SWBAT:
• Define factor
• List the factors of whole numbers up to 50.

How many different rectangular boxes can you design to fit 100 brownies? Students explore the relationship between factors and area models.

5 minutes

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. For this lesson I want students to spend a few minutes looking at this great visual representation of numbers. What do they notice? We will revisit this picture at the end of the lesson so students can build on what they know and notice.

To share about the do now, I have students think, pair, share. See my **Think Pair Share **video in my Strategy Folder for more details. Students will share patterns and connections that they see. It is okay if students aren’t using words like factor, multiple, prime, or composite at this time.

20 minutes

I introduce the brownie task to students. I then go through the requirements and the example. I want students to understand that for this task the brownies must be in one rectangle. I do not mention anything about factors.

I have students work in partners as I walk around and monitor student progress. A common mistake is that students create the same rectangle, but one is rotated (like 20 x 5 and 5 x 20) and think they have two designs. Sometimes students struggle to come up with the 1x100 rectangle, and actually there are not enough squares so students have to continue the rectangle off the grid. Other students leave out the 10x10 box, since I indicated that the box needed to be a rectangle. If this is the case, I ask them for the characteristics of a rectangle and whether or not a square satisfies those characteristics.

If students are struggling with coming up with the designs, I may ask some of these questions:

- How many brownies need to fit in the box?
- What shape is the box?
- What if the box only had 12 brownies, what are some possible boxes? (I may give this student square tiles to show me)
- What do your box designs have in common?

If students successfully find all of the box designs, they will work on the next brownie project which introduces multiple layers.

Once groups have completed the first brownie task, we come together as a class. Here are some questions I may ask students:

- How many different designs did you come up with for the first task?
- What do your designs have in common? How are they different?
- Could you design a box that had a width of 40 brownies? Why or why not?
- Which box do you think would be the best for the company? (I want students to rule out the long and cumbersome boxes as impractical)
- Can someone who worked on the second task explain it?
- What do you notice about the box designs for the second task?

I want students to share that all of the designs in the first task involved numbers that could be multiplied to get 100. Students may use the word factor here, which serves as a great segue to the next section.

20 minutes

Students take notes about the vocabulary words. I ask students to list the factors of 100. Is it a prime or composite number? I introduce the rainbow method as one way of listing factors. You always start with 1 x ____ and then work towards the inside. Then you think about whether you can multiply 2 by another factor to get your number, etc. The other method involves area. We go through this page quickly because it is directly related to the brownie task.

Students work independently on the practice page. For students who struggle with their multiplication facts I give them a multiplication chart of the Factors 1-60 reference sheet. This way these students can engage in the task and hopefully start to notice patterns.

As I am walking around, I am looking to see what students put for the “prove it” section. Some students may list a multiplication fact that proves it’s a factor. Other students may list a couple multiplication facts that show the number is not a factor of the given product.

If students successfully complete the practice, they can go back to work on the second brownie task, or spend a few minutes revisiting the picture from the do now to see what else they notice. Another fun task is to challenge students create a dot diagram for 50, 51, etc.!

15 minutes

For the Closure, I will hand out the HW Brownies & Factors.

As students worked, I walked around and observed students work. Some students drew boxes that had the same dimensions, but were rotated. I asked these students if this box was a different design, and most were able to explain that it was not. Other students stopped after finding a few designs and I pushed them to find even more.

This student designed a Tetris-like brownie box that had 100 brownies. See Unit 1.6 Student Work.jpg. There were a few other students who I observed drawing boxes that would hold 100 brownies but were not a rectangle. I asked this student about this design, but did not tell him that it did not meet the requirement of being a rectangle.

After about 10 minutes, I asked for volunteers to come up and share one design with the class. Volunteers displayed their work under the document camera and then explained their design. I then picked a volunteer to respond to the student’s explanation with a statement or a question. I gave students sentence starter examples like “I agree because…” or “I disagree because…”. I also mentioned that it would also be appropriate to ask, “How does your design meet the requirements?” Students were doing the talking, commenting, and debating, rather than me. They enjoyed sharing their work and responding to each other.

In this class, I had the student from above come and present his Tetris-like design. After his explanation, a student raised her hand and acknowledged that the design held 100 brownies, but that it wasn’t a rectangle, so the design would not work. He was able to realize this mistake without feeling embarrassed.

Another great debate was whether or not the 10 brownies by 10 brownies design met the requirements. I had some students saying that the design did not work because the box was a square, not a rectangle. Other students said that the box was a square and a square can be also called a rectangle. A student in each class was able to explain that a square can also be considered a rectangle since it has the properties of a rectangle (4 right angles and 2 sets of opposite, equal, and parallel sides). After students were convinced that a square could also be called a rectangle I asked, “Can a rectangle also be called a square? Why or why not?” Students were able to explain that a rectangle cannot be called a square because it does not have four congruent sides.