Wednesday, August 17, 2022

Post It Notes Part 1

I love using post it notes in my math classes.  For some things, using the digital ones are great but for other activities, the physical ones are great.  I am going to focus on the physical ones first and digital second.  I love that post it notes came in both the plain variety and ones with graphs printed on them.  I got a bunch from Amazon and my students used the ones with graphs on them.

My students loved the post it notes that had the coordinate planes on them because they could use them as part of their notes when they needed to draw the graph say for logarithmic functions or linear.  The other way they loved using them was as part of their homework when they had to graph anything.  They could just put the sticky note on the page with the problem and draw the graph.  If they made a mistake they could just take the messed up on off and replace it.  It also meant the graph was much more accurate.

I also love using sticky notes as a manipulative to help students learn the processes needed to solve one, two, and multistep equations, or for combining like terms.  For like terms, I give students different colored notes so they can write each term on one but all the x's would be on green, all the x^2 would be blue, xy would be pink, and they would write the term with the plus or minus side.  This way they can rearrange the terms so they are grouped by color since all the same variables have the same color.  This allows students to see visually what constitutes like terms.  I've had students who never moved on from using the colored notes and some who did.  I didn't worry, I just handed out the post-it notes as needed.

Another way I used sticky notes was teaching students to solve one, two, and multi step equations.  If we were working on say 2x + 3 = 9.  I had students write the 2x on one color note and the +3 and 9 on a noter color but they would both be the same color so students would know they were both constants and different from the variables.  Then I'd have a third color to show the process.  So when I showed students to subtract three from both sides, I'd write the -3 on two different notes of the same color and place them under the original equation.  I'd draw a line below that, move the 2x down, talk about the 3 -3 is zero, and 9-3 is 6 and I'd write the 6 on a note on the same color as the +3 and 9.

So now I have the 2x = 6 and using the same color I used for -3, I"d write /2 on two different notes so I could put them below each term resulting in 2x/2 = 6/2. the final line would be the x on the same color as the variable, and the 3 would be on the same color as the +3 and 9.  Then for every problem I used this method for, I'd use the same colors because the colors help students remember and once they completed the problem using this, they could copy the problem onto their paper.  

Other ways to use sticky notes are using them to fill in the real number system graphic organizer, each type of number is assigned a color.  For instance, integers might be orange, irrational numbers might be in pink, decimals in blue, fractions in green, so as the numbers are placed in the chart, the students have the color coding to remind them of the type of numbers.

One thing I see students struggle with is substitution. Doesn't matter if it is a straight substitution of 3x + 2y  with x = 2 and y = 1/2.  You'd write down 3 with space + 2 followed by space. Students would write 2 on one note and 1/2 on the other and place the notes in the blank spaces to get 3 (2) + 2 (1/2) and they can solve it this way.  This can also be done when teaching solving systems of equations using substitution so instead of using numbers, you use equations.  

I'll finish off the physical post-it notes on Friday and address using digital sticky notes in the math class.  Both can be used by in different ways.  Let me know what you think, I'd love to hear.  Have a great day.


 

Monday, August 15, 2022

Geomagic Squares

 

We've all heard of magic squares but not as many have heard of geomagic squares. One person, Lee Swallows, took the idea of magic squares further by investigating geometric magic squares, aka geomagic squares for short. A geomagic square is defined as whose cells contain spacial elements of certain dimension.  Magic squares are considered to be a special case of geomagic squares where all elements are one dimensional.

Think of it this way.  The numbers in the magic square represent the segment lengths in a straight line that add up to a total distance. In addition, a geomatic square might be made up of 2 dimensional areas that when put together create the same area with the same shape such as squares.

The same applies to if the square is made up of individual volumes that when combined create the same 3 dimensional shape with the same total volume. 

In regard to two dimensional shapes, they might be geometric shapes both regular and irregular or sections of a circle that when combined in any direction will form the same circle.  The circle ones will be based on the standard magic square.  Think of it this way.  360 /15 = 24 degrees.  The 15 is from the total that each row, column, or diagonal adds up to.  Thus if you look at the magic square, the first row first cell 

would have part of a circle that is 8 x 24 or 192 degrees, the second cell would show one segment or one that was 24 degrees while the last would be 6 x 24 or 144 degrees.  If you add them all up, you get 192 + 24 + 144 which adds up to 360 degrees,  If you repeat this for for the second and third rows, you get a full complement of segments that do add up to full circles.  


There is a general formula that can be used to describe magic squares is :



Now we can use it with numbers such as C=5, A = 3 and B = 1 you end up with the magic square as referenced above but what if you assigned A, B, and C to shapes.  Think of A = a small rectangle.  The C represents a square with sides equal to the length of A and B is a semicircle of diameter equal to the length of A, you can use those shapes to form a visual representation of magic squares. The idea is that you may have to rotate the shapes to get them to match up to form the shape.  

Now there are so many applications of this basic idea that I don't have the space to go into it, so you might want to read this article by Lee Swallows.  I was impressed with the different ways one could represent the basic magic square referenced above.  Read the article then check out the gallery at  this site by Lee Swallows to see what all one can do with the geomagic squares.

Sunday, August 14, 2022

Warm up


What is the population of New Zealand if the population is 5 million people spread over 103,484 square miles? 

Saturday, August 13, 2022

Warm-up


 This New Zealand tree can live to a thousand years. If each ring is about 6 to 8 months growth, use the formula to find the number of rings for an 80 year old tree. The formula for age is to divide the number of rings by 2 and add 20.

Friday, August 12, 2022

Algebra Touch

 

Algebra Touch is an app that has been around for a few years but the people who originally released it, have released an updated version that has some really nice additions.  In case you're never checked this app out, it is different than many other math apps because it focuses on teaching students how to do the process by actually having them do it.

Algebra Touch has the student use their fingers to combine terms, move terms around, or other movement so they actually do the process.  The old version had this facet but something new has been added which I really like.  I'll tell you about that later on.

When you open the app, you get a list of 8 topics that can be explored and practiced. Topics are divided into beginner, intermediate and advanced.  The beginning topics are like terms and order of operations, intermediate cover factorization and elimination, while advanced is equations, distributions, exponents, and logarithms.  Each topic has a certain number of subtopics to practice and at the end of the topic, there is a challenge practice for students to try.

 The topic on the main page, tells you how many subtopics it has.  When  you click on a topic, the submenu comes up so you know what you will be practicing. In addition, there is an introduction to give you a short description of what is happening.  Then you work work your way through each subtopic.  At the bottom of each subtopic is a create your own choice where students can input their own problems.  When you type in a problem of the same type, it allows you to solve that problem.  This means students can type in assigned problems, work through them, copy down the steps to show their work, and they are done.  

When you click on a sub topic, in this case I went to the like terms one, 
and chose the first subtopic.  I got a demonstration screen to show me how to do this and then it went to the first problem.  to do this I would place my finger on the -4a and move it over by the 19a so they are next to each other.  To combine the terms, you touch the + button and it combines the 19a + -4a to make 15a.  Then touch the + between the 12 and -10 to get 2 so the answer is 15a + 2.  It doesn't matter whether the terms with the variables are first or last.  In fact, the app does not make you put the variables first.

Now if the student has trouble figuring out how to do this, every problem has a video link at the bottom of the page.  If you look above, that little tab that say something like 0 of 6 which tells you how many problems in the section and how many have been done.  If you click on that tab, it rises and has an embedded link to a video so the student has video help available for every single problem including those in the challenge but not the do it yourself problems.

As far as I can tell this app is free and is available for the iPad and iPhone.  I love playing with it myself.  I do like that even if the students choose to use the create your own option for problems, they still have to work them out and that is awesome.  Give it a look and check it out.  Let me know what you think, I'd love to hear.  Have a great day.


Wednesday, August 10, 2022

Magic Squares and Observations.


If you look at the instructions for how to do a magic square they have a specific progression on how to fill them in. Today, I"m only looking at the 3 x 3 grids but I plan to look to see if these observations for a 3 x 3 work for larger square and could lead to a interesting activity.  I used numbers for this but it could be done by hand or by excel.  

To create a 3 x 3 while using the digits of 1 to 9, start by putting the 1 in the middle cell of the top row.  



The next digit 2, is located one pace up and over but that puts you outside the grid so you end up placing the 2 in the bottom right most square.  The digit 3 is then placed right one and up one so again, you are off the grid and end up placing it in the first cell of the second row.  







The digit 4 is one cell down, just below the 3. 5 is then one cell to the right and one up and the 6 is also one right and one up so the 4, 5, 6 are on a diagonal from lowest left to highest right. To get to 7, you move one cell down.









To place the digit 8, you go right one cell and up one so you have to move to the top row, first cell.  The digit 9 ends up being placed in the last open cell between the 4 and the 2.

The next step would be to have students add up all the rows, columns, and diagonals, to make sure they all add up to 15.  If you are using numbers or excel, have the students create the math for the cells. 

Once this is set up, for all the cells, students can see they all add up to 15.


Now for the fun part, ask students if you rotate the magic square 90 degrees, 180 degrees, or 270 degrees, will it still work out properly?  I did it and yes, the magic square still adds up to 15 in all directions for all those rotations.





We all know it is but many students wouldn't and it makes a nice extension for the activity and it brings in the idea of rotation in an area other than geometry.  Let me know what you think, I'd love to hear.  Have a great day.

Monday, August 8, 2022

One Less Real Life Grid Example

 

As I write this, I am in my hotel room located in Sydney, Australia enjoying a nice laid back day.  I spent some of the day online looking for some tours so I could see more of town.  Many of the tours do not pick you up at a hotel so I was on google maps trying to see if the point of departure or return was within walking distance of my accommodations or if there was another hotel near mine so I don't have far to walk. 

Google maps are so different from their predecessors because they automatically identify where you are and you input your location.  It gives routing from where you are to where you want to go so you could go by car, bus, or walking.  In the old days, we'd use a variety of maps which used grids.  You might check out the map section in the phone book, a map, or those books used by real estate agents.  

All of these use the same process.  You look the address up in the index, go to the correct page and the focus on the part of the grid containing the street.  It is at this point, you actually start your search. For instance, you might look up the road in the phone book map section and it sends you to page 45 to look for section I-5 and that was all you did to find the location.  

I mentioned a book because a friend of mine who lived in Los Angeles had this huge book used by real estate agents, listing every road, every street, every anything so if she had to go to a new place, she could find her way there.  Every page in the book was part of a larger grid, so if you took every page out and put them together, you'd get a huge map but it didn't suggest routes at all.  Instead you were expected to flip pages forwards or backwards looking for the previous or next part of the grid, until you figured it out.

Since I've never really lived in large places, so I could get by using the map section in the phone book for local things and a regular map when I needed to cover a longer distance since it contained enough info but maps did have grids to help narrow the search on a map.  

I've been using Google Maps quite a lot recently and just the other day I realized that google doesn't use grids at all.  I suspect this is because google is set up to show your location and you choose the other location and voila, you see it all.  There is no need to search through the phone book, a map book, or a paper map because it's done for you.

I'm a bit sad about this because students no longer have to learn to read the usual maps with their grid systems.  In fact, I'm not sure many of our high school students know how to read a map or even identify a map.  It is a skill that is no longer needed but that means it is one less real life examples I've used in the past but I really can't use it anymore. 

It's like watching one of the original Superman episodes where Clark Kent would dash into a telephone booth to change into his alter ego Superman.  By the 1970's or so, the full length telephone booth had morphed into those new ones with the phone attached to a half sized one and then by the time cell phones took over, no more phone booths of any sort.  In fact, we can't even talk about cell phones or rental cars with the base rate plus so much per text or call or mile so those are out because we don't operate that way anymore.  

I'm still working on finding a replacement example but it's slow.  Let me know what you think, I'd love to hear.  Have a great day.