Improving Algebraic Understanding

 I have been at a loss figuring out how to help my students learn and retain their ability to understand and work out Algebraic problems.  One day, they can do it, the next they act like they are reading Greek.

The other night I came across an article focusing on improving algebraic understanding in middle school and high school.

The paper has three major suggestions for ways to improve algebraic understanding and one of the ways mentioned, I've never actually heard but it makes sense.  After reading it, it made so much sense but I also know I have to teach my students how to do it.

The first area to concentrate on is analyzing solved problems that show all the steps.  The idea is they look at the steps and discuss them to find the reasoning and strategies used.  In other words, they are finding connections.  When they find these connections, it makes it easier for them to transfer the information.

Furthermore, the selected problems should apply to the objective of the lesson and some of the examples should include common mistakes so students learn to watch out for them and to look for them in their own work.  This should be done using whole groups, small groups, and independent practice to teach them how to do this.

One big reason for showing solved problems is it allows students to see the the strategy used in total context rather than each step.  When they discuss the reasoning behind the steps, it helps students develop a deeper understanding of the logic involved in solving the problem.  Looking at incorrectly done problems help students learn to think critically.

The second area is to teach students about mathematical structure and algebraic representations by using proper language, teaching them a self reflective method they can use to notice structure as they solve the problems and help them understand that different algebraic representations provide information about the problem.

This process helps students connections between problems, strategies, and representations that at first may seem different yet turn out to be the same.  When they understand the structure of problems, they have time to focus more on the similarities between problems.  In addition, if they understand what makes an algebraic expression an algebraic expression, they will see the connections regardless of how they are presented.

The last area is to work with students to choose from several strategies to solve problems.  It is suggested students make a list of strategies they can use to solve a problem, explain why they chose the strategy, and they need to evaluate a variety of strategies they could use to solve the problem.

Its important to teach students that strategies are different from algorithms.  Algorithms are a set of steps one always follows to get a specific result where as a strategy requires students to make a choice based on the problem and the end goal.  Getting them to move past the algorithm means they can move past memorization to true understanding.

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

Universal Design Learning in Math

  We are always looking for ways to improve our teaching so students learn the material better. One way to do that is to utilize Universal Design for Learning or UDL when planning lessons.  UDL allows for more flexible ways to teach lessons and more ways for students to show they've learned the material.

Behind the UDL is that when used, it can reach all students.  There are three basic principals UDL is built on.  First one must provide multiple ways of representation, second provide multiple ways of actions, and third provide multiple ways of engagement.

The teaching practices that meet UDL design includes video captioning, multiple online resources, observation, organize your board, and use multiple forms of assessment.  Some of the recommended ways to organize your board are to cover the board left to right, underline important information, recap the previous day's material, and always include the outline of a schedule for that day's lesson.  My schedule usually looks something like warm-up, notes, practice problems, and a game.  Sometimes instead of notes, I might use a video or even use a video for practice problems.

One reason for making sure the video captioning is on is to help students focus on the material instead of just watching it.  I know for myself, I always put the closed captioning on so I can read the what is being said in addition to listening to it.  I find, I pay more attention to what is going on.  It also allows hearing impaired students a chance to follow the videos without feeling as if they are different.

As far a online resources go, there are so many sites out there filled with video's, tools, examples, quizzes, visual aides to help students.  Some of these can be assigned to help students do the work at home.  There are multiple types of online calculators students can use to check their work for various types of problems.

When. observing students, watch for facial expressions indicating their confusion, ask questions, make them feel as if you care about how well they do.  Not every student is going to ask for help because they might be too shy, too insecure, or too lost.

So you may be wondering what a UDL lesson plan should look like.  The teacher should anticipate student needs when building the lesson plan.  When creating the lesson, build in differentiation so the need of each student is met.  The lesson itself should be filled with well supported activities which engage students in a safe learning environment.  Include multiple ways to learn and show knowledge and skills.  Assess student learning and adjust often.

Some ways to increase engagement include using alternate seating, offering choice boards, giving brain breaks, think-pair-share, peer tutoring, timers, and exit tickets.  Some ways to represent the mathematical concept would be a Three Act Math activity, visual cuing, text to speech options, manipulatives, word walls, or flash cards.  Suggested ways to show what they know might be using online tools, acting something out, record a video, use manipulatives, make an oral presentation, or think-pair-share.

This is just an introduction to give people an idea of UDL and its application in math but I'm planning on addressing this topic later on in more detail.  Thank you for reading this.  Let me know what you think, I'd love to hear.  Have a great day.

The Mathematics of Game Shows

My mother loves watching Wheel of Fortune while my father is a Jeopardy lover.  Both play the game from the comfort of their chairs hoping they beat the contestant with the correct answer.  I figure its good for their brains because it exercises the brains and keeps them active.  When I watch, I think of the odds each player has of winning the game.

I'd love to teach a statistics class focused on the odds of winning various television games, especially the more popular ones. Most every game on television is based on the statistics of spinning the right space or choosing the right letter or number, or space.  All game shows are based on probability.

If you've ever seen Let's Make a Deal, you know they have one activity in which a person has a choice of selecting one door for a prize.  If they get the goat, it's over.  So the person chooses a door and the MC shows the goat  behind one of the other doors, indicating that one of the two left has the prize.  The person is then given the choice of keeping the door they chose or switch to the other one.  Most people stay with their choice because they feel they have a 50% chance of being right but mathematics indicates, it is better to switch because it increases your odds of winning.

I found a paper that sprinkles game show probabilities with information on regular probability. Its actually notes from a class but I like the way the author discusses various games throughout the paper.  The first two shows he discusses are Deal or No Deal, and the Price is Right with a short summary of each game.

As he introduces probability beginning with the basics such as sample space, followed by  expectations, counting, poker, inference, competition, backwards induction, and special topics, he sprinkles bits about various game shows and how certain parts apply to the topic.

For instance, when he introduces sample spaces, he discusses the squeeze play in the Price is Right.  In this game, you remove one digit, usually the first or last, to find the correct price.  He shows how the sample space applies to this particular situation.

As the author discusses various topics, he continues to apply probability to each situation which makes it more interesting because its not usual examples.  Back to the Price is Right and its 10 chances game.  The small prize has a two digit price but the contestant is given three digits,  the medium prize has a three digit price but given four digits and the big prize is made up of five digit price with no extra digits.  The object is to guess the correct price to win as many prizes as possible.

The author analyzes the probability of getting all the prizes and how to figure out the most likely prices based on mathematics.  The cool thing is that every example he sites, he's included a video clip so you can actually watch the activity he's talking about.

This 146 page paper is actually class notes form a class taught on The Mathematics of Game Shows from 2018.  I had a blast reading it but I plan to go back and read it in more detail.  Let me know what you think, I'd love to hear.  Have a great day.

Game Templets

 I try to integrate games into my classroom as I can because my students would rather play a game than do anything else.  I think part of it may be due to the fact they all play a variety of games on their devices outside of school.

I've been using Kahoot and Jeopardy a lot but there are other games I can use.  The other day, I stumbled across a list of templets for a variety of games I could easily use in the classroom.

I"m sharing sites you can make power point type versions of several different game shows.

One of my favorite sites for finding Jeopardy games is at JeopardyLabs.com where I could also make a game if I chose but they have tons of done ones.  On the other hand, you can use power point templets to create Jeopardy games on your computer.  This site lists 12 places you can go to find templets to create your own games including one that uses Google Slides.

To shake things up, how about a family feud game for a change.  It uses two teams with 5 people on each side.  The question is asked and people answer.  It is possible to make one for math but you need the templet to create a game.  If you check here, you'll find 6 sites which will allow you to create one for your math class.  Each is a bit different but they all work about the same.  The one game I saw referred to as Family Feud, had students answer questions and show their work in order to get the point.

If neither Jeopardy or Family Feud is something you like, you could try Wheel of Fortune.  I realize it works mostly using words but it could be used with mathematical vocabulary.  This one is considered quite realistic while this one is definitely a power point presentation set up as a Wheel of Fortune Game.  I think it would be a great way to practice vocabulary, operation words, even mathematicians.

I've often wondered how one could use The Price is Right in the classroom but it's taken me a while to figure that out.  Its a great avenue for having students practice cost per ounce for various items.  Rather than giving a product to students to guess the price, why not provide two versions of the same product and have students calculate which one is the better buy?  This site offers a power point with 8 slides that could easily be edited. This one on the other hand, is simple and has only one set of slides but you can easily make copies.

If you've ever seen the Cash Cab show, there is even a templet for creating one of your own.  The templet is set up for 12 questions which could easily be math questions of any level.  Students need to work through the problems before they come up with an answer.

Last but not least is the "Who Wants to be a Millionaire" with three different templets.  The first has space for 12 questions, easily edited.  This Google slides version is ready to be edited for your topic while this one comes with music, lifeline, etc.  The questions can be set up from easy to harder to hardest which is the way normal games are set up.

So all these templets are downloadable and you can create multiple games to use in your class.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

Mars is 140 million miles from the Earth.  If it takes 3888 hours to fly to Mars, how fast is the space ship flying per hour?

Warm-up

The moon is about 238,900 miles from the earth. It takes four days to get to the moon. What is the rate of travel per hour?

Matching in Math?

Note: Sorry, today's column is running a bit late but due to attending a conference, traveling back in time for Parent Teacher Conference, I didn't get things done ahead of time.

Back to the regularly scheduled column.   Last night, my mind woke me up in the middle of the night thinking about matching games based on the concentration model.  The one where a person chooses two cards to see if they match.  If they match, the person gets the pair, if not they are flipped over for the other person to try.

The conference I just finished attending, said that sometimes you need to unplug and do things without digital devices.  These can be done using index cards and pens.  You can have students prepare sets so that you do not have to make them.

1.  Vocabulary - Prepare two cards per word.  The first card has the word while the second card has the definition.
           Make a set with the vocabulary word and an example
           Make one with the vocabulary word and a picture,

2.  Word sentences - Prepare two cards per word, the first card has the written sentence while  the second card has the algebraic or arithmetic equation.
           This can be done the reverse way from the equation to the word sentence.

3.  GCF or LCF.  The first card has a pair of numbers while the second card has the GCF or LCM of the two numbers.  You just have to make sure none of them repeat so they do not share answers.

4. Process - The first card can have the equation while the second card has the next step to solve the equation.  Its quite possible to set it up so it takes a couple rounds to solve the equation from start to finish.  If an equation takes four steps to finish, it requires the student to find two matching pairs.

5.  Geometry - the first card has the shape while  the second card has the the name of the shape.
             The first card has the shape while the second card has the formula for area, volume, or perimeter.
             The first card has a description of the shape while the second has a picture.
             The first card has a picture of angles and the second card has the name or vice versa.

6.  Decimals, Fractions, and Percents.  The first card has a decimal and the second a fraction or the first a fraction with the second being a decimal, or a decimal and a percent.

So many possibilities to create games.  If you would rather use digital devices, there are some websites you can go to to create actual games.

This site allows you to create your own matching game but they show both options at once you just have to match them from a crowd of choices.  They also have a gallery of games made by others so you don't always have to begin from scratch.  It is free and allows one to sign in through your google I.D.  It also gives you a code to let people access the game.

This is a site that makes the game more of a concentration with cards facedown.  You can create for free but it costs if you want to download it to your computer to print. 

Both look easy to put together. The only reason, I'm more into the cards rather than the digital for this game is that I can have students talk a bit more rather than focusing on their devices.  Next week, where do you find templets for other games and google home use in the classroom.

Have a great day and let me know what you think.

Alexa? What is 2 + 3?

I made it home safely from the technology conference.  My flight for the last stage ended up being delayed 2 hours but I got in on the only plane of the day.

I am glad I went to the technology conference because the last keynote speaker turned me onto something I didn't know about.

Did you know that people have written lesson plans for all topics using the Alexa?  I didn't until he shared it with us. I'll start with normal uses and save the lesson plans for another day.

To begin with,  Alexa can be used to set a timer for those activities run for a specific length.  Instead of physically setting a timer, you can ask Alexa to do it and you can focus on other things.

Instead of keeping multiple pairs of dice in the classroom, you can ask Alexa to roll the dice for any activity requiring the use of dice.  Students can keep track of the response and then analyze the data to see if the numbers ended up meeting the 1/6 probability.  The dice results can also be used to create math problems for students to complete.

In addition, Alexa can do addition, subtraction, multiplication, and division.  For students who struggle with math fluency,  they can use Alexa to check their work after completing a worksheet.You can also use the Alexa to find data such as temperatures in another place such as the daily high and low in London.  Students can take the data, organize it into charts, or even compare it with historical data to see if it follows past trends.

Amazon has skills which others have written for math.  Many of the skills already written and free to download to your Alexa.  Some are easy fact type skills and others are more complex but there are also skills to help factor quadratics, learn more about the Pythagorean theorem and the Fibonacci theorem.


You can also go to Alexa Skill Builder which allows you to go and create your own Alexa program.  They have the basics done, all you do is fill in the blanks and then publish it.  For me, the quiz one under education is perfect because it allows you to create short questions, provide an answer, add in a small fact.

You can also create a flash briefing which can be used by students to catch up on what they missed when they are out sick.  You can also create flash cards so students can practice their multiplication and division problems.  Work on the AC method so students learn the possibilities when factoring trinomials.

For students who need music, Alexa can be used to play nice background music gently in the background.

I'll be revisiting this topic another day, once I've caught up on my sleep.  Let me know what you think, I'd love to hear.  Have a great day.

Heading Home

Timelines in Math

 One of the workshops I attended gave a list of websites which can easily be used in classes and of course are free.  Teachers love free and are always searching for apps that meet the basic criteria of easy to use, free, and is a great tool.

One of the areas of apps that always appears in these types of presentations are timelines.  Everyone knows timelines are easier to use in certain subjects such as Social Studies, History, or English.  Even Science can find uses for timelines but math is much harder.

I've never thought about using it in math until I realized that we no longer teach students any mathematical history.  When I was in school, teachers took time to discuss the mathematicians who contributed to the development of this topic.  In fact, most of my teachers  hung pictures of mathematicians around the room so we knew the names.  Now, we don't.  My students don't know who Decartes is or any one else who contributed to the field.

I propose, we build in time to explore the history of mathematics the people who contributed to it, or the development of various branches.  Perhaps look at the women who made contribution to the field so girls see that mathematics is not strictly male dominated.

It wouldn't be hard to a create timeline for the life of a mathematician giving information on their birth, death, marriage, schooling, and accomplishments. This type of assignment requires research to find the information needed for the time line.  If its an interactive time line, it could include a map with their place of birth, death, where they lived, etc , a picture of the person,  and any books they published.

Students could create a time line showing the development of Boolean Algebra, Topography, Logic, or other branch of mathematics.  A student could even create a timeline of the development of numbers.

Of course, one could have the class create a timeline of the history of math so each student gets a segment of time and when they are done, they've completed a timeline from  50,000 BC to the present.  The timeline can include mathematicians, their important theorems, etc so the final project is interesting rather than dry with only dates.

Today's timeline software allows people to create products that are more interactive, contain more informations, pictures, maps, rather than just dates and one line descriptions.  If we start adding this into classes, perhaps can show that its been around for a long time and developed to explain the world around us.

I think, I'll be doing this in class when we have a short week and can't do much or during testing when students are tired and need a change.  Let me know what you think, I'd love to hear. Have a great day.

Lightbulb went off on X-Y-Z axis

 I came to Anchorage to attend and present at the Alaska State Technology in Education conference.  I love going because I always learn something new every time I attend.

The first day of the conference, they offered several three hour workshops so I chose the one on 3 D printing.  The speaker took us from start to finish including the fact they us a 3 dimensional axis for printing.

The speaker explained the x axis as going left and right, the y axis is front and back, while the z axis is the height.  I've never had this grid explained this way and it made so much sense. Much more than theoretical drawn grids.  In addition, since three dimensional grids are extremely difficult to visualize or draw, this explanation puts a context on it and when drawing on an app or graphing website, it makes it easier for them to construct the graph.

It also makes it easier to create three dimensional shapes on autocad or similar software because they know where to place the base or height.  I can see the pyramid in my mind as I draw it since I got that application and his explanation gave me what I needed to "see" the process.

  It also throws a different perspective on the regular x and y axis. We teach it as horizontal and vertical but it seems to the the left and right for x is still the same but why not express the y as front and back so it can be explained in terms of maps.

I don't know if they still have those books filled with maps for cities.  The ones with the numbers and letters to tell you what area the street is in but it operates on a flat x and y grid except one is numbers and one is letters.  I'm also thinking of the maps found in the middle of telephone books.

I've often explained the coordinate system using directions such as north, south, east, and west because that is something my students relate to.  My example puts the hotel at (0,0) and the coordinates give directions to the local McDonalds, or the store, or where ever because they need a way of understanding the coordinates are a location.

I realize there are situations where we want the x and y axis to run horizontally and vertically depending on the situation or context. Now I can communicate these contexts to my students and perhaps widen their knowledge base.

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

Warm-up

Warm-up

The Montgomery Alabama Bus Boycott

Most math teachers are looking for ways to make math more interesting and relatable to students.  Teachers also look at ways to incorporate other subjects into math.  So the other day, Dr Who gave me an idea of a topic that would be perfect in both Social Studies and Math.

We know that Rosa Parks refused to give up her seat when requested to move so there'd be another row for "Whites".  We know this lead to a community boycott of the bus, and a couple of court cases ending bus segregation in Montgomery Alabama but what do we know of the economic repercussions of this event?

The boycott lasted 381 days, just over 1 year.  The boycott did some economic damage by denying the city income from transportation because 75% of the ridership were African American.  Due to the decreased ridership, many drivers lost jobs because fewer buses ran.  It is estimated that about half of the 44,000 African American residents used the bus daily.

The transit company lost 30,000 to 40,000 fares every single day amounting to at a loss of at least  $3000 per day since the cost was only ten cents per ride.  Instead of riding the bus people carpooled, took rolling taxi's run by churches, or walked up to 8 miles to work.  Overall, revenue for the bus company dropped 69% leading to drivers being laid off and routes being cut. By the end of the boycott the Montgomery city lines lost $750,000

Over 300 cars were used to transport people around Montgomery during the boycott so these cars increased the purchase of gas and oil as they were running more often.  This meant gas stations started to make more money but the cars underwent increased wear and tear.

There is enough information here for students to put it all together.

1.  If the bus company lost $3000 every day, that means its based on 30,000 fares at ten cents per day.
2.  Again based on losing $3000 per day with a total loss of $750,000 that means it accounts for 250 days out of the total 381 days.  What might account for the differences between the 250 days and the 381?
3.  If they lost $3000 per day and that represented 69% of their normal income, how much did they normally make?
4.  If only half of the 44,000 people rode the bus normally, what might explain the loss of $3000 per day?

There are other questions which could be asked but I'd let students see what mathematical questions they could come up with from all the information provided in this entry.  Then I'd share the four questions I've provided.

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

Happy Valentine's Day

Improving Questioning in the Math Class

  As a teacher, its easy to get in the habit of answering all student questions rather than taking time to determine if answering the question is the best solution.  I need to take time to think about the question being asked and my response.  I'm going to share something I discovered in regard to questions and answering them.

Let's look at the questions so many students love asking the teacher.  "Is this right?"  In a way its asking for immediate feedback but it also means that many students rely on the teacher's response rather than checking the answer themselves.  It is quite possible they rely too much on the teacher responding and not enough on learning the process or concept themselves. 

When a child asks this, it might be better to respond with "Are you sure?" weather its right or wrong.  At first students may panic but they will eventually get used to it and might start responding with their thinking.  This can lead to the next question of "How do you know?".  This question requires them to explain their thinking.  Often as they state their logic out loud, mistakes will become obvious and they can fix it.

If you write two equations on the board, then ask "What do you notice?" it gives students a chance to comment on similarities, differences,  and helps build understanding of variations in procedures in addition to assessing their understanding.  If students have limited English, you might ask "What's the same?" or "What's different?" which is a more specific set of directions.

Another way of making students think is to have them convince you of something such as all multiples of 8 are multiples of 16 or taking a square root is the opposite of squaring a number.  This activity helps students develop generalization of mathematical ideas.  You could also ask students if there is another way to work a problem because it shows students there is more than one way to do a problem.  It also allows them to see they don't have to do it like everyone else.

If you want to improve generalizations, help develop reasoning skills, or deepen understanding, ask questions that are NEVER, SOMETIMES or ALWAYS True.  A question like "Is it always, never, or sometimes true that the four angles in a quadrilateral add up to 360?  This makes them think about all situations before answering.

These are a few ways to make questioning more effective in the classroom.  Let me know what you think, I'd love to hear.  Have a great day.

4 Ways to Make Math Class More Interesting.

One thing about math, is that students often arrive to class saying "They hate math." or "They aren't good at math" and some have already given up, convinced they can't do it.  One way to help them get past that attitude is to make class more interesting.  If you can capture their interest, they are willing to work.

One way is to create a hook to real them in much like a fisherman who uses a fly or bait that all fish find attractive.  "Teach Like A Pirate" by Dave Burgess has a book filled with tons of suggestions for creating hooks to grab student attention.  I've used some of the suggestions to create ones for math.

Another way is to relate the math being taught to student's hobbies or interests.  For instance, relate discounts to shopping because most people are into saving money as often as possible.  Or surface area to buying paint and tiles for redecorating a bedroom.  I've assigned a project in the past where students designed their dream bedrooms complete with movie theaters, basketball courts, swimming pools etc and then they calculated how much it would cost to finish it using a list of prices I provided.  Students loved this one because they loved the idea of making their bedroom the size of the classroom or bigger.

Of course, as the teacher you can change up the order in which class is taught.  Instead of lecturing right after the warm-up, do some sort of activity that allows them to explore the current topic before the lecture.  Throw in a game to stir things up.  That way, they don't get stuck in a routine of same old same old. 

Another suggestion is to give students a choice on how they will show they've learned the material.  Will you let your rapper create a rap on the different types of quadrilaterals?  Let the video guy make a movie of how linear equations are used in real life?  Will the budding classic musician be allowed to explain the time signatures in terms of mathematics? 

If students are given choices, make sure the choices stays true to the goal of the lesson.  All choices must have the same end goal.  Before giving students the list of choices, its important to make sure they have a good understanding of the topic and its concepts.  When setting up choices, the choices should all be rigorous and you should know how long each option will take.  They should all take about the same amount of time.

Choice can be offered via a menu, a choice board, a Think tac toe board. If you wanted to offer something a bit different, off students the option of making a game of some sort to practice the topic when played.

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

Change in Plans

Due to the limited bandwidth at the school, we've had to discontinue all computer based programs which is making it challenging.  As a result, the department heads had to figure out how the alternative students will finish their classes.

The majority of these kids are there because they have difficulty in regular class and honestly, the majority of them have not been doing well with these computer based classes.

As a response, we've put together units which we hope students will find interesting.  The majority of students taking math were enrolled in a 6th or 7th grade math.  I believe the idea was for them to get through a math class so they could graduate but few have finished more than 5% of the class.

 Most of the teachers are hoping students will do better if they can choose what they want to study.  I spent Saturday putting four different units together for 15 students.  The units only contain the first week of work but I have more to organize for the next two to three weeks.

One unit covers aviation math which will look at simple vectors, converting C to F, calculating distance, speed, and time, calculating amount of fuel needed, ground speed, take off and landing, and a few other topics.  The first week has them reading a general introduction to the topic so they have a better idea of what the math is for each item.  They will do a worksheet on vectors, converting C to F, and calculating airspeed. 

Another unit is on carpentry math.  The first assignment is focused on learning about angles, Pythagorean theorem, and roof angles.  There are four reading assignments with problems, and identifying angles as right, acute, or obtuse.  Carpentry is a skill they can use out here in the village so its something they might be interested in.

Of course, I threw in a Cryptography Unit because I think its fun.  They begin by working some simple codes using the Caesar shift. The next activity explores this shift in even more detail followed by the reading and trying to work a worksheet introducing each type of common cipher.  Each week, they'll do more coding and breaking codes while learning about various codes in detail.

The final unit is on Forensic Math.  For the first week, they will use ratios to calculate a perpetrators height based on their foot length and they have to identify if the person walked, jogged, or ran away.  The last assignment will deal with bullet trajectory.  Next week they will be introduced to the math of blood splatter.  Over the next few weeks, they'll calculate how long since the person died, learn more about calculating height from the bones, and if the victim is male or female etc.

The hope is that these students will do more math if it is not taught in the traditional way and if its something that will peak their interest.  The science teacher is offering a forensic unit from a science point of view which is why I chose to look at the math. We are hoping students might get interested and do the work.

I'd love to get feedback from people on this idea.  These students don't work in regular class. They are absent or late to class to much and only seem to come to school for the socialization.  We hope this sparks their desire to work.  Have a great day.

Warm-up

Warm-up

No! NoTo Timed Tests!

  The other day, our principal gave his report on the workshops he attended at the last RTI conference.  In one breath he told us he'd been told that its not good to do timed multiplication drill tests and then said we need to use those to teach the kids their tables.

Unfortunately, the belief exists,  the faster a student is at completing their addition or multiplication tables, the better they are.  It sends the message that faster means you are more intelligent but that is not always true.

I've seen it manifest itself in high school to the point that students race to finish an assignment as fast as they can but they don't take the care to make sure their work is correctly done.  When these students finish their assignment in the shortest amount of time, it makes others feel as if they are not good in mathematics even though they make fewer mistakes.

The other day I gave my Algebra II class a test.  The first two tests were turned in well before the period ended.  I glanced at the tests, promptly handing them back before telling those students they needed to check their work because I'd seen quite a few mistakes.  These students checked their work and found several errors but I've had other students tell me their work was fine and they didn't need to check it.

There is concern that timing these types of test may create anxiety in the person taking it which in turn can cause them to forget the facts they've already learned.  Furthermore, this type of testing may interfere with their learning and it does  not necessarily give a true picture of who knows their facts.  Some students love competition and want to be the first done but that is not true of everyone.  Those students who are not competitive may know all their facts but are unable to answer problems in this situation.

There is some evidence out there indicating that timed fact fluency tests may actually cause students to learn their facts slower. create a life long anxiety and  make it harder for them in the future.  The ability to do your facts in under one minute does not indicate a person can apply the knowledge in other situations. 

There are other ways to test for fluency without involving timing.  One way might be to "interview" students by asking them what "5 x 6" mean?  "What is the answer to 5 x 6?"  "How did you find the answer to 5 x 6?"  "Could you have found the answer another way?" "If your friend has trouble remembering this, what might you suggest to help them learn the fact?"  Another way might be to ask them things like "Can you use 3 x 8 to help find the answer to 6 x 8?

These questions require more thought to answer and they require more understanding of the concept.  Timed test are great for checking for rote but they do not check for understanding and comprehension of the multiplication.  I believe it is important to know the facts but you also have to know what they mean and how to use them.  I am working on my students to improve their fact fluency through games etc rather than using timed fact fluency tests.

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

Comparing and Contrasting in Math.

  There are so many times I let an opportunity pass by for students to compare and contrast in Math.  They see this exercise as one you only use in English class but it is one that works well in Math.

Compare and contrast offers so many advantages to math students in terms of learning the topic. This technique helps develop higher order thinking skills, improves their memory of the material, it improves their understanding, helps them organize new material, and it helps develop their minds.  

Currently, I am using this in Geometry.  I have students doing four different compare and contrasts with the notes they took on five quadrilaterals.  I've asked them to do one for rhombus and squares, kites and trapezoids, rectangles and squares and rectangles and trapezoids.

They are finding it challenging because they prefer to just copy notes into their notebooks rather than trying to make sense of the notes.  I like the way its working out because it is making students talk to each other as they work in groups trying to find all the similarities and differences between each set of figures.

This is not the only place compare and contrast can be used.  I've discovered several other ways which can be used in different levels of high school and middle schools.

1.  Compare and contrast long division and synthetic division.

2.  Compare and contrast different ways of multiplying binomials.

3.  Compare and contract GCF's and LCM's.

4. Compare and contrast regular fractions and algebraic fractions.

5. Compare and contrast decimals with place value.

6.  Compare and contrast student methods used to solve a problem.

7. Compare and contrast the ways to solve systems of equations.

8. Compare and contrast solving systems of equations and matrix.

There are so many other possibilities for using comparison and contrast in the math class.   Let me know what you think, I'd love to hear.  Have a great day.

Yahtzee and Math

Computer based testing just started and that means that only the computers in the Library and Computer lab are allowed to be used.  All other computers and digital devices cannot so I have to plan alternative activities for my computer based credit recover class.

Most of the students in that class have tried the regular classroom but cannot hack it so they are here.  Some are trying to make it while others don't do much.  So I have to prepare lessons for students as if its a regular class.

So on Tuesday, I had to do something that didn't involve a computer.  I printed off two worksheets to practice order of operations and two of the three who made it, did quite well.  They listened to me, they worked through the problems doing it step by step until the sheet was completed. 

Rather than have them do another worksheet, I broke out the Yahtzee game for them to play.  I figured it would strengthen their adding and multiplication skills, have them predict what they needed to get a small or large straight, full house, or other combination.  The boys helped each other, discussed what would be better for the other to aim for and had a good time communicating.

Later in the week, I think I'll teach a probability lesson using Yahtzee and dice.  I found this activity which has pairs of students roll two dice until they each roll a six on one die.  They have to write down the numbers they rolled each roll and the number of rolls they made.  On the whiteboard, students will write down the total number of rolls it took to get the six.  At this point, have the class calculate the average number of rolls to get a six.  It should be about 6 rolls.

Repeat the experiment, only have each student roll both dice to see how long it takes to roll two sixes at once.  They need to write down the combination each time the dice are rolled and keep track of the roll it is.  Have students write the total number of rolls it took to get double sixes on the board.  Once all the data is on, they need to find the average of rolls which should be around 36. 

At this point, its time to have a discussion with the students to see if they can figure out the mathematical probability of rolling a six with one die (1/6) and the probability of rolling double sixes (1/36 or 1/6 * 1/6).  Its possible to take this even further.  If you want to know all the probabilities for three of a kind, four of a kind, a full house, etc check here because they have the probabilities with explanations and graphs.

Speaking of graphs, one can take the number of rolls and arrangement of the die or dice and create a histographs or other graph from the completed data collection.  I am going to try this later in the week and we'll see how it goes.  Let me know what you think, I'd love to hear.  have a great day.

Using Origami to Teach Math.

Origami is so much easier to include in classes since people have created apps that show step by step how to fold different animals, and objects. In the past, I've given students my iPad with an app on it, the paper and let them loose. There is usually one student who takes the lead and instructs everyone else step by step until the item is completed. 

Origami in and of itself teaches the following skills.

1.  When making a fish, or other origami item, a student has to follow directions in the proper order to get the final product.

2. The directions involve manipulation of paper spatially  because the final product is three dimensional and if the paper is not folded properly,  with regard to its spatial position, it may not come out correctly.

3. Many of the origami figures begin the same way but it is the way it is finished that determines the final product.  So the origami offers variations from one basic set of instructions, much like many mathematical procedures are based off a set of instructions which can be applied to different situations with a bit of adjustment.

4. Origami is made by following a set of directions which forms the basis of communications.  If the directions are not communicated well, the student might not be able to finish it properly.  The young lady who took the lead in my class communicated the directions to everyone else so they'd get the correct finished product.

5. It is also student centered, requires cooperation as students work together to follow directions, and its a form of applied mathematics.

Origami is one of those fun activities where you can divide the students into several groups.  Each group is giving one figure to learn to create so later in the class, students go to other groups to teach others to create the same figure.  This allows students to practice communications and vocabulary because its easy to incorporate geometric vocabulary into the activity in addition to general vocabulary.

Furthermore, you can have students create directions for the origami figure they learned so others can attempt to follow these directions.  The directions should include both words and diagrams.  This gives them a chance to work on communications.

Other things you can do is have students identify types of angles, types of triangles, or geometric shapes they see as they are folding their creation.  This helps them see geometry is a part of origami and their knowledge is transferable.

So if you want to incorporate origami into your mathematics course, you now have the justification to use it.  Let me know what you think, I'd love to hear.  Have a great day.

Pattern Block Use In MIddle School and HIgh School Math.

Every school I've taught at, my math room has had a set of pattern blocks. The science teacher just borrowed my set because she wants to work with the 9th graders learn to graph and these blocks offer an easy opportunity.

One way to do this is to hand students a handful of pattern pieces which they sort and note the number of each type of block.  Using the data they are able to create a bar graph. 

In addition, pattern pieces can be used in geometry to help students name the shapes, find the equation for area and for perimeter.  The pieces can also be used to practice finding lines of symmetry.  You can also have students use certain pieces to create squares, parallelograms,  pentagons, etc instead of using tangrams.

Of course, pattern pieces can be used to illustrate fractions because the pieces are set up so students can see that the rhombus is 1/3rd of a hexagon, a triangle is 1/6 of a hexagon, while a trapezoid is 1/2 of a hexagon.   Furthermore, students can determine how many triangles in a rhombus or the number of  triangles in a trapezoid.  It gives students the chance to understand that fractions are not always slices of pizza, either circular or square. 

Students can use pattern blocks to create their own version of which one does not belong or odd man out so that one is different. Rather than having the teacher set them up, let students do it because it allows them to develop critical thinking skills to look deeper than just color or length of sides.

Another activity for students is to create clues so a person can place the shapes in the correct order from one to four.  The clues might be "The shape with four equal sides but with no 90 degree angles is in space 3. This activity helps students think logically.

Furthermore, pattern blocks placed on a coordinate plane can be used to help students learn about transformations such as translations, rotation, and symmetry.  Students can mark the points of a pieces vertex in the original place and the new place, then discuss the translation in terms of unit and direction.  Rotation and symmetry would work the same way.

These are just a few ways pattern blocks can be used in middle school or high school.  Let me know what you think, I'd love to hear.  Have a great dayl.

Warm-up

Warm-up

The Importance of Mathematics.

  I love challenging my students when they tell me there is no real reason to learn math because they'll never use it.  I ask them to name one profession that does not use math.

I'd get guesses from rock star to grave digger.  For every job, I'd point out the math needed.  I pointed out to the guy who wanted to be a rock star that he needed to know math so he could make sure his manager was not ripping him off.  To the grave digger suggestion, I pointed out that graves had to be dug to a certain depth, have a certain length and width so the coffins could fit in.  One young person finally commented that he thought the only way you could escape math was to be dead.

There are so many reasons one needs math in life.  Its been found that high school students who take demanding classes in high school including math courses beyond Algebra II had a much better chance of graduating from college.

In addition, math is good for the development of the brain.  Certain studies indicate that students who know math are able to access certain areas of the brain more reliably and those areas have more gray matter.  These same areas appear to be associated with decision making and visual attention.

Although digital time pieces are on the rise, there are still quite a few analog clocks around.  My school is filled with them but studies have found that more and more children are unable to read the time on analog clocks.  Analog clocks use fractions as part of the process of reading them because you have quarter hours, half hours, and whole hours.

Unfortunately with the rise of credit card use, so many students do not get the opportunity to learn about finances. They don't know the math involved in paying down or paying off a credit card, versus the deposits made into a bank account.  They don't know how to plan a budget, figure out how much they've spent on things, etc.  Last semester, several students took a basic finance class which included students learning to balance a check book with deposits, service charges, etc and they struggled because most come from homes where people cash checks as soon as they get them and use cash around the village.

On the other hand, if you know math, you can expand or cut recipes easily so the final product turns out well. Its not easy for many students to half or double 1 2/3 cup or 1 tblsp.   In addition, math helps us with problem solving because it helps us learn to think analytically and improves our reasoning.  All those "stupid" word problems (I'm quoting my kids) actually do help people identify known and unknown while applying strategies to solving problems or situations.

As stated above, you'll find math in every job out there from stocking shelves, calculating taxes, to commission, to everything.  There are jobs out there that require people to take basic math tests and certain long term care insurance policies that test you on your mathematical ability as part of testing prospective insurees.

In addition, when NASA launched space craft to outer space looking for life, the language they chose was mathematics because it's considered a universal language.  After all, the process and written format is the same no matter the language spoken.

Its always nice to explain why math is important to students who only see it as drudgery.  Let me know what you think, I'd love to hear.