Geometry Scavenger Hunt.

 

I will have about 5 students this coming year in Geometry. Rather than beginning the year with the standard definitions, I think I'm going to give them a list of shapes to find but it will be done with certain restrictions. 

What I hope to do is to give them a list of shapes such as circle, square, rectangle, trapezoid, and others.  Their job is to find examples of the shapes around the classroom, on the playground, or even around their house.  

They will be required to snap photos of the examples (all my students have a camera) and then transfer the pictures to their computers.  Once the phots are downloaded, they are required to highlight the shape and then write a short description of the shapes they found in each photo.

For instance, if I were to talk about the picture above, I might say that there are three triangles on the front of the house.  The three triangles are actually iscoceles triangles.  There are also two windows that look as if they are square and one of the garage doors is rectangle but the smaller one is almost a square but not quite.  The front door is a rectangle. 

I feel by doing this, I am encouraging them to work on their mathematical vocabulary while I use this to assess their knowledge of geometrical shapes.  If they can only tell me there are three triangles but can't tell me the type of triangles they are, I know I'll have to cover that.  

Another reason for this activity is to show them that they can find geometric shapes all around them.  I am hoping they take photos of cars to show circles, squares, and rectangles.  I already know, they might have trouble finding an example of a right trapezoid and a regular trapezoid.  Those can be hard to find.  I am also hoping they will do a quick search on the internet to figure out what a rhombus is, or an equilateral triangle.  

I've never done anything like this before but I want to try it.  I want to see if I can move the students from passive learners to more active learners. Unfortunately, Covid caused many students to be less independent as far as learning, so I've been working on activities that will have them learning but will help them be more independent.

Let me know if you think this might work.  I'll report back in a couple of week after I've assigned it to them.  This way I can tell you how I'd change it to make it better.  Have a great day.

Warmup

 Douglas Fir burns at 662 degrees F while Western Red Cedar burns at 670 degrees F.  A cord of Douglas Fir costs $415 while a cord of Western Red Cedar costs $500.  You need to buy a cord of wood to burn over the winter.  Would you buy a cord of Douglas Fir or Western Red Cedar?  Justify your choice.

Warm-up

 

The maximum temperature that spruce burns at is 1148 degrees F while oak burns at 1652 degrees F.  How much hotter (in percents) does oak burn than spruce?

Computer Assisted Proof Of Packing Color Problem.

I think this is cool that someone used a computer to help solve the packing color problem.  The packing color problem is also known as rectangular packing problem, or the geometric packing problem.  The problem looks at how many numbers are needed to fill an infinite grid so that no two identical numbers end up either next to each other or too close to each other.

An undergraduate at the University of Chile spotted a question posed at an online forum and he fell in love with it, wanting to find the answer.  The question asked about filling an infinite grid with numbers and in order to solve it, he had to leave Chile to attend Carnegie Mellon University in the United States.

In August, this young man met with a computer scientist who loves using the computer to solve hard problems and they began collaborating to find an answer to the question.  This past January, they finally came up with the solution.  15! A nice beautifully simple answer. 

 Now let's go back in history a bit.  In 2002, some folks were looking at the question about coloring maps with certain constraints applied such as no two contiguous areas could have the same color. To work the problem, they relied on the premise that a grid that goes on infinitely in both directions is a good place to start. They made the constraint that the distance between two numbers could be closer than the number itself.  Distance was defined as adding the vertical and horizontal separation together. 

An example of this would be the number one.  Ones cannot be next to each other because their distance is one apart but they can be placed diagonally from each other since the distance is two ( 1 over + 1 up). When they finally published their results, they'd calculated the numbers needed as 22 in 2008.  One thing about this problem is that they didn't have to do it for the whole infinite grid.  Instead, they could use a small subset such as a 10 by 10 grid to show how many numbers needed because it is thought that the smaller grid repeats itself infinitely.

The young man from Chile, Bernardo Subercaseaux, originally attacked this problem by trying to sketch out various scenarios using pencil and paper.  When he got tired of this he asked a friend to design a web-based application he could use and play with like a game. Eventually, he hit a point where he couldn't move any further because it is harder to prove can't be covered by a certain group of numbers than it is to prove they can.  

Once he began collaborating with the computer person, he was able to expand his search and discovered 15 numbers is the perfect set and they also were able to rule out 12 numbers in the subset.  Shortly after they realized that others had concluded that the answer could not be less than 13 or more than 15.  In fact, there was 20 years of research on this particular problem.  

In order to effectively use the computer, they needed to find ways of limiting the number of combinations the computer had to try.  They didn't want to use brute force but find a way that would be more elegant. They were able to acknowledge that many of the combinations are pretty much the same.  For instance, certain locations are essentially the same such as one up and one over or one down and one the other direction. These two are symmetrical to each other so it's not necessary to check both. 

Using this and a couple of other constraints, they were able to rule out 13 as a possibility in January of 2022. Then redoing a couple of things, they were able to rule out 14 in September of 2022 and thus came to the conclusion that 15 was the proper answer.  If you want to read the article it's here. 

I think this is cool.  Let me know what you think, I'd love to hear.  Have a great day.

Different Types Of Choice Boards

 

For the past several years, there has been a movement to give students the opportunity to take more control over learning.  This can be done in tests, or assignments but not always in what they want to learn. However, they can be given a choice in how they practice the topic or concept.  One way is through the use of choice boards which are also known as menus or tic-tac-toe boards.

Choice boards are usually a subdivided board filled with activities that students choose from.  It might bar 3 by 3, 3 by 4, or even larger and each square contains a suggested activity .  

The choice board could contain the option of creating a rap song, making a visual representation, a poem, play a board game, do a few real world projects but these activities are designed to help the students learn. The board you create will have a specific focus so the choice in it will be different based on the topic or concept. 

When we use a choice board, we offer a powerful learning tool to our students because choice is a powerful motivator.  In addition, it allows students to do activities they want to do and are more comfortable working.  Students have more choice over how they progress through the unit, and it provides more time with teachers to conference with students. 

There are various types of choice boards.  There is the standards aligned choice board with choices that help students meet the standards.  Each column focuses on a specific standard while the choices within the column allow the student to choose an activity which helps them meet the standard. 

Another type of choice board is one that focuses on strategies such as problem solving.  The type of problem solving might be focused on word problems, figuring out the first step, converting words to numbers, or any other strategy.  The strategy board might not focus soley on math, it might focus on reading or vocabulary skills which is just as important in the math classroom as math skills.

Then there is a thematic choice board.  This one might be used more often in elementary school but could be used in the upper grades.  With a thematic choice board, one looks at a topic that is important but is not one found on the standards.  In elementary school, that topic might be looking at a holiday but in middle school, high school, it might be used to provide students with a brain break for students who finish first.  It might also cover real world activities, or just why show work.  

Instead of a study guide, use a study guide, make a traditional review and practice choice board.  This is the one we think of when we think about choice boards.  This board helps prepare students for the upcoming assessment by targeting vocabulary, concepts, and skills.  Encourage them to select one activity from each column so they choose the one that is most interesting to them.

You might create a performance task board that allows students to communicate their learning more effectively.  This is the one where you have students create the larger project that looks more at the overall amount learned.  It might be having a student create a TED talk, Infographic, create a slide show, etc.

Once you've decided which type of choice board you want to use, think about creating three different versions (advanced, regular, and scaffolded) for the different levels of learners in your classroom. This way you provide the correct level of rigor for all students in your class.

Next time, We'll look at the types of things you can use in a choice board.  Let me know what you think, I'd love to hear.  Have a great day.

Manufacturers Suggested Retail Price, Discounts, Markups, etc.

 

I was talking to someone the other day about how the had trouble budgeting because they kept receiving emails offering them things for discounted prices.  You know the ones!  This is your last chance before the price goes up, or hey for the next 24 hours, we'll give you this product for 35% off so instead of $47, you can buy it for $17.

Since COVID, I don't think we've done much to teach discounts, mark-ups, and even sales tax.  In addition, when the topic is taught, it doesn't cover things like MSRP or Manufacturers suggested retail price or the price that manufacturers suggest the price be set at. This suggested price is about 2 to 3 times the wholesale price but stores set their own prices.

This is when stores use it to their advantage by using a little sales psychology.  When I was quite young, I asked my dad why prices ended in 9 rather than 0.  He said when the price was $9.99 rather than $10.00 so the customer would feel as if they were getting a bargain.  The same thing with stores that list the MSRP and the price the item is actually being sold for.  This makes it seem as if you are getting a bargain.  Some stores use the term Rollback to persuade you, this is a deal.

The other way is to tell. there is some sort of sale and what you want to buy is marked down a specific amount so instead of paying $52.00, it is only $39 because they've marked it down 25%.  This is the discount they take when you check out.  I don't know if the price printed on the tag is the MSRP or the price the store set but you feel good when you get the deal.    I love shopping the clearance because the discount is being applied to things that have already been discounted.  This is an area that needs clarification for many students because some think that the discounts should be added together.  In other words, if the item is marked down 50% and then reduced another %25, the kids think the item should be marked down a total of 75% rather than the last discount applied to the new price.  

On the other hand, most students are unaware that most merchandise is sold at a lower price by suppliers referred to as wholesale.  Then suppliers markup the price to the amount they sell it for.  Most of us do not know what the markup price is and that markup is not always the same.  It depends on the product.  Usually gas is only marked up some 10 to 15 cents a gallon but out of the markup they only make about 2 cents net profit. 

It is important to take time to teach students about markups, discounts, even profit and net profit so they understand these real world situations, they will end up in at some point.  It is important to know what these terms mean and their application because it is something they run into regularly.  In fact the applications and such are a bit different when in the bush.

Where I live, the stores often order products from the local Costco, have them shipped out and the markup is on top of that. I usually include information on what the markup covers such as the building, salaries, etc.  I try to include one week of this information so students are exposed to it.  Let me know what ou think, I'd love to hear. 

Warm-up

 A cord of wood is defined as being 128 inches cubed.  What different ways can you arrange the wood to create a stack measures 128 inches cubed?