Making and Measuring Boxes

25 min

Teacher Prep
Setup
Pre-make 3 boxes from square paper with side lengths 6 inches, 8 inches, and 12 inches. Students in groups of 3–4. Access to metric rulers or tape measures marked in millimeters. 3–4 different sizes of square sheets of paper per group. 10 minutes to answer questions, followed by a folding demonstration and time for students to fold their paper.
Required Preparation

Choose at least three different sizes of origami paper for students to use. Common length and width sizes of square origami paper include 6 inch, 7 inch, 8 inch, 9 inch, and 9.75 inch. Though less common, larger sizes such as 10 inches and 12 inches are also available. To see the mathematical structure more clearly, using 6-inch paper for the smallest size and 12-inch paper for the largest is recommended. If origami paper is not accessible, cut squares of paper from available paper (thinner is better). Prepare at least 1 sheet for each student.

Pre-make sample boxes of different sizes to show students. To prepare to demonstrate the folding, consider practicing the steps and the verbal instructions.

To help students fold their own origami boxes, both an embedded video and printed instructions are provided. If using the printed instructions from the blackline master, prepare 1 copy for every 2 students. The instructions can be re-used with multiple classes.

Narrative

In this activity, students are given square sheets of paper to measure and then fold into open-top boxes. Students work with decimals as they measure the side lengths of paper of different sizes, and estimate how the heights and surface areas of the resulting boxes would compare.

As students take measurements and make estimates to describe the relationship between two measurements, they have opportunities to attend to precision (MP6). For example:

  • If one sheet of square paper is very close to twice the length of another, it is reasonable to predict that the height of the box folded from the former to be 2 times the height of the box folded from the latter given the possible error in measurement.
  • If the relationship is very close to a fraction, such as 32\frac32 for the 8-inch and 12-inch squares, it makes sense to report the number as a fraction.
  • If a decimal is used to describe an estimated quotient of two lengths or surface areas, then using tenths may be appropriate.

Launch

Ask students if they have ever done origami or folded sheets of paper into three-dimensional objects. Ask a few students to share their experience. 

Tell students that in this activity they will measure pieces of paper and make some predictions about the measurements of the boxes to be created from the paper. Later, they will fold the paper into boxes and measure the boxes to check their predictions.

Arrange students in groups of 3–4. Provide each group with at least three different sizes of paper and metric rulers that can measure in millimeters. Make some extra squares available for each group, in case they are needed. 

Read the prompts as a class and answer any clarifying questions. Then give students 5–7 minutes to measure their paper and complete the activity. Leave at least 10 minutes for discussion and paper folding.

MLR2 Collect and Display. Circulate to listen for and collect the language that students use as they communicate about attributes of the paper or the folded box, as well as about the measurements and their relationships. On a visible display, record words and phrases such as “length,” “height,” “surface area,” “right angle,” “twice,” “approximate,” “estimate,” “rounding,” “precise,” “as long (or tall, or large, or much) as,” “units,” “centimeters,” and “millimeters.” Invite students to borrow language from the display as needed, and update it throughout the lesson.
Advances: Conversing, Reading
Action and Expression: Provide Access for Physical Action. Give students who need support with fine-motor skills the option of representing the experience in the activity kinesthetically on a larger scale. For example, cut squares from large, easily folded paper such as chart paper or newsprint for students to use in place of origami paper.
Supports accessibility for: Fine Motor Skills, Visual-Spatial Processing

Student Task

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

  1. Measure the side length of each sheet of paper to the nearest tenth of a centimeter. Then record the lengths, from the smallest to the largest.
    side length of paper (cm)
    Box 1
    Box 2
    Box 3

  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 ‾\underline{\hspace{1in}} times the side length of the paper for Box 1.

    2. The side length of the paper for Box 3 is ‾\underline{\hspace{1in}} 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:

    1. The surface area of Box 3 will be ‾\underline{\hspace{1in}} times as large as that of Box 1.

    2. Box 2 will be ‾\underline{\hspace{1in}} times as tall as Box 1.

    3. Box 3 will be ‾\underline{\hspace{1in}} times as tall as Box 1.

Now you are ready to fold your paper into a box!

Sample Response

Sample responses based on 6-inch, 8-inch, and 12-inch square sheets of paper:

  1. side length of paper (cm)
    Box 1 15.2
    Box 2 20.8
    Box 3 30.5
    1. The paper for Box 2 is about 1.3 times as long as the paper for Box 1, because 20.3÷15.220.3 \div 15.2 is a little less than 1.5. In inches, 8÷68 \div 6 is 43\frac{4}{3} or 1131\frac{1}{3}.
    2. The paper for Box 3 is 2 times as long as the paper for Box 1 because 12 inches is twice 6 inches, or 30.5÷15.230.5 \div 15.2 is about 2.
    1. About 4 times as large
    2. About 1.3 times as tall
    3. About 2 times as tall
Activity Synthesis (Teacher Notes)

The goal of this discussion is for students to think critically about the accuracy of their measurements and predictions. Consider asking the following questions (assuming use of paper squares with side lengths 6 inches, 8 inches, and 12 inches):

  • “What did you find for the side length of the smallest square?” (15.2 cm or 15.3 cm. Expect a range of values.)
  • “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.)
  • “When you make predictions about how some measurements of the boxes would compare, what numbers did you use?” (Whole numbers such as 2 or 3, fractions such as 32\frac{3}{2} or 1121\frac{1}{2}, and decimals such as 1.5 or 1.3.)
  • “You predicted some measurements of one box to be 2 times those of another box. In those cases, how confident are you that the size of one box will be exactly twice the size of the other when measured? Why is that?” (Confident, because the length of one paper is exactly twice the length of the other, and if the folding is very precise, then the result will also be exact. Not confident, because the measurement and the folding may not be very precise.)
  • “Did you use numbers such as 1.95 or 2.08 in predicting how the boxes would compare? Why or why not?” (No, because the measuring wasn’t that precise, and we don’t have the measurements of the boxes yet to calculate the relationship accurately.)

After discussion, demonstrate how to fold a paper square into a box, explaining each step so students can follow along. Alternatively, demonstrate the folding once, and then give each group a copy of the printed instructions from the blackline master or provide access to the demonstration video. Encourage students to make strong creases when folding their paper. Suggest that they use the side of a thumbnail or a ruler to flatten the crease after making each initial fold.

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 statement will need to be adjusted.

A box is created using a sheet of square paper.  It is folded using an origami method.

Anticipated Misconceptions

When measuring the side length of their paper, students might not remember to align the ruler to the edge of the paper or to start at the 0 mark of the ruler. Remind them to do so or demonstrate as needed.

If students round their measurements to the nearest centimeter, ask them how the measurements would change if the lengths are measured to the nearest tenth of a centimeter (or the nearest millimeter). Urge them to repeat the measurement at the specified level of precision.

Standards
Addressing
  • 6.NS.3·Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
  • 6.NS.B.3·Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

20 min