This was one of the first images I've seen in a long time that inspired me to actually investigate and try to explain what was happening. I didn't just try to use logic, reason, estimation, etc. I actually got out graph paper, scissors, tape and MADE a paper chocolate bar. If it can inspire me to build and play, I'm sure it will do the same for students.
I got this gif from this site.
Grade Level: 7-10
Course: Algebra
Standards: 7.G.1, 7.G.4, 7.G.6,G-CO.6, G-CO.7, G-CO.12, G-MG.1, G-MG.3
SMP: SMP.1, SMP.2, SMP.3, SMP.4, SMP.5, SMP.6
Skills: Geometry, Area, Constructions, Modeling, Problem Solving, Ratio, Proportion, Scale
How to use this as a mad minute:
You have 60 seconds. Give me one reason you think this DOES or does NOT work.
How to use this as a warm up:
You could ask the students to consider one of the following:
1. What is the area of the original chocolate bar in generic "units"?
2. What is the area of the new chocolate bar in generic "units"?
3. When the candy bar is broken up, there are 5 pieces. Describe each piece using a correct geometric name and explain what properties each piece has to categorize them.
4. Draw the 5 pieces on your paper (sketch!) and then label the angles. (Acute, obtuse, right, straight.)
5. What is the perimeter of the original candy bar? Is the final perimeter the same or different?
How to use this as a mini-lesson:
0:00 Look at this awesome gif! Have you seen this online? If so, what did you think? Watch and then turn and discuss with a partner.
1:00 If you haven't already discussed with a partner, focus on whether or not you think this works and why.
2:00 Ok, let's talk. Who thought it worked? Who didn't? Why?
4:00 How could you prove your side? What would you do? (If they don't know, gently guide them toward making their own model) What supplies would you need? What information would you need?
6:00 Here's what I can give you: Graph paper, rulers, scissors and markers. You have 5 minutes to create your own ACCURATE candy bar. (For your information, the side length ratio is 3.5:6, you can decide if this is helpful for your students, I think it would be, but could make construction challenging!)11:00 Now that you have this candy bar, you want to "break" it accurately. How could you "cut" this candy bar accurately? (The bar, if students watch carefully, is cut on a diagonal from 1.5 "squares" up on the left through 1.5 "squares" down on the right. But I would encourage kids to measure angles as well.)
13:00 Next, we need to break the top piece into three smaller pieces. How should we do that? (This is a much easier "cut" since they are clear vertical and horizontal lines.)15:00 Finally, we need to take out the extra square. (I would have the kids label the pieces either by number, letter, or size. I'll call them "single", "double", Small, medium and Large for my explanations.)
16:00 Now slide your medium piece up and your small piece over and down. Fill the gap with your double piece. Discuss what you see with your partner!
18:00 What did you see? How do you explain the extra piece? (Hopefully they see that the "squares" are not the right size or dimensions and to get them to "line up, the students need to shift the small and medium pieces "up" a bit leaving a long thin "gap" between the top and bottom. Almost like the photo to the right.)
19:00 Does your extra piece "fill the gap"? Is this real "infinite" chocolate? Why or why not?
How to use this as a full lesson?
I would definitely use the mini lesson above, allowing for more freedom if students are enjoying the exploration and discussion. Depending on the grade level I would also ask appropriate questions and use appropriate vocabulary.
For example, if you repeated this experiment without the "squares" of chocolate and one large bar, could you prove congruence? Why or why not? If you can prove congruence, explain the criteria for congruence and back up your answer mathematically. If not, what mathematical proof (not just modeling) could you use to justify why these are not congruent?
Why are the "small" and "Medium" pieces NOT similar? Use definitions, properties, and measurements to back up your claim.
What is the ACTUAL area of the original? What is the actual area of the final candy bar (minus the extra square)? Does this prove that they are or or not congruent?
After these explorations, I would show the video linked here. This is a similar optical illusion, trick, or manipulation. Encourage your students to watch, either as a class, or on their individual devices.
I apologize in advance for the ad that precedes the video, but you can skip it after 5 seconds.
I would ask students to watch out and consider these questions as they watch:
What are the original dimensions?
What is the original area?
What do you notice about the space in the "box" as he shakes it?
What do you notice about the cuts of the pieces?
What do you notice as he lays out the tiles the first time?
As he moves them, what do you see?
As he places them back the box what do you see?
As he "repeats" or "reverses" the trick, what do you see?
Can you explain his "trick" mathematically?
Use your number sense. (I think you can see more "wiggle"room once the 3 squares are removed, and 3/63 is such a small percentage of change, it's not too obvious. Add to that the fact that he has a very hard time at the end of the video making them all fit again!)
Ask the students to justify, model, draw, explain, etc. They should use correct mathematical vocabulary, appropriate skill and relationships to their learning. For example, can they discuss area and congruence? Can they name shapes and angles? Can they formally prove or disprove congruence?
How to use this as an assessment?
To use this as an assessment, I'd do the mini lesson at the beginning of a unit, refer to it throughout the unit as we are using vocabulary and talking about proof, and then I would show the video at the end. I would assess the students on their explanation of the "trick" and how well they used what they had learned.
You would definitely want to create your own rubric before the assignment. You would also want ample supplies for students, as well as multiple devices, as students will want to watch the video over and over as they work. (Isn't it awesome that the video is nearly silent??)
Please feel free to use any of these ideas and modify them to meet your needs. However, please acknowledge the original source of the items and my own lesson outlines. ©NatalieRSprigg 2013
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