact. The box is therefore not exactly a rectangular prism with a square base, and 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, making it challenging to trace where they originated.

### Launch

Keep students in the same groups. Provide access to rulers. Give groups 8–10 minutes to complete the first table collaboratively. There are three sets of measurements and a surface area calculation for each box. Each student can fill out the row for the box that they made. If a group has more than three paper sizes, adjust the tables in the activity accordingly. Ask students to pause for a class discussion after they have completed the table.

Select a couple of groups to share their measurements and surface area calculations. Display their responses for all to see. Ask other groups if their responses are close, and if not, ask what their answers are. Come to a general agreement about the approximate measurements and areas.

Then, ask students to complete the remaining questions.

Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts to support students who benefit from support with organization and problem solving. For example, present one question at a time and monitor students to ensure they are making progress throughout the activity.
Supports accessibility for: Organization; Attention

### Student Facing

Now that you have made your boxes, you will measure them and check your predictions about how their heights and surface areas compare.

1. Measure the length and height of each box to the nearest tenth of a centimeter. Record the measurements in the table.

side length
of paper (cm)
length of
box (cm)
height of
box (cm)
surface area
(sq cm)
Box 1
Box 2
Box 3
2. Calculate the surface area of each box. Show your reasoning and decide on an appropriate level of precision for describing the surface area (Is it the nearest 10 square centimeters, nearest square centimeter, or something else?). Record your answers in the table.
1. To see how many times as large one measurement is when compared to another, we can compute their quotient. Divide each measurement of Box 2 by the corresponding measurement for Box 1 to complete the following statements.

1. The length of Box 2 is ________ times the length of Box 1.
2. The height of Box 2 is ________ times the height of Box 1.
3. The surface area of Box 2 is ________ times the surface area of Box 1.

2. Find out how the dimensions of Box 3 compare to those of Box 1 by computing quotients of their lengths, heights, and surface areas. Show your reasoning.

1. The length of Box 3 is ________ times the length of Box 1.
2. The height of Box 3 is ________ times the height of Box 1.
3. The surface area of Box 3 is ________ times the surface area of Box 1.
3. Record your results in the table.

side length
of paper
length
of box
height
of box
surface area
Box 2 compared
to Box 1
Box 3 compared
to Box 1
4. Earlier, in the first activity, you made predictions about how the heights and surface areas of the two larger boxes would compare to those of the smallest box. Discuss with your group:

• How accurate were your predictions? Were they close to the results you found by performing calculations?

• Let’s say you had another piece of square paper to make Box 4. If the side length of this paper is 4 times the side length of the paper for Box 1, predict how the length, height, and surface area of Box 4 would compare to those of Box 1. How did you make your prediction?

### Anticipated Misconceptions

Students may neglect to attend to units of measurement when calculating area. Encourage them to attend to the units being used.

### Activity Synthesis

(Note: the discussion questions below assume sheets of square paper of side lengths 6 inches, 8 inches, and 12 inches were used).

For each size of paper used, ask students to report their measurements for the volume of the box and record for all to see. Ask:

• “Is there variation in the measurements? Which ones?” (Yes, possibly for the dimensions of the squares; almost certainly for the dimensions of the boxes.)
• “Why do you think the measurements were not all the same?” (It was difficult to line up the edges of the paper with the ruler. It was also challenging to measure to the nearest millimeter.)
• “Were there extra difficulties measuring the boxes, as opposed to measuring the squares of paper?” (Yes, e.g, the sides of the box were not completely flat; the sides of the box were not identical; the height of the box was different at different places.)
• “How accurate do you think your measurements were?” (In each case, within 1 or 2 millimeters.)
• “How does this influence the way you record a result? For example, for the surface area, did anyone record their answer to hundredths of a square centimeter? Why or why not?” (Some students might have done this, but it is overly precise and does not take into account the imprecision in measurements mentioned above.)
• “In the table for the third question, did you record units for your measurements? Is it important to record units?” (It is important that the measurements of the differences between boxes must be in the same units.)

Also discuss with the class their predictions and measurements for how the heights and surface area of the boxes compare to one another. Ask:

• “How much larger did you think the surface area of Box 3 would be than the surface area of Box 1?” (4 times because it takes 4 of the smaller sheet of paper to make one of the larger ones. Some students may also say 2 times, because the height, length, and width are about 2 times as long as those of Box 1.)
• “Did your measurements match your prediction for the surface areas? Why might that be?” (Students who did not think the surface area of Box 3 would be 4 times as large as the surface area of Box 1 can discuss their old thinking and new. Because area is found by taking a product of length measurements and all measurements (length, width, height) of Box 3 are twice the corresponding measurements of Box 1, Box 3 will have 4 times as much surface area as Box 1.)
Listening, Speaking, Conversing: MLR8 Discussion Supports. Before the whole-class discussion, give students time to converse with their group about their predictions for how the heights and surface area of the boxes compare to one another. Provide this sentence frame for students to use: “My measurements matched/did not match my predictions because __. ” Invite group members to press for details in their peers’ explanations and challenging ideas by asking, “Why did you make that prediction?”, “Could you explain that using a different example?”, or “What evidence do you have to support your reasoning?” This will help students solidify their reasoning, and improve their explanations before the whole-class discussion.
Design Principle(s): Cultivate conversation

## Lesson Synthesis

### Lesson Synthesis

The discussions of student work at the end of each activity provides opportunities to summarize takeaways from this lesson. Students can use this optional lesson to practice fluency with calculations involving decimals. Highlight instances where students have to make an estimate in order to proceed, figure out how to determine significant digits, or apply their understanding of units when measuring and recording information.