Friday, June 30, 2017


Rocket Launch, Night, Trajectory, Spacex  Do you realize how often students use an instinctual knowledge of math to calculate the trajectory path for the game they are playing?  Watch them!  They use it so much.

I've done it in when playing Angry Birds.  I aim and hope but depending on whether its too high or low, I readjust by eyeballing it.  After a while I become quite good at it but not perfect.

My students do the same thing with their games involving some sort of projectile.  They try again and again until they get it.  If the score is too low, they redo it just like me.

My first thought is they are using some sort of quadratic but programmers may not think that way when they create an app.  So if they don't use the quadratic, what do they use?  Think physics with the two vectors that help locate the position of the object along its trajectory.  These games use the projectile motion formulas and trajectory formulas.

The various formulas are as follows:
Horizontal distance = the velocity along the x-axis * time.
Horizontal velocity = the initial velocity along the x - axis.
Vertical distance = the velocity along the y-axis * time minus 1/2 gt^2
Vertical velocity = the initial velocity along the y-axis minus gt.

Then there are these more specific ones:
Time of flight is t = (2*Vo*Sinx)/g  where Vo is the initial velocity and sinx is component along the y axis and is usually in degrees. G is gravity or 9.8 m/s/s
Maximum Height is H = (Vo^2* Sin^2x)/2g
Range is R = (Vo^2*Sin2x)/g.

This site has a great example of the coding to show how programmers code projectile motion in games such as Angry Birds.  This type of activity, especially if you provide the coding and the math, students can compare and see how the two relate.  A programmer I know is always telling me that one should "borrow" already written code rather than starting from scratch as it is faster.

I would love to incorporate this type of project in my trig class so they experience a real life situation that uses trig.  In addition, it would be good to show substitutions and such but unfortunately, we are not allowed to teach coding in school because it is not seen as an important skill. I disagree but can't do anything about it.

Let me know your thoughts on the topic.  I love hearing from people.

Thursday, June 29, 2017

Student Interaction.

Man, 3D, Background, Beard, Concept  As teachers we are having to look at changing the type of lessons we use in class.  If you watch students outside of class, they are usually sitting down, against a wall, playing some sort of games.  It might be a mideaval protect a town type game, one of the angry bird manifestations, or something like Zombies vs Plants. 

A game in which they are active.  They are not listening to a lecture. They are not filling out a worksheet.  They are not passive.

They are thinking about strategy.  They are active.  They are working towards a goal.  Unfortunately, most of us do not teach in a way that capitalizes on their abilities.  I started google classroom the final quarter of school but I hadn't figured out how to make things more interactive than using digitized worksheets.

Unfortunately, most of the interactive activities I found on the internet are all basically the same.  It involves printed material a student reads in order to fill out a worksheet.  More of the same passive style work but students need something more.  A few activities require the use of manipulatives but it still requires filling out a worksheet.

This summer, I'm taking several different classes.  One is designed to create interactive lessons with videos, activities, and fewer read in order to fill out the worksheet activities.  The others are so I can use Google Suite for Education and create a classroom that is more student centered than teacher centered.

In addition, I'm reading books and getting ideas such as creating trailers to tease students with the next topic just like movie trailers tease the population into wanting to see the movies.  Another book is helping me think about incorporating gaming into the classroom but not just using badges.  Setting the unit up as a Mission Impossible experience with a mission so they feel as if they are in charge.

So many ideas from so many different books.  I am realistic enough to know, I cannot implement everything I've learned but I can choose a few ideas and implement them at the beginning of school. Add a couple more ideas each quarter, through the year.  The next year, add a few more.  I hope the ideas I implement will help students take ownership of their learning so I do less and let them do more.  Will it work?  I don't know.

Let me know what you think.  I'd love to hear.

Wednesday, June 28, 2017


Puzzle, Learn, Arrangement, Components  I recently got a chance to play with EDPuzzle, a web based site which allows a teacher to select a video, shorten it, add questions, assessments, etc so the student has to respond while watching the video.

 I logged on the other day using a google account and since I'd not used it before, it provided a tutorial.  In just a few minutes I had my first interactive video completed.  I thought I would share it with everyone.

After watching the final product, I noticed I could have put a couple questions at bit later but on the other hand, the first question is a good one to explore their prior knowledge.  Questions can require an answer, open ended or multiple choice.

I love the idea of assigning an interactive video to the students.  When students watch the video, it will automatically stop at the question.  It does not give them a choice to skip.  Students must submit an answer before it allows them to continue the video.  In addition, they do have the option to rewatch the part of the video, the question applies to.

This is a screen shot of the question.  It asks about the material just presented in the video.  As you see the only two choices are to submit an answer or rewatch.  If they push the submit button in the hopes of moving on, it won't let them.  It provides the message "Please, answer all the questions."  It will not let them go on without putting even IDK ("I don't know") in.

This means the students will do more than just "watch" a video.  They have to really pay attention to the material and they can work at their own pace rather than at someone else's including mine. The questions can be used to signal the information is important so perhaps they might want to make it a part of their notes.

It allows you to assign the activity to your students via google classroom.  In addition, its easy to see who has watched the video, their grade, when they last worked on the activity and when it was turned in.  This information can be exported as a Numbers document.

 You can also look at the individual questions to see who answered each question and how it was answered so you know who needs additional scaffolding, who should redo the assignment.  The information can also be exported in Numbers so you can see all the results.
I am impressed with this website.  It does not take long to find a video, cut it to the exact length, add the questions and assign it.  Just a few minutes.  Something I can accomplish in my prep period.  In fact, I can get a few done during my prep period.

I made the video using my iPad so I could not use the audio track or audio notes due to my lack of flash but I can use them on my computer.   There is an app for students to use to access any activities.  As long as they have a Google ID, they can sign in.  Not a problem.

I've been taking a class this summer where I learn to create interactive lessons and this is a perfect addition to the lessons because they are expected to be active in their own learning and this is a great way to do it.

Let me know what you think.  I'd love to hear from you.

Tuesday, June 27, 2017

Visual Note Taking

Computer, Laptop, Notebook, Pen, Table  I've been seeing quite a few things on visual note taking.  I'm impressed with the drawings and the scribbled words that make visual notes.  I've seen some wonderful examples but not for math.

Of course, I'm always interested in how to use something like this in my classroom.  The idea is to attach both images and text together to utilize dual coding.

The first step is to look at visual note taking itself. Visual note taking is a way to take notes using pictures, doodles, shapes, lines, and text on paper.  This type of note taking can personalize learning, synthesis information, and helps create meaning for the students when they use their own ideas.

The value is in making the notes, not in being a perfect artist because the notes are designed by the individual.  Encourage students to use sketches, pictures, doodles, shapes, line, and text to create their notes.  It is important to practice before having students begin creating notes for class. 

Start students off with simple topics so they can practice taking visual notes.  As they practice, move them towards taking math notes so they have the chance to develop their own style.

 According to this blog entry, visual note taking is good for lessons with categories and subcategories or topics with layers, or relationships between ideas and topics.  There are three different possibilities for doing this.

The first is for putting together the pieces showing relationships with parts and connections.  One can use pieces/puzzles or gears, webs, Vann Diagrams, or clusters.  They are great for showing how parts connect.

The second is for showing steps or processes. Ladders, steps, pyramids and stacks are great for showing this type of topic.  The shapes help students remember better.

Third is telling stories to connect material using metaphors, call outs or posters. 

I think students have to learn to create their own visual notes for math, including being able to look at a textbook and change the written words into pictures which make sense to them.  I plan to revisit this topic later on.  I'm leaving Friday night for Phoenix and then San Juan, Puerto Rico before visiting my parents.  I'll keep posting but you might get some information as I stumble across it at my conferences.

Let me know what you think.  Have a good day.

Monday, June 26, 2017

Self Reflection.

Calm, Lake, Mirror, Reflection, Water  As a teacher, it is strongly recommended we stop and think about our lessons.  How they went? What could I have done differently?  How can I make it better?  That is one reason I always read books and try to find ways to engage my students more.  I do not take time to have my students reflect on their understanding of the current material.

What would it do if we had students reflect on their learning?  Would it help us clarify and reteach material sooner than if we waited for a quiz or test?  Unfortunately, if I do not provide guidance to help them learn to write their thoughts down, I'll get many who will just copy things from their notes.

This site has a wonderful question with ideas on how to answer the question.  The question she used "How did you improve as a Mathematician today?" is so much better than "What did you learn?" or "What do you remember?".  Both questions would get me "nothing" as the answer.  She offers five suggestions for students to write on.

1. Describe a new strategy you learned today.
2. Tell a math word you learned today and what it means.
3. Describe a mistake you made and what you learned from it.
4. Explain how you challenged yourself today.
5. Tell about something you noticed today and how it helped you solve a problem.

These are great questions to help guide students in answering the overall question.  The self reflection can actually help students change their mindset from fixed to one of growth. 

Other ways to encourage self reflection are:

1.  Have students keep math journals based on certain math prompts.  The above five suggestions would work quite well with the journal.  In addition, it is quite easy to find an assortment of math prompts.  I figure it is quite easy to do via technology by setting up a google doc for each student and then checking each doc to grade.

2.  Implement Student Led Conferences in which students bring their assessments and reflect on. The idea is to have the student determine where their weaknesses and strengths lay so they can focus on the topics they need to work on.  This is actually a nice way to work on figuring out where you need to differentiate so as to meet the student needs.

3.  Anchor charts.  After a formative assessment or test, have a classroom discussion to get student feedback on problems they had trouble with or questions on.  The concerns form the basis of the discussion on the process to solve each problem and the mathematical process itself.

4. Blogging.  Use blogging to have students explain how they solved a problem, their thoughts, and problems they had completing the problem.  When they blog, they have to slow down and think about the words they need. 

These are some great ways to encourage students to reflect on their learning. I will tell you, I don't do it because I have not had any idea of how to do it.   Let me know if you have any suggestions on this topic.

Friday, June 23, 2017


Police, Crime Scene, Blue Light  As you know, I find the math in forensics quite interesting.  It is a way of finding information from the evidence to find the perpetrator.

On all those shows, they find the guilty person within one hour, well 45 minutes give or take a couple minutes.

In reality it can take up to a year or longer.  There was a murder in the village I live in.  A young lady was stabbed multiple times and left behind the clinic.  The police came out, interviewed everyone, gathered data, found evidence, and took forever but almost a year later, they arrested a young man for the crime. I shared one way I'd used math to prove a person not guilty of reckless driving.

Lets look at some other ways math is used in forensics to help solve crimes.

1.  One can tell an animal hair from a human hair by calculating the ratio of the diameter of medulla to the diameter of the whole hair.  If it is .5 or higher, it is animal hair.  If it is lower than .5, it is human hair.

2. Estimating the size of an individual grain of pollen using the magnification of a microscope.  The idea is to estimate how many grains of pollen will fit across and divide the size of the field of view by that and voila, you have the individual size.

3. Blood splatter uses trigonometric calculations using the height and distance to find the angle of impact.  height is opposite while distance to the splatter is adjacent.

4. Using angles, they can tell if the pelvic bones are from a male or female.  If the angle beneath the ischia bones is less than 90 degrees (forms an acute angle), it is male.  If it is greater than 90 degrees ( forms an obtuse angle), its a female. 

5. Time of death, looking at the drop in the body temperature.  For the first 12 hours, the body cools by 1.4 degrees F each hour.  After 12 hours, heat loss is calculated at .7 degrees F each hour.

6. Time of death based on insect development.  They can use the development of certain insects to approximate a time of death because insects require a certain temperature to hatch and progress from one stage to another.

If the math is packaged in something as exciting as helping to solve a crime, students are more willing to do the calculations because it is a fun and applicable situation.  I'll give some sites for creating a unit next week. 

Let me know what you think.

Thursday, June 22, 2017

Board Foot

Abstract, Antique, Backdrop, Background  I am currently working with a family member to finish off the basement.  He and his wife asked to move into the basement so they could save money and help finish the basement.

All I can say is the basement has a cement floor, lots of studs and pipes for things.  It does have a few power boxes but I need more in there.

While looking over some of those magazines on finishing this or that, I stumbled across a comment on board feet.  I'd forgotten about that measurement.  My father, being a a shop teacher, spoke about board foot and linear foot.

Technically, a board foot is a volume of 144 cubed units, such as 12 in by 12 in by 1 in while a linear foot is just the length regardless of width or depth.  Honestly, I'm never sure which unit hardware stores sell the lumber by.  The last ones I bought were sold for a flat rate, the board already cut to a certain length.  In addition, if the boards thickness is less than 3/4 inch, it is sold by the linear foot.

After some research, I'm told some stores do sell lumber by the board foot while others sell lumber by the linear foot.   Using this information and other pieces, its possible to come up with some questions students can explore.

1.  If the volume of a board foot is 144 inches square, how many different measurements can a board foot have?  Which ones are more likely than other?  Explain your choices.

2.  If you have a board that is 8 inches wide, 8 feet long, and 1 inch thick, how many board feet is that?

3.  If you have a board that is 2 inches wide, 3 inches deep, 8 feet long and costs $5.00 per board foot, how much will the board cost?

4.  If the board you have is 8 inches wide, 3/4 inch thick and 8 feet long,  how many board feet is it?  What would it cost if the hardware store charges $4.75 per board inch.

So a real life application of board feet.  let me know what you think?  Have a good day.

Wednesday, June 21, 2017

Calculating Speed from Skid Marks.

Skid Mark, Asphalt, RoadI have always loved watching television shows which involve some sort of forensics.  CSI and all its variations including NCIS captured my attention because of solving a mystery based only on evidence.

One topic they don't usually discuss is determining the speed of a vehicle based on the length of the skid marks left behind.

I have a friend who was driving home from church one day.  He came around the curve, just across the train tracks, when he hit a street sweeper that was making a U-turn in the middle of the road.  What's worse, there was a sign posted before the curve advising people to look for the flagman who was absent.

He was issued a ticket for speeding.  He came back as soon as he got a tape measure to determine the length of the marks.  He brought the information to me so I could check the officer's conclusion.  So after a bit of research and calculations, I discovered he was only going about 36.5 mph in a 35 zone. 
I used the calculation S = sqrt(30*D*f*n).  S means speed, 30 is a constant, D is the distance of the drag marks, f refers to the drag factor based on type of road, and n is the breaking efficiency in a percent.

The officer was notified of his calculation error.  I thought that would be the end of it but the officer came back with a charge of reckless driving, a charge that could result in my friend spending time in jail and loosing his license.   Add injury to insult, no lawyer would touch the case because they said the case was too absurd to be prosecuted.

The city refused to provide photos of the damage to the street cleaner.  They would not talk to the insurance company, or do much at all.  He supplied everything he could from my calculations to photos of the damage to his car, to drawings and anything else he could think of.  The insurance company was using my calculations, his pictures but the city refused to discuss it at all.

He finally went in to talk to a District Attorney to discuss his plea of not guilty with the damaged bumper in his hand. Fortunately, the D.A. had a enough classes in physics to understand my friends argument on why he was not recklessly driving.  The DA basically threw the charges out due.

This is my real life example of how my use of math in real life saved a friend from getting convicted of a fairly serious charge to being freed.  Mathematical equations do work within a real life context.  I am going to have fun having students do this in class in the fall.

Let me know what you think.  Have a good day.

Tuesday, June 20, 2017

Mapping Patterns in Crime.

Map, Attacks, Terror, Belgium, Brussels  Mathematicians love the beauty of patterns.  There are patterns in traffic flow, in nature, in crime and math is used to map those patterns showing where the crimes have been committed.

Crime analysis looks at the patterns of crimes being committed in which areas to determine the best response.
According to an article published by UCLA,  criminals are hunter gathers who hunt for criminal activities.  They follow certain patterns. 

Using mathematical modeling, the police are able to find hot spots, determine the type of hot spot, and determine the best reaction to the activity. There are two types of hot spots, the first is characterized by a small rise in activity that grows while the second is a large spike in a central location.  By knowing which type of hot spot, the police can respond appropriately so they do not cause the hot spot to move to another area without suppressing it.

In addition, they can determine if the it is a hot spot of violent crimes, burglary, or auto.  Mathematical modeling that provides this detailed information.  Apparently when a hot spot occurs for a specific crime, the chances of it occurring increase because the criminals appear to be comfortable working that area.

There are others who are investigating this type of analysis.  In the Boston area, three people including a person at MIT created a program to look at trends in crimes and discovered trends that had previously been unidentified using traditional methods.  In addition, they found several crimes that met the criteria but had not been classified prior to this.

Crime mapping provides information for three type of analysis. The first, tactical analysis, looks at the short term such as  a crime spree because they want to stop what is going on. It looks at one criminal with many targets or one target with many criminals.  It is used when an immediate response is needed.

The second, strategic crime analysis, looks at both long term and on going events.  It often focuses on areas with high crime rates and tries to find ways to decrease the crime rates. The final, administrative crime analysis, looks at the police and their deployment to determine if they are being used effectively.

The data used for these crime analysis usually come from 911 call records.  The crime is entered into the data base, if the perpetrator is arrested, if he is convicted, or put in jail, all of this is put into the data base.  The information is analyzed using mathematical modeling so one of the three type of analysis can be applied to the data.

Let me know what you think.  Have a good day and enjoy yourselves.

Monday, June 19, 2017

Cartography and Math

Italy, Alpine, Alpine Region, Map  Cartographers or map makers use quite a bit of math in the creation of maps.  I suspect if you asked most students "What math is used in map making?" they'd respond with a shrug, or its only used in the key in the corner.

In truth, cartographers use quite a bit of math.  They use math in map scales, coordinate systems, and map projection to begin with.  The math scales shows the relationship between distance on a map and distance in real life as a fraction or ratio.

The coordinate systems refers to the numerical representation of locations of places on the planet while map projection is a mathematical transformation of points from a curved surface to a flat surface.  Did you know there are at least 18 different map projections including the Mercator which is the one most people are familiar with.  They type of map projection chosen depends on what needs to be shown.  This site has a great explanation of all the different types of map projections. 

Back to the coordinate systems used in cartography.  One is the geographic coordinate system which is based on longitude and latitude to pinpoint the exact location of any place on earth.    The other type is a projected coordinate plane which takes the earths curved surface and projects it onto a coordinate system.

New Zealand Maths has a nice unit created which has students creating their own maps of the classroom complete with scale and coordinate planes to mark the location of an object on the map. The nice thing about this activity relates magnetic north to true north. 

I can hear my students telling me that paper maps are out of fashion.  Maps are on their phones, so that information is not up to date but contrary to that opinion, they are wrong.  Math is even more important because mathematical equations referred to as mathematical exact visualization are what allow you to move your view of the map around, check out streets as if you are driving down them or keep track of the various labels of building, hotels, etc.

These mathematical equations are needed so the viewer can move digital maps around and still return back to your location.  Tomorrow, I'm going to look into the use of maps and math to find patterns in crime. 

Let me know what you think.

Friday, June 16, 2017

Plumbing and The Real World.

Bathroom, Bad, Toilet, Bathroom Sink  I got back Tuesday morning from a trip to Hawaii.  When I got back, I discovered the toilet was not working.  Actually, the inside piece broke so when you flushed the toilet, water sprayed out all over the place and would not stop running unless you turned the water off.

So on a trip to town, I purchased a new ring set and a new internal flush mechanism but when it came time to remove the screws at the base, I made several discoveries:

1.  There were cracks on the part of the toilet where the water flushes down.
2.  The bolts holding the toilet down had rusted completely.
3.  They don't make the old fashioned floater insides any more.

At this point I went shopping for a new toilet and discovered you can't buy one that flushes more than 1.6 gallons of water?  What is that in ounces or cups?  Most were either 1.1 or 1.28 gallons per flush.  I looked at duel flush which do 1.1 gpf if you only have liquids or the 1.6 gpf if you drop a solid in where as others used 1.28 gpf no matter what.

Then you had the ones which sprayed water out the bottom which supposedly increases the power of the flush or one that had a larger whole so water flushes with a higher pressure.  Most toilets give the gallons per flush but not the pounds per square inch of pressure.  I got some funny looks when I asked the sales people at the local hardware chains.

The price of toilets ranged from about $95 to $350 depending on type and brand.  I ended up paying $107 for a duel flush toilet.  It was easy to install because it came in pieces. The base, the water reserve, and the seat. 

With a bit of imagination, I can come up with math problems for replacing a toilet.  Now as far as all ready created problems, Robert Kaplinsky has one for comparing toilets.  Its well done and very open ended.  I went through trying to figure out which one would do what I needed.  I'd never thought about doing it this way.

This is the only one I found dealing directly with the toilets.  The other problems I found dealt with which toilet is the best if you have lots of port-a-potties at an event.  

Let me know what you think.  I love real world problems.  Have a great weekend.

Thursday, June 15, 2017

Creating Open Ended Questions

Question, Mark, Question Mark, Surprise We know it is important to ask more open ended questions in math but we can't always find the right ones for the topic we are teaching.  So how does one go about creating this type of question.

One way is to give the answer and ask students to find all possible combinations which give the answer. An example would be "Find as many ways to find an area of 48 square feet."  There are multiple possibilities.

Another way of creating open ended is to select two different answers to the same question, one right, one wrong or two incorrect answers and ask the students to explain why the answer they selected is wrong.  Then they should correct it.

Find math problems to the real world and relate the information to something they know such as providing restaurant menus and a school menu.  Have the students find the cost of the same meal from a restaurant.   You could ask students to discuss math they see on the way home from school.  Their answer might include a sale for books so they are 50% off or others they see.

Ask students to write down everything they know about a topic before beginning the unit and at the end.  This way they only have to share what they know. In addition, it is good to have students include vocabulary words they know.  One could even go so far as to have students brainstorm their ideas.

Take textbook questions and adjust them so students have to explain their thinking.  Rather than have students round 23 to either 20 or 30, ask them to explain when you want to round to 20 or when its better to round to 30.

Have students create a problem based on the information given such as the answer is 18 and the problem must include subtraction.  The student might write 36-18 while another student might come up with 18 - 0.  Both questions are correct.

Think about these things as you create the questions:
1. Does it focus on essential concepts?
2. Does it lead to other questions?
3. Does it tap real world situations?
4. Does it allow students to work together to find solutions?
5. Does it allow for multiple pathways or multiple solutions?

Have fun creating your own open ended questions.  Let me know what you think. 

Wednesday, June 14, 2017


Mindset, Mindfulness, Meditation There have been multiple books and articles published recently on the general topic of mindset.  It comes down to mindset are said to either being fixed or one of growth.

As you probably know, a person with a fixed mindset believes certain attributes such as intelligence, talent, or even ability to do math is fixed and cannot be changed.  They believe talent is set and you cannot develop it. This is the mindset I often run across in both students and adults when they talk about math.

Its expressed often as "I can't do math." or "I don't have the ability."  So often if the parent says this enough, students arrive at school with the same attitude.  Its the standard excuse for why someone is not good at math.

On the other hand, a growth mindset indicates the person believes attributes such as intelligence or ability can be learned with hard work and dedication.  Talent is not enough.  I see students who will practice basketball for hours on end but are unwilling to try in math because they are convinced they do not have the talent for it.

Is it possible the closed mindset of "I can't do math" does not mean they are unwilling to learn but its more they don't know how to find the answer a problem?  Is it possible they don't know how to ask the right questions? Perhaps we need to teach students to articulate their questions rather than remaining silent.

So how does one go about helping students change their mindset from fixed to open?  There are ways.  One thing is to change the classroom environment to one of encouragement where they can work on complex and interesting problems.  Allow them to try their ideas, fail, and retry.  Where they learn it is acceptable to fail because failure is a part of learning.

According to something I read, teachers who encourage multiple attempts at solving a problem are more likely to have students with the growth mindset. It has also been suggested a teacher only provide help when they are asked and students should have the opportunity to resubmit work.

Too many students who struggle solving problems, are likely to decide they are not a math person.  I have students who would rather fail than try because they've decided they are unable to do the problem.  I have two signs hanging on my wall.  One sign says "I haven't practiced this enough." while the other states "I haven't learned this yet."  Both important phrases are important in my opinion because they reinforce the idea they can learn.

In addition, people often think mathematics is composed of calculations rather than seeing it as a topic filled with patterns.  Consequently, they think those who can complete calculations the fastest are the best in math.  In fact, most good mathematicians are slow because they stop and think about the problem.  I think if we can convince students its O.K. to take it slow and do a good job, this might help students change their thinking.

It is important to teach mathematics using open questions such as find as many rectangles as you can with an area of 24 square inches rather than asking what is the area of a rectangle that is 12 by 2.  The first problem has multiple answers and it requires students to think deeply, using their own ideas rather than if they have to answer the second one.

These are a few things to think about when trying to help students change their mindset.  I don't know if we can change everyones mindset but we can try.

Let me know what you think.  I'd love to hear from everyone.

Tuesday, June 13, 2017


Love, Feeling, Affection, Friendship  I am taking a class designed to teach people how to create Hyperdocs.  Hyperdocs is a document containing the lesson complete with all the links needed so the student can work through everything at his or her own pace.

Reading over other students first assignment introduced me to a website called Thinglink where you can create interactive documents.  This site allows you to annotate videos and images so when you click on something, you get a link or definition.

It turns a simple video or image into an interactive document which is easily explored by the student.  In addition, there is a library of already prepared activities or you can make one yourself.

Since math is one of those odd ducks, I've found many places do not offer as many opportunities in the library of shared activities but this site  has quite a few I can use in class.  I looked up slope, linear equations, quadratic formula, trig ratios, polynomials, end behavior, congruent triangles, finding the common denominator for fractions.  Every topic I put in, I found something already created I can easily use.

It's easy to check out any of the activities to see if it has what you want. If not, its just as easy to make a new activity.  There are tutorials in addition to lots of available help so you can do it yourself.  I tried making one on slope using a roller coaster and I didn't even look at a tutorial.  I just put it together and it worked.

This is the one I made in about 8 minutes.  I got the basic picture from pixabay because they are not copyrighted and are so easy to use.  I seldom actually  check tutorials, preferring to play with the program as I learn.  It is quite intuitive and I have no fears creating anything because it does not take too much time.

The basic educators license is free.  For me, its perfect because it allows me to have up to 100 students per account.  It has everything I need but if I wanted to enroll more students or needed more icons or custom features, I can easily upgrade at any time.

I like that I can set up groups of students and share activities via a code so I don't have to upload all their emails.  That makes it so much easier for me as I do not always have time to do that.  If you are looking for a new way of presenting information, check this site out.  It is great.

If you have any comments, or experiences, please share it. I'd love to hear from you.

Monday, June 12, 2017

Formative Assessments

Quiz, Test, Exam, Questionnaire  Last week while at the Kamehameha Schools Educational Technology conference, I had the pleasure of attending a session devoted to formative assessments.  This is one of my weak spots because I hate creating and "grading" assessments in general.

The speaker sent us to five different sites containing online tools designed to engage the student while providing assessments.

The first is Plickers.  I'd heard of it but hadn't ever used it.  No one at school is familiar with it. It is free and easy to use.  It uses a paper printout with a number and the letters A to D.  You post the question, the students hold up the paper with their answer at the top and the teacher scans it using an app on their phone or ipad  to get the number of correct and incorrect answers. 

I got to play with it and it was fun because I didn't have to do anything other than determine the answer and hold up the form.  The speaker indicated it is not good to laminate the cards because the light can be reflected making it harder for the scanner to record the answer.  In addition, you can get reports at the end so you don't have to grade it.

Next was Quizzizz which can be setup as a game with a code so students can use the code to enter the classroom.  I gather each student gets questions but not the same questions at the same time.  There is a library of quizzes created by others or you can create one from scratch.  The quizzizz can be done in class or as "homework" depending on what you want.  This is another one that provides reports and it has an ipad app available.

Then we explored Go Formative, another online formative assessment site. The instructor set up a quiz to take using the class code created by the site. This one allows the instructor to create an assignment with questions for the student to answer.  Once the students have completed the assignment, the teacher gets live results and feedback is provided.  Sweet quick and easy.  What is nice about this one, is questions can require students to show their work.  Its rather nice

The last online site is Google Forms in which the teacher creates the quiz.  The link includes step by step directions for doing it.  I prefer the other ones but if you are using google, this is just another facet of the Gsuite.

The final form is called "My Favorite No".  Put a problem on the board.  Pass out small pieces of paper or 3 by 5 inch note cards.  The students work out the problem.  The teacher collects them and sorts the answers into two piles for yes or no.  Then the teacher goes through the no stack choosing his or her favorite no.  The teacher then places it on the board and the class first discusses what is right about the problem before looking at where the mistake was made.  This gives the teacher a quick grasp of the number of students who understand the problem. 

I knew about the last one but none of the others so I now have a few more tricks in my bag which should make life easier next year when I teach.  I know what classes I'm teaching next fall so I am starting to create my units.  This way when school starts I'll have them done and I'll be ready to go.

Let me know what you think.  Have a great day.

Friday, June 9, 2017

Gui, Interface, Internet, Program  On Wednesday, I learned more about the offerings at  If you don't know them, they provide a wonderful graphing calculator which does not require students to rewrite equations into the y= format.

They offer activities which can be downloaded, personalized and come complete with a teachers guide to help prepare to use it.

In addition, they offer something called polygraphs, activities composed of three rounds.  In the first round, the student plays a round against the computer.  This is where they learn to play the game.  In the second round, they are paired with another student.  One student makes a choice while the other student asks questions which can only be answered with a yes or no.  In between rounds, students answer more questions designed to help them learn vocabulary and strategy. 

I'd never heard of these but they sound really great and I can hardly wait to try some in the fall.  In addition, I several activities are bundled together based on topics such as systems of linear equations, linear equations, conic sections etc.  I've already checked out one activity in the linear equation section, my students will love.  It requires students to place the plane so its on the correct approach to land between the lights.  It also requires them to find the equation for a plane to take off. 

Furthermore, you can preview the activity and try it just like a student would.  I tried the plane one and loved it.  The math is based on topic rather than grade level but if you need to adjust the activities, it is easy to do so. 

If you want to create a custom activity, you have the ability to do that.  You can include graphs, math equations, notes, movies, and so much more.  No matter whether you use one of their activities or create your own, students do not need an account because all activities use an entrance code. 

I have not looked at this site for a few years and I'm impressed with everything it has to offer.  Please check it out here and have a blast.  Let me know what you think.  Sorry its not very long but I'm exhausted from attending the conference.  I'll share more next week.  Monday, I'll share the five assessment tools I learned about which seem to work well. 

Have a nice weekend.

Thursday, June 8, 2017

Other Uses Of Infographics.

We think of infographics as a way of conveying data in a visual form.  Math is full of data but what if we also used it to help introduce topics.
 What if we had students research the way the upcoming topic is used in real life rather than waiting for them to ask the questions of "When will I ever use this?"  Why not let them answer the question before we begin teaching the topic.

Since the slope is rate of change, we can take the examples found by students, discuss what element in the example is the rate of change or have students identify the rate of change and provide justification for their answer.

They could even provide actual examples for each situation.  In my examples, I start with variable cost for renting a taxi.  Students can check with local taxi companies for rates so they can calculate how much it would cost to travel to different places such as the store, the pool, or the airport.

What about party costs.  Students could call around to several restaurants or party providers to find out how much it would cost to rent a room and the cost of feeding people.  They can use that information to create the linear equation needed to plan a party.

Students could check the local sales flyers to find out the cost of various printers.  They can check the stores or online to find the cost of replacement cartridges.  They now have the information needed to figure out which is the best buy.

By doing this, they have already learned a lot and they have built a foundation they can relate the abstract to the general.

I can hardly wait to try it this fall.  I got the idea from the session on infographics at the Kamehameha Schools Educational Technology Conference.  Let me know what you think.

Wednesday, June 7, 2017

Bright Idea.

I've been thinking about those two tours.  I wrote about them the other day.  I said I wanted to take the tours some day but I really don't have to wait.  I can use Google Maps, Google Street View and Google Earth to actually see the buildings one would see on both tours. 

In fact, I could take a copy of the map on the website or use Google Maps to find the routing for the tour.  Then using that map, I can navigate my way down the streets, turning where they turn so I can see the same things the visitors see.

Imagine being able to take that tour without ever leaving home.

Or how about taking a day to visit the Great Pyramid of Giza, the Taj Mahal, the Parthanon, the Gherkin in London, The Guggenheim Museum in Spain or any other mathematically interesting building or bridge.

This is the Millennium Bridge in London which opened in 2000.  On the day it opened 90,000 people walked across it and it wobbled so much, they shut it down 2 days later.  It was not reopened until 2002 because they had to install 91 dampers designed to absorb vertical and lateral oscillations.

Imagine being able to have students walk across the bridge using Google Earth.  It makes it more real and they get a better feel for the structure because Google Earth allows the person to travel along the bridge.

This would make a cool project scheduled for those times during the year when you can't do as much instruction due to it being the end of school or its the middle of testing, or other item.  You'd have students research mathematically interesting buildings and bridges. 

In addition providing actual statistics and data, they can use Google Earth and Street View.  Street view often gives a better perspective than Earth so the two used with each other is great.  Who said we can't take time to show real world applications of mathematics.

Have a good day and let me know what you think.  I think it would make things more real and interest students.

Tuesday, June 6, 2017

Attending conference

I am attending a conference which started yesterday. Last night I went to a 2 hour session filled with lots of hula and local music. I will give a report tomorrow. Yes, I am in Hawaii.

Monday, June 5, 2017

Math Tours

Photographer, Camera, Recordings  Saturday night, I came across this cool web page offering Math tours in both Oxford and London England.  Yes, you read that correctly!  Math Tours.

The idea behind the tours is to take you around town and show you real life applications of mathematical concepts.  Isn't that cool?

The tour is apply named Maths in the City and is absolutely free but requires groups of 10 to 20 people and takes about an hour.  If your group is less than 10 people, they put you on a waiting list until they can create a group.

The Oxford tour begins at Rewley house which is home to the maths department.  From there, they head to Sackler Library which was designed as a cylinder rather than the traditional building.  Next is the Ashmolean Museum with its symmetries, followed by The Beehive, a hexagonal building.  Then comes a short piece on GPS and how it shows you are where you are. Afterwards, you wander over to the Wadham College to check out the Penrose tiles.  These tiles have a pattern which never repeats.  These tiles also explain the structure of certain metallic crystals before finishing at The Sheldonian Theater to concentrate on the roof. 

If you'd like to read more about the maths on this tour, follow the link and check it out. I found the description quite interesting.

The London Tour begins at the Tate Modern where people learn about a certain part of mathematics.  From there, you head to 30 St. Mary Axe also known as the Gherkin whose tapered shape and construction is based on the triangle.  The next stop is at Tate Modern to observe Jackson Pollocks artwork.  His art is based on fractals. In addition, mathematical analysis can be used to tell the original from fakes.  From here, you head to the Catanery chains over the Thames.  This principal is used in architecture to form arcs.  Then comes the Millennium Bridge which opened in 2000 and wobbled as people walked across it.  You learn why it does that.  The dome of St. Paul's cathedral is next with its excellent interplay between math and architecture. The final stop is to look at the topology of the London Underground. 

Want to know more about this tour?  Check here and learn more about the maths involved at each stop.

By the time you read this, I'll be in Honolulu attending the Kamehameha Schools Unconference.  Tomorrow, the regular conference begins and I'll be presenting on using Google Maps, Google Street View and Google Earth in the Math Classroom.  I'll share interesting things I learn with you in future columns.

Let me know what you think.  Have a good day.

Saturday, June 3, 2017


This came about because I'm helping a relative move this weekend.  Enjoy.

Friday, June 2, 2017

Dress Sizing.

Girl In Green Dress, Pattern In Vector I bet you are wondering why I'm covering the topic of dress sizes in a math column?  In fact, I suspect you believe there is no reason it should be covered here since you take whatever size you take.

Prior to the 1940's dress sizes were not standardized as they are today.  For children, their sizes were based upon their age so a 12 year old was expected to take a size 12 while women purchased patterns based on their bust size.

In the 1940's sizes became more standardized when the WPA measured 100,000 women to create the first real standardized charts since ready to wear still did not fit as well as home made.  Even as late at the 1930's the size of the dress was determined by the bust measurement except the size listed was half the bust measurement.  In other words, a size 18 was made for a 36 inch bust.

By 1958 the Bureau of standards published an official guide to pattern sizing.   At this point a size 8 was made for someone with a 31 inch bust, 23.5 inch waist, and a 32.5 inch hips.  By 2008, a size 8 increased five to six inches for each of the three measurements.  That is quite a large increase over 30 years.  In addition, Marilyn Monroe was a perfect size 12 which is now a size 6.

So lets look at the changes in size from 1958 to the present.  I've put the numbers on a spread sheet.  The top are measurements for the bust while the bottom are measurements for the hips.  Both are in inches. 

So now what to do with this information?
1.  Create line graphs out of the numbers. 
2.  Determine percent increases to see if the increases were constant.  They might need to determine a yearly increase to make this work.
3. Have students note that a size 12 in 1958 was not the same as a size 8 in 1970 and ask them why?
4. Have students look at the size 0 and determine if it matches any earlier sizes.
5.  Ask students why the measurements assigned to each size got larger over time. (Its a sales tactic.  If a women can buy a garment with a smaller size, she feels better about herself and she is more likely to buy it.)

This is a real world look at sizes and marketing.  Let me know what you think.  Have a good day.

Thursday, June 1, 2017

Some Thoughts on Absolute Value

Book, Learning, Study, Homework, Read  Its amazing how students come to me only knowing the absolute value is always going to be positive but they don't always know why.  Due to this lack of understanding, they sometimes have difficulty solving |x-7| = 5.  Most of my students know x = 12 in |12 - 7| = 5 but they have difficulty with the other possibility.

Unfortunately, this carries over to solving absolute value inequalities such as |3x-3|> 6.  The students who do not have the basic understanding down, will have difficulty solving this type of problem because they do not realize they have to solve two separate equations.

They have to look at this as 3x-3< -6 and 3x-3> 6.  My students are able to solve for the positive value but not the second value.  They have almost the same problem if the problem is |3x-3| < 6.  I think its because they do not grasp the concept fully.  I believe my students arrive in high school without a solid foundation in absolute values and inequalities.

So far, I've not had any students ask me when absolute value is used in real life.  The most obvious one is distance.  The distance is the same whether you go to the next town over or come back from the next town over.  I tell the students, the positive or negative represents the direction.

Another situation which uses absolute value has to do with the speed limit.  In most cases you are allowed to go x miles per hour over the speed limit or below the speed limit.  Remember the signs on the freeway which state you may not go below.

Another absolute value application is the commission charged when converting from one country's money to another or back.  The commission is the same regardless of which way the value is converted.

These three are the examples which pop up again and again but examples of absolute inequalities are much harder to find.  Or at least real examples rather than seeming contrived.  The best examples I found had to do with eggs or fruit or something similar where their weight varies a bit and the total can be no more than a certain amount. 

The other reasonable example is situations when things are plus or minus a certain range such as plus or minus 1.5%.  Statistical deviations fall into this category but those are really the only ones I found that seemed believable.  I think the final example dealt with sales commissions and similar situations where the amount you receive is based on selling less than or equal to a certain amount.

Let me know what you think.  I'd love to hear from people with real examples for absolute inequalities because I want to be ready for the day I'm asked the usual question.  I also want to make sure my examples for inequalities are correct.  Please feel free to comment.