Homeschool Lesson Plans: Math Puzzle: How can this be true?

How can this be true?

I saw this math puzzle in my Facebook newsfeed. I've seen a similar one before regarding splitting up a bar of chocolate. Similar, in some ways, but different. First, the problem:

Some students might be clever enough to solve this without any help. Wonderful to have such a keen mathematically savvy mind. For those who don't, we don't want to frustrate them. The page from which I've drawn this puzzle provides a hint. If that is not enough, they also provide a more specific second hint.

However, if we are using this puzzle as a homeschool math assignment, we might want to milk it for educational value.

What is the name of this shape?

Presumably, the child will say it is a triangle. (Which is wrong, but we'll address that later.)

What are the dimensions of the red triangle?

3 by 8

The green?
 2 by 5

Are the upper and lower red and green triangles the same? 
The dimensions are the same. The upper and lower red and green triangles are coincident.

Are the orange and lime green irregular shapes also coincident?

Yes.

How many square units does the orange shape contain?

7.

How many square unites does the lime green shape contain?

8.

How do you calculate the area of a rectangle?

A rectangle's area, the number of squares, is the height times the length.

How, then, do you calculate the area of a triangle?

A triangle's area is half the area of a rectangle with the same length and height. Basically, a triangle is a rectangle split evenly in half along the diagonal.

What is the area of the red triangle?

3 * 8 /2 = 12 cubic units. (Could be feet or inches, we don't know what the boxes represent.)

The green?

2 * 5 / 2 = 5 cubic units.

What would the top shape, ABC's area be if we used the formula for the area of a triangle?

13 * 5 / 2 = 32.5

What would the top shape, DEF's area be if we used the formula for the area of a triangle and subtracted one cubic unit for the gap?

13 * 5 / 2 = 32.5 - 1 = 31.5

Add together the individual areas previously calculated previously. What is the true total area?

Orange = 7
Lime = 8
Red = 12
Green = 5
Total: 32

The Associative Law says that it doesn't matter in what order we add number together. Therefore, it shouldn't matter how we shuffle the shapes around. The actual area of the shapes as determined by adding the areas of the component parts is 32. Both of our calculations for the area on top and on bottom in the diagram are off by .5 cubic units, one greater and one lesser.

Assuming the formula to calculate the volume of a triangle is correct, and it can be shown to be correct by logic, we have a situation where the correct formula for the volume of a triangle does not result in the value calculated for the area. Therefor, the shape is not a triangle even though it looks like one.

Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth. ~~ Sherlock Holmes, written by Arthur Conan Doyle  

The name of the diagonal line of a triangle is called the hypotenuse. We can express the hypotenuse as a ratio of the values of length and height. Therefore the hypotenuse of the red triangle is 3 high / 8 long. The hypotenuse of the green triangle is 2 high by 5 long. Dividing it out you get a ratio of 0.375 for the hypotenuse of the red triangle and 0.4 for the green triangle.

Confirmation Bias causes people to see what they expect to see. In the case above, we saw two shapes which were nearly triangular and assumed they were triangular. In actuality, the "hypotenuse" bows downward in the top figure and upward in the second every so slightly. This explains where the "hole" in the second figure comes from.

Of course the website from which I've pulled the puzzle provides their own solution, but I think it is nice to draw the children through the logic slowly. :) Hope your students enjoy! Now that they've worked through this logic problem, they might want to visit the infinite chocolate bar problem referenced previously.

Problem

Solution

Home School Lesson Plan: Comets: 3+ hours of video.

Today, Aug 6th, 2014, the Rosetta Mission reached comet #67P. I learned about this from a Facebook Post by the Jet Propulsion Laboratory.

"Comet countdown! Just a few hours until ESA's Rosetta Mission arrives at‪#‎67P‬. Live coverage starts 1 a.m. PT (4 a.m. ET, 0800 UTC)http://rosetta.esa.int/" ~~ JPL

Being a little late to the party, I wanted to promote a possible home school lesson plan involving the event. I will begin by visiting the site for The Rosetta Mission from the Jet Propulsion Laboratory's Facebook post.

Source: Rosetta Mission page

Along the left side of the image you see above, there are a sequence of circles on the Rosetta Mission page. The first circle displays the image above, which is a beautiful and clear images of the comet called #67P. The next circle will take you to a video stream of the media event as Rosetta reached its destination. This is an hour and a 40 minutes of Core Science credit.

However, I think it would be a nicer introduction to the children to begin with the cartoon videos lower in the list. The third button from the top introduces the mission in a cartoon format running 4 minutes. It should be easily digested by even very young children. This provides background and a frame of reference for why we should be interested in the mission.

The fourth button will provide another two and a half minutes. This describes the mounting excitement as Rosetta wakes from her long slumber and prepared for her work. The process of aiming the satellite and adjusting speed and trajectory to match the comet must have required a lot of planing and precision from the European Space Agency. Watching this video helps build a sense of anticipation and importance.

 For a little more detail, the following YouTube video can illustrate some of the degree of planning that goes into such missions. (2 minutes, no words.)

The Rosetta website also provides a link to a visualization of the flight of the Rosetta Mission. It is tucked away under a heading that asks, "Where is Rosetta?" which doesn't reveal the detail found by visiting the link. That is to say, it draws out the flight path of the mission which can further demonstrate the planning and, one hopes, inspire a student with the notion that some great things require a great amount of planning and patience to accomplish. (The child can look for astronomical navigation tricks like what appears to be a gravitational slingshot occurring in the early part of 2005. It might be interesting to ask they child if they can guess what is happening and why.)

The full progress of the visualization takes some time to complete. I don't have a timer to consult, but it appears as if the visualization runs at 1 day per second. The date range from 3/3/2004 to 8/4/2014. This web page allows calculation of a date range. If we trust the results, we come up with 3,806 days. Dividing by 60 results in about 63 minutes to watch the entire path. We should probably assume even the most ardent student will not possess that much patience. I'd time them and give them credit for the time they chose to spend on it, however.

If one wants to hurry past this part (reasonable, I think) you should still check out the Milestones. One of those milestones is Rosetta going into the long sleep mentioned in the video found at the fourth circle on the Rosetta home page referenced above. The following milestone occurs when "Sleeping Beauty" awakes.

The fifth circle continues to build expectation. One could skip this two minutes of "Are we there yet?" if one felt so inclined. I think there is some value in building expectation. For an enjoyable break, you might follow the link labeled "Competition Voting Open." Here you will see the photographs people who were eagerly anticipating Rosetta's arrival submitted for the "Rosetta, are we there yet?" campaign. Some should make you smile. You might also pique the student's interest in a small, related, art project.

Returning to the second circle down from the top, we have now should have sufficient interest raised. With all of the videos added together, we are getting pretty close to a full two hours of video time, all Core Science credit. I will admit that it seems as if the servers may be having some trouble today as I am writing this lesson plan. This might have to do with streaming video over from Europe or it might be an example of demand for the video exceeding the ESA's web server capacity. An alternate link to watch the video (it appears to be the same video) can be found at a LiveStream url. (If you skip ahead to 12:15 seconds in the LiveStream video you can skip a boring waiting period.)

You might be thinking that this is a lot of material to cover. However, from the "Mission" page there is so much more yet to explore.

• Rosetta at comet: First images & science results (1 hr 16 minutes)
• Postcards from Rosetta (Some pictures)
• Hello Comet (an Article)
• The Rosetta Blog

(If this all seems nice, but a little advanced for your young star-gazer, you might consider allowing them to explore the ESA's general kids website.)

Additional Viewing:

Investigating Comets I Rosetta Mission I Exploratorium (26 minutes)

An entire YouTube Video list related to the Rosetta Mission.

Homeschool Lesson Plan: Social Studies: J. K. Rowling: 3.5 hrs

Amazon Instant Video currently has "Magic Beyond Words: The J. K. Rowling Story" available for streaming for free with Amazon Prime membership. As such, this seems like a fairly good time to have a Social Studies unit on the author.

( http://amazon.com/gp/product/B005DD7SQS  )





The movie will run for 1.5 hours, approximately. The movie has received less than favorable reviews, but one can overcome this with some patience and tolerance. Certainly, it is at an appropriate level for children who wish to see a glimmer into the life of the author of one of their favorite series of books and movies.

J.K. Rowling has her own website which might also bear a review adjacent to or subsequent to the watching of the movie depending on the setup of the home. A parent can load the website and navigate with the left arrow to the beginning of the timeline view. This presents a sequence of stories originating in 1965 when her parents met on a train. Details can be found on the timeline not apparent in the movie.

If your child, like mine, has voiced an interest in being a writer later on in life, they may have a degree of interest yet remaining for this topic. For a slightly different approach, we also have a British Documentary from 2007: J. K. Rowling: A year in the life..." This documentary might be a bit dry for younger audiences. One cannot, however, fault their dedication. (1 hr)

At the end of these three efforts at learning about the author, the student should have a better understanding of the author as a person. I like the idea of assigning a little creative writing as a review. This doesn't have to be particularly well defined. Ideas for the essay might be as follows:

What interesting details of J. K. Rowling's life most appealed to you? Specifically, what do you admire about her?

Rowling's life, like most lives, involves struggles. Her mother's illness. Pressure to pursue a practical or pragmatic career over her love of writing. How one confronts the difficulties in life builds their character. In what ways would you wish to be more like Rowling? 

Have you learned anything about the life of an author or the process of writing? Would a career producing fiction appeal to you based upon what you've seen here?

Not everyone can be a successful writer, the job requires creativity, ability, and persistence. Would you be able to write a full story? Would you worry if people will like it? If so, would those fears keep you from persisting? Even with a good book, selling the idea to a publisher can be challenging and require great patience. Can you see yourself sticking with your dreams despite setbacks? 


The first serious relationship a woman has upon leaving her parent's home can seriously impact her life. She moved away from England to Portugal, why do you think she chose such a move? Her romance and marriage to her first husband went quickly. What caused the relationship to move so quickly and why did it fall apart? 

People are imperfect, they do not always make ideal choices. The consequences of our decisions are not always evident while we are in the process of making life-choices. A gypsy wind blows J. K. Rowling from England to Portugal seeking for some change in life that will make her happy. She has experiences there; Memories are made. Yet, she also finds herself separated from family and friends who might have been able to advise or help. 

Returning to England, she tries to rebuild her life. What obstacles created difficulties for her? What are your thoughts about the public aid system in place? How would you feel if you were in a similar situation? 

Into every life some rain must fall. Some people could have become complacent and resigned to being a part of the system of public assistance. One woman suggests having a second child to make it easier to live 'on the dole.' Certain hurdles made it difficult for Joanne Rowling to work. Her persistence, however, made it work. 

Comparing the style of the movie, the website, and the documentary, which of the three appealed most to you? 

The timeline view allows more personal choice in which areas to explore more deeply. This may be far more interesting because it empowers the learner to focus where their interest is greatest. The the movie tries to present some of the facts with a sense of imagination and drama. The emotional content of such story telling can be more appealing and allow one to have better recall of details after the fact. The documentary will provide the most details, but can seem dry and a bit boring in the slow and methodical approach. It can be important to identify what seems to work best for the individual student. 

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