Unit Cubes and Volume

See my Do Now in my Strategy folder that explains my beginning of class routines.

Often, I create do nows that have problems that connect to the task that students will be working on that day.  Today I want students to review the difference between surface area and area.  I want students to recognize that area is covering a two-dimensional figure while surface area is covering a three-dimensional figure.  I also want students to start using their spatial reasoning to determine the number of cubes that make up a figure.  Some students may struggle to count the cubes that they cannot see.  Other students may count the faces of the cubes without realizing that some cubes have multiple exposed faces.  If students struggle, I encourage them to think about the solid as a number of layers. 

I ask students what strategies they are thinking about to answer the questions.  I call on students to share what they think and why.  If there is disagreement, build the first cube out of connecting cubes.  Students are engaging in MP3: Construct viable arguments and critique the reasoning of others.

Notes:

• For this lesson I have collected 12 open boxes that can be filled with unit cubes.  I use place value cubes and wooden cubes depending on the size of the box.  I only give students a few unit cubes, not enough to fill a complete layer of the box.
• If you don’t have boxes, you can use inch grid paper to create boxes.  Here are some examples. I show students that we can create a net for the rectangular prism.  Some students struggle to use the three labeled dimensions to label the other sides of the net.  I explain that each rectangular prism is composed of three matching faces.  Two faces that fold together to make one edge must share the same measurement. 

I explain the task to the students and review expectations.  I pass out materials.  As students work I walk around to monitor student progress and behavior.  Students are engaging in MP5: Use appropriate tools strategically, MP6: Attend to precision, and MP8: Look for and express regularity in repeated reasoning.

 If students are struggling, I may ask them one or more of the following questions:

• What is your job?
• How many unit cubes do you have?
• How many unit cubes would it take to fill the bottom of the box?  How do you know?
• Is there another way to find the number of unit cubes it would take to fill the box?

If students complete the problem and explain their thinking on their paper I may ask them:

• How many unit cubes would it take to fill half of the box?
• How many unit cubes would it take fill 1/3 of the box?
• Another box takes up 75% of the space of your box.  How many cubes would it take to fill it?

We fill in the notes and complete the examples together.  I want students to look for patterns when finding the volume of the boxes.  Some students may notice that if you multiply the length, width, and height the result is the total number of cubes it takes to fill the box.  Other students may notice that if you multiply the number of cubes it takes to fill the bottom of the box by the height the result is the total number of unit cubes that it takes to fill the box.  I push students to see whether these ideas work for each of the different boxes.  Students are engaging in MP8: Look for and express regularity in repeated reasoning.

We work on these questions together.  I ask students, “How can we apply the work we just did to answer these questions?”  I call students up to the front to show their ideas and explain their thinking under the document camera.

Note:

• I Post a Key for these problems around the room.

We go over directions and expectations.  As students work I walk around to monitor student progress and behavior.  Students are engaging in MP6: Attend to precision and MP8: Look for and make sense of repeated reasoning.

 If students are struggling, I may ask them one or more of the following questions:

• What do you know? What are you trying to figure out?
• Make an estimate for your answer. 
• What strategies do you have for finding volume?
• Does your answer make sense?

When students complete their work, they raise their hands.  I quickly scan their work.  If they are on track, I send them to check with the key.  If there are problems, I tell students what they need to revise.  If students successfully complete the problems they can move onto the challenge problems. 

For the Closure, I have students turn to the first challenge problem.  Students participate in a Think Pair Share.  I call on students to share their answer and explain their thinking.  I want students to apply their knowledge of triangles to this situation.  Some students may mistakenly think that they can multiply the base of the triangle by the height of the triangle and then by the height of the prism.  If this comes up I ask, “How do we find the area of a triangle?”  I want students to see that we can find the area of the triangle base and then multiply it by 25 cm.  This connects to the V = Bh method for a rectangular prism.  Students are engaging with MP3: Construct viable arguments and critique the reasoning of others and MP8: Look for and make sense of repeated reasoning.

I pass out the Ticket to Go and the Homework.