Google Expedition Lessons For Math

I've been interested in using Google Expeditions in the math class but I really didn't know where to start.  I've seen expeditions for most every subject but Math so its a matter of starting from scratch.

The other day, I stumbled across a website by Kathi Smith that has several Math based Google Expeditions.

The lesson plans range from grades 3 to 12 and the topics range from pyramids to volume of recycling materials to the New York Transit system. Each lesson plan has the title, standards, worksheets and everything needed to carry out the lessons including the Google Expedition.

From another website by Sheil Spiel, talked about using Google Expeditions in a Geometry class.  Use the Expedition of Machu Picchu to discuss the geometric characteristics of the attraction while exploring the area. Then have students calculate the angle of elevation from the base of the Eiffel Tower in Paris and Big Ben in London introducing both via High Points of Europe:Tour of the Towers.  This requires students to do a bit of research to find their heights and use the height plus distance and right triangles to find the elevation.

I took time to check out Expeditions that directly deal with math and there are several.

AR based Expeditions.
1.  Geometry : Types of triangles explores right, acute, obtuse, isosceles, and equilateral triangles.

2. Math in Structures which looks at the Pyramids, Chichen Itza, Eiffel Tower, Spider Web, and Honeycomb.

3. Probability covers rolling a dice, flipping coins, cards, bingo, and the lottery.

4. Rotation and axis which is on the seesaw, baseball, rotational orbit vs spin, and flight dynamics.

VR based Expeditions

For any of these, it would be looking at the mathematical application based on which expedition you wanted to do.  For instance, you might show one on the art of Egypt as a way of introducing the thirds rule for bodies.  Or look at the Leaning Tower of Pisa to introduce cylinders in real life.  These virtual ones are more about whats.

One thing I discovered while researching this topic is something called Google Arts and Culture where they look at artwork in galleries.  The cool thing is they have a bunch of artwork that falls under the topic of math.  This section looks at mathematical applications in art including symmetry, tessellations, perspective, grids, fractions, proportions, rotations, shapes, and the golden ratio.

The first page shows the paintings but if you click on the painting, another page comes up with the picture and provides more detail on the painting itself from size to general information about the topic of the painting.  Something new I can add to my repertoire for teaching come fall.

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

AR Geogebra

I downloaded this app about 2 months ago but wasn't able to do much with it because I couldn't figure out how to use it but the other night, after a quick search, I found an instruction manual with step by step directions. 

 First things first.  You can get the Geogebra Augmented Reality app from the App Store.   It is free to download and use and the app allows you to take photos with your camera.  

The picture to the left, I used two equations:  z = sqrt(9-x^2-y^2) and z=-sqrt(9-x^2-y^2) which I got from the manual.

The manual, Augmented Reality: Ideas for Student Exploration, begins by having students input certain equations that form a circle, ellipse, parabola, and hyperbola so students can view them in 3D.  I used the two equations they gave me to create two different shapes.  The first is above and the second is to the right.

I like the way they've built some play time into the introduction so students can explore a bit while learning to use the app.  In addition, there is a short video showing the same things.

The instructional manual begins with an exploration of conic sections before heading off to explore transformations in 3 dimensions which give students a different way of "seeing" the actual shape.  

The activities have you create a surface of revolution by rotating the function around the x-axis before creating a surface of revolution using the y-axis.  The final activity for this has the student creating a Gabriel horn and more on creating surface of revolutions.

Geogebra has included a section on AR Modeling tips and hints to help students create these awesome shapes.  Some of the topics cover domain restrictions, duplicating objects, going from 2 D to 3D and several more.  The final sections are filled with easy and harder challenges such as building a square prism with heart shaped cutouts.  They provide hints and a silent video showing the completed object in more detail.

The second to last section shows AR demonstrations made by other people with attempts, final equations and questions for people to think about and answer.  One of my favorites in this section is the replication of a Pringles potato chip in 3 dimensions.  The video includes the equations used so students can do it themselves.

The final section is titled "Quick Illustrations" and goes over topics for teachers.  I love this manual because it allows me to familiarize myself with the program before I let my students use it.  I've found it helps if I can get them started or at least provide them with the material to get started.

Download it, give it a shot and have fun playing with it.  Let me know what you think, I'd love to hear.  Have a great day.

2 New VR Apps.

It is the time of year where I check for VR apps for math.  I'd like to integrate some into my math classroom because some of my students are more willing to play a game especially if its VR or on the computer.

I found a recently released game called "Medieval Math", a VR game in which the player is defending the place from invaders.  In order to get more power, you have to answer problems but you as the player can choose from addition/subtraction, multiplication/division, fractions, or Pre-Algebra.

After defending your castle, you can travel to help defend other lands.  This is not a static game, it is designed to be adaptive, provide information on student progress, and suggestions on areas the student needs to work on.  It has great reviews and has versions for both IOS and Android phones. It does require a headset with button to use.

On the other hand, there is the BuckeyeVR 3D plot Viewer created by members of the faculty at Ohio State University.  It allows you to visualize functions, vector fields, and parametric surfaces using virtual reality.  This is actually the viewer because you have to create the plot here before reading the generated associated QR code with your phone.  Once it comes up in this app, you check it out using Google Cardboard or other VR reader.

In addition, this computer based creation site includes a users guide dated January 2018 with everything you need to use the site.  Furthermore, there are versions for both Android and IOS and the best thing is that it is free.  Even the computer based creation program is free.

These are two new VR math apps, well new to me, that I've not seen before.  I have not actually tried either app because my google cardboard is packed on its way to my new school district so I can't try it but I plan to check both out in August, once I've unpacked, I plan to try both.  If you have used either one, please let me know, I'd love to hear.  Have a great day.

Designing The Ideal Bedroom.

In geometry, I love having students create their idea bedroom before calculating the cost of buying paint, carpeting, tile, ceiling tiles, wood flooring to finish the room.

They design the room, figure out where windows, doors, fans, furniture, closets, bathroom, etc go.  Then they must calculate surface area of each wall, floor and ceiling so they know how much of each product they want to buy.

If they want to put in a fan, hanging mirror, painting, set in bookcases, window seats, or anything like that, they are required to research the cost for each item so after they've designed it all, they are required to write out a proper estimate for the cost of finishing the room.

Although I have often had them do this on graph paper, I've since discovered there are apps that allow people to create three dimensional representations of their rooms.  Some of the possible apps for iPad include Home Design 3D or  Live Home 3D or even Keyplan 3D Lite which allow people to design and create 3 D renditions of the plans.

For students who have difficulty visualizing their ideas, these are great because they are set up to help the student draw the scale drawing, furnish it with lights, furniture, everything you need and then the apps turn the 2 dimensional into a 3 dimensional representation so a student can get a better idea of size.  In the past I've had students put small basket ball courts or movie theaters into their bedrooms and often time their furniture was out of proportion with the rest of things.  For instance, the bed might be done so it was 3 times larger than the actual space but they couldn't see it.

I chose this project to end my unit on surface area and area because it gives students the opportunity to apply rounding, surface area, visualization to a real life situation.  They also get a taste of what the interior decorator, architect, and contractor jobs require as they design and create a bill for the cost of their rooms.

There are so many places online they can go to get costs for every piece of furniture, bedding, bookcases they want so they get a realistic idea of the actual cost.  They will discover if the items they want will even ship to their bush addresses. Some are too large or have too much of a shipping cost.  Sometimes, I'll have students add a 25% shipping cost to large items and I'll do some research to get basic costs for paint, carpet, tile, and a variety of flooring.

At the end, they turn in pictures of their room with a itemized bill listing everything and associated cost.  They can provide pictures of the plan and the three dimensional representations so I know what they see.  They enjoy it and I give the whole period for a solid week to this project.  Let me know what you think, I'd love to hear.  Have a great day.

More Census Math

Last week, I began exploring the Census Bureau and shared the first activities on predicting and growth.  Today, I'll be sharing the activities they've designed to use real life data with statistics.  The activities are designed for grades from Kindergarten to Twelfth so even the younger ones  have the opportunity to explore statistics.

For grades K to 5, there are 10 different activities ranging from using fractions to compare amusement parks by state to learning more about college degrees and lifetime earnings.

The amusement park activity, suggested for grade 3, begins with students brainstorming the number of amusement parks in their state which for Alaska is zero.  The only amusement rides we get are from visiting companies for the state fair and other things.  The second step is to have students use the data from 2016 to create a visual representation of the number of amusement parks in each of the 50 states in fractional form.  Students use the visual representation to answer questions based on the data. The activity comes with both the teacher and student versions for this activity.

For grades 6 to 8, there are 15 activities available ranging from fitting line to data for education and earnings to two way tables for biking or walking to work, to interpreting dots and box plots to creating and interpreting histograms.  These 15 are more focused on the learning more about the different types of graphs, etc.

I checked out the exercise on understanding the distribution of data in regard to  pet food manufacturers.  The suggested grade level is sixth and is expected to take one class period.   The students look at sorted and unsorted data, histograms, dot plots, box plots, and one more to learn to read different forms of data so they can answer questions about pet manufacturing.  Most of us just look at one form of data when teaching probability and statistics to students and this gives them the chance to read multiple forms at once and interpret what they've read.  A very good exercise.

For grades 9 to 12, there are nine activities ranging from educational level and marriage age, to aging, to differences in earnings and marriage age based on age and sex.  Each topic has a suggested grade level and time.  The aging activity provides students with data from 136 counties throughout the United States for them to interpret and use to estimate populations of people  65 and older.

All lessons come with objectives, materials, suggested grade level, time, everything you need to teach a lesson.  This site also offers warm-up exercises, a variety of resources, and so much more.  If you want some realistic statistic activities designed with real life data collected by the Census bureau, give this a shot and have fun.

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

Warm-up

If two people need 30 feet for an RV, how many feet should 6 people have in an RV?  Is that reasonable?  Explain your answer. 

Warm-up

If each person needs 49.5 square feet of space in a tent, how large a tent will you need for 8 people?  Give its measurements in addition to the amount of space needed.

Celebrating a 90th Birthday.

This is a picture of a woman who just celebrated her 990th birthday.  Her husband who is sitting next to her turned 95 back in February.  They have been married about 65 years.

The statistics of anyone living that long is rather interesting.  According to a document released in 2011 by the census bureau indicates that between 1980 and 2010, the number of people in the 90 and over group increased from 720,000 to 1.9 million people or an increase of almost three times.

This group now makes up 4.7 percent of the 65 and older population, up from 2.8 percent in 1980.  It is predicted that this age group will make up 10 percent or 7.6 million of those over 65 by 2050.  The majority of people in this group are women in a ratio of 3 to 1 but they have a higher rate of disabilities than any other group.

Due to the growth in this group, the lower limit of the "Oldest Old" has increased from 85 to 90.  It its noted that 20 percent of the 90 year olds are in nursing homes but the number increases to almost 40 percent for those who are 100 or older.

The couple who are in this picture are released to me and I wondered how many people managed to live this old.  I found the above information in various documents from the Census Bureau.  They offer lesson plans for grades K to 12th for teachers to use in the classroom.  They have a whole set designed to teach students about the 2010 census taken just 9 years ago.

I checked out the lesson plans for grades 9 to 12 dealing with sampling and data.  One lesson discusses the three types of sampling - random, cluster, and systematic - along with examples.  It goes on to talk about finding the average heights of high schoolers between 1940 and 1970 before asking students to design their own sampling activity.

The other activity deals with estimating populations and predicting future growth.  The lesson begins with covering the basics including vocabulary and methodology before getting to the actual activity.  The first activity has students calculate population both numerically and percent for the growth between 1970 and 1990. The last entry is one the student has to find the populations for 1970 and 1990 before doing the calculations.

The second part requires students to calculate the estimated population for each state in 2010 based upon the information from the first activity.  At the end there are questions such as why is the percent increase more than the numerical increase and more.  Questions that require some serious thinking to answer the "why".  This is nice because its important to do more than just calculate increase.  It's important to interpret the results and provide an explanation.

Check back Monday for more on the statistical activities offered by the Census Bureau.  Let me know what you think, I'd love to hear.  Have a great day.

Universal Lesson Design in the Math Classroom.

 I am using the summer to catch up on learning more about lesson planning so I can improve my lessons.  I know I have to work with certain things but I can still make better lesson plans.  I'm working my way through Matt Millers "Ditch That Textbook" Lesson Plans.  One thing he's discussing is Universal Lesson Design as part of lesson plans.  I've heard of it, read some about it but I have stumbled when it comes to using it in Math.

In Universal Lesson Design, it asks the teacher to provide multiple means of representation which is fairly easy in mathematics because you can use drawings, manipulative, symbols, words, use apps to provide the representation, or use real world examples.  The material can be presented via the textbook, via edited videos with questions, and quizzes, or it could be read out using adaptive choices on the computer itself.  So this section is designed to provide a variety of ways to acquire both information and knowledge.

This is also the part of the lesson to clarify vocabulary and its well known that mathematical vocabulary can be general or have meanings that are both non mathematical and mathematical such as product which means the end result of a multiplication problem or something you make to sell or it could be mathematically specific such as a torrid shape.  It is used to activate prior knowledge, translating mathematical symbols into useable form, and maximizing transferral of knowledge.

A second element is to provide multiple means of actions and expressions. This can cover everything from worksheets, to completing problems on google slides, flip grid, or other digital manner.  This is the section to utilize appropriate technology, build fluency, and improve goal setting.  It is the section that provides a variety of ways of demonstrating what they know and what they've learned.

The final element is using multiple methods of engagement by providing choices, minimizing distractions, encourage collaboration and community, and increase appropriate feedback.  This is designed to both motivate and challenge them.  One way to provide motivation is to provide both choice of what to do and choice of rewards.

Personally, I understand all the parts. but I'm never sure what it should look like when I write lesson plans so I found an example here with each part labeled either representation, engagement, or expression, each in a different color code.  This goes with the idea of color coding your lesson by using the colors for the activities themselves.  This lesson is for Algebra but it gives a great idea of how to form a lesson with the UDL elements.

Its easy to find examples of lesson plans for high school math but the one I choice has a better example of showing the individual parts identified.  For me, this gives me what I need to start doing better in writing lesson plans.  I hope you find this helpful.  Let me know what you think, I'd love to hear.  Have a great day.

Math Wire

As a teacher, I rely heavily on material I find on the internet.  I don't usually have a library available to visit for material, I don't have other high school math teachers to collaborate, and I seldom get professional development I can use so I go to the internet.

The other day I came across the Mathwire website. This site had tons of activities, links, and information to make a teachers job easier.

The site allows people to go through material either by alphabetical order, standards, or a inactive blog.

Before I go any further, I'm going to let you know that much of the material is dated back  around the early 21st century with comments about putting the templates in a plastic sleeve so students can use dry erase markers to fill out the templates.  When done, they can be cleared up.  These templetes can be digitized and opened in a variety of apps which allow students to annotate them.

One blank templet is the triangle used to show how multiplication or addition facts relate to division or subtraction.  There are frequency tables, spinners, grids, function machines, and all sorts of other templates that can be used in a classroom.

Another suggestion is to have students use white boards but those are becoming a thing of the past because they are being replaced with apps.  You can still do white board activities on either a white board app or a doodling app.  The nice thing about using mobile devices is that you do not use a ton of dry erase markers.

There are pages devoted to specific topics such as coordinate Geometry.  Although many of the activities are actually designed for elementary students, with a bit of adjustment, they can be used in middle school or high school.  There is a penguins game which uses letters and numbers in the first quadrant but with a minor adjustment, make it both numbers on the X and Y axis.

Basically, two people fill a grid with penguins.  They then roll two dice and based on the results of the roll, the student will remove a penguin from that location.  A player gets 10 points if they remove their own penguin, 20 if they remove the other person's penguin, and the first person to reach 100 points win.

Another activity is called Who Has?  The examples are set up to use multiplication tables such as "I have 0, Who has 5 x 7?"  and the person who has the answer speaks up and reads their Who Has question.  This means all students are involved and it gives them a chance to practice their multiplication but this game could easily be extended to practicing division,

 The site has a "Who has" which uses expressions in written and verbal form.  It might go like "I have 3x, Who has two times a number? but it would be quite easy to make one for solving one, two, or multi step equations. This might go as "I have  x = 3, who has 2x + 1 = 13" to give students actual practice.  If I do something requiring more thought, I give students time to work the problem read out so they get practice but only one person is going to have the answer.

I like sites like this because it sometimes has activities I can easily use or made adjustments to or it reminds me about activities I've done before but let slip to the side.  Later in the summer I"m going to make some Who Has decks for my algebra classes.

As stated earlier, many activities are for elementary students but if your students are well below grade level, you can utilize many of these activities in class.  Most students like playing games or doing things that allow them to stand up and move around. Let me know what you think, I'd love to hear.  Have a great day.

Hotel Impossible and Stats

I have spent the last couple days watching the show "Hotel Impossible".  The premise is that someone goes into failing hotels that in  horrible shape, figures out why it's failing and then helps it by renovating the lobby and one room, giving it professional help, all new linens, and an updated website.

Yes, its a modern day reality show with all the drama  of the unaware owners, the employees doing a job they'd never been trained for and all the rest but the one saving grace it has is the use of statistics.

At various points in each episode, the host announces the occupancy rate, often in comparison with other hotels.  Sometimes he discusses the amount of cost connected with food served in restaurants which should't run more than 30 to 35% but at one place it was a whopping 55%.  Other times, he discusses setting the price of rooms based on interpreting data.

That is a reason I'm sitting through four seasons of the show.  In just about every single show, he discusses something mathematical about the running of the hotel.  In one episode, he had someone create a graph comparing the hotel's income versus other hotels in the same area.  While every other hotel had gone down about 3%, this hotel dropped almost 80%.  A graphical representation of the information.

In another episode, he discussed increasing income by creating regular events for everyone.  Usually, the guy would connect the hotel owners with people who could put packages together to increase occupancy.  For instance, in one episode, the owners of a hotel could gain 30 weekends a year from the local Symphony which was located across the street.

This show uses industry stats, industry vocabulary so after a few episodes you know what occupancy rates are, you know what the STAR report (gives average occupancy rates), you learn how hotel's set their rates and why the rates go up or down.  You learn so much about the real life application of math in the hotel industry.

In addition, students learn so much more about what is involved in running a hotel.  Just by watching a few episodes, I learned what the General Manager's job is, what responsibilities the head of housekeeping or maintenance should have and so much more.  Yes, as I said, there is drama, lots of it and each episode follows the same script but the stats included change from episode to episode depending on what is being emphasized.

You can check out episodes for free on Amazon Prime.  I'd advise you watch some of the episodes to find one with a bit more math in it so you can spend more time discussing the statistics.  You can even go into the reason these stats are important.

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

Math and Quilting

 Quilts have been a part of American history since people came over in the 17th century.  Various patterns have developed, some are said to have been used to direct escaping slaves while others tell us more about the material of the times.  I have relatives who made quilts but we don't always look at the math behind quilting.

One thing about quilting is the number of books on quilt blocks.  A quilt block is one square with one complete pattern.  Each pattern has a unique name and is made out of polygons.

The majority of shapes used are triangles and squares which are put together in different patterns to create trapezoids, rectangles, pentagons, and parallelograms.  Eventually they are connected together to create a quilt block.  These blocks are then connected together into rows and the rows are sewn together until the whole quilt is built.  Usually, its big enough to cover a bed.

Regular quilting pattern.

A mathematics professor, Irena Swanson, created a new method of quilting which uses fewer seams and much quicker than the traditional method.  Usually, you cut the individual shapes, sew the pieces together, slowly.  Rather than doing this, she created "Tube Piecing".  In this method, she sews together strips of alternating light and dark fabrics that are slightly offset to create parallelograms.  Then she sews the strips of parallelograms together to form tubes.  As the last step, she cuts the tubes into rings so the cuts are perpendicular to the seams forming quadrilaterals.  These quadrilaterals are repositioned and sewn together to make quilts.

If you look past the standard forms, you can find patterns made out of hexagons, or hex rings made out of 18 equilateral triangles.  In fact, some quilters have not restricted themselves to traditional patterns.  If you look, you'll see this beautiful Mobius quilt that has no backing because of the way it was designed.  Its classified under topology.

Some artists have used tessellation's to create beautiful repeating patterns such as this one.  Furthermore, the same page has identified quilting material with tessellation's on an isometric grid.  If you take time to explore the Math Quilts site, you find quilts based on factals such as the one inspired by a Martin Gardener article from Scientific American, or even one based on the Pythagorean theorem.

Of course one can always find quilts created as optical illusions so the two dimensional blocks appear as three dimensional.  At mathwire, they offer some wonderful quilting activities you can easily include in your classroom. If you don't have material handy, you can use colored paper.  In my geometry class I had students create their own quilting pattern made out of polygons.  Unfortunately, most wanted to take a large square with like 8 triangles of 8 different colors and glue them to the paper.  This lead to a long discussion on numerical patterns versus quilting patterns.  Both have repeating parts.

One nice thing about this topic is the math part can be done in math class while the actual construction of quilts can be done in the home economics class. Check it out, Let me know what you think, I'd love to hear.  Have a great day.

If it takes 6.2 hours to travel 3470 miles from New York City to London, England, how fast did you go?

Warm-up

On average it takes 6.5 hours to drive from Budapest, Hungary to Munich, Germany.  How far is it if you are traveling 62.5 mph?

Why Memorize?

There is a lot of discussion concerning the memorization of multiplication facts.  Many elementary teachers I know think that if they haven't learned their multiplication facts by a certain age, it really doesn't matter because there are calculators and charts.  Unfortunately, as a high school math teacher I've seen the effects.

I've seen students sit there skip counting their way through 8 x 4  because they don't have fact fluency.  When they have to skip count or stop to use a calculator, it disrupts their flow and they often have trouble learning the process to solve a problem.  They have difficulty factoring, finding GCF and LCM, and fractions which makes it harder for them to pass the math class.

In addition, many students who do not know their multiplication tables, cannot divide, and when tested end up well below grade level.  This makes it more likely students will get further and further behind until they are unable to do high school math.

According to one study done by the Stanford School of Medicine, its the hippocampus in the brain that helps provide scaffolding for learning math fluency when learning moves from counting on fingers to pulling facts from the brain.

There are other studies which indicate that students who do not develop math fluency will continue using less efficient strategies such as counting on fingers thus finding it difficult to move to automatic recall.  There is a developmental sequence when children move from beginning fluency to full fluency.

1.  Finger counting where they add or subtract on their fingers.

2.  Verbal counting strategies where they add and subtract using fingers from the starting number to the answer.  I include skip counting for multiplication and division with this because many of them learned to skip count well but did not move to fact fluency.

3. Decomposing or splitting strategies which allow them to break down facts they might not already know such as 8 x 12 is the same as 2 x 12 x 4.  Too many of my students who arrive in high school do not have this strategy down and haven't been taught to do it.

4.  They are able to retrieve facts automatically from their long term memory.  When they get to this stage, they are able to focus on learning the new material rather than forgetting where they are because they just spent several minutes figuring out 5 x 8.

In addition, if students do not develop fact fluency, they have a higher chance of dropping out, rather than graduating because they struggle and eventually give up but there are things we as teachers can do to help students become more fluent.

One recommended methods to help increase fluency is by using  drill and practice via flash cards, software, or even an app on their computers.  From personal experience, older students tend to prefer the drill practice in the form of a game because they are used to games.

Another method is to use the Cover-Copy-Compare.  The teacher prepares a worksheet with no more than 10 completed problems on the left side of the sheet.  The student studies the problems, then folds the left side over so the problems are now hidden and writes down the work from memory one problem at a time.  They then check their work by comparing it with the original.  If it is correct, they move on, otherwise, they study the problem they missed, cover it and write it down before checking the answer when done.  This step is repeated as often as needed until they get it right.  At this point they move on.  They continue until they have done all the problems correctly.  A student is considered proficient when they can repeat do the same problems three times.

Students can make their own flash cards out of 3 x 5 inch index cards.  If they do it this way, they can work in pairs where one student holds the card up with the answer facing themselves while the other student tries to answer the problem.  They can also access one of those apps which allows them to create their own virtual flash cards and the app tests them so they can do it whenever they want.

I like to set up races where a paper has say 8 problems.  The students are in rows, one behind the other.  I give a paper to the first student in each row face down. When I say go, the first person turns it over and answers the first problem before passing it to the second student who answers the second problems and so on to the end of the row.  If I have fewer students than problems, I sometimes have them send it back down the row to the front until all the problems are answered.  The row with the most correct answers in the quickest time, wins a small prize.

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

Magic Squares

At some point, math teachers talk about magic squares.  A magic square is a 3 by 3, 4 by 4, or 5 by 5 square where the numbers total the same number vertically, horizontally, or diagonally.  It is claimed the first magic puzzled appeared in China on the back of a turtle.

The story goes that China had been experiencing a lot of flooding. One day, a turtle appeared with something on its back.  The emperor discovered that the markings were numbers and each row, column, diagonal added up to the same number.  He found the pattern.  He named the turtle Lo Shu which translates to Book of the Lo River.  This is the first reference to magic puzzles ever found.  Over time, different people played with magic squares including our own Benjamin Franklin and it has appeared in at least one painting from 1514.

Benjamin Franklin created a 16 by 16 magic square whose sum totaled to 2056 and is referred to as Franklin's Magic Square but most magic squares range from 3 by 3 to 5 by 5.  How do you create a magic square?  Here's how.

Lets start with a 3 by 3 grid.  The first thing you must do is find the magic constant by using the formula n(n^2+1)/2 where n = the number of rows and columns.  For this square it is 3(3^2 + 1)/2 or 3(10)/2 = 15 or the total for each row or column.

If the square as an odd number of rows/columns, you put the number 1 in the middle of the top most row.  The rest of the grids are filled in using a one up and one to the right pattern.  So if one is at the top, the one up and one to the right actually takes you to the bottom right corner where you write 2.  Continue the one up and one to the right and place 3 there.

At this point, you might notice if you continue the one up and one to the right, you end up at your starting point, so you move one down and write 4 there and repeat the up one and right one for 5 and 6.

Again, if you went up one and right one you'd end up at 4 so instead you go down one and write 7.  From there you can finish the rest of the numbers using the up one and right one pattern until you have finished your magic square.

The bottom line is use the up one, right one pattern until you start to repeat, then go down one and begin the whole pattern again.


This process works on any odd numbered magic square.

What if you want to make a 4 by 4 or other even magic square.  How would you do that one?  Well that actually is a bit easier.  Lets look at doing a 4 by 4.  First,  we find the magic number which would be 4(4^2+1)/2 or 4(17)/2 or 34.  Then you would place the  1,4, 6,7, 10, 11, 13, and 16 in counting order so it looks like this.

When you are done, you should have the square so it looks like the one above.  Then you fill in the remaining spaces with the remaining numbers starting at the bottom right next the the 16 and count up so your grid looks like the one below.

and all the rows and columns should add up to 34, the magic number.  For multiples of 4 such as 8 by 8, instead of using one square at each corner, you'd use 4 at each corner with the middle 16 for the main numbers just as you would as you were counting.  Then begin at the bottom and do it in reverse.

For a 6 by 6 which as 36 squares you actually end up doing 4 different 3 by 3 columns only using slightly different numbers.  The top left grid uses the digits 1 to 9, the top right grid uses the numbers 19 to 27 while the bottom left uses the digits 28 to 36 and the bottom right uses 10 to 18 and they all follow the one up and one right pattern with the beginning number in the top middle square.

These are a great way to introduce patterns because to complete the magic squares, you have to follow general patterns which can be applied to larger squares depending which type they are.  Let me know what you think, I'd love to hear.  Have a great day.

P.S.  Sorry this is so late but I've been without internet for a few days and it died before I could write this one over the weekend.

Traveling Today

I am heading out today which means I will listen to the VHF for my flight after checking in with the agent.  As soon as the pilot radios how far out he is, we grab a side by side 4 or 6 wheeler, throw the luggage in and head off for the airport.  Once to the dirt strip, we board the Navajo 206 or similar while they load the plane and we fly to Bethel where I"ll change to a major airline because I trust them more.  I should arrive late tonight.

Math Behind The Tallest Buildings.

I was checking out the tallest buildings in the world which got me to thinking about the math behind them.  There has to be some, otherwise they wouldn't having buildings as tall as the Burj Khalifa in Dubai, towering 2,717 feet above the ground.  The building itself has 163 floors made up of a hotel, offices, and homes.

Traditionally, the ratio for height to width is in the 8:1 or 9:1 range but with taller buildings it could be 15:1 such as 432 Park Place.  To put this in perspective for children, you might want to stand a ruler on the table so its smallest width is on the table with the longer edge going up to show a 12:1 ratio.

The problem with building that have a ratio higher than 9:1 is the cost increases due to needing thicker walls and additional technology to reduce the sway of buildings.

The biggest enemy of tall buildings is wind.  When a new skyscraper is designed, a model is created and put into a wind tunnel to see how it will react to the occasional 100 mph winds.

According to research, they've discovered they do not want smooth buildings because the a design with vertices, rounded edges, and layers such as in the picture break the wind so it is not as forceful.  Furthermore, they incorporate dampers which act as counterweights to shift and stabilize buildings.  This is done by building in a  300 to 800 ton piece of steel or concrete on a floor near the top.

Engineers use two different types of dampers in today's buildings.  The first is a tuned mass damper that works much like a swinging pendulum in a clock, and has been used since 1958, while the other is a slosh damper  or slosh tank that uses water to absorbs vibrations.    Tuned mass dampers often use small motors to adjust the building in the opposite direction to the wind and some use magnets to do the same thing.  These dampers can add up to $5 million to the cost of the project.

In addition, over the years, engineers have made concrete both lighter and stronger by changing the additives to include fly ash, slag, and micro-silica thus improving it.  Furthermore, they are playing with the way concrete and steel are put together to improve its strength without adding extra weight.  There are buildings being built right now that are expected to be higher than the current 2,717 feet because people want to be known as the one with the highest building.

The current title holder used a buttressed core design which is sort of a three wing spear due to its stability, usable space, and less loss of building to structural necessities.  The company that uses this particular design feels that buildings could be build so they are two to three times the current record.  However, engineers and designers are working on other technologies that might allow additional heights while using the current technologies.  Architects have come up with other designs but the technology has not caught up yet with the designs.

Unfortunately, elevators are one of the current limiting factors on tall buildings because there is a limit to the length of the steel cable that moves elevators up and down.  The current limitation for rope is about 500 meters or just over 1600 feet.  An elevator can weigh up to 60,000 pounds which puts quite a strain on the steel cable and uses over 130,000 kilowatts of energy each year.

There is a company in Finland working on creating a new carbon based elevator rope but its still in development as far as I can tell.  Once they've got it up and working, it will remove the current limit and allow elevators to travel longer distances.

Due to travel, I won't be here tomorrow, other than having an in transit picture.  As you can see tall buildings do require quite a bit of math.  Let me know what you think, I'd love to hear.

Warm-up

This is the tallest building in the world.  It is 828 meters tall with 168 floors.  What is the average height for each floor?

Warm-up

If it takes an elevator 58 seconds to reach the top of the CN Tower and the elevator's average speed is 15mph, how tall is the tower?

Mathematical Habits of the Mind.

One advantage to having internet access is that one can explore various Master's and PhD papers and dissertations.  I love finding research to help me learn to become a better teacher.

This paper "Five Processes of Mathematical Thinking" by Toni Scusa caught my attention because I'd never considered the processes connected with mathematical thinking.

The author identified the five key areas as representation, reasoning and proof, communication, problem solving and connections.  These are also referred to as "Mathematical Habits of the Mind."

Let's look at each area in more detail and why it's important.  When a student is able to create representations designed to show mathematical concepts or relationships, they have gained tools to expand their ability to model and interpret a variety of situations.  Furthermore it shows they understand the material better.  Representations can range from simple drawings to the use of manipulative including tiles, and Legos.

When a student is able to reason well, they find it easier to think analytically so they can "see" patterns, and structure in both real world and mathematical situations.  With good reasoning skills, students can determine is patterns occur regularly or if they happened by accident.  Furthermore, they are able to create and investigate conjectures, and create and argue for their proofs.  They know why.

As far as problem solving, students need to have persistence because if one method does not work, they need to stand back, review, and try again.  When a student knows multiple ways to problem solve, it allows them to be curious about things and it builds confidence to attack problems both in and out of the classroom.

When students are able to communicate clearly they are able to use mathematical language concisely in either verbal or written form.  These four lead to the student being able to connect prior knowledge to the current situation, concepts to the abstract application, and connect similarities among the processes used to solve problems such as the distributive property and its application to multiplying binomials in Algebra.

All of these processes of mathematical thinking or mathematical habits of the mind allow students to think about mathematics in the same way that professional mathematicians do.  The nice thing about habits is they are automatic which from a brain point of view means students are able to focus more on the concept rather than simple calculations.

Another thing about habits is that they can be good or bad.  Some bad habits mathematically is a person looks for a fast answer, gives up quickly because its seen as "too hard", prefers to memorize things rather than have true understanding, guesses a lot.  On the other hand, good mathematical habits are a willingness to stick to a problem until the answer is found, a desire to explore a curiosity, visualizes, makes comparisons, and noticing the patterns, etc around us.

A simple paper sent me off to an exploration of a topic.  I'm currently exploring speech to text for mathematical equations, etc.  I hope to report on this next week because I'd like to have students talk out solving a problem and have it show up in written form.  Let me know what you think about either topic, I'd love to hear.  Have a great day.

Revisiting Khan Academy

  We now have students who have had access to technology all their lives so they think nothing of using technology for snapping photos, checking out the latest on You Tube, texting or posting on various social platforms.  They are used to immediate feedback with this internet lifestyle.  I read somewhere that they have an attention span of eight seconds which is really not very long.

One of the resources I've used in the past is the Pixar in a Box course with Khan Academy.  I had quite a few low performing students who needed a math credit and didn't do well in a regular math class.  We had fun with the topics and I integrated more in-depth math problems because my superintendent told me it had to have a lot of math, so we just explored the associated math in greater detail.  At the end, they worked in groups to produce animated shorts.  They results were cool and we made a movie with all the shorts for the parents.

I haven't visited the site much this past year due to internet issues but I was surprised to find more material for math teachers and students.  They now have middle school math from Illustrative Mathematics which is a problem based curriculum.  From their web page, they say it is designed to connect procedure with concepts.   I checked out a couple of lessons to see how they worked.

I checked out an 8th grade quiz on rotation around a central point.  There were four questions to answer, some asked for clockwise while others asked for rotation in a counterclockwise direction.  The nice thing is the immediate feedback with information on why the incorrect answers are wrong.  That is really nice because its a step more than "It's Wrong".  Most lessons have a short video with an available transcript so a student can read the material before viewing it.

Another one they have is Eureka Math and Engage New York which have some great lessons.  One provides remedial material for grades 3 to 8 while the other has grade level material for grades 3 to Pre-Calculus.  The remedial material for grade 8 is broken down into modules with a few topics that the student works through.  Once they've completed the module, they take a unit test to see how much they learned.

For grade level material, I checked out the Pre-calculus class.  It is also divided into modules with topics covering around 20 lessons.  I checked out module 1 which covers complex numbers and transformations including matrices.  This unit does not yet have lessons 1 to 3 live so you actually begin with lesson 4. There are practice questions available for each topic with quizzes sprinkled at the end of each set of lessons.  The module ends with a unit test.  This is set up so students can work at their own pace which is nice, especially if students end up traveling a lot.

I like it as a teacher because if I have to teach say statistics which is one of my weaker areas, I can brush up on the material before teaching it.  I realize You Tube has a ton of videos but you can't always access them during the school day.  Our school blocks it as part of its routine because of our very limited band width.

I found one cool feature I'd been unaware of before today.  Khan academy has a feature which allows the teacher to enter a student's MAP scores to generate a "play list" or set of customized skills they need work on.  I think this is cool because many of our students arrive below grade level and need scaffolding. This allows us to do this. We use MAP testing at the school I'm at and the one I'll be at next year so I can use this to help in class.

I think this is cool.  Let me know what you think, I'd love to hear.  Have a great day.  Beginning next week, I'll be cutting down to three times a week and possibly weekends because I'm scheduled to travel about every other week and will not have the time to do five a week.

Cha Cha Slide and Transformations

The other night at prom, one of the dances the DJ played was the ever popular "Cha Cha Slide".  It's a dance that can be done individually in rows, aka "Line Dance".  A line dance by definition, is choreographed set of steps performed by people in lines.

In the song "Cha Cha Slide" the artists tells dancers to step to the right, left, forward, back, add stomps reverses, etc in a particular order so they move around in a set space.  Normally, line dances are associated with Country Western music due to dances like Cotton Eyed Joe or The Electric Slide.

As I watched the kids participate in the Cha Cha Slide, I realized the dance was composed of transformations.  Moving back and forth, left or right are translations along the x or y axis.

If they spin around, it is a rotation and if they flip to change the direction they are facing such as you face forward and then turn so you are facing back while moving forwards, it could be looked at as a reflection.

I think it might be fun to watch a video showing the Cha Cha Slide so one could map or document the movement on a graph before analyzing the transformations contained within the dance.  The same could be done with other line dances to see if they have more transformations or fewer.  If you look at the Electric Slide, there is a quarter turn as part of the steps and a quarter turn is a 90 degree rotation around the origin or you.

This website has the steps for several popular line dances from Cha Cha Slide to The Electric Slide to  Boot Scootin Boogie.  It doesn't matter that the step you do is a grapevine, what is important is that you are moving along the x axis three steps or one.  Just think of the actual foot movement as the unit.  So if you grapevine left once, you've covered three units in the negative direction.

There are also a ton of videos on Youtube available showing the dances so its easy to watch the videos and write down the number of steps and direction along with rotations, reflections and translations or combinations of movement.

I like this as a real life application of transformations. Its something students can relate to and it builds on prior knowledge since most of the students at my school know Cha Cha Slide.  I simply asked them about that song at prom, the one where everyone moved forward and backwards and the kids knew the dance.  They really like it a lot.

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

The Brain On Math.

I was off following a lead on Math and the brain and stumbled across an article I found quite interesting.  According to the article, neuroscientists have wondered which part of the brain is used for complex mathematical thinking.  Traditional thinking said that solving mathematical problems requires a complex manipulation of symbols and relationships, consequently, it must use the same part of the brain as language is processed.

Not everyone agreed with this view including Albert Einstein.  So two researchers designed a study to test this out.  The study involved 15 high level mathematicians and 15 high level academics in other fields.  All were asked to listen to  short sentences made up of both mathematical and non-mathematical statements under a working MRI.

The subjects listened to 90 questions, 72 were high level math statements divided into topology and three other areas while 18 were mostly historical in nature.  They had four seconds to decide if the material was true, false, or meaningless

 The results showed certain parts of the brain lit us as the mathematicians thought about the math.  This includes the same area identified in babies for basic number sense.  The MRI showed three parts of the brain lit up when the high level mathematicians worked on the mathematical problems.  These three areas did not light up for the other academic people until they were asked more general mathematical questions. Furthermore, they did not light up when the mathematicians read non-math problems.

What this shows is that the parietal, the prefrontal and inferior temporal areas process math regardless of the complexity of the problems.  This means that when a high level mathematician is balancing his checkbook he is using the same part of his brain he uses as when he is trying to find proof of Riemann Hypothesis.

In addition, none of the areas that light up are associated with language processing indicating that a different part of the brain is used to process math regardless of the type of math.   Furthermore even though language is associated with learning math, math still uses its own part of the brain.

At this point they do not know how innate number sense and higher level processing occurs. They also don't know if studying enough math to become an expert changes the way you do arithmetic or if learning arithmetic lays the foundation that allows you to learn higher level mathematical concepts.

This opens up a new avenue of research. One where they hope to explore to try teaching someone enough math while monitoring the brain to see what happens.  Let me know what you think, I'd love to hear.  Have a great day.

Learning from Videos

Unfortunately, most of our students do not know how to watch videos actively so they learn.  They are used to being entertained by videos so they just observe the material passively and at the end, they are ready to move on, not to answer questions.  

There are ways to make watching the video more active so students are learning.  I love Edupuzzle because they have access to so many videos and its easy to edit them so students have to pay attention due to quizzes, etc.  

I can't always access Edupuzzle because the internet is too slow so sometimes I have to download a video directly to my computer right before school.  I can't annotate it, can't add questions so there are other things I can do to make the experience more active than passive.  

The biggest thing is to review the video before you show it so you know if it meets your needs.  Next you should cut the video down into smaller chunks to make it easier for students to pay attention.  I read that the modern child's attention span is quite short so chunking the material into shorter segments is better.  Finally, create guided questions that students have to answer as they watch the movie.  Sometimes, I have my students watch the video, then rewatch it so they can answer the questions better.

Before introducing the video, have the students do a quick write or a pair-share on the topic to activate prior knowledge.  If students think about what they already know on the subject, it makes it easier for them to connect the new material with their prior knowledge.  Furthermore, give them a reason for watching the video such as it has a different way of multiplying binomials, or this has a fun real world example.  It's similar to telling students who their audience is when they write a paper.

Take time to pause the video so students can write down answer, process the information, discuss the material, or ask questions.  This also takes the watching from passive to active.  In addition, take advantage of the pauses by implementing a watch-think-write strategy.  When students watch the segment, they may not write anything down. Once the segment is finished, students discuss the material and they still may not write anything down.  When the discussion is done, they may then answer the questions.

There is a way to take the guided notes several steps further to help increase their understanding.  They can break into groups of two so they can explain the material to each other.  Students can create small voice bits starting with "Did you know....?". What about creating concept maps that tie previous knowledge with newly acquired information?  

Other things to consider doing prior to showing any videos is to check for a transcript. If there is a transcript available for the video, have students read the transcript first.  This way they can create questions they have about the material before watching the video.  Its like previewing the material.  

Furthermore, you could set up a backchat channel so students can discuss the material in reference to the essential questions you've posted prior to the video.  This way they can focus on the material and interact with each other.  If you do a simple search you'll see several backchat sites such as Go Soap Box, or Back Channel Chat.  I'd consider the backchannel chat when rewatching the video after the guided questions have been answered so they can focus on the discussion.

This is just one way to utilize videos actively in the math classroom.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

The horses average 27 mph for 4 hours, how far did they run?

Warm-up

These dogs run at about 12 miles per hour.  If they ran for 9 hours, how far would they go?

Interesting Statistics from World War II

 The other day, I discovered a connection between Los Alamos, New Mexico and Oak Ridge Tennessee.  Most people know that Los Alamos is where they build the first plutonium based atomic bomb, the one later dropped on Hiroshima and Nagasaki but did you know that Oak Ridge produced the plutonium for the bombs?

Both places during World War II did not appear on maps, nor did they really "exist".  Los Alamos began by taking over a huge school complete with over 50 buildings and encompassing close to 50,000 acres of land but Oak Ridge had to start from scratch.

The government had to get 60,000 acres of land, move about 3,000 people from the area so they could erect the facility in as short a period of time. Over a period of about 2.5 to 3 years, they built enough housing for 75,000 people.  So they went from a population of 0 to one of 75,000 in say three years. 

Furthermore, to build the houses, they contracted a company to build single family houses in such a way that trucks would move them, half a house at a time.  At the peak of growth, a new house was assembled every 30 minutes.  That is impressive, especially at that time. 

This site has some great graphs on Los Alamos, Oak Ridge, and Hanford sites from World War II.  These sites were all part of The Manhattan Project.  The cost for the whole project was just over $2 billion in 1945 dollars but the government spent about $300 billion for the whole war. 

Furthermore, most places went from zero to full running within 22 months.  This meant they built the facilities and housing, hired and trained people and were producing and making the bombs.  At its peak, it is estimated around 125,000 were employed as part of The Manhattan Project but over the course of the project, it is estimated over 600,000 people worked on it.  The reason for so many people working on the Manhattan Project is due to turn over.

The Hanford facility had a 26% turnover rate with a 3 to 1 ratio of resignations to discharges.  Many of the resignations were due to illness, working conditions, being drafted, moving, or getting a job elsewhere.  At Oak Ridge the turnover rate was 17% split in a 2 to 1 ration for resignations to discharges.

There is also a graph showing the total hires and terminations, hires and terminations for operations and research  and for design and construction.  There is enough information included in this article for students to create a variety of graphs to display the information.

I love finding information like this because it is real world and its fun to see how it all looks in a variety of graphs.  Let me know what you think, I'd love to hear.  Have a great day.

Mathivate!

A couple days ago, I attended a webinar lead by Kim Thomas, author of Mathivate.  Although the webinar lasted only one hour, it was filled with so many ideas that I'll have to go back and rewatch the video because I need to check it out again to make sure I know the ideas.

The ideas she shares are fun and can be used for most ages although she focused mostly on middle school.  One idea she uses at the beginning of the year, is to have students introduce them self mathematically.

They give their birthdate, home address, information about siblings, favorite number, and other choice tidbits using mathematical equations.  Students are not allowed to use straight numbers, they have to use math instead.  It makes them think and makes math more fun.  Another activity she uses at the beginning of the year is where she has students provide information inside certain shapes.

When she has taught similar figures, she has students create similar figures which they have recorded the size and the scale factor before combining the shapes into letters that form their name.  This activity allows them to practice similar figures while creating something they can relate to and take ownership of.

I love the way she created activities for the whole year from the first day to Christmas, to Easter and all sorts of activities in between.  Every activity requires the students to practice their math in a fun way and it personalizes the activity so students want to do it.  She reinforces the idea that "Math is FUN!"

Kim Thomas wrote a book called "Mathivate" filled with the activities she shared in the webinar and more.  I decided to get the book because I'm always on the lookout to make math more fun and relatable to the students.  Even though she has geared the book for middle school age, its easy to adjust this for high school students.

The Global Math Department sponsored the webinar and I'm so happy I was able to attend.  I wish I'd been able to make more of these but after attending this one, I'm going to try to rewatch the video when I can stop it and write the details in more detail.  One nice thing is that they record the videos so people can go back and watch these later.

I love when I get more material to use in my classes because the more engaged I can get my students, the more likely they are to learn. I also wish my teachers had used things like this when I was in school because it would have made things more interesting.....LOL. Let me know what you think, I'd love to hear.  Have a great day.

Situations + Calculator = Not Always!

 We all have that one student who wants to use a calculator to find the answer to a problem not realizing there are situations where you do not want the answer to have decimals.  Unfortunately, many students want to turn everything including fractions into decimals so they don't have to find a common denominator. They don't realize there are situations where having the required amount in decimal form is going to make it difficult to purchase.

For instance, I sew much of my clothing but they do not sell material in a decimal form.  Usually you have to buy material in units such as 4 1/3 yard or 3 1/2 yard.  I've never had anyone ask me "Was that 4.3333333333333 yards you wanted?".  They use the fractions all the time.

Another instance is when purchasing lumber.  All lumber is sold using whole numbers and fractions.  I know my father has never said he needed 6.45 feet of wood.  He might say he needs 6 1/2 feet of wood for the project.   There are so many things sold by the foot or yard that are always done with fractions.

Furthermore, when buying certain things like paint, you end up using whole numbers or numbers broken down to smaller sizes but you don't use decimals.  When you plan to paint a room, you usually find the area of all four surfaces of the room.  Once you get the total, you'll have to divide it by the number of square feet one gallon of paint covers to find out the number of gallons you'll need.  Unfortunately, many students do not think about needing to round up so they will give an answer like 5.78 gallons, perhaps 5.8 instead of 6 gallons.  Sometimes you might get 5 gallons and 3 quarts but most people find it easier to round to the next whole.

This also applies to things like carpet, tiles, flooring, even ceiling tiles.  I priced flooring, wooden type, so I could redo the living room and it came in boxes that covered a certain area.  I had to calculate the number of boxes I needed and I had to round up to make sure I had enough.

Most recipes in the United States are listed using fractions.  I've never cooked using a recipe that asked for 1.3333333 cups of sugar.  It called for 1 1/3 cups of sugar or 1/2 teaspoon of salt.  Check out most American cookbooks and you'll see them filled with fractions.

The other day, I explained to one young man that he needed to do the division to include a remainder because there are times it is necessary to know if there is a left over.  For instance, your grandmother gave you $358 to share among you and your three siblings.  How much would each person get and how much would be left over.  He discovered he had a remainder of $2.00 so I asked him what he would do with the remainder.  He stated he'd give it back to his grandmother.

The same applies to splitting up candy or other items similar to that.  You will not give 6.56 pieces of candy to each person but you might give 6 pieces to each person and keep the remainder for yourself so you get more.  This is real life since most things do not split equally among people be it candy or money.

These are instances where calculators are not good because most give you a decimal remainder rather than a fractional list.  Let me know what you think, I'd love to hear.  Have a great day.