The Big Cheese - A Lesson On Number Sense

I was using Feedly to read Christopher Danielson's Blog and came across his observations around the comparison between regular Cheez-Its and Big Cheez-Its. My sense is that Danielson's observations are perfect for any study in irrational numbers and are also a great opportunity for teaching the growth mindset. I will share some of Danielson's observations and mix in a few of my own. I wanted to share some ideas for how this lesson might work and if I ever assemble the complete lesson, you will find it on http://betterlesson.com (I am working on their Master Teacher Project).

There is so much to gain from this investigation, but the main take away is for students to be aware of the existence and need for irrational numbers, which are a part of these two standards from the common core catalogue:

You can adjust this investigation to reach almost any take away related to number sense and groups and scales and ratios, but I love finding intuitive ways to discover the idea and need for irrational numbers. Students often feel that irrational numbers are just another abstract construct with no purpose or meaning. I think this investigation can show them that irrational numbers are the result of observation of the world we live in. 

How would this lesson go? I am not exactly sure (yet). But I started to run through the process as if I were a student. When I plan and create a lesson, I take the time to do the investigation. Not just think about it, but actively go through the process. This helps me catch things I might have otherwise missed. This has become an essential practice for me. I play the role of both teacher and student and try to understand the lesson from both perspectives. 

After reading Danielson's post, I went shopping and bought Cheez-Its:

I might have looked a bit crazy, but I purchased enough Cheez-Its to run the lesson in my four classes (with plenty of extra).  I bought 4 of the regular Cheez-Its (351 crackers per box):

I bought 5 of the Big Cheez-Its (154 crackers per box):

The mathematical concepts we are aiming for revolve around the claim on the cover of the Big Cheez-It box, claiming that the crackers are "Twice the Size":

I think I would start my lesson with this simple intro. I would tell them about my shopping experience, show the photos and start a conversation around the claim on the box. 

"What do you think they mean by Twice the Size?" 

I also like Danielson's approach of asking, "did they really mean 4 times as large?" And then proceeding in the lesson by showing that it isn't 4 times as large and then progressing to the tougher question: "how much longer did they make each side?"

In order to help students compare these crackers, I plan on giving my classes the Cheez-Its and possibly a ruler, but I might only offer tools that they request. You don't need a ruler to understand how these crackers are related. 

I tried the approach that Danielson's children worked out by comparing the two crackers visually and then with a sketch. I started by putting the two crackers next to each other and thought how I might compare them and articulate why I know the larger cracker isn't four times larger than the smaller cracker:

I think the way I would do this is by tiling four of the smaller crackers and showing that the area of these four smaller crackers is larger that the "Big" cracker. This type of visual proof is perfect because it allows all students to access the problem and is something they could argue to the whole class. 

Another approach might be to sketch the crackers (Danielson's son's strategy). 

I overlapped the crackers and color coded them (students don't need to overlap them in this way, I was just working through it in a way that made sense to me). Here the regular sized Cheez-It is drawn in green and shaded in. The gap between the Big Cheez-It and the regular sized Cheez-It forms a nice "L" shaped area:

As Danielson suggests, you can cut this space out to compare whats happening. Here I cut out the "L" shape and took a photo on my red table:

If your students do this, be prepared for the "L" shape to be larger than the green square (even though they should be about equal). Here the two white strips are larger than the green square (the regular Cheez-It). You can see a tiny region of green not covered, but this would not hold the overlapping portion of the "L" shape (I wish the photo could have been better):

I was curious about this and started to measure the dimensions of the Cheez-Its. The regular size were relatively square at 2.1cm by 2.1cm:

The Big Cheez-Its were not square. I measured several to be sure. In this Big Cheez-It, length was about 3 cm and the width about 3.3cm

In another Big Cheez-It the length was about 2.8 cm and the width was about 3.2 cm:

As students start to use the rulers, I might ask them why this is happening. Did they make an error in their calculations? Is it a machine error? Is something else going on here? 

Students could use a sketch with these measurements, the next progression in problem solving might be to try and make some precise comparisons. I chose to take the average side lengths of the two Big Cheez-Its, but students might take other approaches. I would use these different strategies as a source for conversation in the summary. This could especially lead toward conversations around similarity as well. 

Here are my measurements and my sketch (which I might make into a hand out if I decide to not use actual crackers in a lesson):

Here I used the overlapping regions to compare.  This shows that the regular Cheez-It is not equal in size to the "L" shape and that the Big Cheez-It is more than twice the area:

This whole process prepares students for the summary. Whatever their approach, they will find that the Big Cheez-It is about twice as large in area as the regular Cheez-It. But the conversation needs to focus on why we aren't getting exact amounts here. The question becomes "How large should the Big Cheez-It be if we want it to be a square cracker that is twice as large?" I might not say "square" since I want students to play with this. They might create other rectangles that have twice the area, but I would say to them, "does that still look like the regular Cheez-It?" I might even make that the challenge from the start (I am still thinking how this lesson will pan out). It could be fun to say, "I want you to redesign the Big-Cheez It. If needs to look like the regular Cheez-It, but actually twice the are of the regular Cheez-It." This would also tap into the process of approximating square roots, another common core standard:

The fun part here is that students are now trying to "fix" the Big Cheez-It. I believe that this process of trying to "fix" the dimensions will help them realize something incredible: there is no rational number that works here.

The discussion at this point would be critical and I am not exactly sure how to structure it, but I know students have to realize that they are looking for a scale factor that makes the over all area twice as large.

I might use other scale factors to help them make some conclusions about what must be happening. They might use the fact that an area four times as large has a scale factor of 2, which is the square root of 4. I might present this and other simple scale factors in a string or list to help them make the connection that (scale factor)^2 = change in area or that the scale factor = (change in area)^(1/2). 

I imagine that students will come up with two possible changes for the Big Cheez-It. They might suggest multiplying each side by the square root of 2 or multiplying each side by a decimal approximation close to the square root of 2. 

My question will be "what is the connection between the decimal that some people found and the idea of the square root of 2?" This would lead them into the next lesson where we define this relationship and the concept of irrational numbers. 

Danielson points out a few other wonderful avenues to explore with the crackers. Since the scale factor for each side is about the square root of 2, we can line up the crackers to show decimal approximations for the square root of 2. Just be aware that the orientation of the larger crackers matter, since they are not squares. If you line up the Big Crackers by their shorter dimension, then you get the 7:5 approximation of the square root of 2:

If you line them up the long way, you will not get the 7:5 ratio that approximates the square root of 2:

In this case students can line up the crackers to get the 3:2 ratio, which is the first rational approximation of the square root of 2:

I think these cracker comparisons are at the heart of understanding the irrational number. I would love to challenge students to find how many crackers it would take to get them to perfectly line up. 

The idea of irrational numbers will show in the way that the crackers never seem to line up perfectly. Of course, this is the idea of an irrational number: there is no rational fraction to represent the ratio between the side lengths of the regular and Big Cheez-Its. Even those these crackers will line up perfectly at some point, they are a nice way to represent the idea of an irrational number. I might design the whole lesson around this component. I am still not sure. 

Danielson also discusses the mass of each Cheez-It. Here is my image from the boxes I bought:

This approach might work nicely in the lesson as well, since 14 of the Big Cheez-Its = 30 grams and 27 of the regular Cheez-Its = 30 grams. I didn't confirm this with a scale, but if I were to analyze the mass, I would probably have my students use a scale. 

There are so many ways to approach this lesson and I am still not sure how I will develop it, but I am glad Danielson posted this connection. I am wondering what other foods we could use?