Friday, September 22, 2017

Explaining Why in Math

Parents And Children, Curiosity  Recently, I've been requiring students to write down explanations of what they did or why they did something.  My students fight me on this but I'm noticing it is making them stop and think.  It shows me if they really understand what they are doing.

In geometry, my students are classifying triangles by angles and sides. In addition to stating its a right scalene, they have to add that it as a 90 degree angle with three sides of different lengths. 

Since my Pre-algebra class is still struggling reading signs during addition and subtraction, I started having them write down if there are two negative signs, a double negative, or different signs before doing the addition or subtraction.  They need to slow down and read it carefully.

In Algebra II, we are just starting finding solutions to systems of equations using the elimination method.  They are going to have to explain what they are going to multiply by to change the coefficients before actually doing it. They will also have to write down why they chose that. 

In another class I have students working on solving one step equations.  I have them write down what the one step is so they can solve it.  For instance, if they have an equation like x -3 = 8, they have to write down "You have to add three to both sides to isolate the variable." Or since I've got a negative three, I have to add three to both sides so I have the X alone."

For one class, I am not asking them to explain what they are doing because they are still learning the basics of multiplying decimals.  When they get to adding and subtracting fractions, they are going to start explaining how they found common denominators and created equivalent fractions so they can calculate the answer.

Although they hate doing this, I am seeing an improvement in their understanding of why they do certain things to solve problems.   It sometimes takes me a while to decide what the best way to ask them to explain what they are doing but its worth the time.

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

Thursday, September 21, 2017

Money and Rounding

Dollar, Bank Note, Money, Finance, Funds  I've noticed that students in this location have  tremendous difficulty in rounding money when shopping or deciding when the value should be rounded up or down.

I've noticed many people who shop at the local store tend to just grab food, watch the total as its all being rung up and if it goes over a certain amount, they add and take off food until the total matches the amount they can spend.

Most of my students do not know they should round the prices to make it easier to determine the total amount they are spending at the store.

So I send students to the store with a list of products they need to find the prices of.  I use this to put together an activity that requires them to round the prices so they can estimate how much they might spend.  Unfortunately, when they round anything, they carry out the math first and then round the answer.

I've also taken time to discuss why you might round something up or down, depending on who you are.  For instance, if you are a tax collector, you might round a taxpayer's amount up so you get a bit more but if you are paying money to someone, you might round it down so you pay less.  When I brought that up, the kids looked at me strangely because they've only been told the usual rule of 4 or under round down or 5 and above, round up.

I actually took time to show how if $32.5447 is rounded up to $32.55 when collecting it.  The $.0053 adds up to quite a bit of money when you are talking 100,000 people.  On the other hand, by rounding it down to $32.54 when paying someone money.  I showed they could save quite a bit if they had to pay 100,000 people. 

This is not a topic covered in elementary or middle school math.  I wish they'd take time to talk about it so students are exposed to it.  Sometimes, I think nuances are left out when teaching mathematics.  I don't know if they teach rounding of money in real life situations or are the situations contrived.

I try to provide real life situations and experiences for my students so they learn more because some of the situations are not as obvious living in the wilds of Alaska.

As usual let me know what you think.  Have a great day.

Wednesday, September 20, 2017

Math Games

Super Mario, Nintendo, Mushroom, Toys  The other day, I discovered a web based math site whose games work on the iPad.  Its Math Games!

Too often I'll find a site but it won't work on my iPad because its java based.  It is frustrating when that happens because my students are disappointed.  Even if the site states the games will work on the iPad, I check it on an iPad.

I found this when I was looking for a game for my pre-algebra students could play to practice adding and subtracting integers.

The site is actually designed for grades K to 8 but much of the 7th and 8th grade material can be used to reinforce skills in high school pre-algebra or algebra classes. Although it was not actually a game with things to shoot, it did work my students hard and they enjoyed it. Even some of my students who do not normally work, did so.  One even asked if we were going to do this again today.

I had them practicing the adding and subtracting integers from grade 7 material.  The exercise had four levels with 10 questions for each level.  I told my students they had to get all 10 right before moving to the next level.  This insured they slowed down.  For several students, the activity clarified the rules and they were able to do it.

I like they have several ways to find material.  You can either go looking at grade levels or you can check the skills page.  The skills page has everything listed by topic such as addition, fractions, geometry, or number properties.  When you click on a skill, they list the appropriate exercise by grade level so its much easier to find.

Each problem is set up as a multiple choice with possible answers.  The program provides immediate feedback so they know if the problem is right or wrong and there is a progress button so they see how many problems they've done and which ones they got correct.

In addition, there is a virtual scratch pad a student can use to do the work.  When they click the scratch pad button, an opaque sheet covers it but you can still see the problem to work it.  Once they have the answer, they can click the get rid of the scratch pad and select the answer.

As I said earlier, each topic has multiple levels which take them from easy to hard.  If a student repeats a level, the problems are different each time, so they cannot write down the answers to use again.

I plan to make this a regular part of my instruction. 

Check it out.  Let me know what you think.

Tuesday, September 19, 2017

Creating Islamic Art

Persian, Art, Tradiotional, Islamic  Yesterday while researching muqarnas I discovered a great unit on Islamic art and geometric design put out by The Metropolitan Museum of Art.  It is ready to go and is perfect for a geometry unit.

It has several activities which require the use of a compass to construct the designs.  My state standards still requires that skill so something like this adds a real life element to the unit.

The first activity begins by having students create a drawing using 7 overlapping circles. It sounds easy but the circles have to be equidistant so as to be even. 

The second activity has students find shapes within the drawings from activity one.  They find there is a 6 pointed star, 12 pointed star, hexagons, and triangles all within the 7 overlapping circles.  The third activity again takes the rosette from the first activity to create triangular and hexagonal grids.

The fourth activity has students go from one circle to five overlapping circles used in the next couple of activities where students find 4 and 8 pointed stars and octagons. Activity 6 has students finding square grids from the circles.

Activities 7 to 10 look at finding patterns for triangular, diagonal, 5 and 7 overlapping circle grids. 
The eleventh activity is the one with relevance to yesterday's topic.  It has students create six and eight pointed stars out of a circle.  The finished product looks quite similar to the shapes used in the Introduction to Muqarnas video from yesterday.

Each activity has great directions and good drawings so its not hard to follow things and end up with a great finished product.  I also know that more and more people are relying on apps for geometric constructions because you can find free apps to do the job.  I have a few compasses in my class but mostly I rely on the apps myself because the physical compasses can easily break.  In addition, they cannot stab each other with an app.

Let me know what you think.  I'm off to try a new app I found that looks quite interesting and is free.

Monday, September 18, 2017


Granada, Alhambra, Generalife, Rural  I suspect you saw the title and wondered about it?  I did when I stumbled across the topic in the latest Make magazine. 

The simplest way to describe muqarnas is they are three dimensional renderings of two dimensional geometric design.  In addition, there are two types - North African (Middle Eastern) or Iranian Style.  The School of Islamic Geometric Design has a short explanation on the differences between the two styles.

The idea behind muqarnas is that they smooth transitional zones between one area to the next.  The interesting thing about these is they can be made out of cement or wood.

This slideshare provides a great introduction to the topic.  It includes information on muqarnas themselves, the people who helped create them, types, and even the history of them. It is well done and shows lots of examples.

So where does one go to find instructions for use in your classroom.  This 13 minute is a great introduction to making one out of cardboard.  The creator takes people through the creation process showing everything step by step.  Unfortunately, the measurements are a bit vague in that he states its what he chose but he takes time to show everything in detail.

In addition, this 24 page pdf has greater detail with hand drawn patterns so a person can see how certain ideas are created. Some of the information appears in the slide share but the reason I recommend this one is there is a whole chapter beginning on page 11 which goes into great detail on going from design to the muqarna itself. 

It discusses how the arrows meet with shapes, the 5 basic rules to keep in mind when creating a muqarna, and even reading muqarna graphs and subgraphs.  Everything you need to create a unit in geometry.  I could not find any lessons already created but this is a fascinating topic. 

Let me know what you think.  I'd love to hear from people.  Thanks for reading.