Hooda Math

I stumbled across a Hooda Math app for the Hooda Math website.  I admit, I've not bothered checking it out in the past because many of the games looked a bit young for my high school students but I found two games I could easily use in Algebra or Geometry.

The first game that looks good for my high schoolers is Slope Intercept Surround which has students identify the m and b for two lines that surround the colored area.  In essence, the graph represents the answer for two inequalities.

However the way the game is set up, you identify the two lines that border the colored area.  when you click on the first slope, you are given a choice of  four possible slopes and it is your job to select the correct one.  Then you move to finding the intercept for the first line.

 Once you have the slope and intercept for the first line entered, the line appears on the graph and you can see if you are correct.  You might have to go back and make changes to get everything correct.

The process is repeated for the other line.  Once the student has both lines correct, they move on to the next graph.  As far as I can tell, the lines all match up with the type of graph you see to the right.

The second game is transformation golf.

 This game requires students to identify the type or types of transformations needed to move the white circle  to where the black circle is located.

 Notice each transformation has its own button.  I would choose translation first, then a choice of directions appear, so I'd choose up, then select 2 off the next screen because I have to go up two units.

The app moves the white circle up two. I'd choose translation again, right, one saying move it to the right one space.  I'm done because the white circle has met the black circle.






Finally, Hooda Math has certain math manipulatives that can be used on the iPad.  Its got Base Ten Blocks, Algebra Balance Equations, Algebra Tiles, Number Line Addition and A Hundreds Chart. 

This is great because I have a few activities that need Algebra Tiles but I've not found a free app that allows access to either an algebra balance or tiles and most of the web based ones do not work on the iPad.  I'm still exploring the other games available to see which I can use on my iPads in class but these will fit. 

Real World Uses of Exponents

  We usually teach students all the rules for working with exponents after we teach the basic definition.  I don't think I start teaching anything real world until I get to the higher level maths when I teach radioactive decay, population growth and decline, and compound interest.  But what about working it in with the other classes?  How do you do that?  What are some ways exponents are used in real life that we could easily implement in other math classes?

Most of the sites I found that give information on uses in real life only give no more than 4 or 5 examples and even then the examples are just a mention such as exponents are used to determine the spread of disease but its just a mention.

If you go to this site, it has tons of examples of where exponents are used in real life ranging from computers to the size of silicon atoms.  Each example includes details on the way the exponents are used and  includes a visual illustrating the example.  This site even includes an activity for making an exponential decay graph using M & M's.  Most students love doing any type of activity using candy, especially if they know they can eat it when they are done.  At the end, there is a link to a slide presentation you can use in your classroom which contains the same material.  This site is one of the more thorough ones' I've seen.

It has so many examples, the teacher could assign a topic from the presentation to a pair of students who could further research the topic and then the groups could prepare two to three slides in the class presentation.  For instance, the author mentioned the pH scale uses exponents.  I check it out and yes it does indeed.  It actually looks like its more likely a negative exponent because the pH of one means 0.1, a pH of two is 0.01 etc. 

The slide show makes a good starting point for an introduction to the real world uses for exponents.  If you aren't sure how to teach a unit on exponents or what a student should know there is a lovely mind map at this blog.  Although it is focused on junior high, it does have a great layout to see what students should know by the time they hit high school and gives an idea of what to review.

I'm off to do some planning.

Multiplying Binomials, Factoring Polynomials and Real Life.

When I teach multiplying binomials, I teach five different methods because many of my students do not relate to the FOIL method.  They sometimes ask "How do I use this in real life"? and I have no answer for it.  When I got my degree in math (Yes I'm one of those weird ones) I specialized in theoretical math and never bothered learning real life applications for what I learned.  Unfortunately, most of my students always want to know why they need it.

Multiplying binomials and factoring polynomials go hand in hand, so when discussing multiplying binomials, you have to discuss factoring polynomials and the same question arises with factoring polynomials.  "How is this used in real life"?  Its going to be on the test will stop their questions but it doesn't satisfy their desire to know why its important to learn the material.

This prezi has some nice examples to answer this question. It covers dropped balls, architecture, and painting. Its short and sweet but a nice introduction.  On the other hand, Mathematics Stack Exchange has a great answer that explains its use in engineering. This PDF has three really nice examples including one that looks at accidentally dropping a camera off a bridge into the water.  That is one that I have seen happen in real life when someone was taking a picture of something and a bee came bye and surprised her.  There went the camera.

This site gives a list of occupations which require people to factor polynomials.  The list has 31 jobs including Funeral Directors!  There are several jobs on the list I'm not familiar with.  I assume you could easily check the site for other mathematical topics to see which jobs use them. This list is a great starting point to look up the individual job to find out how they use polynomials.  I looked up funeral directors + polynomials and came up with this prezi that talks about cost of multiple funerals and includes three other areas.

Math is Fun has the best real world examples I've seen for factoring polynomials.  They give a situation and they show the math to prove how factoring is used.  One example is designed to tell how high you threw the ball, two examples focus on selling items, one example even discusses resistors in parallel.  I like the variety of examples shown and the fact they include the mathematics. 

I think I am better prepared to answer the questions from students on "Who uses this?"  I am always glad to be able to answer something other than you need it to graduate or for the next high stakes test.

Teaching Algebraic Fractions

I need to find a better way to teach algebraic fractions in my math classes.  I start by reviewing ordinary fractions because it builds on their previous knowledge but too many students are not fluent in the basics of fractions.

I follow a process when teaching algebraic fractions once I've reviewed basic fractions and focus on the most important rule for adding and subtracting fractions!  The denominators must be exactly the same!

After reviewing basic fractions, I move on to having them solve simple fractions with the variable in the numerator so they can practice adding or subtracting fractions with a familiar denominator.  I might give a problem like 3n/2 + 1n/2  so they gain experience.  The next set of problems might be something like 3x/2 + 5x/7 so they have to find the common denominator.

At this point, I move the variable to the denominator and I make it a simple problem like 3/2m - 4/3m.  This is where they start having issues because they do not like the variable in the denominator.  It uses everything they've learned.  Once they've become comfortable with this, I start throwing in more complex variables in the denominator such as 4/x + 3/x+1.

This last problem is the one that shows me who is really shaky on their algebraic expressions because they try to add one to the first term to make it 4 + 1/x + 1 = 5/x + 1 + 3/x + 1 = 8/x + 1.  These are usually the same ones as the ones who do the 1/2 + 2/3 = 3/5.

If I have the time, I branch out to x+1/x + x/x+1 type problems with variables in both the numerator and denominator.  I save solving actual algebraic equations for later once they have binomial multiplication down because I tend to teach that by multiplying through the whole equation using a common denominator to get rid of fractions.

If you want a couple of videos that are not YouTube - these sites offer videos that are accessible.  I don't have access to Youtube at school so I've had to find alternative sites.  Explaining Maths is a site out of the UK that has 7 different videos on everything from what are algebraic fractions to combining and simplifying algebraic fractions.


Finally, check out this power point presentation that talks about algebraic fractions and the four rules of fractions.  It uses the multiply the numerator by the other denominator method of finding a common denominator but for algebraic fractions, that is the quickest way to do it.  At the end, the author goes on to show how this is used to solve problems.  This would be a great presentation to use as a review just before beginning to teach solving equations with algebraic fractions.

Monopoly and Math

  I'd like to thank David Bernstein from the google group "Mathematics Education" for today's topic. He shared one of his posts on using Monopoly to help teach probability. Its titled "Teaching probability with seven questions from the game of Monopoly".  In it each question has a link to previous blog that discusses the topic in detail.  This is an activity I could easily integrate into my classroom without much work.  It also provides everything I need to create a SmartBoard activity with little effort.  Thank you David.

Quick commercial note - Contrary to the story that it was created early in the 1930's, it was actually created back in 1902 by a woman who called it the "Landlord Game" and patent it in 1903.  It took off when a man in the 1930's shared "his game" and eventually sold it to Parker Brothers who marketed it and gave credit to the man for inventing it.

So what other activities are available to use the Monopoly Game as the basis for teaching some real life probability.  Even Business Insider created a slide show titled "How to use math to crush your friends at Monopoly".  The presentation starts with the probability of landing on a piece of land when rolling two dice and starting at Go.  It looks at the probability of rolling doubles, of where to build houses to have your opponents land on your square and how the community chest, chance cards, and go to jail cards change probabilities.  It is a very through presentation.

This resource library has a great lesson plan that uses Monopoly to collect data and use an Excel spread sheet.  It comes with a three page lesson plan, all the worksheets you need and has students use Excel to enter their data so they can create graphs to show their results.  It is geared for 8th grade Algebra I students but could easily be extended up or down.

MathCoachblog has a great entry on this topic but he approaches it from the cost of landing on certain squares in terms of rent.  Most of the other sites look at the probability but this one looks at the rental cost based on location.  He has excellent pictures to illustrate his point.  In addition Mr Ward at MrWardteaches has a great visual breakdown on the break even point for each group of property, railroads and utilities.  These two blogs fit together quite well.

The last resource for today comes from Math Concentration with a list of 5 links showing how Monopoly can be used in the classroom including directions for creating Mathpoly. Mathpoly is a great review game and could be individualized for various classes.

Real World Application of Algebraic Expressions

Today while researching information on teaching algebraic fractions, I stumbled across this cool lesson plan from Open Learn Works on introducing algebraic expressions with a real life context.  This is one of the first lesson plans I've seen that invokes a situation that does not sound contrived to explain algebraic expressions.

Too often the only time I see algebraic expressions is associated with verbal expressions without any mathematical context but this lesson plans puts the expressions in context and the questions tend to be open so there is no single correct answer.

The first activity in the first section has students look at a picture and a situation to determine which items are constant or variables.  Some things like parking are constant because the number of spaces available never change but the number of people using the car parks vary every day.  I like this because it shows how real life items can be classified as variables or constants.  Although the situation is in India, it would be quite easy to change it to a local shopping mall.

The second activity in the same section requires students to choose four variables to create algebraic expressions with.  The process chosen uses mind maps, selecting quantifiers, and putting it all together into expressions that could be used to model a facet of the situation.  These two activities tie the pieces together.

The second section focuses on real life applications of substitution and keeps in mind alternatives and possibilities while keeping limitations and restrictions in mind.  Activity three has students make up stories to go with algebraic expressions such as 2x + 4 might represent that you bought four more twice the number of pastries than Sally bought the day before for work.  Or the double decker bus can carry twice more than a regular bus and still has room for four more people.  Students choose the context of the expression. The fourth section has students take the expression and generalize it.

The lesson is actually a unit that could take a week to do, exposes students to real life thinking for a mathematical situation that they have experience with.  This activity for me requires some serious thought because the town only has a population of about 1000 people and its extremely isolated.  Most students have been to Anchorage but there are a few who have never left the village. So I might have to choose a mall in Anchorage.

I like the lesson because provides specific details of a nice open ended situation which will encourage higher order thinking to complete the activity.  Check it out and enjoy.

Algebra - Study Cards

If we assume algebra is a way of using variables and operations to represent quantities in equations, then at least one of the apps I  downloaded is actually one that has students solving problems using mental math within a specific time period.  In a sense this goes against what we try to teach our students which is show all your steps.

  So today I'm looking at Algebra Study Cards, an app that allows you to select which operation you want to practice.  They use the variable X in all equations.  Most of the problems I've seen are all one step or combining constants and then finding the value of X

 I played the game and chose subtraction.  The first few problems were simple and straight forward with the problem and suggested answers.  This one the x = 9 but not all problems are as simple as this.  If the wrong answer is selected, the correct answer is flashed and then you get another problem.  You are told the level and your score for each step.

As you progress through this, the problems get harder and more complex.  Notice the one to the left which has you figure out the 5 - 10 before you can solve for x within a 10 second time limit.

If a student needs more time to mentally solve the problem, they might get frustrated with the 10 second and give up.  I still need to figure out if you can change the time limit on the game.

It does require combining like terms before you can solve.  It also has both positive and negative numbers for what the problem equals.   Most of my students are scared of negative numbers and need the practice. 

Due to when I learned algebra, I still have problems with the idea that a person should practice solving for the value of x without any written work, especially since many of the new tests require students to explain or justify their answers.  As mentioned earlier, the time limit might frustrate students who need more time and if their answer is wrong, they won't know why because the program flashes the correct answer. 

The bottom line, is that I'm not sure its one I'd use in class.  I have to think about it.

Bug Hunter - The Secret Of Algebra

  Since I'm back in town, I am able to download math apps to explore and enjoy.  I came across this nice one that introduces the algebraic process of isolating and solving for the variable but it uses bugs called Bug Hunter - the secret of Algebra. 

You choose an avatar for the game and you travel from planet to planet "hunting and capturing" bugs.  Your space ship goes to the first planet (the blue and purple one) to work your way through the 20 levels.  Each level introduces a skill and starts with only bugs.

It takes place in space where your avatar travels to various planets to hunt bugs.  Each planet represents a different level and you must complete each sublevel for each planet before you can move on and you must go complete the sub level to unlock the next. 

You cannot unlock a level until you've completed the level before.  If you do a good job, you can get up to three stars depending how well you accomplish your goal.  Every level you complete unlocks the next level.

At various points you receive instructions on what you do next.  Each set of instructions builds on what you've been doing.  You do not always capture bugs because they do introduce variables and constants in addition to the bugs. 

 If you look at the picture to the left, it is from the second planet.  Notice how you have a left side and a right side with a pathway inbetween.  Right now, the red middle indicates I cannot drag anything across, I have to do the opposite to both sides with the button on the bottom. 

It slowly builds algebraic skills one step at a time.  I will admit, I am visiting the second planet out of five and I'm enjoying it tremendously.  I plan to reach the end of the fifth planet before I head off traveling.  I think students who need scaffolding will enjoy this to help them gain algebraic skills. 

The best thing about this app?  Its free.  You get more than just a sampling, it seems to be a full app with a lot of questions.  Check it out, give it a try.

Factors Factory App

 I am finally back in town where I have access to reliable internet so I can download apps to try and report on.  Today's app is Factors Factory, a free app that focuses on helping students learn to factor numbers using prime factorization and then apply the skill to find LCM and GCF. 

I realize students should know how to do this by the time they hit middle school but so many of my students have no idea on how to factor so it makes life a bit more difficult for reducing problems. 

The app gives immediate feed back if you are wrong but it won't give you the answer, you have to figure out the correct number.  This means a student cannot just whiz through it without learning.

 

 When the student goes into the app, there are four levels to choose from beginning with the Youngling one for those who need the most scaffolding on up to Master which has a person practicing their factoring with larger numbers.  At the bottom, students can practice finding GCF and LCM.  Each of these require the student to use prime factorization for the first number and check it before doing the same for the second number.  Once the factorization is correctly done for both numbers, the student moves on to the GCF or LCM. 


I took a picture of the Youngling page.  It shows a picture of 30 objects and has the number on top of it.  A person is most likely to choose the three so the app shows the number being cut into three pieces and it suggests the next number to use.  Once the next number is chosen such as two, the app cuts it into two more pieces.  So the picture gets broken down based on the suggested choices. 

The Padowan level still has the hits but it shows the step by step break down.  For instance you start with 36.  You select 2 and it cuts the picture in half showing 18 and the math shows 36 = 2 x 18, 18 = ? x ?, so you choose 2 and it cuts the 18 in half and the equation becomes 18 = 2 x 9.  It continues until the problem is totally factored.

The visual and the prompting is great for students who need the extra scaffolding to help them learn prime factorization.  By the time you get to the Knight level, the app assumes you know what you are doing and it does not provide a prompt.  Students are expected to be able to factor directly without the help.

I plan to add this one to my choices next year since most of my students arrive in high school without being able to use prime factorization properly.  

Real Life Applications of Inverse Functions.

  My students often ask how do I use this in real life.  Its a good question because I've never really thought of where it is used.  Its just not something I ever think about.

So to start out, this prezi introduces functions and their inverses by defining them and then focuses on showing how it works through currency conversion.  I've always thought of currency conversion as the use of ratios but in the strictest sense it is a function.

For the most part, just about everything I find talks about undoing functions and trig pops up most in regard to inverse functions because you have to use the inverse to find length or the angles.  So with that in mind, I discovered that Sophia.org has 5 tutorials on real world examples of inverse trigonometric functions.

Other than that, the most common explanation I find is you are undoing the function.  Or as the prezi states, if you convert from the Dollar to the Euro, that is the function and when you convert back from Euro's to the Dollar, you are doing the inverse.  Still other than composition of functions to prove two equations are inverses of each other, I'm hard pressed to find solid real life examples.

I've found problems with families buying hot dogs and drinks for $10 and then finding the solution but most people don't calculate inverses per say.  I think the other issue is that I focus on the mathematical aspect of inverses rather than inverse being a process of reversing things.

While I was off looking for information on inverse functions, I checked out real world applications of composite functions.  I found a slide show with a cool definition that makes total sense and which I plan to use in class.  A composite function is nothing more than combining two functions into one to model something.  This places composite functions into a framework so I could find lots of examples and use those to introduce this topic.

So the example here is great for showing how to use composite functions to determine which deal is better in the situation.  It is something that is quite possible and doesn't sound too artificial.  I am off to find more examples.  Have a good day.

Inverses

I love inverses but its kind of hard to teach them to students because they don't always understand precisely what inverse means.  So I'm always on the lookout for ideas to use when teaching the topic.

I came across this nice website with comics that operate as a hook for the topic. In addition, she offers ideas for introducing and teaching the topic so its a step by step process.  Her suggestions are quite clear and I like the examples she's used. She has it going from simple to more complex which is great.

I found this nice introduction on inverses that use map directions as a way of introducing inverses. The author gives the directions to a place and the student has to write the directions going from the place back to the origin.  I really like that the activity builds on prior knowledge.  Unfortunately, out in the village, they don't operate on blocks and streets.  Its more like ohhh the house with the new porch, or the one next to where the old chicken coop used to be.

The author then introduces inverses using a verbal number problem that requires the student to write it in the reverse way.  It then goes on to show how inverses appear in graph form.  Its done in a nice logical manner.

This site has a great partner share activity on inverse functions. The activity had pairs receiving one function and one inverse but not of the same original problem.  They had to find the student with the matching inverse for the function and function for the inverse.  This meant students had to know how to find the inverse and it generated discussion.  I wonder if I can extend this to include graphs to match the functions and inverses?  

So now back the the blog Math Equals Love.  She creates units for interactive notebooks.  This one discusses domain, range, and at the very end, she includes inverse function notes. She looks at inverses from points, equation, and graph which is really nice. I know from personal experience the usual way the topic is covered is to only look at the equation.  Students do need all three.  This is a second way she covers the same material for inverses. 

So once we know how to find inverses, how are they used in real life?  Stay tuned for tomorrow.

The Navy and STEM Education

  If you read yesterday's entry, you found information on the Navy STEM lesson on GPS and Navigation, but what I didn't tell you is that they have a whole website on additional topics that come complete with the power point and the lesson plans.

These lessons were created in conjunction with discovery education.  There are 8 different lessons of which two are labeled math, however all the ones I investigated addressed at least one math standard.

This makes it possible to use them in the math classroom and at the same time showing how the math is not isolated from other topics.  The one on Submarines and Aircraft carriers addresses the standard on creating equations with two or more variables to represent relationships between quantities.  This puts a real world spin on something that is usually presented as a boring math problem.

Although the unit is mostly about nuclear power, it does address half lives which is a topic usually covered in Algebra II or college prep.  In the set up, students are required to use google maps to plot a course from Washington state to the Arctic, followed by how long will it take going the top speed of 20 knots per hour.  The fourth explore addresses the question of how long is radioactive waste dangerous.  This is an appropriate application of the half life formula.  After learning about radioactive decay, they have to predict the time for a radioactive decay scenario.

I'll be honest.  I'd probably use the parts of the lesson plan that focus directly on radioactive decay for use in the math classroom.  I love the lessons are set up so you know what to ask students for each slide in the power point and the activities/explores are pretty easy to do and the supplies are not difficult to find.

So this is a site I plan to bookmark and come back to when I plan next years classes.  Go explore and have fun.  Best of all, its free.

Navigation and Math

  Locally, navigation usually means knowing where the markers are on the trail.  Sometimes you can use your fingers and a knowledge of the terrain to find your way.  Of course if you are in the city, you use various land markers to get you there but how does math fit into it all. 

I found this page of lesson plan links for math in navigation spanning 4th to 12th grades. Finally a site with more high school things than elementary.  Yeah! 

Middle school has one lesson on measurement and degrees of movement which is a great introduction to navigation being a 360 degree circle. using physical movement.  It might bore older students but if you have some who need to move around, the activity is great.

For high school there is a two day lesson on using vectors to navigate which shows real life applications.  It includes vocabulary, hints on using the Smart Board and requires a map for a lake. The first day is an introduction to vectors and maps while the second day has students applying the knowledge gained from the first day to a classroom activity.

I love the plotting a course through school lesson where students plot a course to get them through the school building.  It would be easy to change out the map provided if you wanted them to do your school but I think I'd have them use the one provided as practice.

There are 17 lesson plans listed. The material is for Michigan but it is easily adjusted for your own state or you could use them as listed.

From the Mathematical Magazine is a lovely 10 page pdf on the Mathematics of GPS.    The author goes into explaining how the GPS calculates your location using satellites and math.  He includes a simplified explanation but it is good and well thought out.  Although, this is more of a paper, it can easily be used in a trigonometry class as a real world application.

Finally from the Navy comes this 34 page STEM activity called GPS and Navigation.  This has everything needed to teach the lessons including essential questions and the standards it meets.  The first lesson addresses how the GPS works and includes information on how triangulation works.  Lesson two looks at applying GPS principals on the ground while lesson three focuses on GPS and naval navigation and includes an arctic ice race.  It appears this activity accompanies a discovery education power point however, the power point can be found by doing a quick search.

So know we have more indepth activities using GPS and math.

Solar Energy

  This is the last day on Solar Energy.  I just found so much good material that I needed to do one more day.  I would love to have my students work through a couple of these packets because no one has solar energy out here and I'd like them to determine if it would even be viable option.  The village has several wind generators which help keep the electrical cost down.

First is this great 16 page packet on the Maths of Solar Panels and positioning them on buildings.  Although it is from Australia, it could easily be adjusted to apply anywhere.  It is actually a cross curricular unit for math, science, and geography.  Although it requires a specific multi-meter, it shouldn't be hard to obtain alternatives so students can do this.  The activity points out that the path of the annual movement of the sun is much like a sine wave.  It has all the information you need to carry it out.

This site shows how to calculate tiled array spacing for solar energy.  It includes a link to the NOAA Solar Calculator and explains how to use the site.  The site also has nice clear directions on using it.  For some places, it may only have certain cities but it gives a chance to practice.  According to the map, they only provide information on Anchorage and I'm West and North of the city so the information might not help me but I can still learn how to do it.  It provides the equations necessary to complete the calculations.  I'd like to divide my students up into groups and assign each group a different city so they can report back on their findings.

Finally is a 47 page passive solar module from Rutgers University which uses trigonometry.  This five lesson module comes complete with everything needed to run a two week unit on passive heating and cooling.  It even looks at determining if the cost is worth it to use passive solar.  Lesson 2 focuses on location and which locations are better for solar than others.  Lesson three looks at thermal mass in detail.  Lesson 4 focuses on solar angles while lesson 5 looks at how overhangs effect solar gain.

Each lesson has an activity that reinforces the topic.  homework, math, and could be used as a cross curricular unit for math and science.  It comes with standards and was published in 2014 so the information is quite recent.  I downloaded the pdf so I could explore it in more detail so I know when I'm going to use it.

Next topic is navigation.

Trigonometric Ratios Cont.

  The thing about trying to find real world examples of trigonometric ratios is simply that too many situations have a unreal feel about them.  How many times do you find people who want to find the distance a ladder is propped against the wall, or the height of a flagpole?  I know I've never needed to use trig for anything like that.  My challenge is looking for examples that my students see as "real".

I found a few things that have a feeling of being "real".  For instance this presentation discusses its uses in a variety of disciplines from chemistry to meteorology.  I love the indepth discussion which shows way's I'd never even thought of.  This would be a good introduction and with a bit of research, its possible to find worksheets with problems from these disciplines that give students a chance to explore these jobs.

This short Prezi explains how trigonometry is used in navigation, including a bit of the history.  I enjoyed it because I learned more about how the sextant works which I'd never realized.  It is a general introduction but if you add it to this page, you learn how its used in surveying, navigation, and solar power.  The solar power example explains how to calculate the amount of power being produced in a certain situation.  This could easily be extended to covering solar energy as a two or three day activity.

This site has indepth information for calculating where solar panels should end up to get the power.  It has excellent examples that are extremely clear.  This page gives the information needed to plan a system of solar panels and covers everything you need.

Teach Engineering has a great lesson on tracking the sun throughout a year.  Something that has to be done if you are planning any solar so you get things right. From what I've read, you have to know the yearly sun cycle so you can build a house that does not get too hot in the summer and allows the right amount of solar gain during the winter.  It has everything you need.

Check back tomorrow for more on this topic.  I can't believe the amount of cool information I've found on this.

Trigonometric Ratios

I now have to teach trigonometric ratios in geometry, in addition to college prep math. I have apps on the iPad but I'm wondering about some hands on activities that would get students moving around.  I like giving them an excuse to move around and get some oxygen into their brains.

I found this great blog by Dan Percy that provides a wonderful activity to introduce trigonometric ratios in the classroom. He opens with a picture of a crane outside his window and has students guess its height. He shows how you cannot use the Pythagorean Theorem to find the height because you only know its distance along the ground.  He goes on to provide a worksheet with a variety of triangles  of 20 to 70  degrees and students have to measure the lengths and record them on a data sheet.  Once the measurements are filled in, they find the ratios in both fraction and decimal form.  The next step requires the use of geogebra to verify the data.  The whole activity allows students to see that Sin 30 is the same regardless of the measurements.  Check it out.

The Teaching Channel has a video that has a great idea for helping students learn where the opposite side is or the adjacent using several cut out triangles and a bit of water sprayed across the triangle.  The teacher highlighted in this video uses a drum to beat out SOH - CAH - TOA while she says it and has the students say it along with her.  She noted that later in the year students will beat the rhythm of SOH - CAH - TOA on their desk.

I like the idea of using the triangles but I'm wondering about writing the words opposite or adjacent on the triangles and then holding the triangles above a heat source to make the words "magically appear" so it catches their attention.

Over at CPalms, there is another lovely activity that has students comparing triangles with the same angles. Its a worksheet that has them measure the lengths of the triangles and then calculate various ratios without knowing they are trigonometric ratios at first.  The trigonometric ratios are introduced in the second part of this activity and comparing the data with the earlier calculations.  Its a nice way to introduce the procedure without using the language quite yet.  I like this one because it comes with accommodation version that separates the triangles better.  This activity also comes with an applications page so students can apply the ratios to a page.

So three nice introductory activities for trigonometric ratios.

Manga and High School Math

  For anyone who does not follow trends, Manga is a type of Japanese graphic novel.  Its so popular here that the library has several popular issues on the shelves they regularly check out.

I found several math books in Manga style for a variety of grades.  Learner Books publishes several Manga style books for grades 3 to 5.  The books cover probability, fractions, geometry, multiplication and division, money, distance and measurement, and time and temperature.

Although they are not for high school aged students, I have a few students who have missed certain basic skills and might need scaffolding.  They books might be the right reading level for many of these students and they might help them catch up.  Nothing wrong with providing interesting material.

For high school, there is a series of books from No Starch Press called the Manga guide to _____________.  There are editions for statistics, calculus, linear algebra, regression analysis, or tons of science topics.  Each is drawn in the manga style with a story designed to teach people about the mathematical topic.  For instance in the linear algebra book a guy is exchanging teaching a girl linear algebra in exchange for martial arts lessons from the girl's brother.  There is a love line but the math is solidly embedded in the book.  The books all have good reviews on Amazon.


I'm revisiting Manga High because they now claim they work on tablets but when I tried the games on my iPads, a adobe flash needed page popped up and so I couldn't go any further.  I tried it on a Computer and it works well on the computer.  I tried the wrecks factor game, easiest level and it was both fun and challenging.

The idea behind the game is that a trinomial ship wrecks and you have to cover an area of the grid so the trinomial is factored.  Say you have x^2 + 3x + 2 is factored into x + 1 and x+ 2.  The x is the corner so you go up one block and over 2 so the whole area is in green.  If you are right, the helicopter flies in to rescue the people.

In addition, as a teacher, you can set up classes to assign games, quizzes, etc and the site keeps track of the their results.  The results give you all types of information so you can see how they are doing, what they need to work on and other information.  As I said earlier it does not work on my iPad but I do not have the latest up to date OS on it and I have to check on the adobe aspect.  The site does say it will work so I'm going to investigate that.  I'll keep you posted as I figure this out.

Math 911 and Class

  I stumbled across a cool resource that explains math in a different way.  It has comics!  It has great pictures!  It has humor!  It is not Superman! It is Professor Weissman's Introduction to Algebra! site.

There are eight modules that are ready to go in pdf format.  Each module is between 10 and 15 pages long with lots of examples, pictures to illustrate the material, and comics.

The topics cover topics from whole numbers to combining like terms but they include variables and algebraic information.  I've looked at several modules and love the way it is set up like a newsletter or newspaper.  The material is chunked into easily read segments.  A student will not be overwhelmed and there is humor included in the material.

Some modules include newspaper articles with questions for the student to answer but all have practice problems in addition to jokes and brainteasers.  These are great and I like the comic section which uses the actual topic and has some humor spread throughout it.

There appears to be software available to just download and place on your computer.  I don't know what it does because I'm not allowed to download any new software for my computer but you could check it out and let me know.  I am not sure how old this site is.

So where am I leading with something  like this?  Well, I like the way the material is arranged and presented.  I'm wondering if I could take the important material from a section and at least create a  newsletter type that students can easily read and which could be kept in a pocket at the back of their composition books. 

My ELL students are overwhelmed when they do have to read the textbook since its over 400 pages long and its not easy to chunk.  I can create comics on programs and then put them into the newsletter so those are included.  The math 911 has a nice format for inspiration.  As soon as I create my first newsletter, I'll share it here.

Using Comics In Math Class

  I have quite a few students who love reading comics.  I know I have several students who would rather draw than do the work and so I've found a couple ways to use comic strips in the class in  way that should engage the artists and have them do the work.

Resourceaholic has a great way to use comics in class.  The author got the idea from another blog and I'm highlighting it here.  The first way is to create a comic strip showing the process used to answer a question such as finding the solution to a system of equations by graphing.  So a student chooses one of three problems to do and then creates the comic strip explaining how they solved it.  The assignment can be used either in class or as a homework assignment.

Flummery has information on comic strips, drawing them and using them in various content areas including several templates.  The suggested math activity has students selecting a math element as the main character and creating a 3 frame strip. As I read the activity, I pictured having a plus sign as the main character who talks about combining numbers to make a bigger number.  So many possibilities.  It also contains an evaluation rubric and a worksheet to compare drawing styles between comics. 

Although the next two are actual studies they each have activities in them that would be easily applied to the classroom.  The first one from Australia suggests having students select two algebraic expressions and create comic strips for them.  The study has examples so you get an idea of what should be done.

I'm not sure if this is a study but it explains the process one person went through to create an actual math comic book.  I like that it shows how this author used a super hero with two students to create the lesson while having the elements of a comic book. 

Now comes the part about creating the comics.  Do you have students use an app or program to create the final product or do you have students create on paper.  I think  it depends on the student.  For those of us who do not really draw, the app or program allows us to create without getting lost in the act of drawing.  For students who enjoy drawing and are always doodling, even during the lecture, it may be best to let them physically create the comic.   There is always the possibility of letting students draw the frame, snap a picture of it and then insert it into the app. 

I've made a few comic strips and I plan to make more over the summer so I can put them up on google drive.  I want to have several examples ready for students so they know what I want.  Does anyone have suggestions they can contribute to this?  Please let me know your experiences.

Keeping Students Engaged!

  Our district is getting a new superintendent.  When the three finalists came out, we asked them questions.  One person asked if they could think outside the box to meet the needs of the students.  Two of them stated that the teachers need to make their lessons so students want to be there.  The third recognized that many of our students often miss school because they have to go home and babysit because their parents are working or leave town for a while.

We also have students who live in homes where they do not get enough sleep at night for a variety of reasons.  This has lead to a whole discussion on how do we create lessons that will keep them engaged while allowing them to learn the same material if they are in class everyday?  This is something we struggle with.

This site has five suggestions to personalize the lessons so students are more engaged and perhaps relate to the material.  The first is to personalize the material by making it into a story.  Create a hockey player when you are teaching probability is one example.  The second thing is to open with a hook.  I'll be honest, I have trouble with the idea of creating a hook. It is recommended that the hook reappear throughout the lesson in different forms.

Third is to emphasize your main points and make sure the main points appear throughout the lesson.  Fourth is to choose images over words.  Use diagrams, arrows, images, color coding, to connect ideas to visual reminders.  I love SmartBoards and computers because they allow me to easily create this type of presentation.  Finally, don't forget the why of the math being taught.  Take time to explain why they are learning this material.

Harvard Graduate School of Education strongly recommends that students be exposed to multiple approaches because comparing methods or approaches helps students thing about what works in addition to how and why it works. In addition, they have created materials that expose students to multiple approaches to certain problems.  The material is done in a cartoon format so its more enjoyable for students to do.  

These materials are from Temple university and have 11 chapters available on line to download and use.  It covers topics from order of operations to systems of equations to quadratics.  I like the material because it has everything needed to use and is set up to encourage conversation among students who work in groups.  I plan to download and use these next year.  The files are pdf and can be used with the SmartBoard.

Encouraging Collaboration

  As teachers we know collaboration is needed to help students learn and to prepare them for certain future endeavors.  There is an engineering program for Alaska Native students but most of our students who go have trouble because they do not offer tutoring and students are not used to real collaboration.

Most of the collaboration my students engage in, is the one person does the problem and helps the others complete the problem.  They do not like to really collaborate.  The engineering program wants students to already know how to collaborate before they get to college either for their college career or as a high school student involved in the summer program.

So how do we encourage true collaboration.  I remember doing the thing with everyone getting a job and we'd work as a group that way butt even in high school I felt the whole process was kind of hinky and it didn't really teach us to work together.  What helped teach it to us was when I had a teacher who asked us to be prepared to justify the answers we got.  We had to be prepared as a group and we ended up debating so much in Math.  I think this helped foster true collaboration.

I have the Kagan Books for various math classes but I need to extend the material so students actually work together rather than expect to tell them if each answer is "correct".  It still falls comes down to how do you help high school students develop into collaborative learning units.  I prefer the term collaborative learning unit to collaborative group since it indicates a more specific focus.

For those who want to create collaborative learning units, here are a few things to help do that.

1.  Keep groups to between 4 and 5 members.
2. Use collaboration when introducing new material.
3. Break up the material into chunks so each group is responsible for presenting that bit.  Or use a version of Jigsaw puzzle here.
4.  Change the mindset of students when using technology. Use the technology to help encourage and develop collaboration, rather than using it for only internet research.  Set up Google Docs for instance.
5. Enable students to teach each other as part of the collaboration.
6. Build in accountability.  Have the group work on the assignment together but collect only one piece of work (teacher's choice) to grade for the group.
7.  Set up the room to encourage collaboration.

I like suggestion number 6 because it creates accountability in a way that the students do not know whose paper will be corrected. So I have some work to do this summer to prepare to encourage collaboration in my classroom next year.  I'd love to hear from people on their experiences.

In Class Versus Take Home Tests.

  Out where I teach, we seem to loose time every month for various things from a death in the village which requires 3 days of mourning then the funeral, to flooding in the school, to visitors who need to meet with the students. 

I feel like I spend all my class time working on trying to cover the material and my tests and quizzes fall to the wayside and I end up with one major final at the end of the semester.  I want to do more assessments and have a way to do the quizzes but I'm wondering if a take home test can be as good as an in class one.

There are advantages to take home tests such as:
1.  The time restraint is lifted especially for students who do not do well under pressure.

2.  Students have time to think through questions and let their minds process material before starting to answer the question.

3.  Questions can be tougher than those given in an in class exam because of the extra time available.

4. It can give them a chance to learn the material.

Unfortunately, there are several huge disadvantages.
1.  The internet where student can look up problems or pay someone to do the work.

2.  The student does not develop additional understanding when they just copy.

A way to check to see if the student did it, is simply explain their work to you.  That gives a chance for them to prove it is their work. 

I stumbled across someone who stated they graded the take home questions are geared more for rigor and completeness of answers along with the quality of their writing.  The author does use an honor code but he does look at the writing.  To me this suggests that part of the process of the take home exam requires students to do more than just work out the problem.  I like that, especially if the material is discussed in class.  He also allows students to resubmit the work if it is wrong.

Unfortunately, I was unable to find much material on this topic for high school.  It was pretty much all for college.  So my thinking on this boils down to I plan to give the practice test as a take home but give the actual test in class.  Since there are two study hall nights a week, students can come in either as a group or as individuals.  I can also set up a site for them to type in their answers and they can get feedback on which are right or wrong so they know where they stand but I will not discuss this in class.

If they are willing to expand the energy to make sure their practice test is correct, they should do well on the final.  If not, they won't.  I plan to give them the practice one week before the scheduled test so they have time to work on it.  During this time period, they will not get any homework. 

The online site will give me instantaneous results so I know who is doing the work and how prepared they are.  Honestly, I need the day or two saved to teach material in my classroom.  Its a never ending challenge.

Different Math Activities.

  I get tired of trying to think of activities for my math classroom.  To quote a lot of friends, "Borrow what is already done and tweak it."  So I do that and I found a site with great activities for a variety of classes and topics.

There are 15 different activities with instructions and worksheets.  Some I will use, some I won't because of my student population but they all look fun.

For instance, there is one on Stylometry where you analyze the word length, the sentence length, and word repetition.  This information is often used to settle disputes over ownership of material.  This exercise has students graph distribution of words, sentence length, etc for an author.  It is suggested students do this for an author who writes under their own name and a pseudonym to see if the writing styles change or remain constant.  This is a cool application of math and it connects literature to math.

Another uses Voronoi Diagrams which are often used to create boundaries such as you have a town with 5 schools  and you need to establish boundaries so all students will go to the school closest to their residence.  This activity makes full use of perpendicular bisectors so it would be a good activity to use in Geometry. 

I love the tangent experiment which looks at finding the slope of the line of a ramp, or stairs.  It has students moving around to the steps or a ramp, measuring, and calculating the slope.  My school has a ramp but it goes down and part way down changes direction.  I have steps in the front and back of school but I'd have to get permission to go to the steps that head up to the server room so they can measure those.

All activities come with the worksheets, instructions, information on the activity itself and where it come from along with some suggestions of what to use.  Just looking at the activities shows me three I can use in Geometry, a couple in other math classes for statistics and probability.  There is also information on using Excel to crunch some of the math.  Go check it out and enjoy.

Cinco de Mayo Math

Since I live in the boondocks of Alaska, Cinco de Mayo is not normally celebrated out here but that doesn't have to stop me from doing something.  The 5/6 grade math teacher had a pinata but being the math teacher, I look for math based activities.

I found a couple of nice activities that can easily be done. This activity links Geometry and Probability using the Mexican flag.  It has students analyze how much of the flag each color takes up which leads to determining the probability of certain things. In addition, it has questions to find the area and perimeter.  Its a nicely balance worksheet.  You do need  color printer for this one.

This site suggests students use the Census Bureau to find out where there are Hispanic communities so they can determine which states have the largest influx of Hispanic immigrants. This activity could be extended to determine percentages of state population, of the percent of total Hispanic population, and do some research to determine Hispanic business owners, etc.

Although this is a paid site, it does have a few thematic worksheets for grades K to 8.  Grade 6 has a unit ratio worksheet that has nice practice problems using taco, tomatoes and other foods.  Grade 7 has a shopping spree worksheet in which students calculate the cost of items for a big Cinco de Mayo party.

Unfortunately, there really is not a lot of material available for Cinco de Mayo but it is possible to create a few activities for the students.

1. Research the battle it celebrates and calculate the odds of winning based on the size of the armies.
2. Calculate the distance between Veracruz and Mexico City, determine the average speed the army would have traveled and calculate the time it took them.
3. How long would it take the ships to travel from Spain, France, and Britain to Veracruz Mexico.  Could they go straight or did they have to go around something?

This are simple but require research and provide a connection to social studies.  Use your imagination and have fun.

Thoughts on Review Math

  It's the end of the year and I'm working on ways to help students review for their final.  We all know games and jeopardy are great ways to review but it relies more on those students who always know the answer?  What about the students who try not to volunteer an answer, are always quiet, who never give you a clue as to how much they actually know?  How do we arrange a review to help them prepare for the final.

One way is to create regular quizzes to assess their understanding but what about those who are regularly absent or refuse to work. How do you help them prepare for a test.   I used to give notes and then the test but this time I gave the notes, a practice test and next week students get the final.

The difference is that once students finish the practice test, I have them pair up to compare answers.  If the answers are the same, it should be correct.  If they have different answers then they need to compare their steps to see who might have made a mistake.  It was awesome!

Most students sat quietly and worked hard.  First thing they did was compare answers.  I over heard one person asking the other to "Explain what they did".  The other one started explaining and the first person showed him why there was a mistake.

I did a quick poll to see if they felt it was beneficial.  All of them said they learned something, even if it was as simple as "Watching their signs".  Others realized why they made the mistakes.  I see it as a profound learning experience.  Research does show that peer tutoring is an excellent instructional method. 

This particular technique I think I'll use in my classroom next year.  I can't always take the time to do this in class but I can assign the "practice" as homework and have them go over their work in class.  At the end of class I can pass out the answers so they can check their answers at home.  I usually have students correct their tests by filling out a form with the problem they did incorrectly, the corrected problem plus a line or two on what they did wrong.  In other words, I have them do a type of error analysis because too many of my students are unable to recognize a step done incorrectly.

The use of games and jeopardy are fine but most students need that little bit extra.  

Math Baseball

My neighbor, who teaches science, introduced me to the idea of playing a version of baseball to review material for a test.  One day when she ran one of these games, I heard them through the walls having fun. 

I found lots of computer based versions of math baseball but it took longer to find the rules for the same so I could organize physical movement in the classroom.

The great thing about starting with the rules is you make the questions so you are not restricted by someone elses games.  This is important because many times the available questions are not exactly what you want to test or do not meet the needs of your students.

I found this set of rules for math baseball that has you drawing a diamond on the board. I would rather set up the diamond in the room itself.  As far as the time limit, I might add a minute to the 2 minute limit for more complex problems. Its rather simplified and does not cover a few situations.

This set of rules covers penalties, the number of bases the hit made, or how many outs.  The number of runs and outs are based on a roll of the dice.  These rules add a complexity to the game that the other set of rules do not have.

Imagine being able to use this game for any mathematical topic.  Most of the ready made questions I found covered multiplication, division, addition, subtraction, fractions, decimals, and percentages.  That does not mean you can't make your own to cover:
Trigonometry,
2 Dimensional shapes, area, and perimeter or circumference.
3 Dimensional shapes, volume, surface area, edges, faces, vertices.
4. Solving one, two, or multistep equations.
5. Multiplying binomials.
6. Factoring polynomials
7. Unit circles
8. Logs and ln.
 9.  And lots more.

All you have to do is make sure you have at least 20 questions available for the game and have a tie breaker or two should it become necessary.  To make this more fun, I found a couple places that have a free online baseball scoreboard you can flash up on the Smart board. 
1. This is only a beta version but its worth a check.
 Otherwise, check for scoreboard apps for apple or android and have someone keep track of it on their mobile device.  If none of the above work, try setting up something basic using Numbers or Excel.  Here is a tutorial for creating a baseball scorecard using Excel with step by step directions.

Now you and I are set up to play math baseball in our classrooms.  Have fun and enjoy.

Magic Squares and Math

  I've always wondered how the magic square got its name and how it applies to math, especially with the changing demands for teaching mathematics.  I actually found a good reason to have students learn about magic squares.

A magic square is simply a larger square that is subdivided into smaller squares such as the one to the left where the big square was subdivided into 16 smaller squares.  In addition, the rows, columns, and diagonals all add up to the same value.  It sounds simple but it really isn't.

One of the biggest things it teaches is perseverance which is one of the big things in math right now.  Students have to develop the ability to continue working on a problem until they reach a solution. They also provide a link between math and history and encourages problem solving. 

Although this activity from the University of Cambridge, requires  specific software for part of it but the introductory section is great because it offers a partially filled out square that students finish so it works.  This provides a good guided practice before students are expected to put it together.  The lesson plans are clear and it includes the necessary worksheets for the activity. 

NCTM has a great activity in their Middle School Magazine from February 2015 on Magical Squares.  It includes a good chunk of history and an activity that has  students creating  magic squares.  It is well written and has the worksheets that are needed to complete the activity.

This article from Wolfram Math gives an in depth mathematical explanation  of magic squares. It includes the formulas, the pattern for filling the squares out successfully.  It gives multiple examples of both simple and the more complex  squares so you can see how the pattern works for each one.  This would work for either an AP class or a college prep math class because it does involve the higher level mathematics.

For students who are not as advanced, this PBS lesson provides some wonderful instructions and drawings so you know what to do.  This lesson includes historical information plus all the worksheets needed and it has great step by step directions to fill out a simple 3 x 3 grid.  I like the clarity the instructions have so I can see how to do it.  The pattern decreases the chances of students becoming as frustrated.

I can see popping this in at the beginning of the day, on a shortened day due to weather issues, half days or any other time you need a quick something that will still meet standards and keep student interest.

Dice Games

  Since it is almost the end of the year, I continue to search for games my students can play once the last few days of school when we have to turn in all the electronic devices.

In my hunt, I found this webpage which has games that work for high school students.  The first dice game  that caught my eye is called number sense call out.  The teacher rolls two dice either using dice on the Smartboard or using real dice and then asks questions about the results.  The site suggests several questions such as write the sum of the dice in words.  This could easily be adjusted to creating a verbal sentence using the two numbers and a variable to create an algebraic equation.

The other game that could easily be adjusted for high school students is the array multiplication. The original game has students roll two dice, then draw a multiplication array on graph paper.  I can see taking this a step farther by rolling two dice such as 3, 4 becoming 3x + 4 and rolling a second time to create the second binomial.  The students could easily practice multiplying binomials using a graph paper.

I've used dice in my classroom with my pre-algebra class for practicing basic operations using integers.  For instance, I've rolled two dice on  a paper divided into quadrants, each designated as positive or negative.  Then using those two numbers and an operation, my students try to find the answer.  If you use four dice, you can create fractions that could be added, subtracted, divided, or multiplied.  Its all up to your imagination.

This 20 page pfd which has math games for middle school and filled with everything you need for seven games but one is particularly interesting as it uses special dice.  The game is FractWards which uses special dice to create fractions which are added together to determine the next move on the game board.  I like the way everything for this game is included including the netting for the special dice and the game board.  The directions are clear and this looks like a good game that could be used to help students become more comfortable with fractions.

I have so many students who struggle with math because they need additional work in their knowledge of mathematics as they pass from grade to grade until they hit high school where they either fail or struggle with the demands.  I need these games for those students who are weak with the basics and starting high school in Algebra I because Pre-Algebra is no longer being offered.  I am hoping that using these games will provide the scaffolding needed to help students gain the needed foundation.

Math Card Games

  As the end of the year approaches and its getting harder for students to sit still, this is a perfect time to integrate card games.  I admit, I don't know any good card games for high school students but there is the internet which has good information.

For students who are not good at their multiplication there is a variation on war that allows them to practice.  The variation is provided by Denise Gaskins in her blog.  She suggests giving each player a double deck with the aces, deuces and tens removed.   As each player turns over a card, the first person with the product gets the two cards.  Continue until done.

This 69 page pdf from The Positive Engagement Project  which has tons of mathematically based games for grades K to 8. I've included it because many times we get students who are not strong in certain basic skills that we have to find time to work on but can't always fit in the actual lesson.   There are several based on fractions, several that require the use of integers, several strategy games, and enough games for the 5 to 8 group that students will not be easily bored and have a chance to work on multiple skills.

This site offers several variations of war which can be used for students who need extra help. The variations focus on fractions, integers, absolute value, plus a few more that will help student practice their skills.  Although the directions tend to be short, they are enough to determine what should be done.

This is a list of 50 fun and interesting middle school games which covers games from apps, to dice, card, computer, or a variety of other forms.  I focused primarily on the card and dice games as those are the ones I am looking for right now.  One of the suggestions is the Algebra Tic-tac-toe game in which students must solve the two step equations before they can put a X or O in the grid. This is a way for them to practice solving two step equations instead of filling out worksheets.  I like many of the games suggested in the card and dice sections.  There is also an Algebra Taboo which focuses on mathematical vocabulary which is an area my students are weak in since they are classified as ELL.


Finally is this huge site of card games covering fractions, prime numbers, order of operations, exponents, and multiplication which can easily be used in a high school classroom.  There is one called exponent battle.  The players turn over the first card which represents the base and the second card is the exponent.  The idea is each student figures out the value of his/her base with exponent and the one with the highest value wins. 

Give the above a look.  Since the school year is coming to an end and many of us have to turn in our technology before the end of the year, some of these games might be fun to play.  I'm going to check out dice games next.