1.1: Which One Doesn’t Belong: Boxes (5 minutes)
This warm-up prompts students to compare four open-top boxes with specific dimensions. It gives students a reason to use language precisely (MP6) in addition to recalling how volume is calculated and considering units of measurement in preparation for the following activities. It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another.
Arrange students in groups of 2–4. Display the information about the 4 boxes for all to see. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, ask each student to share their reasoning as to why a particular item does not belong, and together, find at least one reason each item doesn’t belong.
Which one doesn’t belong?
length: 4 cm
width: 8 cm
height: 10 cm
Ask each group to share one reason why a particular item does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question asking which one does not belong, attend to students’ explanations and ensure the reasons given are correct.
During the discussion, ask students to explain the meaning of any terminology they use, such as cubic units. Also, press students on unsubstantiated claims.
1.2: Building Boxes (20 minutes)
This activity is a hands-on introduction to the mathematical work of modeling the volume of a box using a polynomial function. It is not important that students develop an equation with variables for the volume of the box or identify the greatest possible volume at this time, since that is the focus of the next activity.
Monitor for students using logical reasoning to figure out the volume of their boxes instead of, or in addition to, measuring directly to share during the whole-class discussion.
Arrange students in groups of 2. Display for all to see the table from the task statement and a labeled set of axes like the one shown here.
Assign groups a side length between 0.5 inches and 4 inches in half-inch increments (if using non-standard pieces of paper, adjust these values).
Your teacher will give you some supplies.
- Construct an open-top box from a sheet of paper by cutting out a square from each corner and then folding up the sides.
- Calculate the volume of your box, and complete the table with your information.
|side length of square cutout (in)||length (in)||width (in)||height (in)||volume of box (in3)|
Students who measure the dimensions of the box may read the ruler marks incorrectly. Remind them that inches are divided into sixteenths and centimeters are divided into tenths.
The goal of this discussion is to make sure students understand how they can calculate the volume of the box without measuring each dimension, which will help them understand how to write an equation for the volume in the following activity. Start the discussion by inviting students to describe what the different points mean in this situation. (Each point is a different box created by making a square cutout using the value of the input. The point with the greatest output represents the box with the greatest volume and is somewhere around cutouts with sides of 1.5 to 2 inches.)
Select previously identified students to share how they calculated the volume of their box without having to measure directly.
Finish the discussion by asking students if they think the volume of a box where the side length of the square cutout is 1.75 inches has greater volume than 1.5- or 2-inch cutouts, then have students use the non-measuring method to calculate the volume (65.625 cubic inches).
Ask students to keep their boxes out as a visual aid for the remainder of the lesson.
Design Principle(s): Support sense-making
Supports accessibility for: Organization; Attention
1.3: Building the Biggest Box (10 minutes)
The goal of this activity is to write an expression that models the volume of a box as a function of the side length \(x\) of the square cutouts. Students then graph the function, use the graph to estimate the value of \(x\) that will produce the box with the largest volume, and establish a reasonable domain for the function by considering the context.
During the discussion, students interpret their expression in context (MP2). Of particular interest, values of \(x\) greater than 5.5 inches result in positive volumes even though cutouts of this size do not make sense when constructing a box from an 8.5 inch by 11 inch sheet of paper. There is no need for students to rewrite the expression for \(V(x)\) at this time in standard form. In later lessons, students will practice rewriting equations for polynomials in different forms.
Arrange students in groups of 2. Tell students there are many possible answers for the first question. After quiet work time, ask students to compare their response to their partner’s and decide which plan they like best. Select 2–3 groups to share their plans. Provide access to devices that can run Desmos or other graphing technology.
Supports accessibility for: Visual-spatial processing
The volume \(V(x)\) in cubic inches of the open-top box is a function of the side length \(x\) in inches of the square cutouts. Make a plan to figure out how to construct the box with the largest volume.
Pause here so your teacher can review your plan.
- Write an expression for \(V(x)\).
- Use graphing technology to create a graph representing \(V(x)\). Approximate the value of \(x\) that would allow you to construct an open-top box with the largest volume possible from one piece of paper.
Are you ready for more?
The surface area \(A(x)\) in square inches of the open-top box is also a function of the side length \(x\) in inches of the square cutouts.
- Find one expression for \(A(x)\) by summing the area of the five faces of our open-top box.
- Find another expression for \(A(x)\) by subtracting the area of the cutouts from the area of the paper.
- Show algebraically that these two expressions are equivalent.
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Some students may forget that the width and length of the paper are based on removing length \(x\) twice due to cutting out squares at each corner. Encourage these students to revisit their work from the previous activity and think about how they can calculate length and width without measuring directly.
If students have trouble finding a general expression for the volume of the box, encourage them to revisit their results from the previous activity and write the volumes of the boxes without simplifying their answers (e.g., 6.5 in \(\times\) 9 in \(\times\) 1 in instead of 58.5 in3) in order to see the pattern more easily.
The purpose of this discussion is for students to think critically about the equation they’ve written to model the volume of the box. Here are some questions for discussion:
- “What is a value of \(x\) that wouldn’t make sense?” (\(x=6\) would not make sense, because the sheet of paper is only 8.5 inches long on one side, so cutting out 6 inch by 6 inch squares from each corner is not possible.)
- “What is the smallest value that would work for \(x\)? The largest?” (The smallest value is some number greater than 0 inches that you could still use as a side length to cut out the squares and fold up the box sides to make a very short box. The largest value is some number less than half of 8.5 inches that you could still use as a side length to cut out the squares and fold up the box sides to make a very tall box.)
- “What is the range of \(x\) values where the volume is increasing? Decreasing?” (The volume increases as the cutouts get larger, until \(x\) is about 1.6 inches. Then the volume decreases until \(x\) is 4.25 inches, which creates a box with a volume of 0. The volume also starts increasing again when \(x\) is about 4.9 inches, but these values of \(x\) don’t make sense because we can’t make cutouts that large.)
If no students bring up the word domain during the discussion, remind students of the meaning of the word and select students to recommend a reasonable domain for the function \(V\).
Conclude the discussion by telling students that \(V\) is a type of polynomial function and that they will learn about other situations polynomial functions can model along with what graphs of polynomial functions can look like in future lessons.
Design Principle(s): Maximize meta-awareness
The purpose of this discussion is for students to create a polynomial function for an open-top box made from a sheet of paper and then estimate if the paper is large enough for the box to have a specific volume. Students are not expected to solve this algebraically, but rather by strategically testing values within the domain of the function or graphing.
Ask students to consider an open-top box made from an 18 inch by 24 inch sheet of paper. Is it possible to make a box with a volume greater than 650 cubic inches from the paper? If time allows, consider asking students to answer with a show of hands if they think it is possible to make such a box before giving work time.
Here are some questions for discussion.
- “What strategy did you use to answer the question?” (I figured out the equation for the volume of the box, \(V(x)=(18-2x)(24-2x)x\), and then graphed it, and I could see the volume is greater than 650 cubic inches for square cutouts with side lengths of about 3.4 inches.)
- “What are some side lengths for the square cutouts that don’t make sense?” (Since the shortest side is 18 inches, the largest side length for the cutout is 9 inches.)
1.4: Cool-down - A Box’s Domain (5 minutes)
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Student Lesson Summary
Polynomials can be used to model lots of situations. One example is to model the volume of a box created by removing squares from each corner of a rectangle of paper.
Let \(V(x)\) be the volume of the box in cubic inches where \(x\) is the side length in inches of each square removed from the four corners.
To define \(V\) using an expression, we can use the fact that the volume of a cube is \((length)(width)(height)\). If the piece of paper we start with is 3 inches by 8 inches, then:
\(\displaystyle V(x) = (3-2x)(8-2x)(x)\)
What are some reasonable values for \(x\)? Cutting out squares with side lengths less than 0 inches doesn’t make sense, and similarly, we can’t cut out squares larger than 1.5 inches, since the short side of the paper is only 3 inches (since \(3-1.5 \boldcdot 2=0\)). You may remember that the name for the set of all the input values that make sense to use with a function is the domain. Here, a reasonable domain is somewhere larger than 0 inches but less than 1.5 inches, depending on how well we can cut and fold!
By graphing this function, it is possible to find the maximum value within a specific domain. Here is a graph of \(y=V(x)\). It looks like the largest volume we can get for a box made this way from a 3 inch by 8 inch piece of paper is about 7.4 in3.