Monday, December 30, 2019

Celcius vs Fahrenheit. Formulas vs Graphs

I usually teach converting temperatures from Celsius to Fahrenheit and the other way using the standard formulas.

If you want to go from Celsius to Fahrenheit you use the C = 9/5 F + 32   while F = 5/9(C-32).  These work ok until you ask students at what temperature are both Celsius and Fahrenheit the same.

This is when I promptly lose them because they don't realize they can determine the answer in one of two ways.  They can either set the two formulas to be equal or they can graph.

As you can see if you replace the C or F with x and y, you can graph the two linear equations and they cross at the point (-40, -40).   This means when the Celsius temperature is -40, the Fahrenheit is -40 and vice versa.    I got the idea the other day as I was looking at the formulas and realized the formulas are basically linear and I could easily create a visual representation using Desmos.  You see the results in the photo above.

The other way is to set them equal to each other and solve:

9/5 x + 32 = 5/9(x-32)

9/5 x + 32 = 5/9 x - 160/9

9/5 x + 288/9 = 5/9 - 160/9

9/5 x = 5/9 x - 448/9

9/5 x - 5/9 x = -448/9

81/45 x - 25/45 x = -448/9

56/45 x = -448/9

56 x = -2240

x = -40

This means the value has to be -40 for the two temperatures to be equal.

You could have done each equation separately by subbing x in so it looks this way

x = 9/5 x + 32
-4/5 x = 32
x = -40

or

x = 5/9( x - 32)
x = 5/9 x - 160/9
-4/9 x = -160/9
(-9/4) (-4/9x) = -160/9 * -9/4
x = -40

So there you have it.  -40 Celsius is -40 Fahrenheit and a way to prove it.  Have a great day and enjoy yourselves.  Let me know what you think, I'd love to hear.



Sunday, December 29, 2019

Warm-up

Orange Tree, Oranges, Tree, Fruit

If it takes 16 oranges to make 59 ounces of juice, how many oranges do you need to make 1 gallon of juice?

Saturday, December 28, 2019

Warm-up

Oranges, Fruits, Orange Tree

The average tree produces 325 oranges per year.  If your orchard has 90 trees in it, what is the total crop for the year?

Friday, December 27, 2019

Slowmo, stop motion, and Animation in Math.

Explosion, Exploding, Pop Art, Boom I have been trying to think of ways to improve instruction and learning in my math class.  In math, we say if a student can explain their thoughts and how they came up with the answer, they are showing understanding but it seems to me that if they can create a video showing their thinking, that works.

Animation is a great way to create equations and move things around without you standing there and explaining every step. Instead, you can have the terms move on screen by themselves, or move to the proper spot.

I have used keynote to create animation showing how two shapes were congruent using transformations.  I used keynote because that is the presentation software I know well.  Right now, I am working on learning to use Google slides for the same thing.  I have several ideas I want to bring to life using animation.

1.  Instead of talking about slope, why not create animation so the word positive climbs diagonally up towards the top, then the word negative is going down to the starting level, the zero slope is a flat area and undefined is a cliff but I'm seeing the words appearing one letter at a time.



I made a quick animation showing the idea for slopes.  My students do not always stop to think about which slope is associated with upwards, downwards, flat, or straight down.  My art isn't the best so forgive me.  I used animation HD for kids.  It was free and so easy to use.


2.  Showing how to solve various equations using animation.  You can have the terms move or you could arrange for a train or other vehicle to move the terms around.  In addition, it is possible to put music in the background or voice over to explain why.

3. Stop motion can be used to show how the area of a circle comes from the circumference of a circle.   If you cut the circle into certain shaped pieces and rearranged, you can rearrange the sections into a parallelogram.

4. Students can create their own characters who talk about a problem on the board, each step done via stop motion or slow motions.





This video was made with Toontastic on my iPad.  I used the characters from it and the three part storyline.  It was an easy app to use.  If you notice, my characters spoke about slope in general as seen or experienced in real life.  It was not that technical in terms of finding the slope itself and it explored the idea that a slope could be positive or negative based on which direction you are going.

The above are just a few suggestions plus two short videos I made using two different apps.  Let me know what you think, I''d love to hear.  I hope these videos work properly.  Have a great day.


Monday, December 23, 2019

Christmas Infographics

Christmas Car With Tree, ReindeerThis is the perfect time of year to sneak in reading or creating infographics.  The holiday season has so many different possibilities for learning to read and interpret infographics.

This is a skill we need in today's society because society has moved to sharing information via infographics rather than the standard graphs.

This site has a something like 12 different Christmas related infographics.  My favorite one is the one that looks at the best selling toy each decade beginning with 1910.  The graphic includes prices for each toy adjusted for inflation and the prices are in pounds so students have to convert the unit of currency.  In addition, the toys are divided into genders.  The other thing is this infographic records the change of consumer tastes over the time.

Another infographic addresses Christmas and pets which looks at the number of families who see pets as members of the family, how many presents they buy their pets, and which gender is more generous.  One of the infographics I find most interesting is the one outlining why being Santa is the most dangerous job out there.  The infographic looked at everything from number of children and stops to how fast Santa has to travel to do it all, and snacks left for Santa in various parts of the world.

This site has 9 infographics which are different from the first one.  There is one infographic on the ugliest sweaters at Christmas.  Although it looks at ugly sweaters, this infographic actually breaks down the number of sweaters purchased by region and around the world from this place.  Another infographic explores the safety of Christmas trees from buying them, to caring for them, to disposing of them.  The infographic even took time to look at what causes Christmas tree fires.  Unfortunately, not all the links in the above two sites are live.

This site is one big infographic on the economics of the holiday season.  It looks at economic growth, mortar growth, global e-commerce growth, target audience and spending,  Towards the end is a section where they give hints to vendors on ways to improve their chances of increasing their sales.  It is a very interesting graphic.

This site has tons of Christmas infographics including the most popular recipes, the history of Christmas trees, Christmas dinners, greatest holiday movies, and so many others.  Most of these infographics are fairly short with 5 dinner infographics, 5 destination infographics, but they all have some nice information.  I checked out the one on Christmas cards.  It looked at the anatomy of a Christmas card, one on showing the insight into holiday marketing for businesses, facts about Christmas cards, the 12 days of Christmas, and breaking down the gift card trend.

I love the way this last site is done because you can break the class up into smaller groups, assign groups to interpret each category, or individual infographics in each category.  Students can present their findings via Flip-grid, Google Slides, or any other digital method.  Let me know what you think, I'd love to hear.  Have a great day.




Sunday, December 22, 2019

Warm-up

Christmas, Tree Decorations

You just paid a Christmas Tree decorator $213 to place your lights, ornaments, and tinsel on the tree.  It took her 3 hours to complete the job.  How much did you pay her per hour?


Saturday, December 21, 2019

Warm-up

Christmas Tree, Lights, Christmas, Tree

If the average cost of a real Christmas tree in 2019 is $81, and they sold 32.5 million trees nationwide, how much did people spend nationally on this many trees?

Friday, December 20, 2019

The Cost of The 12 Days of Christmas.

Wedding Rings, Engagement Rings, Wedding The 12 days of Christmas is one of the more popular songs heard at this time of the year and every year at least one media outlet announces the current cost of obtaining all the items.

The overall cost of the 12 items have increased over the past few years.  In 2016, the cost ran $34,363.49 which went up to $34,558.65 in 2017, and in 2018, they were $38,926.03.  This year, the cost will run just a bit more.


The break down this year is as follows:

1 partridge in a pear tree is 210.17 down 4.5 percent.

2 turtle doves is $300.00 down 20 percent.

3 french hens is still running $181.50 which has not changed in price since last year.

4 calling birds didn't change price since last year and is still $599.96.

5 golden rings will set you back $825.00 which is a 10 percent increase.

6 geese-a-laying is $420.00 up 7.7 percent from the previous year.

7 swans-a-swimming is one of the most expensive items at $13,125 which didn't increase at all.

8 maids a milking didn't change and still costs only $58.00

9 ladies dancing didn't change from $7,552.84.

10 lords-a-leaping still runs $10,000, the same as the previous year.

11 pipers pipping $2748.87, up must 8/10th of a percent from last year.

12. drummers drumming will set you back only $2972.25, up just 0.8 percent.

The total is $38,993.59 for all of these items but if you gave the number sang in the song with all the repeats, it would cost you $170,298.03 or $781.94 more than the previous year.  The numbers quoted are for obtaining all the items locally.  If you had to order them online, it would cost you $42.258.91.

It wouldn't take much to turn this information into a graphing excursive.  The data from 2018 and 2019 can be found here for both local and online purchases. Students can translate the information into a graph using a spreadsheet and then produce various graphs from the program. This page has the total cost for all the gifts beginning in 1984.

The same graph for the total cost also allows people to look at the change in the price of each individual items.  The price of a partridge in the pear tree ran $32.52 in 1984 and is now $210 in 2019.  This graph makes it possible for students to calculate the percent increase or decrease each year in order to create graphs showing the information.  I think it would be interesting to see how the cost of each item changes over time.

I hope you explore the second site, it is cool.  Let me know what you think, I'd love to hear.  Have a great day.



Wednesday, December 18, 2019

PBS Learning Media + Math


Sunflower, Flower, Petals, Bloom
 The other day, I discovered the PBS Learning Media site filled with so many cool things you can use in your math class.  This site has videos, interactive lessons, interactive, lesson plans, documents, and all sorts of other things.  This site has a search engine to find lessons on a specific topic or grade.  Most of the material has information on the grades each activity is meant for. I searched mostly for high school material since that is what I teach but there is material for all grades from prek and up.

I looked at the video section and found a nice video on using probability in search and rescue operations.  Search and rescue is something quite real out in the bush of Alaska because sudden storms blow in causing people to get lost, or it warms up and the ice thins so people fall into the cold water and die.  This is a relevant topic for us.

The unit includes the video complete with teaching tips so the instructor knows the best way to teach this.  In addition, there is a page listing all standards associated with the material.  The teaching tips include questions for students to answer as they watch the video so they have to focus on the material rather than enjoyment.  Furthermore, it is possible to download the unit as a zip file.

Next I checked out their interactive lessons and found two that immediately attracted my attention.  One was on Math in Fashion which included Project Runway, and Math in Special Effects. The Math in Special Effects is geared for grades 7 to 10 and includes a teachers guide, transcript, and says it is a student directed lesson.  Students sign into the site to do their work and have it saved.  The lesson also has a page listing common core standards this unit meets.

The site also has several activities that are interactive such as the one where students get to engineer a jetliner.  The activity has the student how to adjust the wings and location of the engine for the best performance for four different situations.  In addition to the usual teaching tips, there is material to provide both background and further reading, and the answer key.  Of course, the activity does include the necessary common core standards to make life easier.

The lesson plans are focused on specific mathematical topics.  Some of the lessons are targeted to teach a topic such as ratios,  or a general lesson on dance.  I checked out the everyday algebra targeted lesson which begins by giving a summary, estimated time to complete, and the learning objectives.  The next page divides the lesson up into three sections, the introductory activity, the learning activity, and the culminating activity, each section has a video with information on what the teacher should focus on.  It includes further information and lists the standards.

The documents section has several interesting topics including Math's hidden women, and one on the risk of flying.  I looked at the document on the risk of flying and found it quite interesting because it had both graphics and written words.  The author even took time to explain how the risk is calculated, in two different ways, per mile, per flight, or average risk for a person.  The explanations are quite clear and easy to follow. 

Last there are galleries which focus on one topic.  I looked at the one on living wage vs minimum wage.  The gallery has five videos, a lesson plan with information on each clip and questions to go with each clip.  One activity has students find the cost for several items in their community so they can calculate the living wage for their location.  I like this because it is everything costs more in the villages of Alaska so having figures based on local prices makes the exercise more realistic.

If you need more material for your class, check this site out.  It is well organized, has great support materials, and topics that are interesting.  Let me know what you think, I'd love to hear.  Have a great day.

Monday, December 16, 2019

Digital Compass for Geometry

Compass, Ruler, Circle, Straight-EdgeI do not like using compasses and straightedges in my Geometry classes because most compasses I end up with break, or the pencil holder dies, or something else so I can't use them.  I've also had some students who would see them as a good weapon so I'd rather not provide them with that temptation.

Unfortunately, I had not found anything I could easily use to  recreate certain geometric drawings.  I looked for digital possibilities but didn't find any until the other day when someone on twitter recommended Robocompass.  It is a web based program and comes highly recommended by a variety of people.

As soon as I opened the program, it took me to a blank slate.  On the left side is the space where you type in the commands.  At the top left side there are several examples such as tessellation's using rotations or translation, bisecting an angle and several other things.  On the right side, there is a help button which lists all the commands, form, and what they do.

As you type in one of the standard commands, a help box pops up, showing you the general command and a real example with numbers so you can figure out what to do.  I had fun exploring it and I created a small piece of pie.

I made a video of the program I made so you could see it in action.  Once you have finished your project, you hit the play button and it shows each step of the process by showing each line and highlighting it so you can connect the command with the action.

It is easy to use.  The help page was clear with the generic commands and specific examples to show how it works.  I could readjust the numbers until I actually had the final product I wanted.  I plan to play with it more.

This free program is fun and best of all, it works on my iPads too so it can be used on computers or iPads.  I hope you check it out.  Let me now what you think, I'd love to hear.  Have a great day.

Sunday, December 15, 2019

Warm-up

Horses, Icelandic Horses, Iceland Horse

An Icelandic horse weighs about 350 kg while a regular horse weighs about 475 kg.  The Icelandic horse weighs what percent of a regular horse?

Saturday, December 14, 2019

Warm-up

Iceland Horses, Winter, Frosty, Winidg

The Icelandic horse has a height of 140 cm versus the 165 cm of a regular horse.  What percent smaller is the Icelandic horse vs the regular horse.

Friday, December 13, 2019

Ways to Show "Applied Mathematics."

Catapult, Sea, Blue Sky, CatapultThe other day, I started class by showing a clip from a new station showing a trebuchet they built to chuck pumpkins.  I talked about how much of the math taught in class is more theoretical but that we see math being applied in real life.

I decided it is time to show students actual situations of applied math in situations without discussing the actual mathematical equations, just the applications in general terms.

I've been trying to promote the idea that the equations they study in class are ways to describe everything we see or experience in the world.  The kids really liked being the Trebuchet send the pumpkin over 2400 feet.  The person in charge spoke about how they built model, how they failed, tried again, eventually getting it to work.  Everything we talk about in class.

In the future, I plan to show clips of various sports, especially basketball, as a way of having students name where they see math, or look at a clip from various car races, dog sled races, or snow machine races to identify the math.  At this point it is not important to identify the actual equations but it is important to look at identifying situations.

Another thing to do would be look at various buildings to show how architects use geometric shapes, parallel and perpendicular lines, arcs, etc as they design buildings.  This would be a perfect situation to bring up historical buildings such as the pyramid of Giza, the Roman Colosseum, or other ancient buildings to look at the math behind them.  Aside from showing the application of mathematics, it also ties other subjects to math, reading a cross curricular connection.

Since the only way in and out of the village is by plane, I could easily bring up the idea of the math involved in scheduling the flight from the cost of the plane, calculating the price of the individual ticket, employee salaries, benefits, the cost of fuel, and scheduling planes because one plane flies several times everyday.

This would be a perfect place to also discuss how professionals prepare estimates when you take in your car or other vehicle in for repair, or you are having the house painted, the roof redone, the floors done or anything like that.  I don't know if the local businesses use them here but I know in most places they do.  This activity also introduces the importance of having accurate estimations so you land the job.

I found a site on the internet with some free videos and some that require a subscription to view but all take time to discuss math and various situations such as designing sunglasses, playing the drums,  designing buildings in earthquake areas, making faster bikes, dancing, and so many other topics.  In addition, each video has worksheets to accompany each video and there are quite a few educator resources.

I'm hoping this helps students connect mathematical equations with real life.  Let me know what you think, I'd love to hear.  Have a great day.


Wednesday, December 11, 2019

Visualization and Word Problems in Math

Graph, Data, Line Graph, StatisticsVisualization is an important skill in mathematics because it helps people see problems more clearly.  Although we teach students to draw pictures or diagrams but we don't always stress the diagrams can clarify the situation so much better than using only words.

In addition, visualization helps students turn mathematical language into something they can understand and apply.  It turns situations into understanding and students can process their learning.

Perhaps when we introduce word problems we might ask students how we'd represent the situation visually.  The problem with concentrating on key words is that some words such as per can mean multiplication or division depending on the context.  If students look at word problems as if they do when reading a book, they might do better.  In reading, they are told to picture the scene in their minds, why don't we ask them to produce a drawing of it on paper, on their iPads, or even create something in VR/AR or in animation.

Rather than working through umpteen word problems, why don't assign each student or group of students one or two word problems to do.  Before they start, ask them how they could represent the word problem visually, where have they run across this type of situation in real life, and how could they present the problem and its solution through video, Desmos, Geogebra, or a sequence of images or pictures.

Recent research indicates that it is important for the brain to use both symbols such as numbers, and visualizations because mathematical learning is optimized.  When students use symbols, they access one part of the brain and when they create visualizations, they are using a different part of the brain, so when they use both, the two parts of the brain communicate and their learning improves.

Furthermore, creating visualizations also makes math more exciting and improved student performance.  In addition, creating visualizations involves the brain using higher level thinking skills, and shows people there is creativity in math and math is not just numbers.

So back to word problems.  It is important for the student to create a visualization of the problem first so they understand the problem and the context of the mathematics.  The visualization can be as simple as drawing a picture or it could be more complex using stop motion animation, creating a cartoon, a comic strip, or a video.  Just some way for students to "see" the problem.

Once students are comfortable with creating visualizations for word problems, they can take the next step of creating their own word problems with associated visualization to show others the problem.  Visualization is the key to learning to do word problems.  Let me know what you think, I'd love to hear.  Have a great day.



Monday, December 9, 2019

How Lava Lamps Create Random Numbers

Lava, Lamp, 1960S, 1990S, Green, SilverI watched a NCIS episode the other day where one of the people had to destroy a bunch of lava lamps used for computing to save the day.  I honestly thought that was a plot device but I checked it out and indeed there is at least one company who uses lava lamps for this purpose.

This company acts as a gate keeper for data on the internet.  In order to maintain security of the data, they have to produce a bunch of random numbers, completely random and totally unpredictable.  To do this, they use the "lava" in lava lamps to help randomly generate numbers.

They do this by recording the bubbling of over 100 lava lamps in the lobby of their main office in San Francisco.  This video is then fed into an advanced algorithmic program which takes the bubbling and changes it into random numbers that are more random than can be produced by humans.

Most random numbers are created by programs written by humans so they are really not that random.  The numbers produced can often be guessed or figured out by hackers but the numbers based on lava lamps is much more random and secure.  There is less chance for these numbers to be cracked.

This lava lamp randomness is actually quite secure.  As stated, the lava lamps are in the lobby of the company's San Francisco office for everyone to see.  By having the lamps out in the lobby for everyone to see, it adds another layer of security to the production of random numbers.  When people walk around, the vibrations change the rate of bubbles or if they block the light it changes the temperature inside the liquid, changing the bubble rate.  Every little change, makes the randomness even more random.

You may wonder where random numbers come into play for computers and the internet.  Well, it's like this.  The minute you log into a website, the computers assign you a random number for identification.  This is to prevent hackers from impersonating people and getting their money, etc. Unfortunately, most random numbers are not random because they are produced by algorithms but the strings can be figured out if the first number is known.  

This is where lava lamps come in.  Their blurps, and bubbles are much more random because the mixture of oil, wax, and water is effected by changes in temperature, vibrations, and other things.  This means the rate that bubbles are produced is never constant.  

Thus the arrangement of bubbles and blurps rising through the liquid is constantly changing making the production of random numbers much more random.  These rates are changed into random numbers which are used in cryptographic keys for extra security.  This unpredictability is the key to randomness in this whole scheme.

I don't think the inventor of the lava lamp ever envisioned his creation being used for computer security.  Let me know what you think, I'd love to hear.  Have a great day.

Sunday, December 8, 2019

Warm-up

Amphibian, Turtle, Animal, Armor, Blur

A turtle can travel at 0.09 m/s.  How many miles per hour is that?

Saturday, December 7, 2019

Friday, December 6, 2019

Chess + Coordinate Plane = Transformation

Chess, Figures, Chess Pieces, StrategyAt one point in my life, I learned chess.  I didn't learn it because I had a burning desire to learn it but my brother decided to learn the game and needed someone to play against.  I didn't care for the game because no one took time to explain one needs to think about the consequences of various moves.  I had no problem with the movement of each piece. because every move could be explained in terms of movement.

Just the other night, I realized chess has so much in common with the coordinate plane.  When my brother taught me chess, he neglected to mention the notation system used in it.  A board is 8 squares by 8 squares or a total of 64 squares.  Assume the chessboard is in quadrant one of the coordinate plane.  The 8 squares along the x-axis are given letters a to h, while the 8 squares lining up along the y-axis are given numbers from one to eight. When you hear someone say "Knight to c3" it means the person is moving his night to the third column along the x axis, then up three units to the third row.  If we were thinking in terms of normal coordinates, we'd say (3,3).

I suspect the use of letters for one coordinate makes it less confusing but this notation system is referred to Algebraic notation.  It is customary to use this system to write down all the moves made during a game so the serious players can go back and review every move, looking for errors and mistakes.  This is the way they improve their playing ability.

Another thing about chess is that all pieces pretty much move in a linear direction.  Some pieces move along a line with a slope of 2 or -2 while others move along a line with a slope of 1 or -1.  Then there are pieces that can move only one square or one jump while others move across the board in one smooth line with a slope of zero or undefined.  Every move can be defined as a transformation.

Chess players use another geometric concept called the "rule of square".  They visualize a square to determine if a pawn will get through the other players defenses.  This has been used since the Middle Ages as a way to make a judgement without using a lot of mathematics.

Furthermore, mathematicians have created two problems  based on chess pieces and a chess board.  One is the Eight Queens problem where people try to place eight queens on a chessboard so that none of the queens threaten each other.  The problem was thought of in 1848 by Max Bezzel in Germany and it wasn't solved until 1972 with the help of computers and lots of work.

Another problem is the Knight's Tour problem where you try to have the knight visit every single square on the board.  This problem is directly related to the Hamiltonian path problem in graph theory.  This problem dates back to Arabic manuscripts from the 9th century.  Mathematicians found several different solutions including Euler which he presented at the 1759 meeting of the Berlin Academy of Sciences.

I love that chess is mathematically related  and I've learned there are several high ranking chess players who are mathematicians.  I wish I'd known more about the chessboard when I first learned chess.  It might have made it easier for me to learn more about it.  Let me now what you think, I'd love to hear.  Have a great day.




Wednesday, December 4, 2019

Glogs and Math

Abstract Background, Abstract MinimalI am just finishing up a flipped learning class for recertification.  One of the chapters in the book mentioned glogs so I had to look them up because I've never heard of them.  Glogs are a shortened form of graphical blogs where the readers can interact with the "poster" on the blog.

The graphical poster is digital and can have videos, pictures, sounds, and so much more.  In other words, instead of reading the blog, you are interacting with the graphical interface.

Think about having students create this interactive digital posters to share information on certain topics.  In addition, creating the interactive poster encourages collaboration when students work together and they use higher order thinking skills to organize the information into a presentation.  It is also a valid method for assessment because it allows students to show their learning.

The cool thing is that there are at least two sites on the web where one can both find and make interactive posters for use in glogs.  Both of these sites have a free level and if you want more you can pay.

1.  Glogster is one possibility.  It allows people to insert text,  graphics, images, walls, audio, video, web , 3-D and VR, along with clip art.  They also have a library of over 40,000 blogs in their content library which includes math and templates teachers or students can use to create something new by simply replacing the blanks with the appropriate videos and text as needed.

Students often ask "How do I start?" and these templates give them a place to start.  It is their work, everything from pictures to text to videos but the arrangement is there.  For students who are good at starting with an empty palette, they can begin with a totally blank canvas and build what they see in their mind.

There is a free version for one teacher and up to 30 students that comes with limited editing and 3-D or media content, no access to premium features and no help desk but for free, it comes with enough to work with.

2. Thinglink is another possibility.  Thinglink allows the teacher to create interactive images or create virtual tours.  I've used this to create interactive pictures for my hyperdocs.  I created an interactive picture on slopes.  There are places on the picture that identify zero, undefined, positive and negative slopes within the ride.

The free version is primarily for the teacher to create interactive pictures for the public mode but for a small amount per year, it is possible to upgrade to the premium account.

The premium allows you to have up to 35 students use the account to create their own interactive posters via assignments using collaborative editing, and allows the teacher to grade the work and provide feedback.  I've never used the premium but I know other teachers who love this and have used it.

If you haven't implemented Glogs into your repertoire, give these sites a shot to determine which one is best for you.  Let me know what you think, I'd love to hear.  Have a great day.




Monday, December 2, 2019

See, Think, Wonder in Math.

Reykjavik, Sculpture, Art, PromenadeI'm finishing off a class on flipped learning and as part of one assignment, I had to create a lesson.  As part of the lesson, I included having students look at a set of graphs, mappings, tables, and coordinates as an introduction to the topic.  They will explain what they see, what they think, and what they wonder as they look at them and then they will record their thoughts in Google Slides.

As I wrote the lesson, I realized it is easy to include the What do I see, What do I think, and What do I wonder in the math lesson.  It is perfect as a way of sparking previous knowledge, mathematical thought, and curiosity.  The picture does not have to be of a math equation, it might be something like the picture above that is filled with arcs.  Lots and lots of arcs used as part of the sculpture.  When I get ready to do slopes, etc I'll post pictures of mountains, or perhaps a picture of the Stockmarket.  Then when it is time to do three dimensional shapes, I can use pictures of buildings like the tower of Pisa, to introduce the topic.

This particular routine encourages careful observations while taking time to think about their interpretations of the picture.  As stated earlier, this routine can be used to introduce a topic or to provide a connection during the unit between real life and the math being studied.  It can also be used toward the end of a unit to make students think about extending the topic or starting the transference of knowledge.

This works best when putting the I see.... I think.... I wonder.... together as one unit and it is important to have them include reasons for their  thinking because one of the math standards is to communicate their thinking.  If you use this routine with visible patterns, start with asking "What do you see?" which has them describing what they see such as shapes, patterns, etc.  Next ask "What do you think?" which could have them thinking about the next step in the pattern, or what math is associated with what they see.  It takes them to the next step in the process of thinking.  The "What do you wonder?" asks them to take things a step further so they might figure out a way to predict the pattern after 10, 20 or 100, or maybe they'll wonder about the mathematics behind something.

Another use of see, think, and wonder is with graphs published in newspapers.  It might be a graph of  the latest pickup truck, the price of the average thanksgiving dinner, or other public graph found in an advertisement.  This activity has students explain what they see, what they think is going on, and what does it make them wonder so they get a better grasp on the idea that sometimes graphs can be misleading.

In addition, this can be used when beginning a new unit.  To do this, have students look through the new material while asking them what they see, what they think they will learn in the unit, and what questions they have about the unit.  If you place a templet online using Google Slides, students can record their thoughts for each question so you can go in later and read their responses. Furthermore, as the teacher, you can go into Desmos and create your own See, Think, and Wonder activities complete with places for students to record their thinking.

It is important to have students discuss their thinking because thinking leads to understanding.  In addition, this activity helps students improve their mathematical communication and increases their critical thinking skills.  Let me know what you think, I'd love to hear.  Have a great day.

Sunday, December 1, 2019

Warm-up

Smartphone, Screen, Android, Technology

If 33.5 percent of the $6.22 billion dollars were made from mobile devices in 2018, how much in sales was that?

Saturday, November 30, 2019

Warm-up.

Data, Business, Growth, Statistics, Sale

Companies sold $6.22 billion in online Black Friday sales in 2018.  That was an increase of 23.6 percent over 2017.  How much did companies do in online sales in 2017?

Friday, November 29, 2019

Paolo Ruffini and Synthetic Division.

Calculator, Three Dimensional, SymbolI love synthetic division.  I love it so much but unfortunately, it doesn't work on anything other than binomial factors. I love it's beauty and simplicity but I've often wondered about it's history.  Who provided the first consolidation of that knowledge?

From what I can tell, the creation of synthetic division is attributed to Paolo Ruffini in 1804.  He is an Italian Mathematician who lived from 1765 to 1822.  The work he did was a forerunner of the Algebraic Theory of Groups and he tried to show there is no solution to a quintic equation without radicals.

In addition to to being a mathematician, he was also a physician and philosopher.  Unfortunately after Napoleon took over, Ruffini lost his teaching position because he refused to take an oath of allegiance but that did not stop him.  He worked as a doctor while continuing his mathematical research but once Napoleon was defeated, Ruffini returned to his university position.

What we refer to as synthetic division in high school is also known as Ruffini's rule which allows people to divide a polynomial by a linear factor to find the zero's of the equation.  As you know, if there is no remainder, you have found a zero but if there is a remainder, you have found the y value of the equation at a certain x value .  It is also much more efficient than using algebraic long division to solve the same problem.

Around 1800 or so, the Italian Scientific Society of Forty opened a competition asking people to provide a method that could be used to find any roots of a polynomial.  In the end, the society received five entries but Ruffini's was declared the winner in 1804 and his paper was published as part of the award.  He refined and republished the paper in 1809 and 1813.

We have students practice it on paper by following certain steps but emathlab  has a really great practice section to help students learn the process.  If you get anything wrong, it will correct the work on the problem so you can see where you went wrong.

In addition, this article looks at ways to extend the use of synthetic division with modifications for x^2 - a and x^2 - bx + c problems.  I had never seen these before because I'd been told you cannot use synthetic division for anything other than linear factors.  I had trouble with the x^2 - a one but not the last one as it made perfect sense.  This site has a wonderful pdf with problems, explanations, a bit of history on synthetic division.

I've always wondered who came up with synthetic division because when I next teach it, I can include a bit of the history so that students will know more about it.  Let me know what you think, I'd love to hear. Have a great day.

Wednesday, November 27, 2019

The Costs of Thanksgiving Dinner.

Christmas Dinner, Christmas, TurkeyI absolutely love finding information that students can turn into graphs.  Information based on real life situations, not something a textbook author dreamed up to perfectly fit the section's topic.

Did you know that a full Thanksgiving dinner cost $5.68 in 1947 but in 2018, it ran $48.90.  If you adjust past prices to account for inflation, the prices have ranged between about $41.00 and $57.00.  If students graphed these, they'd have a better idea of the prices went up and down through time.

The figures can be found here and are based on a meal consisting of a 16 pound turkey with 14 ounces of stuffing, three pounds of sweet potatoes, 12 dinner rolls, a pound of frozen peas, 12 ounces of cranberries, a half pound each of carrots and celery along with everything needed to make two pumpkin pies topped with whipped cream and a gallon of milk.  Enough to feed 10 people.  This site gives a better breakdown of the cost of the turkey, and a combined cost of sweet potatoes, stuffing, and cranberries so students can break the meals down even further.  They can use the information to calculate what percent of the total cost the turkey is or the percent the sweet potatoes, stuffing, and cranberries make up.

If you are only interested in the price of turkey, The Chicago Tribune has multiple ads beginning in 1915 showing turkey sold for $0.28 per pound while as in 2015, it went for $0.48 per pound.  The newspaper also includes the price adjusted for the actual cost in 2015 dollars to give a better idea of how the cost relates.

On the other hand the Business Insider shows the breakdown for the Thanksgiving meal as it cost in 1911 so you know how the $6.81 breaks down but if you include inflation, the cost in 2013 dollars is way, way more.  It is possible to compare prices to 2019 using this article from Moneywise.

The World Economic Forum has a great article on the economics of Thanksgiving.  It looks at everything from the cost of dinner to the average cost of a turkey, to a map showing which states pay the most, the least, and in-between.  It also looks at the number of people traveling, the age grouping of people who end up shopping on Thanksgiving, and the average cost per person including travel.  It is filled with six different graphs so students get a lot of experience reading real graphs.

This site addresses how long a person has to work to afford a full Thanksgiving meal for a large gathering.  It explains where it got the data and how they used it so it is quite educational.

Lots of different ways of enjoying the same information and information one does not usually see associated with Thanksgiving.  Have fun coming up with creative ways to use it in class.  Let me know what you think, I'd love to know.  Have a great day.


Monday, November 25, 2019

Macy's Thanksgiving Day Parade.

Thanksgiving, Parade, Snoopy, NewyorkThe Macy's Thanksgiving Day parade is the parade many people think of immediately because it has strong associations with Thanksgiving Day since the 1920's.

It its also a great event to look at for costs and how costs have changed over the years.  One event that caused an increase of costs was 9/11 because it required all parades have additional police protection.

Lets look at the cost of putting the parade on in 2016.

The parade is 2.5 miles long and takes 3 hours to complete.  The total cost of the parade in 2016 ran between 10.4 and 12. 3 million with an additional 2 millions for costumes, and the property taxes associated with the parade ran another $139,000.

The breakdown of the main amount is as follows:
1. Logistics and coordinations - $1.5 to $3.4 million.  This is the cost of workers, parade supplies, and helium.  Even though the parade is a once a year event, it employs 26 full time workers, and 10 to 15 part time workers.  These workers take a budgeted $1.3 million for salaries.

2. The balloon floats require 50 to 90 people for each one to wrangle them down the parade route.  The balloons also need between 300,000 and 700,000 cubic feet of helium.  The minimum cost is $510,000 to fill the smaller balloons.

3.  It costs $90,000 to sponsor a returning balloon or $190,000 for a new balloon.  If one wants to build a new float, it can cost a lot because the price covers the cost of building a new float over four to nine months and can cost for basic expenses between $30,000 and $100,000.

4.  There is also the cost of hiring police officers, drug sniffing dogs, and rooftop snipers that has to be added in.  No one is sure how much it runs but it is at least $200,000 in overtime costs and that does not include the regular costs.  It is thought the full cost for the police presence is several million.

It is hard to get exact figures because Macy's is extremely tight lipped about how much they spend.  It is hard to even find costs from any of the earlier parades other than 2016 or so.

There are some infographics which contain this information such as the one here or here.  This site has a graphic showing the number of balloons by decade while this site has graphics that compare warmest with coolest days and one on the number of new floats between 2001 and 2016.  This graph shows the breakdown of viewers by percent for the 18 to 64 age groups.

Keep your eyes open here on Friday because I'll be providing links and information on Black Friday Sales.  I hope the information I gave is enough for students to create their own infographics or even graphs.  Let me know what you think, I'd love to hear.  Have a wonderful day.


Sunday, November 24, 2019

Warm-up

Macadamia Nut, Organic, Harvest, Nuts

If farmers are paid $0.82 per pound, how much would one receive for selling 687 pounds of macadamia nuts?

Saturday, November 23, 2019

Warm-up

Australian, Background, Broken, Brown

If a macadamia nut weighs 0.12 ounces, how many make up one pound.

Friday, November 22, 2019

What Does Show Your Work Really Mean?

School, Book, Exercise, Maths, WorkingHaving students show their work is a struggle especially when one is told by the student "But I know how to do it" to explain the answer only.  Since I heard a college professor comment that showing work is part of finding the solution, I've come to the conclusion that having students show their work is important.

I've started telling students that by showing their work, they are actually communicating to both myself and others how they got from the question to the answer.  Showing their work is showing their journey and path.

I remember reading journals in college filled with papers written by various mathematicians.  None of those papers showed the problem with just an answer.  Every paper showed the process the mathematician used to get from the problem to the conclusion.  The work is how they communicated their thoughts, assumptions, etc to others.

Showing your work does not have to be showing every step in written form, it could just as easily include pictures showing how the students get from the problem to the solution.  I use pictures and drawings to "explain" problems to students so I don't see anything wrong with a student providing their work using drawings.

Furthermore,  I've discovered many students who showed only answers, came up with the answers in one of two ways.  First, they did the work in their heads but they were unable to explain how they got from the question to the answer.  They were unable to remember the steps they took and sometimes could not repeat the process on future problems.  They couldn't even provide any sort of illustrations to show this.

The other possibility is they did the work on an app or calculator that provided only the answer so they wrote down the answer but had absolutely no idea how the app got the answer.  This is apparent when they take a quiz or test.  Yes, I know there are apps which show the steps but most of my students tend to use Siri or other app which gives them only an answer.

Unfortunately, it is often easy to do math mentally in Elementary school but when the problems become more complex and requires multiple parts, it becomes more difficult to complete mentally.  For instance, factoring a quadratic with a leading coefficient or performing polynomial long division often requires numerous steps and is so much harder to do in your head.  The other thing, is when students continue doing the work mentally, they sometimes miss certain steps so they are unable to get the correct answer due to that gap.

I do believe it is time to get rid of the phrase "Show your work" because it is not as accurate as it might be to rephrase it into something like "Document your mathematical thinking" or "Write down notes explaining the process", or "Communicate your full solution".  Perhaps we can explain that letting us see how they got from the question to the answer is the same as writing the answer to a short answer question in English.  It is time to get past having students see "Showing your work" as a negative activity and instead see it as a positive form of communication.

I would love to hear what you think about this.  Let me know please.  Have a great day.

Wednesday, November 20, 2019

8 Ways To Transfer Knowledge.

Cactus, Book, Flower, Pot, Read One thing students have difficulty doing is transferring their knowledge from the textbook to other parts of their lives even with real world examples.  Often, students are unable to see the direct link between the textbook and real life but there are ways to help students learn to transfer their knowledge so it becomes habit.

One issue is that memory relies on context which makes it harder for the brain to transfer material learned in a classroom to a work environment.  So one has to learn how to take prior knowledge and give the information a new context so it applies to the new situation.

1.  When teaching new material have students suggest ways it applies to either possible future jobs or life before tying it to their long term or short term goals.  Teachers can suggest connections but if the students do not provide their own connection, they find it more difficult to transfer knowledge because they see no relevance.  In addition, when students find relevance to the material, they are able to apply it to other situations.

2. Incorporate activities where students are able to reflect on the new material to see if they understand the concept in depth.  Then they need to explain it to themselves because this offers them the opportunity to discover if they have misconceptions, correct them which leads to a deeper understanding.  Furthermore, when students work on explaining it to themselves while using their own words, it is easier for them to connect it to previous knowledge while transferring it to a new situation.

3. Have students use a variety of media to learn the material.  This means using text, videos, audio, and other methods so students are able to retain more material and score higher when tested on the material.  Visuals can make it easier for students to learn or at least they perceive it that way.

4. Help students change the way they study so its more random rather than being at the same place, with the same music, at the same time everyday.  Although changing the place, time, and way of studying is more difficult initially, it is easier to retain the information later on.  This change also called "desirable difficulties" actually leads to a deeper learning which is what students want.

5. It is important for students to learn to identify any gaps they have in their foundation because these gaps make it more difficult to transfer knowledge. Furthermore, it allows students to strengthen their knowledge based once these gaps have been identified.  One way to identify gaps is to take practice tests because tests also identify topics students have mastered so they can focus only on the gaps.

6.  Teach students to set both learning goals and success criteria because it tells students what they are going to get out of their learning and how they know when they learned it.  Goals need to be realistic and specific so the goals can be met.  A realistic goal might be learning all the facts above  7 x 7 over the next 6 weeks because you know you struggle with those and you've set a reasonable time period to do this.

7.  Help students learn to generalize the knowledge from the classroom to a different situation.  An example of this might be taking what they learned about inequalities in the classroom and find as many situations in real life that are described by inequalities such as filling a gas tank, the height restriction of a amusement park ride, or buying patterns on sale.

8. Every day, have students find situations to apply what they learned.  These situations should be outside of the classroom such as buying enough pizza's for a party and thinking of this as using an inequality or look at starting a business in town and calculating the startup costs.  Finding real life applications of the material provides that direct connection between the classroom and application to work.

These are some ways for students to learn to transfer what they've learned in the classroom to other situations such as a work environment.  Furthermore, it prepares them so future employers are more likely to feel as if the student is ready to step into the company.  Let me know what you think, I'd love to hear.  Have a great day.

Monday, November 18, 2019

Math, Infographics, and Graphs

Social Media, Media, Board, Networking Most standards are already out of date by the time they are printed.  In fact, we have new ways of communicating information that are not taken into account by standards.  I love reading infographics because they condense information into easy to understand graphical representation.

The great thing is that infographics can be used to teach math while showing real world statistics.  Furthermore, infographics provide students with immediate understanding of certain topics.

One way to use infographics are to have students start with a graph.  It might be a graph on last 50 years and what happens with poverty rates, football statistics for the NFL, divorce rates, or the number of people traveling over Thanksgiving.  Once they have the graph, they might look into what causes the rate to go down or up.  This requires a bit of research possibly  but once the student has determined the reasons for a fall or increase, they are ready to create the infographic that will provide a story to accompany the graph.

Another way is to find two different graphs that are marked in the same unit and graph them on the same coordinate plane.  Once this is done, the student can compare and contrast the two graphs before looking for positive correlation, negative correlation, or there might be no correlation.  It is important to remind students that correlation does not necessarily imply causation.  This site puts two totally different graphs together to show correlation but also takes time to discuss how correlation can imply causation.

Or you could have students investigate how the interval and scale change the way we look at the graph.  For instance, a graph showing unemployment might change the time scale to every month instead of every six months to see how it changes the overall graph.  Another way to look at that same graph is to change the y axis so its every five instead of every one running only from 55 to 60 percent.

What about taking the cost of attending colleges from colleges the students are interested in and creating their own graphs comparing tuition, fees, etc to see how much they would owe after 4 years. The next step would be to present this same information in an infographic.  I know some colleges do not charge Native American students tuition while others do.

Another possible thing would be to have students look at the historical trend of something such as the Alaskan Permanent Fund Dividend and comparing it with the price of oil to see if there is a relationship.  Take this a step further by predicting what might happen to the check over the next few years based on what they conclude has happened so far.

Let students choose a topic, research it before creating a graph on that topic before creating an infographic to accompany it.  It might be something such as the cost of cars over the past years, or the  drop in value from new to 10 years old for a variety of models to see which has a history of the best resale.

There are so many possible topics.  One place to find wonderful graphs to use in some of these exercises is at the New York Times complete with ideas, graphs, and lesson plans.  Let me know what you think, I'd love to hear.  Have a great day.


Sunday, November 17, 2019

Warm-up

Bicycle, Bike, Great Stuff, Large

If the radius of the front wheel of the bicycle is .75 meters, what is the circumference of the wheel?

Saturday, November 16, 2019

Warm-up

Bicycle, Bike, Great Stuff, Large

The radius of the front wheel is .75 meters.  What is the area of the front wheel?

Friday, November 15, 2019

Increasing Positive Emotion in Math.

four green emoticon balls If you read the blog the other day, you'll know that our brains remember more if they have an emotional attachment to the material.  Unfortunately, too many students have negative emotional connections associated with math which creates several problems.

First, many people believe they are not good at math and will never get good at it so they have a mental barrier to learning.


Secondly, if a student does not like math, they are more likely to have lower achievement scores and as soon as they can, they quit taking math or are more likely to take lower level math classes.  This means they are closing themselves to certain career opportunities, especial those of higher earning potential.  It also limits how they will interact with society and it's possibilities.

Right now, if a student struggles with math, one of two things happen.  They are encouraged to work harder or they are moved into a lower level of mathematics.  Unfortunately, neither of these solutions may not help improve student attitude.

So the question becomes, what can we do as teachers to make class more emotionally engaging so students have getter emotional associations with math?  There are ways to do this, some work better with younger students, some with older students but all can help students associate happier emotion with math.

1.  Create authentic inquiry by setting up a situation where students can ask the questions.  It might be drawing a rectangular prism on the board and have students create questions the can be answered from looking at the drawing.  This makes the students operate more as real life mathematicians who guess, question, and narrow their solutions based on repeated results.

2. Always show more than one way to do the math but don't be afraid of asking students to see if they can figure out a way to solve the problem on their own.  This gives them a chance to find counter examples on their own, or set conjectures.  In addition, it motivates students and allows them to develop mathematical creativity.

3. Instead of having students sit in one place the whole period, create opportunities for them to move around.  It could be as simple as asking a multiple choice question, assigning one answer per corner and having students go to the corner they think represents the correct answer.  Or organize a scavenger hunt with at least ten half sheets folded in half.  On the front, write a problem, and on the second page, write an answer but it is not the answer to the problem on the front.  It is the answer to another problem.  These are hung around the room after assigning a letter to each problem. Students have a sheet with boxes.  In the corner of each box is a smaller box where they write the letter of the problem.   The idea is that students start at one paper where they write down the problem, solve it, and then look for the answer.  Once they find the answer they look at the new problem, write it down and it's letter in the box, solve it and repeat it.   I've been known to use QR codes with a problem and answer contained in the code so it uses student devices but they still write the problems down on paper.

4.  When you create an assignment, quiz, or test, set them up so students can choose the problems they want to do.  For instance, you might want to write a quiz with 15 problems and have students answer any 10 problems.  This increases student motivation and makes them feel as if they have some control.

5. Place both examples and non examples on the board before asking students to figure out what makes the topic.  For instance, when discussing spheres, one might post pictures of basketballs, baseballs, or an orange.  For non examples, you could post pictures of bananas, footballs, or boxes.  Let the students use this information to create a list of what characteristics a sphere has.

6. Establish situations where students can collaborate together in a learning environment.  For instance, you can set up Jigsaw activities for sharing vocabulary, create a slide show or book using google slides to have small groups of people share their material with others.

When math becomes more engaging and students feel as if they are taking over more of their learning, they develop positive emotions so they learn the math.  Let me know what you think, I'd love to hear.  Have a great day.  I'll be back Monday with a new topic but I will continue to post warm-ups.

Wednesday, November 13, 2019

Reading and Vocabulary.

Book, Textbook, College, LearningI have recently started doing things differently in my Geometry classes because I'm trying to make students more self sufficient and make the class a bit more student centered. I need to teach a bit less while making the students more independent.

One of the first things I did was create a paper filled with a table of 16 squares.  In each square I wrote one of the vocabulary words for the chapter and nothing else.  I divided the students into small groups of two to three people depending on the size of the class.

Once I passed out the vocabulary sheet, I assigned three words to each group so no other group had the word.  I gave them time to work on defining the terms using their own words. One girl asked if they could draw pictures to help define the word.  I said go ahead because research shows it helps to have pictures to go with words.

After about 10 minutes, I had students get up to new groups so everyone in the group had defined different words.  They shared their vocabulary words with the others so by the end of the time, every person in the group had shared their words and had a chance to fill in all the other vocabulary words.  This is a variation on the Jigsaw activity.

What I liked was the variety of methods used to complete the assignment.  Some groups split the words up so one person defined a word before sharing it with the rest of their group so everyone had their three words defined before sharing with the other groups.  Others looked the words up together, discussed what the definition should be before agreeing on one.

The important thing with this activity is that the words are ones they've used before in their two column proofs so these words are not new but they've not had time to really look at the words in detail.  By waiting this long before doing a serious vocabulary activity, students had a chance to build prior knowledge.

The other thing I've started doing is again dividing students up into small groups and having them read the new material.  After they complete the reading, they have to decide what the three most important ideas or concepts contained in the material.  At the end, I asked each group to give me the one concept they thought was the most important and they had to explain their thinking.

I do this activity on one day and give them 24 hours before taking it to the next step because it gives their brain a chance to process the information.  One day later, I go back and ask them to prepare a more detailed explanation of that concept to share with other students.  This presentation can be done via google slides, a short video, or some sort of animation but they have to explain the concept and why it is important.

I've found that most groups came up with different choices so there is very little overlap.  If I do have two groups choosing the same topic, I often ask one group to think about why a certain other topic is important.  Most of the time, they don't mind the change of topic because it was one on their list of the top three.

Both of these activities encourage dialogue, communication, and explanations so they learn to discuss and communicate math.  I would love to hear what you all think.  Let me know, have a great day and I'll be back with something new on Friday.

Monday, November 11, 2019

Emotions and Teaching Math

Heart, Cord, Suspended, Love, TogetherWe all have those students, the ones who arrive in your math class, convinced they can't do math or don't try because they see no use for it.  They have a negative emotional connection with math which can slow down their learning.  There are ways to help students gain better emotional connections.

Remember, emotions help us think because they pull up memories we have associated emotions with.  The question is then, how do we take advantage of this in our classes.

One thing we are told is to make math fun by incorporating games and prizes but these are considered a quick fix.  What works better is to help students develop a sense of accomplishment by making math interesting, satisfying, and personally fulfilling.  There are some steps each teacher can take to change student dislike into appreciation and wonder.

First, rather than letting students see math as some sort of torture to be tolerated until the math requirement is met, we should explain why these math concepts matter.  Mathematics helps students think both theoretically and logically.  Furthermore, math helps the brain think abstractly which is a skill that can be used in so many different ways.  There is research that indicates students who study complex math topics tend to do better in life.

Second, give students problems where they see how math is used in real life.  If they see connections, they are more likely to participate.  Practical applications range from keep track of their finances, to how fast oil spreads on tissue paper which gives students a real life application of direct variation and understanding of how fast oil spills spread.  Find stories in the news where a need to understand math is important such as in a story on price fixing, or climbing interest rates.  If you have students read an article on climbing. interest rates before setting up a spread sheet to show how much an increase of one-eighth of a precedent can effect the overall amount of money paid on a lone.

Third, take time to discuss mathematicians who helped change the world or make changes to the world such as Pythagorus, Rene Descartes, or people like Alan Turing who helped break Axis codes and allowed the Allies to win the war.  For the sports minded, introduce them to Nate Silver who uses statistical analysis to predict winners in elections and Major League Baseball.

Finally, look at ways to decrease the stress students feel when learning new material or taking tests.  When introducing new material, always try to relate it to previously learned material such as when solving one and two step inequalities, we can related the processes back to solving one and two step equations.  Relating it back to previously learned material also helps students learn the same material better or even get it when they didn't get it the first time.

This is an introduction to the topic.  I will be exploring it in more detail as I have a few students who see absolutely no reason to learn math and do their best to disrupt instruction so others have difficulty learning.  I want ideas to counter this "If I don't want to learn math, why should you?"  Knowing how emotion impacts learning is my first step to countering this.

I am slowing down to three topics a week plus warm-ups on the weekend because of the number of sports events happening at school.  I have been helping out and will continue to do so but it is not leaving me as much time for me.  So the next time I add new material, will be on Wednesday.

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

Sunday, November 10, 2019

Saturday, November 9, 2019

Friday, November 8, 2019

Emotion and Memory

photography of human faceDid you know that emotions help your memory remember information?  Emotion plays an important part in how we remember things.

In general, emotions highlight certain parts of our experiences to make them more memorable.  It works in so many ways with our memories.  Some may be new to you, some may not.

First is attention which helps our minds determine what is most important in our lives.  Emotions such as novelty or surprise, helps us focus on conversations so we are aware of a few items which allows to use our limited attention span to its optimum.

Second is consolidation of memories.  The truth is that most of the information we are exposed to, do not make it into our long term memory.  When we have an emotional connection to something, we are more likely to remember it. For instance, most people of a certain age can tell you about 9/11 because of the emotions it brought up in everyone.  This is because certain stress hormones such as cortisol and epinephrine help us consolidate our memories.

Thirdly, it appears that those memories associated with painful events stay with us longer than memories associated with physical pain.  Contrary to the rhyme about "sticks and stones may break my bones but words will never hurt me" we do remember the words that hurt us.

Next is priming or triggering behavior through the use of unconscious suggestions.  For instance, if people unscrambled sentences that contained suggestions of self control, they would immediately make better choices when it came to eating snacks.

Then we have mood memory which is where we recall memories which have the same emotional feeling as the way we feel at that moment.  So if we are sad, we are more likely to remember things that are sad.

Blanking out is something most of us have experienced at one time or another.  It has nothing to do with drinking but being so stressed out we forget things.  This usually happens during tests.  It turns out that if we are bored or too stressed, our performance is likely to suffer.  When we are bored, our minds are unfocused but if we are too stressed, our minds are too focused and we miss information.  The best place to be is when we are in moderate stimulation.

Finally, we have a duration memory because we do not remember everything involved with the memory.  Instead we remember the best or worst moment and the last moment.  We do not remember everything of the event.

This is why it is important to connect emotion with any learning, otherwise it won't stick.  In the future sometime, I'll cover emotions and learning math.  Let me know what you think, I'd love to hear.  Have a great day.

Thursday, November 7, 2019

Chunking In A Digital Age

Mind, Brain, Mindset, Perception With the way children's brains have been changing due to the use of digital devices, I wondered if chunking was still recommended or if it's been replaced by something else.

Chunking is one of the ways to organize information.  In reading, it is putting together letters and sounds to create meaning while in math it is organizing the information into smaller, more understandable pieces.

According to research, the human brain can only handle seven pieces of information in short-term memory at any one time.  Chunking the information helps the brain avoid traffic jams in short term memory.  Chunking also helps the brain remember more information because they have combined pieces together.

Chunking is also a way of identifying the most important information out of a section or chapter.  This is done so students know what they need to focus on and learn.  Furthermore, when chunked properly, it is well organized and logical and increases student information on what is going on.  It helps them see the bigger picture while remembering the information more effectively.

Furthermore, there is another theory, the cognitive loading theory, that states the more the brain has to remember in a shorter period of time, the more difficult it is to learn because the brain is overloaded.  This supports the use of chunking because chunking does not overload the brain. The chunking also allows them to build on previous chunks thus providing the repetition the brain needs to retain information.

It is known the brain seeks patterns to make sense of material because the brain stores information as patterns.  Furthermore, the information must make sense to the brain otherwise it will not be stored.  This includes new information that is not yet familiar so when we introduce new concepts, our brain does not always store it so we cannot expect our students to remember it after one example.  So it becomes necessary for teachers to help students identify the new patterns while associating them with older patterns in order to formulate new ones. Patterns have been described as the roadways for memories to travel.

In addition to looking for patterns, the brain searches for personal meaning so it is important to help students find a way to relate to the new patterns because that helps students learn the material better and easier.  If the brain cannot find personal meaning, it will lose it and we see it as students who don't "get it."  This often manifests as the "I don't care" attitude which students see as better than a "I don't understand" confession.

Since most of our students are used to working through quite a lot of data, we need to know how to organize multimedia materials into usable chunks.  It is recommended one group related information into each chunk.  This includes buttons, images, graphics, etc as related information.

There are some ways to help students find personal meaning.  One is to ask students to relate to the information in a personal way by writing it down on a KWL chart.  Another is to ask students to share a story with other students about something that happened to them that uses the concept you are teaching.  For instance, they might talk about when they ran out of gas because they didn't check the amount of gas in the tank before heading out.  This story might relate to a compound inequality about filling the tank with gas.

So with chunking and personal meaning, students can learn the material better and for longer periods of time.  let me know what you think, I'd love to hear.  Have a great day.