I can't make this post fit my normal outline. It's just too basic to include in lessons, but too cool not to post or use in some way!
If you are a number nerd, like me, you love "special" numbers. You look for them on your clocks, in page numbers, on receipts and on your odometer! I love palindromic numbers, repetition in numbers, or special sequences, like the one on the right. I have a friend who also loves these special numbers and we often text each other when we hit cool numbers such as 12345, or 12121 or 12321, or 33333. When I saw this photo, I immediately wondered, "How much planning would that take?"
So, the question is pretty simple:
When did this person reset the "Trip A" counter to 0 to make this number appear?
If you want to push students a bit further, you could ask how many tanks of gas, days or weeks or months it would take to drive that far, how old is the car...but I love this one because it's simple. No standards, no lessons, just a quick, fun and easy exploration!
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.
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:
Please note: I included screen shots of the candy bar when it isn't moving for YOU to use, but largely, it would take away some mystery for the kids, so I probably would NOT show the photos to them.
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??)
One of my most favorite places in the whole world (that I've never actually been to) is The Giant's Causeway in northern Ireland.
The summer after college I was a nanny for my cousins in a small town outside of Dublin called Dunboyne. I was too young to get my Irish driver's license and ended up taking the kids to the city on the bus. We did get to take weekend trips with the family to southern Ireland, but while I was there there was just too much unrest to visit up North.
Thus, this is the most amazing place in the world that I've always wanted to go to, but never have. :)
The Giant's Causeway is a natural formation of rocks on the northern coast between Ireland and Scotland. As you can see from the photos, these spires of rock form beautiful polygons, often hexagons, but reports are anything from quadrilaterals to nonagons.
This, to me, is full of opportunities for great instruction. I can see anything from estimation and basic polygon identification (3rd Grade) to tessellations and transformations.
For this reason, I feel the EQUATE model is a perfect opportunity to explore these photos and this location. Rather than focusing on a single grade, I encourage you to use the EQUATE thinking routine to apply appropriate standards at your grade level.
How to use this as a mad minute:
You have 60
seconds. Name all of the shapes you can see. How to use this as a warm up:
You could ask the students to consider one of the following:
1. Name the shapes you see.
2. Does this fit the definition of a tessellation? Why or why not?
3. Are these "regular" polygons? Why or why not? How to use this as a mini-lesson: If I only had 20 minutes, I would use technology to explore this VERY COOL region. This website has an awesome interactive map, some history, and the legend of the Giant's Causeway.
How to use this as a full lesson?
As I mentioned before, I feel that this is an ideal EQUATE lesson. Although there is a ton of math that is obvious to an instructor, this captivates my interest because of the combination of legend, scientific history, and visual appeal. I feel your students will also be drawn to these elements. If you are comfortable, let the students dictate the direction of the lesson and exploration (within reason).
I would show these photos, let the students explore, discuss, etc.
Then I would list all of their questions, encouraging them to "wonder mathematically" about them.
Focused on grade-level appropriate standards, I would ask students to narrow down the questions to make sure they are relevant to things you have already explored or discussed in your class.
I would let the students ask YOU questions and you can provide the answers you feel are appropriate. (How are they formed? How big is the region? How many are there? You can provide as much or as little information as you wish.)
I would settle on a question (or two or three) for your students to apply their knowledge and continue to try to solve. Encourage them to TRY something! Draw on the photo, measure it, get online and do research, look up formulas that might be useful, gather information, start playing with the numbers, rules, formulas, photos, etc.
Finally, ask the students to Explain what they did, what they found, and how they approached the problem.
How to use this as an assessment?
It is up to you if you think your students can use this as an assessment appropriately.
It could be something as simple as providing the first photo and asking students to outline as many different shapes as they can see and explain why they are different and what they are (Elementary School).
It could be more advanced, offering the size of the region, the size of an individual "step" and asking the students to estimate how many are in the entire region. (Upper Elementary to Middle School.)
You could ask the students to find two similar "steps" and justify why they are similar (Middle/High).
You could ask the students to find the volume of two or three different "steps" and justify their solution methods. (Middle/High).
You could ask the students to PROVE that two items are congruent or similar based on transformations such as rotations, reflections, etc.
Not all of my inspiration comes from lame humor sites. :) Just the other day I was watching a rerun of "Friends" (probably while working on a blog post) and saw this intro. I can't embed the video (but a link is included here), but did include screen shots. Basically, Chandler has a Gym Membership that he doesn't use and can't get out of.
Here's the thing: I think that in order to create TRUE mathematicians, kids need to "wonder mathematically" much more often. Sure, they might laugh at the joke, but do they ever "wonder" what those numbers actually mean? Do they have any number intuition? Can we GET kids to wonder mathematically? My EQUATE thinking model asks kids to do just that. Unfortunately, I'm not sure that this is cut out to be an EQUATE type of problem.
Grade Level: 4 & 5 (Though, a fun, quick warm up at nearly any grade!)
Course: 4th &5th Grade Math
Standards: 4.NBT.1, 4.NBT.5, 4.NBT.6, 5.NBT.6 SMP: MP1, MP3, MP4 Skills: Multiplication and Division, Unit Conversion
How to use this as a mad minute:
You have 60
seconds. Estimate how long it has been since Chandler went to the gym. Estimate how much money he has wasted. How to use this as a warm up:
You could ask the students to consider one of the following:
1. How long has it been since Chandler went to the gym? Write your answer in weeks and months. Should your answer be written in years? Why or why not?
2. How much money has Chandler wasted by not going to the gym?
3. List 3 things that Chandler could have purchased with that money.
4. Explain how you could have estimated the answers to problems 1 & 2 using mental math. How to use this as a mini-lesson:
Let's be honest. This isn't going to be a major lesson at any grade level. It's going to be a silly, fun, and quick activity to focus kids on thinking mathematically and to be aware of the math around them every day.
Usually I offer 20 minute mini-lessons. I feel that if you want to use it as a mini-lesson, expand on the questions outlined in the warm up. Give them time to reflect with partners, work with groups, explain their thinking and show their work. REALLY focus on number sense, estimation, and mental math. Consider giving a small prize (pencil? Sticker?) to the student who estimates the length of time and expense most accurately.
How to use this as a full lesson?
See above.
However, here are several other math "fails" in TV and movies that might spark discussion in classes for basic math.
I don't know the name of this movie/show, but it is Ma and Pa Kettle.
Abbott and Costello (Selling vacuum cleaners)
Abott and Costello (Donuts) Note: The math logic is the same as the vacuums!
Abbott and Costello (Two Tens for a Five)
Abbott and Costello (It's Payday!)
How to use this as an assessment?
I wouldn't! If you want to, choose one of the videos above and ask the students to explain the flaws in their thinking!
I can't make this post fit my normal outline. It's just too basic to include in lessons, but too cool not to post or use in some way! 
