TEACHER DIRECTIONS FOR “ORANGE BOX DESIGNS “
Big Ideas: Area/Volume/Multiplication/Division
Embedded Ideas: commutative, associative properties
This is a two to three day lesson plan. On Day 1, students begin working in small groups to investigate the problem. They will represent their reasoning with interlocking cubes and a poster that will include words and numbers to justify their thinking and show the strategies they used to solve the problem. At the end of Day 1, there will be a short discussion at the end of the session to discuss strategies students are using to design their boxes. On Day 2, students continue investigating and preparing for Day 3 presentations. On Day 3, students will share, discuss and examine the work of their peers with the teacher helping facilitate the conversation.
Concepts:
In this problem-solving lesson, students will investigate and make connections between:
· the numbers and equations that describe a physical model representing cubic volume
· the concepts of equal groups (building upon the knowledge of arrays to the concept of layering equal groups or arrays to find volume)
· the patterns in the math equations proving all possible cubic volume arrangements for the number 24 have been used.
· when the commutative and associative properties are being used in their equations
DAY 1 OUTLINE
Developing the Context
· Introduce the box contest and ask students to investigate all the possible designs for a box that contains 24 items arranged in rows, columns and layers (3-dimensional arrays).
· Explain that rectangular prism box designs that look the same when rotated 90 degrees on the same plane (not flipped) are considered the same box design.
· Boxes are to be represented with interlocking cubes.
· Student groups are to justify their thinking with words and numbers on a poster to be shared with the class on Day 2 or Day 3.
Supporting the Investigation
· Listen to students as they develop designs. Encourage them to keep track of their thinking in a systematic way. Also, help students notice when they are using the commutative and associative properties. Ask questions about why these properties work? Is it helpful to know these properties? Why?
Preparing for the Student-Led Sharing Session
· Ask students to make posters of their strategies
· Plan to focus the discussion on the concepts listed above .
DEVELOPING THE CONTEXT:
“The other day I received a box of oranges as a holiday gift. When I opened the box, I noticed the oranges were arranged in arrays that were stacked on top of each other. The top layer looked like this: (draw a 6x2 array on the board or have a cut-out array to show. Also show the array using interlocking cubes or a real box if you have it). Notice the rows and columns. What numbers describe this array? (Write 6x2 on the board. (when held vertically)) What if I turn the box, then how is it described?” (Write 2x6 on the board. (when held horizontally))
“Next, I looked at the layer of oranges directly underneath. What do you think the array for this layer was? (6x2 or 2x6 depending on how the box is turned). Since there are 2 layers of the same array, how many total oranges do you think are in the box? How do you know? (Write the numbers to describe the thinking: (2x6) x 2. Ask: “Why do you think I put parentheses around the 2x6?” (This was the first computation made and represents the number of oranges in the top layer.)
Ask: “What does the second 2 represent?” (2 layers) How many total oranges? (24)
“Then I began to wonder about boxes in general, not just this box but other boxes as well. Sometimes boxes come in one layer, sometimes 2 layers, sometimes more. What if the box of 24 oranges had a bottom layer of 2x2, would it be tall or short? (tall) How do you know? (It would have lots of layers so it would be taller) How many layers would it have? (6) How do you know? What equation would describe this box? (2x2) x 6. How many total oranges? (24)Why do you think the orange company didn’t choose this design? (one possible answer: top-heavy, could fall over.)
Obviously companies must hire people in their company to decide on the size and shape of boxes. I wonder how many possible designs there are? What other arrangements do you think there are for 24 items- that is boxes that have rows, columns and layers holding 24 items?
At this point, establish the fact that in this context, a box rotated 90 degrees on the same plane (not flipped) is really the same box and won’t count as another possibility (just like an array that is turned 90 degrees is considered the same in some contexts).
Example: (2x6) x 2 is the same as (6x2) x 2. Another example: (4x3) x 2 is the same as (3x4) x 2. It has only been rotated 90 degrees without being flipped. On the other hand, a (4x2) x 3 box is a different box. It has been flipped: the bottom is now different and there are three layers of 4x2.
Also make sure they understand that each single cube represents one orange. Once students understand these rules, have them think about the following questions when considering the designs for a box holding 24 items.
Box Design Contest:
For 24 oranges,
· How many different boxes are there and what are the dimensions?
· How do you know that you have all the possibilities?
Students should work on this investigation in pairs or in groups of three. Make sure the work you have done in the opening when setting up the context of the problem is on display for students’ reference.
Materials: Interlocking cubes, markers, poster paper, graph paper, and calculators
SUPPORTING THE INVESTIGATION OR THE “WORK TIME”:
Circulate through the groups, listening and asking questions as necessary. Make sure students are making models of their box designs and keeping track of the numbers and designs they are making. Encourage them to do this if they are not.
CLOSURE:
At the end of Day 1, for a few minutes as a whole group, have student groups share the strategies they are using to build box models and the numbers they are using to describe them. How are they keeping track? How do they know the box designs are not the same?
Journal Quick Write: Explain the strategies you used today as well as your plans for tomorrow.
DAY 2 OUTLINE
Continue the investigation (continued from Day 1)
· Before student begin their work for the day, remind them of the questions they are trying to answer and the evidence they need for the discussion (poster listing the boxes and the explanation of how they know they have all possible designs plus models of all the box designs. Students will continue their work from the day before as the teacher continues to circulate groups.
· Students who finish early should practice their presentation and be ready to present. They should write up their presentation or their “proof” in their journal. Remind them that in the presentation, they must prove that they have found all possible box designs for 24 oranges.
DAY 3 OUTLINE
Preparing for the Student-Led Sharing/Discussion Session
Gallery Walk: Explain that before students start the “math congress”, they will have a gallery walk to look at each other’s posters. Give the small pads of sticky notes and suggest that students that student use them to record comments or questions, and then place them directly on the posters. Give student groups about 2-3 minutes at each poster to read, discuss, and make comments or ask questions. You may want to model a comment. For example, “Your strategy is interesting but how do you know you have all possible box designs?” or “You have convinced me you have all possible designs.” Or “I like you strategy. It seems easier than the one we used.”
Facilitating the Student-Led Sharing/Discussion Session
Once the gallery walk has been completed, students will now meet to discuss as a whole group pieces of work you as a teacher have chosen to highlight. Encourage students to take ownership of the discussion. The presenters should always wait until all students are paying attention. After their presentation, they should ask students: Are there any comments, concerns, or questions? Unless students are familiar with this format, you may need to model questions. Remind the student audience their job is to think about the work being presented. Do they understand what the strategies used? Are the strategies similar to the ones their group used or not? In a powerful discussion students will begin forcing the presenters to justify their thinking when they ask they why they did certain things. As a teacher it will be important for you to make sure the important mathematical concepts are brought out in the discussion. (See the beginning for these concepts) Make sure the students understand the SLEs (student learning expectations) they are investigating and learning.
Math Journals: Students should reflect in a journal what they have learned. You can also give a short assessment if you want to see exactly what they know.
Adapted from: The Box Factory, by Miki Jensen and Catherine Twomey Fosnot, 2007