Happy Halloween.

Where Did i Come From?

When I was in school and I learned about imaginary numbers, I honestly thought it had been dreamed up by a bunch of drunk mathematicians who didn’t have anything better to do. Yes, I know that’s wrong but when you are 16, you think you know it all. Since then, I’ve learned more about I and my story is just that a story.

Let’s rewind history back to the 1500’s in Venice where all the formulas used to solve equations were guarded closely, much like companies guard proprietary recipes. One man, Niccolo Tartaglia, had a real interest in quadratic and cubic functions due to working with ballistics and fortifications. Niccolo and others realized that many of the quadratic and cubic functions had solutions needing the square root of a negative number.

Niccolo and a competitor discovered if they used negative square roots, they would still get proper numerical answers. Thus mathematicians began using I to represent the square root of -1. The one problem with this definition is that it went against the knowledge that a negative times a negative produce a positive number. When you square I you get I^2 = -1 and it took people a long time to be comfortable with that idea. 

Another reason, it took people so long to accept this whole idea had something to do with the fact that the square root of -1 could not be shown via a graph or using geometry.  Mathematician Rafael Bombelli wrote a titled Algebra in 1572 in which he tried to explain math to people who had not studied it. In the book he declared that imaginary numbers were neither positive nor negative and did not follow the normal rules because he got around it by stating it as sqrt(-1) rather than i.  

Descartes called them imaginary numbers because they were imagined solutions.  This lead to other mathematicians accepting the term and eventually they came up with the Fundamental Theorem of Algebra which states that the number of solutions of an equation are equal to the highest degree. This theorem was not proven until 1806 by Jean-Robert Argand who worked on relating imaginary numbers to geometry through the use of complex numbers.  

About the same time, Frederich Gauss declared that imaginary numbers were not made up and could actually be visualized because they made perfect sense.  Once should look at the idea these are not real numbers but exist as an extension to the real number line while Euler came up with the accepted letter we now use.

For the most part, imaginary numbers remained in the range of the theoretical until recently when modern electronic age. Imaginary numbers are used when analyzing any kind of waves from electromagnetic radiation (wi-fi, radios) to audio signals (music, AC).

So now you have a short history of the imaginary numbers.  Let me know what you think, I'd love to hear.  Have a great day.  

Music To Help Teach Math

 The cool thing about the internet is the availability of supplemental materials designed to help teach math.  Over the years, I've explored Youtube and other sites to find music and rap that my students  will like and remember.

For many of our students, if given a chance, they would sit at their desks all day and listen to their music while playing games.  Why not take advantage of it to help our students learn.

Research indicates music can have a positive effect on student scores.  Furthermore, several studies indicate students are able to remember material when it is set to music.  It is also how we learned our alphabet or numbers when we were in elementary school.  It has been found that the region of the brain that stores memories is the same location that processes music and emotion.  In other words, the three things are related.

Due to the strong relations between math and music it is possible to use the lyrics of the music, it can help people learn math better.  When the right type of music is used, students will sing along and enjoy themselves.  In the process, they will begin to like music more.  Furthermore, if they learn the songs, they can use them on tests to help them do better.

There are websites out there that either produce the music or they have links leading to songs.  I'm covering both types and Youtube to give you an idea of what is available out there.

YouTube has a ton of songs and raps dealing with just about any topic but I strongly recommend you check out the videos first to see if you can hear everything through the whole song.  Also determine if it is something your students will enjoy.  I've had videos I loved but my students didn't like as much because it wasn't their style of math.  In addition, I tried to find ones that had animation rather than just people because they liked the motion more.  I use it a lot.

Songs for Teaching has a whole set of music available for a variety of mathematical topics.  It covers everything from elementary level topics to algebra, geometry, and trig.  They offer short snippets of certain songs so you can decide if you want to purchase the whole "album".  It is even possible to get the lyrics for the songs so you know ahead.  The nice thing is you can get the download of the whole "album" in addition to printable lyrics for a reasonable price if you want to purchase it.  I have not used it but I do enjoy what I've heard from various samples.

It doesn't take much to find music on the internet. It takes a simple listing the topic + music or song and you get a list of possibilities.  You just go through the possibilities until you find the right one and go for it.  I have seen students remember lyrics during a test and it helped them so better overall.  If you aren't using music, try it and see how it works.  Let me know what you think, I'd love to hear.

Numbers

 I usually get two questions from students who don't like math.  The first is "Who invented Math?" followed by "Who invented numbers?"  They hate hearing that numbers developed to meet the needs of commerce, government, and business.

Originally, it is said that ancient man understood the concept of one, two, and many but the ability to write down actual symbols representing quantities came much later.  It developed as civilization had need to measure land, tax people and lands, inventory, etc.

It is known that numbers originally began with one since the concept of zero came along much later.  Archeologists found the Ishango Bone dates back about 20,000 years.  It was found in the Congo and seems to be a counting object.  However, it is not sure what it was used to count.  As far as real numbers being used in a civilization, Ancient Sumeria has records of counting back to 4000 BC.  The records indicate some of the earliest arithmetic using addition and subtraction when keeping tract of livestock, crops, or goods.

About 1000 yers later, in 3000 BC, the Egyptians changed the meaning of one from only counting to a unit of measurement.  The Egyptians were responsible for standardizing units with the cubit so they could officially measure pyramids, temples, and other buildings.  The cubit was the length of a man's forearm from elbow to fingertip.  They kept an official cubit in temples so any copies were made off of that one were the same length.  In Greece, Pythagorus made a contribution in the form of identifying even and odd numbers. He felt odd numbers were male and even ones were female.

Another Greek mathematician, Archimedes, loved playing with all sorts of things including turning spheres into cylinders.  It was this discovery that allowed people to take a map from a globe and make a flat copy of it.  The Romans were not particularly interested in mathematics due to their numerals being extremely wieldy.  They used a counting board(an early form of the abacus) to do math because they couldn't do much more than adding and subtracting.  Their math stagnated.

One of the biggest developments in numbers came out of India around 500 AD. They invented Zero which didn't exist up until that point.  They used the basic digits from 1 to 9, added in zero and had the ability to write extremely large numbers.  About 250 years later, the Persians contributed fractions to the number pot.  According to the Koran, all land had to be divided among the children and fractions developed as a way to make this happen.  In addition, Muslins also contributed Algebra and Quadratic equations.

These Arabic numbers spread out across the world to Northern Africa where Fibonacci, ran into them and brought them back to Europe in the 1200's.  He was attracted to them and enjoyed learning more about these numbers and their use in Mathematics.  On the other hand, it took a while before Europeans were willing to accept these numbers instead of the Roman ones they'd been using for quite a while.  The Italians called zero cipha but they viewed it with suspicion so it became the word for secret codes or cipher.  Eventually, people accepted it because it made business so much easier.

Prior to the Catholic Reformation, people did not charge interest on loans because the church said it was a sin to do so but after the Reformation, it became legal and the new system made it easier to calculating interest out 12 places. Roman numerals only allowed them to go out 2 places.

In the late 17th century a German created a way for numbers to be expressed using only zero's and ones. It is now known as the binary system.  He also designed a machine that used binary but didn't build it.  This type of machine came into being in 1944 during World War II.  So now you have a short history of numbers to show how no one person created it but it evolved as the need for things arose.  Let me know what you think, I'd love to hear.  Have a good day.

Warm-up

If you have 158 pounds of wool that you sell for $13.50 per pound, how much will you get if you sell it all?

Warm-up

If there are 32 sheep and each sheep produces an 8 pound fleece, how many pounds of fleece will the whole herd produce?

Flip Grid in Math

 Flipgrid is one of those great programs that allows people to record themselves in video form and is quite useful in the classroom.  This is one of those programs that is much easier to use in Language Arts or History but it can be used in Mathematics, especially since  the Common Core has students "explain themselves".

If you have never used it before, flip grid is easy to sign up for and allows you to use class codes rather than having each student join via their own identity.  In addition, they have an educators startup guide so if you've never used it before, you have a place to start.

The guide has been updated for fall 2019 with everything from the lingo and signing up for an account to using it in class and setting up the assignments.  I like this guide as it is very step by step for the complete beginner.  One of the other items they offer are a variety of grade level integration documents to help teachers use flip grid in their classes appropriately.

The high school integration document has some wonderful suggestions for using Flip Grid in the high school classroom although they have documents for all grades all the way down to kindergarten.  Some of the ideas they give for ways to use Flip grid in the classroom include explaining geometric formulas and how to use them to solve problems, how how to extend the laws of exponents to rational exponents, and so many other ways. But there are other ways to use this program in the math classroom which allow peer tutoring.

Another site suggests assigning each student a problem they explain how to solve using flip grid. The type of video is up to them.  They could make an animated video, talking heads, music video, or even screen capture.  The nice thing about having students show their work in this medium is that they practice communicating their thoughts.  In addition, rather than helping students when they have questions, let them post a video asking for help on a problem, and other students create a video to help the student out.

Students often have difficulty figuring out where mistakes occur.  One way to help students with this is to post a video with a problem and have students post videos with their answers.  The teacher then provides feedback while moderating the answers. The teacher create videos where students post their answers to your "What do you wonder?  What do you notice?  What do you think?".  Or have students respond to your "Which one does not belong?" video.

If you want students more involved with math, let them create problems to share via Flipgrid and let the other students post their answers or place a picture in the video while asking students to estimate something and explain how they came up with their estimations.

In regarding to grading the flip grid products, it is easy to apply rubrics because you can let students know what you are looking for and the criteria you are using for grading such as the thoroughness of the answer, the explanation, mechanics and grammar, etc.

Check it out and have a great time checking it out.  Let me know what you think, I'd love to hear.  Have a great evening.

Thoughts On Solutions Versus Answers.

 After hearing a speaker at the math conference state the solution is showing how you solved a problem and the answer is the last step of the process, I realized most of us do not think that way.

Most of the textbooks I deal with that the word solution, use it to refer to the answer.  This leads to us instructing students to "Show their work" so we see their steps, otherwise, they only want to put the answer down.

If one assumes the solution is showing how they got from the problem to the answer, then we can ask students to show their solution and know it includes the answer.  I've also seen the term solution set used to express the answer.

I took time to look up the definition of solution on several websites.  Some sites define the solution as the value or values that make the equation true while other sites define it as act of solving a problem.  I'm beginning to think that solution is one of those words that has multiple meanings depending on the context.

I think it is a matter of how its phrased.  If one is asked to find the solution, then it is the process of finding the answer and the work involved in going from the problem to the answer but if asked for the solution, then it is those values that make it true.

I actually spoke with a friend on this topic.  He said we should ask students to solve the problem but I pointed out that as a teacher, I usually have to clarify my instructions with "Show your work" because students believe the word solve means give the answer.  If we use "Find the solution" in and of itself means show the whole process from problem to answer.  He thought about it a minute before agreeing but I suspect my students would argue they found the solution when they write only the answer.

Since many words in mathematics have multiple meanings, this is one of those words we should include since depending on its use will determine which meaning is wanted.  It would be nice to introduce students to finding a solution versus the solution or solutions so student learn to the difference.

I don't know if I fully agree with the solution being the whole process but I do know I want to teach my students that "find the solution" means the same as show how you got your answer and hopefully I won't have to remind them to "Show their work" otherwise I'll get answers only.

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

Use Magic As The Warm-up

If you read enough books, you know you need to grab a student's attention and keep it through the whole lesson.  There are a variety of ways to do this but what about throwing in some mathematically based math.  Yes, you read that right, mathematically based math and its possible to take the steps in the magic and convert it to mathematical steps.

One of the easier tricks is to have a student choose a number between 1 and 100. Make sure they don't share it out loud, but do have them write it down on a card.  Have the student multiply the number by 2.  Then have them multiply the product by 5.  The last step is to have them drop the zero and they will have the number they started with.

Original number is 52.
52 x 2 = 104
104 x 5 = 520
520/10 = 52
The original number.

Another trick is to have a student choose a number between 1 and 100.  Have them double the number and then add ten to the result.  Take the new number and divide it in half.  Next subtract the original number and the answer is 5.

The original number is 28.
28 x 2 = 56
56 + 10 = 66
66/2 =33
33 - 28 = 5.

Try this one with a student.  Have them write down their age.  Then have them multiply their age by 1/5th of 100 (20).  Then add today's date but only the date, not the month or year.  Take this result and multiply it by 20% of 25 (5).  Then add the person's shoe size rounded to a whole number if they take a half size.  The final step is to subtract 5 times todays date.  This should leave a 3 digit number.  The first two give the age and the final two gives the shoe size.

The original age is 25
25 x 20 = 500
500 + 23 = 523
523 x 5 =2615
2615 + 8 = 2623
2623 - (5 x 23) = 2623 -115 = 2508

So the age is 25 and the shoe size is 8.

A final trick is to have a student count the value of the coins in their pocket.  Make sure they don't tell you.  Have the student double the value and then add the first odd prime number to the total.  Then have them multiply the result by 1/4th of 20 before subtracting the lowest common multiple of 2 and 3.  Finally drop the last digit and the result is the amount they started with.

Original amount is 52 cents.

52 x 2 = 104
104 + 3 = 107
107 x 5 = 535
535 - 6 =  529.
If you drop the 9, you are left with 52, the original amount.

One day, use one of these tricks as a warm-up for the class.  Its possible to do any of these tricks with all the students at once.  I've done these before and I've had students work out the math that went with the trick so they saw it both as the trick and as a math problem.  Let me know what you think, I'd love to hear.  Have a great day.

STELLA.

A few days, I reported on a app that could used in class to build systems called SPLASH!.  I also learned about another systems dynamic software at the conference.

This software, STELLA, is offered by ISEE Exchange .  It has the same pieces as SPLASH! but it is web based so it works on more devices than the other.  Furthermore, it is free for educational uses.

 Once you've got a free account, you can set up a new model for the account.  You name it, add an URL, with a description.  When you add the model, it opens up with a blank canvas.  This is the place where you put the big square for the item that changes such as bank accounts and interest, or water going in and out of a pail.



At the top, as part of the bar with everything, the first four items used in the setup.  The square for the item that changes, the spigot for inflow and outflow such as interest rate or birth and death rates or anything else that will change the main item.  There is the circle which is the converter or the math part of the process.

This site does offer tutorials on every part of the process from placing a stock or the thing that changes such as money in the bank, to adding inflows and outflows, entering values, graphs, to creating tables, scaling variables, placing and using connectors, and adding converters.  Everything you need to create your own model.

In addition, one can find models to see the possibilities for using the software.  The models include things like college loans, the zombie apocalypse, the spread of measles, housing and supply demand, humanitarian aid, and so many other topics.

Besides using models as a way of seeing how things can be done, they can also be used to see real models in action.  So even if the class doesn't have time to actually build a system, there are models you can use in class.  For instance, if the class is studying exponential functions, use a model to show how it works.

This is mathematical modeling in real life.  It takes the problem and allows students to see how the math connects with the process and puts it all in perspective.  I would say this software is a bit more complex than SPLASH! but I think it is aimed at grades 8 and above.

Check it out and let me know what you think, I'd love to hear.  Have a great day.

Watching Videos Actively

I am taking a college class on flipping my classroom.  The idea is that I have students watch the videos the night before so I can spend more time working with students on the assigned work.  The thing is, most students I've worked with watch videos passively and don't know how to actively watch.

This is my fear when flipping my classroom but there are things I can do to turn students from passive to active viewers.  The good thing about videos is they help build background knowledge while providing another way of seeing the knowledge.  Videos can also enrich and clarify written text and deepen or solidify student learning.

There are several ways to accomplish this. The first is to use a three step process.  The first step is to provide questions to help students focus on the most important points.  Students write down their answers to the prompts or questions on paper or on a digital document.  Next, the class should rewatch the video together with the teacher taking time to stop the video to comment on things students need to be aware of while modeling appropriate behavior.  The final step is to debrief students to make sure they got the important information.

Other suggestions for using videos more actively is to give students a reason for watching the video, taking time to pause the video so students can write down answers to the questions or prompts, and turn on closed caption so students can read along with the narrator or presenter.

There are ways to increase student involvement to make them more active viewers.  Ask students to tweet their comments to each other with the appropriate hashtags or create a poll for students to answer with information from the video.  My favorite way is to use a video from Edpuzzle or use Edpuzzle to curate a video.

I prefer using a curated video because I can include comments on what to watch out for when doing the math, I can include a short answer question, or a multiple choice which students have to answer before they can move on.  I can also include directions for them to copy certain things into their notebooks.  Furthermore, I can check to see how long it took for them to complete the exercise so I know if they whizzed through it or really paid attention.  I can also see how many times students watched each section.

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

Warm-up

If there are 8 flowers per square foot, how many flowers will you find in 9 square yards?

Warm-up

If Water Lillies grow from 2 square feet to 25 square feet in 3 years.  How fast did it grow each month?

Splash!

Splash! is a mathematical software modeling app developed by Creative Learning Exchange to show system dynamics.  In other words, it allows people to model any simple system that increases or decreases.  

This app has versions for iOS and android.  Both versions are free.  It is fairly easy to use but since its for beginners it might be simplified.  

The person who lead the session used the rate of a bucket being filled with water at first and then added a second condition of water exiting.  This process used two different rates.  It was fairly simple to set up.

 The first thing one does is begins a new model by pushing a + button.  It has you name the model and the design page opens up.  Since I took the session at the Northwest Math conference, I had someone guiding me through the set up.

This group has prepared a really nice 43 page instruction booklet which explains how to use the app.  In addition, it shows how some of the different types of systems, this app can be used for.

The left side of the work area has the parts used to create the actual system.  Going from top to bottom, the first one is the container which represents the changing item such as cash, interest, CO2 levels, etc.  The next one is the pipe tool which notes the thing that causes the item in the container to change.  It's location determines if the change is positive or negative.

The circle is the auxiliary button which allows people to select a symbol  to use.  There are so many to choose from.  The arrow on the bottom is the connector which allows you to instruct the program on relating two things together. The buttons on the right provide the basic operations and the ability to add a graph.

I made a quick model for population with only the knowledge I had from the workshop. I can tell you it is not quite correct because I do not have the proper relationship between the population and birthrate.

One nice thing is that I can easily go back, change things, and make adjustments to the basic model until I get it right. There is an undo button when I get something wrong or I can delete things if needed.

Even without putting a graph into the model itself, it automatically makes graphs available.  

According to another presenter, when students set up a system in this type of modeling software, they only have to worry about how the pieces fit together to create the whole system.

Students do need to know rates for things like birth and death rates, or interest, or whatever but they do not need to really know the actual mathematical equations involved because the software provides it.

This app provides a visual of how the pieces interact with each other so they see the whole picture rather than struggling with the math.  As I'm exploring this app, I'm enjoying playing with it.  Creative Learning Exchange also provides lessons and quite a bit of support. I plan to continue playing with this app until I feel comfortable enough to use it in class with my students.

You might want to check it out by downloading the app and playing with it.  I think that students as young as 5th or 6th grade could run easy scenarios using this software.  Let me know what you think, I'd love to hear.

New Way To Teach Square Roots

One of the workshops at the math conference was on teaching students to find and simplify square roots.  It begins using concrete before moving to pictures to abstract.

It was awesome and I learned so much.  The activity uses small square of certain areas.  There are squares ranging from 2 cm squared to 10 cm squared.  These are not 2 x 2 but sqrt 2 by sqrt 2 so the area is 2 cm squared.  The same applies to each square.

The introduction starts with a square that has an area of 6 cm squared.  The diagram shows how each side is the sqrt of 6 cm.  The professor running the workshop, emphasized using the proper language every time including the unit.  He stated if there is no unit listed use units squared and the square root of 6 units because using the language of math properly helps.

After the introduction, students are given a square that is 12 cm squared.  They are expected to use smaller squares to cover the area exactly.  After checking things out, they'll discover they can use four   squares with an area of 3 cm^2.  So one side is 2 sqrt 3 and that is the square root of 12 simplified.  This part of the exercise has every step written out along with questions.

The students repeat the exploration using a square with an area of 45 cm^2.  After a lot of trying different possibilities, they'll discover they need 9 squares of 5 cm^2 to cover the square.  If they count things, they'll find the simplified version is 3 sqrt 5.

As they work through the packet, they move from having to fill the whole square to find the answer, to only needing to do the edges, to understanding it. For each part a step is removed so people are doing one more step during the practice.  About half way through, they are expected to draw in lines so they are creating a picture of the squares rather than using them physically.  By the end, they can do square roots.

In addition, students can use these same squares to learn more about the Pythagorean theorem.  Students take the squares and work on putting them together so they form the edges a right angle triangle.  For instance, if the two smaller squares have sides of area 4 and 4, the larger square is 8, it shows they have a right triangle.  As they play with the area squares, they might notice that the total of the two smaller squares is less than the larger square and it forms an obtuse triangle.  If the total of the two smaller squares is bigger than the larger square, they have an acute triangle.

So cool.  Students see how the Pythagorean theorem is saying when you add the areas of the sides together, you end up with the area of the hypothenuse but that is not shown as often. This led me to wonder if I could use the length of the sides to show the triangle inequality theorem which says that the length of two sides added together must always be longer than the third side to have a triangle.

I really loved this activity and can hardly wait to use it in my classroom.  I'd love to hear what you think about it.  I got the information from a Professor who teaches in North Carolina.  If you want his information, leave a note and I'll be happy to share.  Have a great day.

It Takes That Much?

 I just finished an article on Cochineal.  Its a but that lives and dies on a cactus plant in Mexico. This insect is what was used to produce red dies back in the 17th and 18th centuries in Europe. I read of an incident where the Earl of Essex captured a Spanish Galleon containing 27 tons of this dye.

It takes 70,000 insects to make one pound and another 2000 to make a ton.  So it took 70,000 x 2000 x 27 or 3,780,000,000 bugs for that shipment.

So I wondered about natural dyes in general and what it took to get a pound or how to calculate the amount needed and talk about math and its uses.  It is definitely used here.  According to one site, the amount of dye material (plants, etc) for natural dying is based on the amount of fabric you  wish to dye.  It is referred to as WOF or weight of fabric.

For instance if you want to dye some fabric a medium red using madder, it has a 50% WOF so if you want to dye one pound of cotton, you would need half a pound or 8 oz of madder.  On the other hand if you wanted to use Cochineal (an insect that provides a red dye, you'd use 6 percent Cochineal to one pound of fabric or about one ounce of dye.

According to another site, it is recommended one use equal amounts of plant material as fabric so if you want to dye one pound of fabric, you'd use one pound of plants. The plants are chopped up and  heated in water for a couple of hours.  I saw a suggestion of equal parts of plants and water to get a good dye.

In addition, there are mordants used in dyeing. Mordants help prepare the cloth for dyeing and the final color depends on the mordant.  Some examples are:

1. Aluminum Sulfate or Alum is 12 percent of the weight of the fabric or 2 ounces per pound of fabric.

2. Cream of Tarter about 6 percent of the weight of the fabric or about 1 ounce per pound of fabric.

3.  Iron via old rusty nails is set at 1/2 ounce of iron to 500 grams of fiber.

There are other mordants but many of them are rather toxic and most are 1 ounce per pound of fabric.

There are also fixatives which help set the dye.

1.  Salt is set at one part to 16 parts water.

2. Vinegar is used at one part per four parts of water.  I use this if I think something I just bought might need this treatment.

3. Baking soda requires  1/2 cup per gallon of water.

So dyeing using natural plants, etc use a lot of math.  As you can see, there are ratios and percentages involved.  This is real life math because any dyer who uses natural plants, etc have to use this math.  Let me know what you think, I'd love to hear.  Have a great day.

Easy, Quick, Math Games

One of the sessions I attended at NWMC focused on easy games designed to help students work on basic skills while encouraging higher level thinking skills. Most of these games do not require much more than paper and pencil for the students. The teacher doesn't need anything more than numbers, dice, or markers.


1. "24". is a game where the teacher provides four random numbers between 1 and 9.  The students have to use addition, subtraction, multiplication, division, exponents, and grouping.  The idea is students need to use all four numbers with the appropriate operations to make 24. This game can be done as speed  competitions if you want.  It provides immediate feedback, while creating mathematical dialog.  At the end of the year, hold a tournament with brackets and everything.

2. "Bowling".  Students write the numbers 1 to 10 in a vertical column.  Provide 3 single digit numbers between 0 and 9.  This is the first roll.  Students are to use these numbers to find as many numbers on the list mathematically.
You get a 1, 5, 7
1 = 7 - (5 + 1)
2 = 7 -5 * 1
3 = (7-5) + 1
4 = 5 - 1^7
5 = 5 * 1 ^ 7
6 = 5 + 1^7
7 = 7 * 1^5
8 = 7 + 1^5
9 =
10 =
Once you've done all the numbers you can, you'll roll again and have students use these numbers until they can't find any more.

3. "Hit, Almost, Miss.  You choose a three digit number.  Students make a guess on the digits giving three at a time.  You'll write H for you got the right number, A is Almost in that the number is there but not in that place, and M is for miss or you didn't get close to the number.  It is best to write the H, A, M in a triangle so students don't know exactly which digit it is.

4.  Throw away.  You have students make 6 spaces so you have three in one row and three in the next row to look like this.   ___. ____.  _____
                                    ____. ____  _____
                                   ___________________

You give a target number such as 981.  So you roll one number and call it out.  Students decide if they want to keep the number or throw it out.  If they keep it, they have to write it in a space, they cannot wait.  Then roll another number and repeat.  Eventually, you will roll the dice 9 times and students use 6 numbers while throwing out three.  Numbers can be repeated and students may add or subtract the three digit numbers to get the final one.

Here are four games you can use to spice up your classroom, give students a chance to practice skills and promote higher level thinking and mathematical discourse.  Let me know what you think, I'd love to hear.  Have a great day.

Different Way of Graphing Quadratics.

 I attended the Northwest Math Conference in Tacoma this past weekend and learned so much.  I will be sharing various ideas over the next few days.  Ideas I think are awesome.

One session shared with us a slightly different way of graphing quadratics without completing the square or using a graphing calculator but he started with the idea that a quadratic is composed of two linear equations multiplied together. This is a unique way of looking at quadratics.

In addition he stated any polynomial is made up of a specific number of linear terms multiplied together.  So a 3rd degree is made up of three linear equations multiplied together, etc.

The graphing method is so much simpler and so much clearer than traditional ways while combining finding the zeros and the vertex and line of symmetry.
Lets look at y = x^2 + 3x - 7

Step one: (x^2 + 3x) - 7.  Place the first two terms inside parenthesis.

Step two: x(x+ 3) - 7.  Factor out a common term from the terms inside the parenthesis.

Step three: x = 0 and x+3 = 0.  Set the terms equal to zero and solve.  So we

Step four. x = 0, -3.  These are the two points of the equation that produce the symmetrical points.

Step five: These points give (0, -7) and (-3, -7) when you plug the x values into the equation.

Step six: (0 + -3)/2 = -1.5. Add the two points together and divide by two to find the x value of the vertex.

Step six:  (-1.5)^2 + 3(-1.5) - 7 = 2.25 -4.5 -7 = - 9.25. produces the vertex of (-1.5,-9.25)

Now place dots on (0,-7) (-3, -7) and (-1.5, -9.25)

According to the speaker, this method works on all equations, even the ones with irrational roots.  One does not have to apply the quadratic formula, complete the square, or any other method.  I've not had a chance to investigate these claims but I agree it is much easier to graph the equations this way as long as you are not trying to find the roots.

Give it a try and let me know what you think, I'd love to hear your opinion.  This is just one of the new things I've learned.  Have a great day.

Warm-up

What math questions could you ask about the photo

Warm-up

What math do you see in the picture?

Desmos, Desmos, Desmos

Yesterday,  I attended 2 very good workshop.  Each workshop was 3.5 hours long, packed with awesome information, and we got to learn so much more about using Desmos.  I spent the morning learning more about using various Desmos activities, add pauses, and copy and edit some already done.

I knew the basics but I learned more about using the activities including how to pause it so students had to stop and listen or how to set it so students could only do part of the activity rather than rushing through.  We also learned the Polygraph activities which allow students to chat with each other as they play a type of twenty questions.  The great thing is they are designed so kids cannot ask questions like "Is it the upper left one?" because the arrangement is different on each screen.

In addition, we came out of the workshop with a list of links for list of activities including a list of activities for kindergarten to fifth grades.  There was also a Desmos scavenger hunt with activities to do ranging from beginner to advanced and most have a solution so if you get stuck trying to do it, you can pull up the solution to look at it.  I need the visual examples to help me see how the instructions fit together to make the final product.

In the afternoon, I got to do a workshop with Dan Meyer and it was awesome.  He began with a "What if" activity where he set up a list of numbers and then placed the variable representing the numbers into a coordinate system.  He asked up to make predictions for each of these before trying them on Desmos.  It was so open ended and promoted so much discussion.

Towards the end, he introduced us to the Desmos activity "Graphing Stories".  He had us divide up into groups of two or three people because this arrangement encourages more conversation that having each person doing it individually.  He showed a 15 second video and then asked us what things were qualitative and quantitive.  After some discussion, we were told to create a graph of the distance between his waist and the ground.  He let us know the time was over by pausing everyones screen.

He put a couple graphs up and asked us what was good about the graphs before asking us to identify areas or things that could be looked at to be revised.  We were all given the chance to revise our graphs before he stopped us to compare the original graphs with the revised graphs.  Once we had experience doing this, he asked us to create graphs for two out of four short 15 second videos. AT the end, he went through and shared one or two videos for each one.  We were running out of time, so he had us look at someone's graphs but it was awesome to see that graph against the one we made.

Throughout the hole workshop, Dan provided guidance and leadership without controlling exactly what we learned.  We did the figuring out and it was awesome.  I loved it and I am coming back with new ideas and tools to use in class.  I'll share more with you tomorrow.  Let me know what you think, I'd love to hear.  Have a great evening.

At Math Conference.

I am at a math conference.  Today, I will be learning all about Desmos.  I"ll share tomorrow.

Examples in Math

One of the things I have to do is set a goal to work towards from a list.  I've chosen to work on giving the students ways to help them become more independent learners. Over the past nine weeks, I've realized my students are unwilling to pay careful attention to examples.

Textbook have lots of examples with detailed information explaining how it was solved.  My students prefer being passive as I discuss examples and they really don't want to write things down so beginning next week, I'm going to work with them on learning to "read" examples.

I have one student who takes time to look at his textbook when he does his assignments.  He asks questions when he doesn't understand something and is understanding way more than most who take a passive attitude towards learning.  Most examples include drawings or diagrams to provide the visual key but my students read the problem but won't draw anything so it makes it more difficult for them to understand what is being requested.

I admit, at their age, I didn't want to read my textbook.  I didn't see why I needed to and I'm sure they feel the same.  So it is going to be a bit of a struggle but I think it will be worthwhile.  Over the years of my own education, I've learned how important examples are to learning.  This is something I hope to share with my students.

One way to use examples is to begin with a completed example and having students explain the steps in the example.  This reduces anxiety and memory load so students can focus on understanding the concepts behind the process.  It allows them to explore the equations and process it more deeply because they are not just following a procedure.

There is research to indicate that students learn more accurately from worked examples because they are paying more attention when they concentrate on the examples.  However, students need to be taught to examine examples.  Researchers found there are three components when using worked examples with students.  The first is formulating a problem, the second is examine the steps used to complete the problems.  The third is sharing correct answers with the students. One way to working on the second component is to leave out some of the steps and asking students to supply them.

Research it is not the words the teacher uses to explain the examples but it's getting the students to think about the example as they examined it.  Some of the ways worked examples can be used more effectively in class is to assign a few worked examples to students as part of their homework.  In addition, the teacher should model their thinking as they solve problems.  Finally, create animated worked solutions for students to see solutions develop.  The nice thing about animation is students can rewind and review parts they don't understand.

Another layer to include is to ask students questions like "Why was this strategy used?" or "What principal was applied and why?" because these require deeper thinking.  The question is how do you get students to use worked examples.

One way is to have students explain to themselves as they go through worked examples.  Some of the things they might consider are "where did that number come from?" or "Why did they do that next?"  Another way is to used the faded examples where the first problem is fully worked out, the next example is worked out for the last step which the student fills in.  As the student works through the examples, each one has one less step until they work the complete problem from start to finish.

If you make the problems more complex, its important to show examples with the added complexity so students see how things change.  Let me know what you think, I'd love to hear.  Have a great day.

The Cost of Launching a New Product.

 If you keep your eye on Kickstarter, or Indigogo, you'll see companies trying to raise money for various products they want to bring to fruition.  Yes, I've invested in quite a few things ranging from electronics to clothing.  I bought a pair of shoes that I can wear in water, or in the city, or in the country without a problem.  They are comfortable and I don't need more than one pair when I travel but I've often wondered about the cost involved in bringing the product from conception to the market.

The cost all depends on what the product is being developed.  For instance, the cost of developing a new shampoo from scratch can be upwards of $25,000. Here is a breakdown of the costs.

1. $4000 to $8000 to develop and test the formula for shampoo.

2.  The minimum order for the manufacturer is 400 kg of product which will have a unit cost of $0.80 to $1.20 or a total cost of $8000 to $12,000 for the inventory.

3.  The manufacturer has to make a few samples to be sent to a laboratory to be checked out for stability and testing for microbes for $1100.  It would be more if you want the laboratory to conduct patch testing for allergies.

4. Then the person has to set up an e-commerce or fulfillment site which probably run about $2.40 per order filled only. This cost does not include shipping.

5.  Then add to this the cost of branding and registering your trademark is going to run between $1500 and $2500.

6. There is also the cost of incorporating the company, advertising, sales cost, etc.

What if you prefer to develop something with electronics, how would that change the cost?

The Electronic Development Stage covers the design of the printed board circuits from the preliminary stage to the test and documentation and costs between $15,000 and  $45,000.

The Software Development Stage covers firmware, mobile apps, and P.C. software which will set you back between $15,000 and $100,000.

The Industrial Design where the enclosures are created and runs between $6,000 and $24,000.

The Package Development Stage is where the packaging for the product is created and it can cost between $1,500 and $4,500.

Finally is the Project Management or the person or persons who keep the whole thing together and this costs between $4,000 and $10,000.

So overall you are looking at a cost of between $35,000 to $180,000 for just one products with electronics involved.

So this is another application for real world mathematics.  Let me now what you think, I'd love to hear.  Have a great day.

Conference - Thursday to Saturday of This Week.

 I leave Wednesday  to attend the 58th Northwest Conference in Tacoma, Washington.  I admit, I look forward to it because I've signed up for a whole day of Desmos Workshops including one with Dan Meyer who is known for his three act tasks.

There are so many choices for Friday and Saturday.  Unfortunately, the conference has two types of activities.  There are 50 minute sessions up against 80 minute workshop sessions with hands on activities.

I've seen panels on storytelling in the math classroom, introducing quadratics naturally, the commonality between vertex form and logarithms, Superhero statistics, Math games, creating powerful  moments in the classroom, graphical transformations, similarities in geometry, mathematical discourse, concept circles in math,  spicing up the class with TI-84 activities, Formative Task matrix, proportional relationships, strategies for group work that work, making math meaningful with content integration, modeling our world with math, and so much more.

It is frustrating to have at least two different topics going at the same time with another one or two overlapping which makes it difficult to attend everything I want to know about.  I think I'll talk to others from the district and see if I can get their notes for the sessions I want information on but can't attend.  I know I'm going to undergo math overload but I also know I'll learn so much information to share and use in class.  I hope to take the one on using the TI-84 because I have several I am using in the classroom. I know some things about it but not as much as I should because I've been using Desmos for all my graphing needs.

Although there are several sessions focused on working with ELL students, I won't go this time because the school I'm at only has a few students who are classified as English Language Learners but I'm hoping to get to the session on RTI in middle and high school since many of my students are performing well below state averages.

There is one session on building activities around student lives and identities.  It has the teacher collect information and then create hands on activities using this information so as to make the lessons more interesting and relatable.  This sounds quite interesting but its at the exact same time is a talk on "nudging" students into using their own ideas during numerical talks.

I look forward to this and what I'll learn.  I hope to meet new people and perhaps get a chance to meet any of you who read this blog.  Have a great day.

Warm-up

The average tidal range is 3 feet in most places but in the Bay of Fundy, the tidal range is 53 feet.  What is the percent difference?

Warm-up

The Bay of Fundy takes 6 hours to rise 50 feet.  What is the rate of change?

Bloom's Taxonomy in Math

 One thing we hear about on a regular basis is Bloom's Taxonomy, it's levels and we are told we need to use it to help our students develop higher level skills but we don't always get ideas on how to use it in math.  I am always looking for suggestions to use it in my classes but I don't always no how.

The first or lowest level is simply recalling information such as knowing your multiplication tables, addition and subtraction facts, or knowing the steps to slow an equation without understanding why.  This level is a basic level where they repeat, list, memorize, or define.  Students often confuse memorizing steps with understanding what they are doing.

The next level is understanding or comprehension where they tie basic knowledge to understanding.  Understanding refers to being able to explain the concepts via instructions such as recognize, translate, discuss, or describe.  In other words, students can describe what they are looking for and why. An example might be paraphrasing how to find the area of a triangle or a circle. Once they have developed understanding, they open the door to creating, applying, analyzing, and evaluating.

Applying refers to applying the concept in a new way such as the distributive property to show multiplying two digit numbers or binomial multiplication.  This steps is identified as demonstrate, illustrate, solve, and interpret.  One good way to have students to apply information is to compose an explanation of a topic for others, or create a diagram or photograph of certain fractions.

Creating means students take the understanding and creating something new with it.  If you see an instruction of assemble, construct, design, develop, or formulate.  One way of creating in math is when students create their own word problem or problem for a specific topic.  Another might be asking students how they can figure out the number of dimes in a car without counting them, or create a new monetary system.

Evaluate is the stage where students can justify their position.  This level is indicated by select, support, defend, value, or evaluate.  In other words, if students conclude a product is a better purchase, they can explain why.  An example might be having students consider what criteria they'd use to determine if their answer is correct or come up with a proof and justify each step.

Finally is analyze where students distinguish between the parts using compare, contrast, differentiate or examine.  An example of this might be classifying different functions or comparing and contrasting natural logs with regular logs.  Another example would be to have students figure out what strategies they could use to solve a word problem, or use a Venn diagram to compare and contrast two different topics.

So if you want to apply Bloom's Taxonomy to your math class, you know have some basic information to do so.  Let me know what you think, I'd love to hear. Have a great day.

Using Task Cards In Math.

One of the new teaching strategies I've run across is the use of task cards.  Task cards are defined as a card that has an activity or question on it and is the size of a small card no larger than a 3 by 5 index card.  Some task cards have pictures on them to show what is to be done or they might have written instructions.    Task cards come in two types.  The first asks a question that requires an answer from the student while the second suggests an activity the student should do.

Question task cards are found in several variations.  These cards may be multiple choice, long or short answer, or fill in the blank while the activity cards ask students to do something.  The activity cards require that the teacher have extra materials available for the activity.

Task cards are considered superior to worksheets since they can be more motivating.  They also require less reading which for students who have trouble reading, this means they are more comfortable in completing the task.  Worksheets can have more words than struggling students are willing to read at any one time.

In addition, students feel a sense of accomplishment by completing the task when these same students struggle to finish a worksheet.  Task cards are much easier to differentiate than worksheets and students are able to choose a task they can complete.  For instance, if you have task cards, some with multiple choice, some with long answer, students can select the one they are better able to finish.  Another possibility is to create cards with different levels from easy to hard so students can choose the card they want.

Furthermore, task cards can be designed for individuals, small groups, whole groups, learning centers, or even as part of a game.  When task cards are used for individuals, each student gets a card to do or students can take the task card home to do it at home after school.  Another way is to give every student a card to answer.  Once they've written the answer down on a white board, or paper, the teacher tells them to move one over and they to a new location where they answer a new card.  They continue the movement until they've done every card.

Another use is to divide the class up into pairs who sit or stand back to back.  The teacher reads the task card to the class.  Each student answers the question on a whiteboard and when they both have answers, they compare to see if they came up with the same answer.  Or divide the students into two groups.  Have one group sit in chairs facing inward while the other group faces out so they are face to face.  Pass out one task card to each set of two and they work on the answer together.  After a couple of minutes, ask students to move, one group goes left while the other goes in the opposite direction.

When used with small groups, the teacher can read one card and have all the students work on the same problem.  The ability level of the card is determined by the level of the group.  Furthermore, the tasks can focus on specific skills as needed.  

Task cards are also great bell ringers or warm-up questions, exit ticket tasks, or if your students finish the assignment earlier than expected, have them do some task cards. Keep extra sets in the classroom for anytime you need a filler.  Task cards are handy.

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

That Time Of Year!

  It is the time of year again, when the state of Alaska sends out a check to every qualified resident.  In 2015, they sent out over 680.000 checks for a state with a population of over 725.000.  If you have a bank account and ask fo it to be direct deposited, you get it the first day of October, otherwise you have to wait a few more days.

This year, the dividend sent amounted to $1606, just six dollars more than the dividend issued in 2018.  The State of Alaska of publishes information on the Permanent Fund Dividend from 1982 when they began sending money out to people.  They provide enough information to great a variety of graphs.

For the years 1982 to 2016, the State has a chart with the year, the state population, the number of applications received, the number of applications paid, the amount of the dividend, the percent of change from the previous year's amount, and the total amount of money distributed.

These numbers provide so many opportunities for students to graph the information in a variety of ways.

1.  Students can calculate the percent difference between the  number of applications paid and the state population for each year and graph that.

2.  Students can calculate the percent difference between the number of applications received and the number of applications paid.

3.  Students can graph the amount paid each year vs the price of crude oil to see if there is a correlation between the two.

4.  Students can graph the percent change from amount paid year to year to view how it changes over time.  This information could be used to create a line of best fit.

5.  Students graph the yearly total amounts paid out to all the applicants, to see how much is spent every year.

6.  There are also totals from 1982 to 2016 for applications received, applications paid out, dividend per person, and total amount dispersed to obtain an average per year for each of these categories.

This is real life data for those of us who live in Alaska.  Many people spend their money to buy new snow machines or ATV's, go on trips, or even buy new clothing.  I'm using mine to pay for a new heater for my house.  It should almost cover the cost.  Not quite but almost.

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

Reading Math and Annotation.

Yesterday, I realized my students read the problems but they don't know how to analyze the meaning of the words in terms of mathematics.  My students are learning about complementary, supplementary, adjacent, and vertical angles, and linear pairs.

We defined all the terms, went through the appropriate drawings, discussed each item but when it came time to do the work, they didn't take time to relate the questions to the new material.


So today, I plan to go over the examples in the book but include discussion on taking the vocabulary in the questions and relating it to the new material.  For instance, one question asks students to name pairs of adjacent complementary angles.  The students tried to answer the question by guessing rather than determining exactly what the question wanted.

We are going to rewrite the question after defining the words.  So pairs require two angles or groups of two angles.  Adjacent means they have to be next to each other and complementary means the angles add up to 90 degrees.  So the question is asking students to find at least two angles that add up to 90 degrees.  Then I'll have them look at the diagram to see if there are at least two matching this.

Another question asked if you have two complementary angles and one angle is 32 degrees what is the other angle?  We know that the complementary angles add up to 90 degrees, so 32 + X = 90.  I'm going to introduce algebra to help them see that algebra can help solve geometric problems.

My students tend to score much lower on language and reading than they do on math but they tend not to do as well when language is involved.  Equations yes, language no, so taking time to help them look for meaning in the words, will help them do better on future tests.

From today on, as I work through examples for each section, I plan to model looking at the definitions of mathematical words, relating them to the section's vocabulary, asking myself what they are actually asking.  Students too often want to know how to do the assignment without knowing or wanting  to look at the nuances of the problem.

This will require students to learn to annotate on the side, just like they do in their regular English class. Annotating is a way for the reader to interact with the text to increase a readers understanding, ability to recall information, while creating a connection.  In Math, annotating means looking at the meanings of words, finding information to answer the question.  So that is what I'll be teaching students today.

I'll let you know how it goes.  Let me know what you think, I'd love to hear. Have a great day.