Video Games

  So many of my students are into video games.  They play on their mobile devices, on their televisions, on anything they can put the game on but I wondered what role math plays in the creation of video games.  After a search, I found some great information on exactly how its used.

Quora has a great answer to this question and includes a diagram of a scene with the math scribbled on it.  Mathematics is used to calculate the angle of a jump, how high the creature can go and how far with the angle.  The author gives lots of examples from real games to who where and how the math is used.  Apparently much of the design starts with a word document before moving to a spread sheet.

NRICH has a lovely article which talks about the math used in computer games.  The first part discusses the shooters, the second goes on to show how geometry, vectors, and transformations help create the world.  This is wonderful because it provides examples, exercises and answers so students have a chance to see the whole picture.  I'm going to down load this and share it with some of my students.

This site has a  nice list of the types of maths used most and how they are used.  The author does state that he goes to others who have the math expertise for certain things.  This site includes an actual illustration showing how the math is used to find trajectories of a cannon or something similar and its included in a code sample.

Here is a nice Prezi covering the basics of math used in game designing.  It also includes the levels of education and a bit about salary.  It is a short thing but very well put together.

This place has a nice short article on the importance of knowing geometry inside out for the graphics part of gaming.  I like the discussion of 2 D into 3D objects, congruence , and other aspects of geometry.

The last place is on the  game designing site which states that math is the foundation of game design.  It lists the general topics and gives some details.  Its a nice article.

Armed with this information, I have a real answer to "When will I ever use this?".  I can talk about designing video games or even use a couple things to create an activity for them.

Ratio Infographics

  I am revisiting infographics because I wantedhttps://infogr.am/Basic-Mortgage-Ratios-Explained more information on using them with ratios.  Since ratios are found all around us in activities such as sports, drinking, etc, I wanted examples so I could create an activity on the topic.

The first  example is on ratios in drinks.  I am not a drinker so I have no idea what goes into various drinks but this infographic was done in such a way, I could easily tell.  For instance, I found out that "Heavenly" uses amaretto, pineapple juice, white wine, blue curacao in a high ball glass.  The size of the font indicates the ratio of the ingredients. This particular one was created in a college class.

The second example shows the ratios for creating 38 different coffee drinks. The graphic shows the type of glass or cup, the size of the serving, the ratios of milk, flavoring, or even the type of coffee used.  I learned that a Black Eye is 4 oz of dripped coffee and 2 oz of espresso while the Vienna is 2 oz of espresso, topped off with whipped cream.  I don't drink coffee but there is one on chai tea so that makes me happy.

The next example has to do with explaining the different types of mortgage ratios. It is put out by a mortgage group.  In the infographic the debt to income ratios and the loan to value ratios are explained in detail.  This is nice since most high school students know very little about the mortgage business.

The final example focuses on McDonalds Liquidity Ratios. This would open a discussion on what are liquidity ratios and how are they formbed.

These are some good solid examples of ratios in real life and could easily lead to brainstorming topics that could be used to create infographics on ratios.  The students could then research which ratios are used and how they would fit into the infographic.  This type of discussion could help show which ratios work better than others.

Types of Mistakes

  After discovering what the four types of mistakes are, I went looking for more information.  In the process, I stumbled across a website devoted to looking at student math mistakes.  The site "Math Mistakes" is a place where teachers can submit student errors to share with others.

I like that there are two indexes associated with this site.  The first is listed according to subject and the other is broken down by grades and are aligned to the Common Core Standard.

The latest example shows a circle with the following questions.
c. How many lines of symmetry.    Answer  - 1000
d. Explain why each line of symmetry cuts a circle in half?  Answer - I don't really know
e. Explain why each line of symmetry must go through the center.  Answer - Because it wouldn't all match up.

The owner of the site asks two questions about this.
1. How do kids come to see lines as having no thickness at all?
2. What experiences would support that change?

Questions that really  have me thinking about it.  I'll be teaching this in a couple of months, so this is important for me to think about before then.  It also alerts me to possible problems on this topic.

I checked out the trig problems and the first one was on using the law of sines.
the problem was set up correctly but when the student began solving it, he or she replaced the x in sin x with the value of 10 rather than making it 10 sin x.

He talks about three categories of mistakes that most students make.

1. Mistakes Due To Limited Applicability of Models
2. Mistakes Due To Applying Properties of a Familiar Model in an Less Familiar Situation
3. Mistakes Due to Quickly Associating Something In Place Of Another

He went on to classify this as a mistake of the third type.  Its not that the child did not know the material, just went a bit fast.

I plan to check out more of this site as I prepare to teach binomial multiplication.  I already know most of my students will multiply the first two terms and the last two terms and forget the middle term.

Did You Know?

  Did you know that there are four types of mistakes?  I certainly didn't until this past weekend at a conference.  In fact, it is important that the student know what type of mistake so they can better learn from it.

The first type of mistake is the "Stretch Mistake".  This is the type of mistake we make when we are first learning new skills because we are stretching out our knowledge.  When we first learn new skills or material, we are going to make mistakes.  This type of mistake is best handled by reflecting, identifying what we are learning, and adjust our practice till we learn the material.

The second type of mistake is the "Aha-Moment Mistake.  This is the mistake that sends the light bulb off when we see how all the parts fit together and then we can do the work or skill.  It is a mistake like calling a friend to wish them a happy birthday.  We got the date right but the wrong month.  In math it might be dividing when we should have multiplied and we learn from that mistake.

The third type of mistake is the "Sloppy Mistake".  The mistake we make because we know what we are doing but we don't pay attention to our work so we do something like 2 x 3 = 5.  We add instead of multiplying. 

The final type of mistake is the "High-Stakes Mistakes".  These are the ones people do not want to make in a life or death situation such as the doctors in ER.  If they make a mistake it could result in death.   For many students, these mistakes occur when taking the SAT, ACT, or other high stakes testing which could make all the difference in being accepted into the college of choice.

The big over all picture is simply that the only way to learn from a mistake is to reflect on the mistake made and figuring out how to learn from it.  Just acknowledging the mistake may not be enough to make corrections.  It requires us to make the decision to change and learn from the mistake.

I wonder how many of us ask our students to think about the mistakes they make.  Ask "What did you do wrong?"  "How can you do this right next time?"  I've done it on tests but not on daily work.  We want them to learn to do it correctly and learning to look at the type of mistakes the students make can help us help them improve.

Algebra A+ app

The other day, I found a nice app on iTunes called Algebra A +.  The basic app is free but if you want more than the complementary problems, you'll have to upgrade by paying $3.99.

This app offers exercises covering addition and subtraction of integers, substituting values into an expression, combining like terms, distributive property, binomial multiplication, factoring a monomial, difference of two squares, factoring by grouping and factoring quadratics. 

When the topic is selected, the page comes up stating this is an easy complementary set.  Each set has between 15 and 25 problems that can be worked.

For each problem, there is a pencil up in the right hand corner that when selected, brings up a white board that covers the right side of the page so you can still see the problem.

The white space allows for the student to work out the problems.  Once the answer if gotten, the student can then type in the solution.  If the solution is correct, a green check mark appears in the left corner.  If its wrong, the correct answer appears in a pop up dialog box.

There are two things I don't like about this.  First it has a number line like keyboard to select the answer.  It took so much trial and error to figure out how to work it.

The other thing I don't like is that it does not show a student how to do the problem if they should get it wrong.

I do like the number of problems a student gets with the free app.  I also like that there are some very good topics to choose from.  This app provides good practice problems for students who want the extra practice without hauling a book around with them.

Interactive Number Lines

For many of my students who are coming up from the lower grades, they do not understand adding or subtracting numbers, especially signed integers.  I'm finding that by putting them on the number line, it helps them learn how signed numbers work.

Since I've been off traveling, my students have been using hand drawn number lines. When I get back to school tomorrow, I want to introduce the idea of a variable and then give them time to finish their other assignments.

Fuel the brain has a nice interactive number line which allows you to set the beginning and ending endpoints and the distance between points be it one or five or some other number.  This makes it so students can choose the part of then number line they need.  The bad thing is that it does not appear to work on the iPads.

Interactive sites at weebly has a page full of number lines including one that is easy to use for adding and subtracting integers. They page has at least 9 different number lines to use but at least one is not working so I never got it up.

Helping with math has a number line generator that is easy to use without printing it off. You can set it up so the distance is counted in ones or fives and you can set it up to add numbering every 1, 5, or what you want. In addition, this website has so many more printable number lines to choose from that there is no reason the students cannot use a number line.

As part of teaching using variables, I'm going to create a number line with the variables and coefficients.  I will be also teaching them that 2x is 2 times x or x + x as a way of relating this to the previous.



By using this number line, I am hoping students will see how like terms are combined.  I plan to let them have two different number lines, one with the variables and one with the constants so they have a chance of seeing that the process for combining both is the same.

Balancing Algebraic Equations

  According to a coworker who just attended a workshop on modeling math in elementary school, he said the presenter recommended having the students work with balances to solve algebraic problems.  I know of two free balancing apps for the iPad but not all classes have iPads.

Math Playground has a lovely algebraic balance activity that helps students learn to solve one and two step equations.  This is a lovely virtual manipulative in that it provides a tutorial session when you first begin the interactive balance.  In addition, it provides immediate feedback on if the student is right or wrong for the step.  This works on the mac but not the iPad.

Hooda Math is another website with an algebraic balance activity.  This balance is easy to use but it works strictly with addition, subtraction, multiplication, or division using positive values.  Furthermore, it requires the student to subtract one x at a time rather than the total x value.  This makes it a bit harder for the student to work the problems.  This balance seems to be for younger students to work with.

Home School Math has a great set of illustrations with explanations that show how to balance equations.  It is not neither a balance nor a manipulative but it does provide a great introduction to students so they can observe what happens before they start using the interactive balances.

Math is Fun has one of the best balances I've seen online.  It starts with the equation already set up on the balance so students see that the equal sign does not necessarily indicate the answer but the two sides are equal.  Furthermore, it has a control that allows the student to add or subtract the x's or constants and they see this effecting both sides at once.   In other words, they see the "if you do it to one side, you must do it to the other side" rule.

If a student gets something like -x = -3 it turns out you have to add the x to both sides and then add 3 to both sides to get 3 = x.  Rather than multiplying or dividing by a negative, this  balance has you continue adding x till the x is on the positive side.  The same applies to the constant.  This is a nice way of showing students a second way to get positive values.

Relevent Math Tasks

  Today at the session on cultural context in Math and performance tasks, one of the presenters talked about creating performance tasks that are relevant to students.  She took time to show how to create the tasks that students will understand.

Many places in Alaska heat their houses or saunas with wood.  So she started with a picture of a cord of wood showing it is 4 x 4 x 8.  Surrounding it were four more pictures, each showing a load of wood but in a different mode of transport such as a boat, a pickup truck, a 4 wheeler with a trailer and a snow machine with a sled. The idea is you include enough information so students can find out that not all cords are the same.

Then she showed us a picture of a water tank from one of the villages she'd visited. There were enough items such as a truck that students can make educated guesses as to the measurements of the tank so the students can calculate the volume. 

The discussion made me realized I did a performance task years ago when the high school created a cross curricular unit.

Step one: I had students read an account of the battle of Dunkirk.  I had them note the number of men moved from France back to the UK. 

Step two:  Determine the number of boats they could use from the village and decide how many people each boat could carry.

Step three: Calculate the total number of people who could be carried by all the boats in one round trip.

Step four:  Calculate the time it would take for a round trip, including 20 min to stop, load, and turn around.

Step five: Calculate how many trips would it take to move all the soldiers from the next village over to our village. 

Step six:  Compare the time it would take for the village to move that many people with the time it took originally.

This performance task used a situation they were familiar with so they could picture the operation mounted back in 1940. It turns a historical event into something students could relate to and understand.

Now I know what to do to turn photos, or historical events into performance tasks.

Mortgage rates

  At some point in math, we teach the topic of interest.  Depending on the class, it might be calculating either simple or compound interest.  Usually, I assign appropriate practice problems but that is as far as it goes. 

Today, while watching TV, I watched several ads dealing with buying houses, credit cards, cars and a variety of other items that charge interest because they are bought on time.

So as part of the unit, why not have students research current interest rates for 15, 20, and 30 year house loans, then create a spread sheet with the monthly payments, amount of interest paid on original loan all based on different interest rates.  Although the monthly payments may be higher for a 15 year loan but the amount of total interest paid is lower. 

Econedlink has a wonderful page with the lesson plan and all the information to create the actual spreadsheet for loan amortization including the links, vocabulary, and the template.  This activity can easily be modified to use current interest rates for different length loans.

The next interest to investigate is associated with credit cards.  There are so many different offers out there that students need to understand how various types of interest works.  This spreadsheet lesson from Boise State has students create spreadsheets to look a fixed rate card, a variable rate card and the amount more they end up paying should they miss a payment.

In addition NCTM Illuminations also offers a lesson on credit card interest which explores how long it would cost to pay off a balance of $200 with 22 percent interest.  The activity comes with both a compound interest simulator and a spread sheet for calculating the cost of being late.  This is a nice activity with everything needed to teach the lesson.

Another large purchase which requires a loan that will accrue interest is when a student decides to purchase a car.  Econedlink has a good unit for teaching this topic but you could easily use parts of it to help students calculate the amount of interest for various interest rates and loan times.

The website Teachersnet has an activity for cars, houses, and loans were students will use spreadsheets to calculate compound interest for buying cars and houses.  This activity has them calculate the interest and loan amortization for their purchases.  This takes things one step further and has them decide if they can afford the house or car of their dreams based on a predetermined monthly income.

Once students learn how to calculate interest using a spreadsheet, it would be easy for them to create one to calculate the cost of interest on a huge furniture purchase with two or three scenarios such as no interest payment for the first 6 months and payment including interest beginning the first month.


Most of the students I work with  so not have any experience with the above purchases so when they move to the city and are faced with all those wonderful opportunities they have no idea what the real cost of anything is.

Planning Ahead

  In about two weeks, our school will be hosting a large basketball tournament, well large for us.  There will be 8 to 10 girls teams and another 8 to 10 boys teams and they either fly in or come by snow machines if they are the next village over.

Usually, we have workdays scheduled but due to a glitch, the workdays ended up a week earlier so we are going to have to have classes during the tournament. 

I usually help out throughout the tournament but this year I'll be teaching.  I may still have to help during the day so I've given some thought to how I'm going to have students complete a lesson and learn something if I have to help out.

So I came up with an idea.  If I have to work during school hours, I am going to give students a worksheet so they can choose a player and keep track of rebounds, steals, 3 point shots, free throws, etc.  I want them to collect the data so I can have them analyze it later.

I'm not sure how many students will actually record all the information. I suspect they will want to enjoy the game but I'm going to make it a 50 or 100 point assignment so they have a reason to collect all the information.

Once the games are over, I plan to have students analyze the data, create charts and prepare a report containing graphs, charts, and a summary of the players abilities based on multiple games.  Once the reports are presented, I want to divide the class into groups of four.  Each group is going to represent owners of a professional basketball team.  They are going to look through the reports to decide which player or players their team wants to recruit. 

They will have to complete a report that explains why they want to recruit a certain player.  This report should include the reasons they chose the player based on their stats.  I work with ELL students and this type of assignment helps them develop their language, learn to apply mathematics to real life and learn to do accurate data collection.

In the past when I've asked students to provide me with a short report on their game stats, I've gotten things like  I made a basket in the first quarter and then I intercepted a pass but didn't get a basket out of it.  So this would provide a perfect opportunity to help students learn more about collecting data during a game.

Peer Teaching

I saw something the other day on YouTube where students prepared the lessons in video form for each other.

Several years ago, I would divide the lesson up into parts and have the students prepare a lesson to share with the class.  Now I have iPads in my classroom so I could easily do this again with a technology twist.

So why not assign part of a topic to groups of students who will prepare a video, slide show, stop motion or other method.  They need the rubric with expectations and the grading scale,  a check list to make sure the material is covered, and time.

The projects should include a lesson with examples so students are able see the math being done.  There might even be guided practice problems or some other way students can practice.

Once the projects are finished, they could be uploaded to a school server so other students can access the material to learn.  There could even be a companion worksheet for notes or writing the examples.

This would allow students to check out the material at home or look at it in class.  The nice thing is that it encourages peer learning, helps develop vocabulary, improves their ability to explain math topics and  it helps increase student ownership.

Another project along these lines would be having students create posters online.  When students are restricted to a certain number of words, they have to decide what the most important ideas are.  The language becomes more precise and mathematical.  I've used posters for vocabulary but it could easily be used to summarize an idea of concept.


Just a couple ideas to encourage digital peer teaching.

Reteaching Integers

Yesterday,  I started reteaching adding and subtracting signed numbers to my Pre-Algebra group.  Most of them failed the test on the topic and it struck me that they didn't really understand the concept associated with this topic.

Most of the time, this topic is taught basically with the idea that if the signs are the same, you add the numbers and the answer has the same sign.  If the signs are different, the numbers are subtracted and the sign of the answer matches the sign of the larger number.

They have difficulty with this idea because they don't understand that the signed numbers represent distance and the signs refer to the direction.  So today, I retaught this concept talking about distance and showing how it works on a number line.

I actually saw a few light bulbs flick on during the lesson.  At the end, each student created a number line they could use to answer some simple problems such as 3 - 8.

I showed them how you move 3 units to the right starting at 0 to represent the first number, then I moved left 8 units showing the -8.   I repeated this demonstration with two or three more problems.  Once they had a better grasp on it, I had them use a number line to practice it.  It was amazing how many students wanted to know if they "had" to use the number line.  Couldn't they just put the answer down.

I pointed out that they kept coming up with the wrong answers so that told me they really didn't understand the process and they needed to number to help them do better.  It was great that no one argued with me about that.  It was wonderful that several students actually accomplished more in the time than they had when I tried teaching it the other way.

As a side note:  When I taught square roots in Algebra I, I drew a picture on the coordinate plane in Quadrant 1 to represent the positive area.  Most students only show the positive values of a square root.  I then drew a square in Quadrant III where you can have two negative values which still results in a positive area.   My students saw the reason why taking a square root results in two answers.

Because of yesterdays lesson, I will be teaching the addition and subtraction of integers using distance rather than trying to teach it the usual way.

Stations and Centers part 3.

I went through my book stash and discovered three books on using stations in the classroom.   I'd forgotten I had them.

They are put out by Walsh Education and are geared to meet the  Common Core State Standards. Each book comes with a list of the standards covered by each activity, an introduction which covers everything needed to implement the activities to debriefing the students.  There is also a materials list.

Each activity is designed for four stations and each station has a different activity. The first few pages are the teachers guide with answers, information, etc and then the second set of sheets needed for each station.

Algebra I covers matrices, literal equations, ratios and proportions, linear equations, inequalities, polynomials, systems of equations, solving quadratic equations, functions, and statistics. 

I looked at the ratios and proportions activity.  Station one has students matching cards that have the proper conversion such as 12 inches = 1 foot.  The 12 inches is on one card and the 1 foot is on a different card.  After matching the cards, students answer additional questions and practice converting units.  The second station has students convert temperatures, find area and perimeter.  Station 3 looks at the topic of probability using marbles while station 4 uses tiles to create ratios.  The four stations have students apply ratios and proportions to a variety of situations.

Geometry covers congruence, similarity, right triangles, trigonometry, circles, and geometric sequences.  The congruence section includes activities on traversals, triangles, bisectors, medians, altitudes, and 2 dimensional shapes.

Algebra II has activities on quadratics, functions including piece-wise functions, inverse and logarithmic functions,  conics and probability.

I wish I'd thought about using stations earlier in the year but I still have half the year so I can implement some of these activities.  I am getting ready to teach 2 dimensional figures in geometry and quadratics in different ways to both algebra 1 and 2.

Centers or Work Stations Part 2.

Yesterday I started a piece on using centers or work stations in the high school classroom because it encourages independent student learning.

Too many of my students want to have only the right answer and don't feel they can move on till I check their work.  In addition, I have basketball players who may be gone most of a week so they can travel and play in tournaments.

I thought something like this might  allow me to work with the students while the others are traveling.  I think this might be a solution to help bring up several of the lower preforming students.

 Write Solutions has a nice quick piece on using stations in the classroom.  They suggest using red, yellow, or green folded papers as a way for the student to let the teacher know they either need help because they are stuck, sort of understand, or have it under control.

In addition, it is possible to access materials to set stations up including signs, planning sheets, etc.  Materials that make it easier for a teacher to implement stations in the classroom.

Although this second site has only a short piece on stations, the author includes the idea of using assessment data to help divide groups up into groups and to assign stations.  She shares her experience using stations and provides some very good suggestions.

I Speak Math also offers some wonderful information for creating and using stations in the middle school classroom but the information could easily be applied to high school classrooms.  She discusses how she sets up several stations with two problems each beginning with the easiest problems.  The students are assigned to stations based on how well they understand the material so the students who start with station one, really need the most help.

Her rule is that as each student finishes the problems at a station, they bring their answers to her to be checked and if the answers are correct, the students move on to the next station.  If the student is incorrect, they work with the teacher and then try again.  She also has each student carry an index card with them because if they work with the teacher, they can write the concept down on the card to help them.  She even includes downloadables.

You tube has two videos showing the stations being used in the classroom.  The first one is about using digital stations in math.  It is only 4 minutes long but the teacher actually talks about each station and what the students do.  It is cool.  The other is on using math stations in the middle school.  Its from the students point of view and is awesome.  Sometimes she has students create short items for the iPads which other students use to help learn.  The teacher calls it reciprocal teaching.

Today, I discovered I have three different books on using stations in the Algebra and Geometry classroom.  I'm going to look at those and review them tomorrow.  I'm off to read so look for part 3.

Centers or Work Stations pt 1.

  I wonder why we do not use centers or work stations in high school.   The centers are used successfully in elementary but by middle school they are phased out in favor of whole class teaching.

I see these centers as a way of arranging small group differentiated instruction for many students and its also perfect to use with technology.

Since I am unfamiliar with centers, I did some research on the topic.  Straight off, I discovered a nice site that give full instructions for using them in the algebra classroom along with a sample of 6 centers.  The examples are for solving one step equations with whole, fractions and decimal leading coefficient.

The suggestions given for using stations are:
1.  Have one station per four students.
2.  Determine the length of time it should take students to complete each activity so you know about how long they should be at each station.
3. Create the activity be it a worksheet, web based quiz/activity, etc.
4. Create the answer sheets.
5. Set up the stations.
6. Run the stations and include a timer.
7. Signal when it is time to change to another station
8. Collect the results at the end.

According to another site, there is a difference between the centers and stations.  The center is a place for students to practice or refine a skill while the station is for students to work on tasks at the same time and whose tasks are linked.  Students using centers rotate while those using stations do not and the teacher chooses which station to assign the students to based on need.

The nice thing about stations, is the teacher can have a station so she works with some of the students while the others are working independently on specific material.  Math stations are better for differentiating instruction and students only need visit the stations whose activities help them move towards proficiency.

It is suggested the teacher have an anchor station with activities for students who finish early, get stuck, or do not need to work on the skill.  At the end there is a power point presentation and a suggestion on what a day might look like when using stations.

Even the Teaching Channel has a lovely 5 minute video on using stations.  It shows one of the teachers using this to help his students so you can see it being used.

I found three more lovely articles on the topics of stations and centers that will appear in tomorrow's blog.  Keep an eye open for part two.

Quadratics and Negative Roots

  I've been working on helping my students improve their math literacy and today, I encountered another situation where we choose the answer depending on the situational context.

We were discussing the square root of a number in terms of solving an equation.  I explained in this problem, we need both roots for the solution.

One of the students asked about x^2 in terms of a room.  He wanted to know why we only used the positive root.  This lead to a discussion on rooms using only positive measurements.  So then the question came up when do you use the negative square root?

I had to stop and think a moment.  I know its often used in the standard deviation but I had to do a web search to really answer that question.  I posted my dilemma to Google + Mathematics Education group for help with this.  One of the members pointed out that the negative roots are important in quadratics which lead me to realize that negative numbers are important for parabolic shapes such as mirrors and satellite dishes. 

Then it struck me, the negative is not a subtraction but a direction away from the center of the parabola.  It all relates back to the context of the math.  So now when I introduce my unit on  quadratics, I can reaffirm the negative is a direction. 

I did a quick search of quadratics and found they are applied to the following:

1. Calculating area - You can use it to find the measurement of a room or to find out what your room measurement might be if you know you enlarged the room or made it smaller and knew the area.

2. Finding the rise and fall of profit for a business.

3. Finding the speed of something that has wind or current acting on it.  

4.  Certain parts of vectors.

5. Acceleration.

6. Stopping distance.

7. Air movement.

8.  Aiming missles and rockets.

There are lots more examples but this is a good start.  

Number Lines In High School

  I just tested my Pre-algebra class on integers and they are still struggling with negative signs.  Based on the answers I checked, the kids see the problems in one of two ways.
A.)  They do not recognize the negative sign with a negative number so they treat everything as positive.
B. ) They do not understand what the integer calculations represent.

With this, I am going to reteach it next week before I introduce variables.  I am seriously considering having the students create a number line that resembles a slide rule so they can put the window on the first number and then slide it left or right based on the second number.

I found an activity for making a slider using a zip lock bag but I'm afraid it looks too  "elementary".  I want it to resemble a tool so they won't feel as if they are being treated as elementary students.  I did a search for instructions so that on Monday my students can make their own.

I looked at several images and finally found one that looks like it will work.

 I cut a shoe buckle looking  slider out of an index card.  This will allow students to create the number line they need.  Once they've made the appropriate number line and they can thread it through the slider.  They put the first number in the middle of the slider and then move the strip left or right the appropriate number of places.




You can see how it might work in the picture below.

 I am going to have students make these on Monday so they can use these with a worksheet of problems.  I'll let you know how it goes.

Just in case, I did find a web site with an interactive  number line where you input the ends of the number line and it automatically creates a number line which you can then use to solve a problem. You can choose to count by any number be it one or 20.   Unfortunately, it requires flash so it cannot be used on an iPad. You could use computers.

Reviewing For The Final

  Today, I ran a review in a couple of math classes for the final semester test tomorrow.  It turned out well. The students had a blast and they were totally involved in it.

Step 1:  I helped student create notes for each part of the final so they know the exact material covered, including examples and notes on specific things to watch for.  The notes have the arrows, the stars, everything so they are ready for the final note wise.

Step 2:  Give the students a set amount of time to create a set number of questions per topic and they have to include the answers.  I have them write one question on the front of a 3 by 5 card with the answer on the bottom.  So at the beginning of the activity, I pass out 8 to 10 cards so they have something to write their questions on and I have the cards to read.

Step 3:  I assign students to groups and hand out some blank paper while they get their notes and pencils together.  I ask each group a question and they have a certain amount of time to get the answer.  If the answer are correct, they get a point.  If they are wrong, they lose a point and its available to the others to answer.

It was amazing.  The students got totally involved.  Everyone worked the problems, helped each other and had answers just in case someone didn't have the correct answer.  Sometimes, a student would say, "I have this", give the answer and the others would yell out "Not right" before I had a chance to say anything.

This was awesome because they wrote the questions.  Each student put their name on the card so I wouldn't call on them to answer their own question.  I could hear the students helping each other, explaining how to do it.

This saved me a lot of prep work and each student was forced to check their notes, look at the material, create and answer problems.  I think it worked well as a review. 

Yeah!

Resources from New Zealand

  During one of my searches for best practices, I came across some really nice resources in New Zealand called New Zealand Maths.  The site is actually for teachers in New Zealand but it has some lovely materials that can be used in different grades.

First is the resource finder which has a searchable data base using curricular levels, numeracy, or Pact aspects.
Although there is a section on lesson planning, it requires you be logged into the site to use everything.

The third section is on Numbers and Algebra.  The number part covers strategies, knowledge and a section for the teacher discussing the sequence progressions.  Although many of these activities are for elementary, there are some good ones to use for differentiation and scaffolding.  I checked one out on square roots and cube roots.  It includes a process to see the relationship between squares and square roots, cubes and cube roots.

There are also sections for Geometry, Measurement, and Statistics.  It has the same type of activities for each topic.  I like that there is a section on problem solving and a separate section with rich learning activities.  The rich learning activities provide a context for the math while the problem solving activities helps reinforce both the concept and problem solving

"Take This" are short activity which help springboard students into the mathematical idea and they take something with them.  For instance, for one that uses a cafeteria menu, they include a variety of activities according to strand.  In geometry, they are required to design packaging for the foods while in algebra they are creating problems using the order of operations.

They go so far as to  include a list of picture books for earlier grades that deal with mathematics and some suggestions even have the associated lesson plans.  The first suggested book under geometry middle elementary is "A Cloak for the Dreamer" by Aileen Friedman and published by Scholastic.   This could easily be used in middle school with students who are lower performing.  It could also be used by older students who could go into a lower elementary class, read it and help run the lesson.

So many possibilities.

Pythagorean Theorem. When Is The Best Time To Teach It?

  Today, I reviewed the distance formula in preparation for the semester final.  It got me to thinking when in the semester should I teach the Pythagorean theorem?

If I teach it at the beginning of the year when I do midpoint and distance, then it is quite applicable for distance. 

If I wait till later in the semester when I'm doing trig ratio's then I've missed the chance to relate the distance formula to it. 

I like to teach the Pythagorean theorem just after I teach classification of triangles, congruent and similar triangles because I have students use the  equal, less than or greater than to tell the type of triangle based only on measurements.

I love the theorem because it has so many possible applications from vectors to televisions to physics and it is good for them to know the basic formula.  Perhaps, this needs to be taught at different points throughout the geometry class with different applications so students see the formula as the course progresess.

Some real life applications include:
1. Road trips - finding the shortest route.
2. Painting buildings to help find the right sized ladder.
3. TV's and Computer Monitors
4. Navigation.
5. Surveying.

So if I consider that it is better to teach the theorem at several points throughout the course and I include the appropriate real life examples, then it might help students learn to use the theorem better and become familiar with it outside of the theoretical state.

If you have any suggestions, I'd love to hear from you.

$28 camera

  I was reading one of my google groups and someone posted this cool article on a $28 digital camera.  The camera looks like a retro camera but is only 6 mm thick and made of paper.

You read that correctly, its made out of paper but according to the report, it does so many things.  It can take pictures, videos, and voice recordings.

It comes with an on/off switch, different modes, a 16 GB micro SD card slot, an LED light signalling which mode is active, and a USB cord for transferring videos, photos,, and voice recordings plus the cord can be used to charge the built in battery.   It is so cool.  Just think of the possibilities for a school.

Imagine buying 10 cameras for your classroom for a  cost of $280. There are possible uses for the math classroom.

1.  I could send students out into the village to take photos of real life math examples such as intersections, perpendicular bisectors, right angles, or so many other things.  When the students are down, they return to the classroom and download the photos and create a video complete with voice over.

2.  Students could record each other reading a report on a mathematician,  poetry they created,  ir perhaps even a newscast on real life examples of math seen every day.

3.  What about creating a video inwhich they teach a lesson or explain how to do something mathematical.  Once the video is recorded it could be downloaded and edited.

4.  Create a commercial to sell a mathematical concept with all the information on it.  Perhaps even showing the real life uses of the it.

The advantage to this camera is the low cost so if the camera is lost or damaged, it is not a big loss.  In addition, it offers so much for the price.  The SD micro card is going to cost around $8 to 10 to buy but still the whole cost is quite low.

The disadvantage is that the photo and video quality is not necessarily the best but that would be a trade off. The biggest disadvantage is that it is only being sold in Japan at the moment.  Chances are if it is imported to the United States, the cost will be increased but that may be a while yet.  It is always nice to dream and drool.

Free lesson plans

  I am always spending time looking for information of how to best integrate technology into the classroom.  I find lists of good apps to use but seldom do they actually provide the way to use it in the classroom.

I came across a nice article from Education World on integrating technology into the classroom.   There were suggestions for all the classrooms but I liked the one on mathematics which focused on providing several different sites for mathematics problems.  One thing they suggested was to use one as the problem of the week or month, extra credit or to increase interest in doing their seat work. 

This site had a pop-up to a lesson plan site that offers free lesson plans.  I am always on the lookout for ideas since I tend to default to the lecture/notes style which is what I trained in.  I searched the site for math lessons and found the following:

1. Follow the directions quiz which is a way of helping the students learn to read  directions before taking a quiz.  I love throwing this type of quiz because there is only one question that has to be answered.  I took one back in middle school that had us standing up, turning around, etc and we all groaned when we discovered all we had to do was write "I love math" and our names.  Ever since then I"ve read directions.

2. Conducting a probability experiment and record the data directly into a spread sheet so students can produce a report on the data.  The report could include answers to questions on the activity and allow students to create a conclusion.

3.  Students research triangles after completing a short brainstorming session and direct instruction.  They have a list of questions to help them find theorems etc so they know what is true about triangles.  This is a neat way to introduce the geometric theorems for triangles.

4.  Creating floor plans using Excel.  Although this is rated for grades 4/5, I can use it in my geometry class.  I have a spread sheet on my iPad and I want to play with this at home to see if it will work on the app.  I could have students calculate the area of a house they design.  One of my favorite geometry projects is to have students create their own room from concept to calculating the cost of finishing the room.

5. The last lesson I checked out was "What's a polygon?" activity which is a great introduction to the topic in geometry.  It comes with everything needed to teach it.

Many of the lessons can be used in more than mathematics and has cross curricular applications.  So it is possible to teach the lesson in two different topics such as art and math so students see the relation to both topics.

Trigonometric Golf

  Due to the basketball team traveling my Geometry class was much smaller than normal.  I found a lovey trig game called Trig Golf on the Math Interactive site in Alberta Canada.  Unfortunately, it does not work on the iPad.

So I projected it on my Smart Board so students could see the game.  In order to hit the ball to the next spot on the mini-golf course, students have to answer a set of questions designed to have them find Sin R, Cos R, or Tan R.

For each question, I assigned a finger value of 1, 2 or 3 such as 1 = opposite, 2 = adjacent, 3 = hypotenuse or 1 = Sin, 2 = Cos or 3 = Tan.  It was cool.  I'd show the question, read it out, give them a min or two to think about it and discuss it before asking for a show of fingers.

It was cool.  they really got into it and they were discussing things out loud and holding up fingers or calling out the answer without any prompting.   They loved it so much. As soon as we finished one hole and I shut it down so they could complete some work, they wanted to play another round.  I told them we'd do it Monday. 

It was an awesome use of the materials and the several students had lightbulbs go off in their heads.  That was great.

Teaching Quadratics

  I am getting ready to teach quadratics in about a week and a half.  I"m looking for new ways to teach it so I don't do the same old same old and maybe make it more interesting.

I have two different apps on the iPads I can have students use to practice factoring but I'd like to do something other than teach it in my standard way.  So I did a web search and came across a few nice ideas.

First is Tic Tac Toe: Quadratic Factoring which uses a Tic-Tac-Toe grid to help factor quadratics.  It comes with a complete lesson plan to use and includes all of the necessary papers.  I am not familiar with this method but it looks like it is worth learning and sharing with students.

I believe that it is important to share several ways of doing anything so students find a method that works for them.  Back to the lesson, it includes objectives, a powerpoint to present the method, guided practice sheets and independent practice sheets along with both formative and summnative assessments.

I have to teach completing the square so again I did a search and ended up at the same place as before looking at a Hip to be (completing the ) square.  I only know of one way to do it and it is long and tedious so I needed help finding material that might make it a bit more exciting.  This has a way I've never done it before.  It actually uses a 2 by 2 square and might actually be easier than the way I learned.  Cool.

In fact, I did a search of the cpalms site and there were 188 lessons that popped up when I typed quadratics in the search.  188 complete lessons!  The lessons cover transformations, graphing,  the easy vertex form, real life applications and all sorts of other aspects of quadratics.

I found a lovely Dan Meyers basketball task introduce quadratic/parabolas to the class.  This looks like  a cool hook to use to get the students interested.  This came from a a website that has ideas for introducing the topic, including the vocabulary.  I love the idea of borrowing other people's suggestions to use in my own classroom.

This site, Better Lessons,  has a nice entry and exit ticket in addition to ideas on the order in which to teach the topic.  It has a lovely video with instructions on doing a think, pair, share to discuss the ideas from the video.  It has everything needed to teach this lesson.

Ok, I am set, now all I have to do is sit down and plan. 

What Does X Mean?

  Due to the way a student did a problem I realized that she didn't understand the context of the letter x in the math problem.  This lead to today's entry on this topic.

The students were carefully calculating the surface area of various sized boxes.  The dimensions were listed such as 24 x 1 x 1 meaning 24 by 1 by 1.  So when I spoke about multiplying the area for each surface, instead of writing 2 x 24 x 1 meaning 2 times 24 times 1 she wrote 2 by 24 by 1 and multiplied the numbers together. 

When I asked her about it, she indicated she saw the x meaning by for dimensions as meaning times.  I've used x in both situations as needed and never realized that some students, especially ELL students, may not understand the different contexts as easily as I might.

I stopped and thought about the meaning of X in the math classroom and realized it could be a variable, it could mean times, or it could mean by  as in a 3 dimensional shape or even refer to a matrix.  Furthermore, it really isn't until middle school or high school that students switch from using x to indicate multiplication to using * or parenthesis. 

I get quite a few freshmen who have not been required to switch from x to * and they have difficulties when working with  something like 2 times x because they instinctively write 2xx which to me is 2x^2.  They also have difficulty with 2x meaning 2 times a number.

One way to cut down on the issue is to introduce the idea of using variables down in elementary and perhaps even get kids to change from using x to * when they want to show multiplication.  If this is started down in elementary and reinforced in middle school, I might get students who are aware of the differences in meanings and can recognize the contextual differences.

This is part of being mathematically literate.

Trigonometry Activities

  I am teaching basic trigonometric ratios to my geometry class because its now part of the yearly test.  I just introduced the ratios but I wanted to find a couple of activities that will allow students to see some real uses for it.

So after a quick search I found a couple activities that look like fun and show some decent applications of trig.

The first activity I found is one from Teach Engineering which has students use trig to figure out the width of a river.  The lesson comes complete with everything needed from the learning objectives, to all the worksheets needed, ending with assessments and extensions.  This site is created by the University of Colorado.

I wanted to know if there were other activities from this site I could use in my classroom, so I put trigonometry into the search engine and came up with 102 additional activities that use trig.  Wow, 102 activities that look cool.

1.  There is a hands on activity that uses the Lego Mindstorms Nxt technology to find the actual height of a triangle after the student calculates the height using trig.
2.  Designing a spectroscopy mission which requires the understanding of trig and is done after they build the spectroscopy.
3. Solar Angles and Tracking systems so people can create the proper arrangement of photo voltaic cells for solar power.
4. Flying in Style which uses trig to find the height of the rocket that students built prior to actually doing this activity.

Online Math Learning which has some nice interactive games for students to play.   I checked out the trigonometric ratios game to see how it worked.  It requires people to set up the appropriate ratio for the angle given.  Its a good way for students to reinforce the ratios.

Transum has a whole page of trigonometric activities including one that requires students to come up with the exact values of the ratios from the unit circle. I like the one called "Which Side?" because it requires the students to look at a drawing and classify it as opposite, adjacent, or hypotenuse based on where the arrow is pointing.  This is great for the beginning part of the unit and it works on the iPads.

The Learn Alberta site has a couple nice interactive activities for trig.  The exploring trig ratios has several different parts to it.  The use it section is a mini-golf game where the student answers questions about the triangle in a step by step manner to earn the right to swing the club.  The student works their way through the course until they are done.  This is a nice game to help reinforce the basics.

I plan to uses several of these activities in my geometry class over the next few days.  I'm always thrilled to find activities based on real life or are technologicaly based that I can easily integrate into my classroom.  Check the sites out if you need things for trig.

The Internet

  The internet has been a boon to educators.  It allows students to find information that might not be available in the school library.  It allows teachers to find the latest research on teaching.  It provides so many activities and worksheets that it saves on the district budget.

I use the internet so much in my class. For instance, I show short videos on the topic most every day because my students pay more attention to a video than my lecture.  I can show a video the first day to start building a foundation and then show a different video each day so more pieces of the topic makes sense until it clicks in their head.

I also look for worksheets I can have students use.  I love using worksheets for guided practice and I can find the worksheets for almost everything I need.  Furthermore, I can find worksheets filled with various levels of problems.  If I want, I can even put the students on to take an online quiz which will give them immediate feedback.

I love being able to find activities that fit the topic I'm teaching.  I've found lesson plans, whole units, videos, etc so I don't have to do much planning.  All of this for free!

I can research the best ways to present material to ELL students and find help on line on ways to increase vocabulary development.  There are webinars out there to watch to help me become a better teacher.  There is the teaching channel which has lots of awesome information for teachers who want to be better.

I've come across the futures channel which shows short videos on how math is used in real life and provides worksheets to go with each lesson.  This is cool because I can integrate it into the lesson and help answer the question "When will I ever use this?"

I can find math activities put out by various sports leagues that are based on the math they use in basketball, football, or even baseball.  I can find activities for designing a house, calculating paychecks, finding the best deal, calculating the markdown on a sale item. 

When you live out in the middle of nowhere, it is cool being able to access information to help in class.  One day, years ago when I was teaching science, I had to figure out how to fill the burners so we could heat things.  My dad, who taught till he was older, suggested I use mimeograph fluid.  It actually worked and the school still had a couple cans in storage. 

Now everything I want is available without having to order everything in.  I do know that much of the material out there can be questionable but I can assign research for my students to find statistics from good websites.  This leads to the discussion of what constitutes a good vs a bad website. 

Even today,  I used the internet to find worksheets for teaching, videos, and even the standardized test questions.  I love, love, love it.

Why Am I Teaching These Fractions

  This morning I had an epiphany about fractions.  Why do we teach students fractions with denominators of 5, 7, 9, 11 or possibly even 12.  I tried to think of an occasion I used any of those in my life and couldn't think of a single possibility.  None.  You might use those when you gamble but the context is different.

In the past, I had students make those strips showing equivalent fractions such as 2/2 or 7/7 = 1.  I've had kids adding fractions with denominators of 33 or 75. 

In reality, why don't we just teach fractions using the fractions we run across in real life.  For instance fractions with denominators of 2, 3, 4, 8, 16, and 32 as those are the most common ones we use.  I might even include 10 and 20 for money as there are 10 dimes in a dollar and 1 dime is 1/10th of a dollar or ten cents.

I do see the point of teaching equivalent fractions, of adding, subtracting, multiplying and dividing fractions but do we have to include pages of fractions that may be filled with denominators they will never see in normal life?  In fact, why am I teaching them to multiply or divide mixed numbers by mixed numbers when most of the time we use whole numbers.

Yes, I see that they need to manage all sorts of numbers in a theoretical situation but in reality when will they use it?

All of the situations I can think of only use a limited numbers of fractions.
1. Gas is sold by in increments of 10th or 100ths with a price that is usually a dollar amount with 9/10 at the end.
2.  Recipes use denominators of 2, 3, 4, or 8.  If we increase or decrease a recipe we do it by whole numbers such as double or triple.
3.  Hand tools are usually have sizes with denominators of 8, 16, or 32.
4.  Distance that is listed on roadside signs usually have denominators of 2 or 4 while mileage signs between wholes might be in tenths.
5.  Building materials are in often bought with denominators of 2 or 4.

I could not think of a single example that used some of the odd numbers of 5, 7, 9, 11, 13 etc except for the fifth of liquor.  So I still wonder, why am I spending all this time to teach students to use fractions with denominators that they will never see or use in their lifetimes. 

If anyone has suggestions on where these types of fractions would be used, please let me know.

Math Games

  Sometimes we have occasions where we want students to collaborate or work in groups rather than be working independently.  Unfortunately, most apps are designed to be played individually or if students can work together, they must use  a central game site which may or may not be a place you want your students.

So occasionally, it is good to organize some of the older types of games which have students actually interacting with each other.

Although Learning With Math Games is actually geared for upper elementary and middle school, there are some lovely games that can be used to scaffold instruction for high school students.  I found a lovely game which helps students practice their use of integers. It requires a specially constructed deck for each person in the pair.  Students add two cards together to form a sum and the one with the largest sum gets all four cards.  Another game uses a regular card deck to practice exponents.

East Carolina University created created several Math Middle School Energizers which combines physical activity with math.  One activity I discovered combines line dance with integers. Just think, take a 5 min physical break with a bit of music and holding up integer cards that tell students the number of steps to move left or right.

Another site, Fun Maths have several nice games for use in the classroom.  I looked at the algebra egg race which came with the playing board and directions, everything needed to have students practice substituting values into algebraic expressions.  Another activity requires students to spend exactly $1,000,000.  To show they've spent the money, they have to create a presentation with pictures, prices, and a sheet showing the prices of what they bought.  If you wanted, you could require students to create the presentation digitally or create a hard copy.

Finally is Bright Hub education with a nice collection of games ranging from "Who wants to be a millionaire" to a proportion relay which requires students to move, to a fraction game.  I find that many of my lower performing students are weak in the basics and all of these sites have activities which can be used to strengthen these skills.

Happy New Year