Mathematical conversations.

Thanks to Josh Fisher of the Mathematical Education google plus group for this.  The NCTM has a lovely article on using mathematical conversations to transform the algebra class.  This article offers three suggested conversations starters, each one designed to explore a different aspect of math.  In addition, the author discusses the answers and how most students respond to these questions.


The first conversation deals with the equal sign. I love the way the author set it up so students can discuss what the equal sign represents and opens up the discussion to interpreting the problem.

The second conversation is on representations and the meaning that people put with the representation while the third deals with mathematical language.

I have been working on increasing mathematical conversation in my classroom but my teachers training never actually covered either conversation or writing in the math classroom.  Since I work with ELL students, I've had to create my own activities with little or no help so when I see articles like this, I am thrilled.

I really enjoyed reading  this article because it gives me a starting point for this fall when school begins again in August. It is my opinion that if students can verbally express their thinking process, they develop a better understanding of mathematics.

I would recommend you head over to check out the article, read it and include these three conversations into your math classes because it makes the students use higher level thought processes that are important and it helps pinpoint or eliminate misconceptions.

How many in this?

I like to put a drawing up and have the students count how many total shapes are found within the shape. 

I usually start with a 4 by 4 square and have the students count the total number of squares.  Usually they say 17 because they see 16 - 1 by 1 squares and 1 large square.  They forget the 2 by 2 and 3 by 3 squares.

The next time I do it, I have them look at a larger square such as a 5 by 5 or 6 by 6.  Once they have this down, I move to a simple triangle.

If you start with a triangle such the one to the left, it is much easier for them to count.  I sometimes will put a triangle  and ask them how many quadrilaterals do you see?  Be sure to break down the types of quadrilaterals.  There are trapezoids, parallelograms, and rhombus.  This type of problem can challenge them.

I also like to put triangles like the one to the left with multiple different sized triangles within triangles.  This can take a while to do and students tend to naturally collaborate to make sure they have all the triangles.  Sometimes I will take this a step further and ask them "How do you know you found all the triangles?"  This questions requires them to give an explanation of the process they went through

AFactorTree

I stumbled across AFactorTree by Walter Kissach. Although designed for the iPhone, it works quite well on the iPad.  In fact, it is only found under iPhone apps but not iPad so it can be difficult to find.

This app is one I would use for scaffolding, differentiation and review because it is designed to give small chunks of information followed by examples and a special type of calculator. 

The app uses a roller type list at the bottom so people can choose the topic they need to review.  The photo shows the information on factoring trees.  If you flick the list on the bottom up one, the topic will change to the building factor trees using the common approach.

The picture on the right shows what a student would see if they select building factor trees.  It sets it up so the student knows that it requires 2 numbers. 

If the student flicks to walk through, the answer is given so the student can check their answer before moving to the next problem.

In addition, there are two factor trees which will factor any number the student inputs.  You can change the digits or change the whole number but the app automatically provides a factor tree.  I can see this being used by the student to check their work.

This is not all the app covers.  It also covers the least common multiples either using prime factorization or listing multiples, greatest common divisor, adding fractions, and signed numbers.  This app is one that could easily be used throughout a pre-algebra class to reinforce, review, scaffold, differentiate instruction or to have students check their work. 

Best of all, it is free.  There are several other free apps in this series that I shall review over the next few weeks. 

Common Core Nuggets: Quadratic Zeros

This app is another one in the common core nuggets series.  It specifically works on helping students factor quadratics to find zeros or start with the zeros to find the equation. The app has three parts, each focusing on a slightly different skill to meet the standard.

The directions are rather sketchy but with a bit of play, I figured out how to work each section.

The first section focuses on finding the zeros of a quadratic with a leading coefficient of 1.  The app gives you an equations such as f(x) = x^2 + 4x + 3. You are expected to factor it, set the factors equal to zero and find the zeros.  Then you move the green sliders around till they are on the proper numbers on the number line.  Hit the red show/next button to show where the actual graph is.  If you are correct, the green dots and the graph line up.  If not, you see where they are they should be.



The second section focuses on finding the zeros of a quadratic equation with a leading coefficient of something other than 1. You follow the same process as in part 1 looking for the zeros, set the green dots and check.  If you are correct, the green dots match up with the graph.

The third part gives you the zeros and you figure out the equation that goes with the zeros.  You are expected to put in the values for a, b, c from the equation you calculated and if its right, the graph goes through the two points, if you are wrong, it just gives you the next two points.  It does not show you where the graph should be.

As I stated earlier, the directions on how to use the app were rather sparse but I used the play around with it until you figure it out.  Much like what I tell my kids when I can't answer a question on how to do something in an app. Overall I do like it and plan to use it.

Solving one step equation video from keynote

Yesterday, I shared a "writing prompt" I created to use with students as a way of checking where they are in solving one step problems.  Today I have the video that I made and would follow the writing prompt from yesterday.

Again I used Keynote to create it.  I chose to use stars instead of X's to show the process is the same. I have one I could have posted with X's.






These are the steps.
1. Bring up Keynote.
2. Create the title page
3. Type in the equation.
4. Duplicate the slide and add the next step.
5. Repeat step 4 until its all done.
6. Automate transitions but do not put any fancy transitions in it.
7. Record the sound
8. Export as a video.

I've been playing with things like this so I can model what can be done in math with various programs.  They've had basic computers but they still have trouble figuring out how to keynote to create something like this.

Posters and other ways to grap student attention

I was reading something today about making the class hour into things that are short bits so we don't loose a students attention.  So I did some playing around with Keynote and created this problem I could import into my Smartboard presentation.

This has a visual to catch their interest with the problem at the bottom.  I created the slide and simply exported it as an image when I was finished.  I think the students might find this way a bit more appealing than straight words.

I wondered about other things I could do to engage student interest so they want to work.  The video  was created using Keynote and exported as a quicktime movie.  I plan to use this as a "writing" prompt.

I could import this short video into my Smartboard presentation and show it to the students just before I have them do a writing assignment.  I've tried regular prompts but my students have problems writing using a prompt on the board.  I'm hoping that something like this will inspire them to write.

Let me know what you think.  Tomorrow I'll be including another quick-time video that I did on solving one-step equations and uses the equation from this prompt.

World War II Math

Since this is Memorial Day, I thought I would share the details of the math assignments I created for a cross curricular unit three or four years ago.  To introduce the unit, we had the students assemble in the gym first thing in the morning, right after breakfast.  There were several desks in the gym and the tech guy set the air raid siren off.

The students got quiet fast and then started laughing because the teachers were going under the desk, just like students would have in World War II.  The kids laughed watching the wrestling coach struggle to fit under a desk.

We introduced the World War II unit and shared with the students what they would be doing in each class. The history class would cover the war, the English class would read the appropriate novel and in Math, I had three one week topics planned that focused on World War II.

The first week focused on the Japanese attack of Pearl Harbor.  Students calculated how long it took the aircraft carriers to travel to get close enough to launch the fighters that bombed Pearl Harbor.  They had to figure out how fast the fighters were traveling to cover the distance.  In addition, they figured out how close the submarines got that launched the mini subs around the same time.  They calculated the approximate volume of the mini subs, how long it took them to get to pearl harbor, etc.

The second week focused on Japanese balloons that traveled to the United States and killed a few people in Oregon.  The students calculated how much paper each balloon required and the amount of paper required for all the balloons.  They also researched the distance between Japan and four places the balloons landed, the average speed of the jet stream to calculate how long the balloons took to get to each location.  In addition, they had to figure out the total amount of sand required and other materials used to create the balloons.

The final week focused on Dunkirk.  I had the students read the account, calculate how many people were being moved every hour, figure out how many boats they had and the average number of people per boat.  They understood that some boats carried more people and some carried fewer.  Since the village is on a river, not too far from the coast, I had students figure out how many boats were owned by people in the village, the approximate number each boat could carry if squished and where they would have to go to equal the distance as the people did back in World War II.  Their final calculations required them to figure out how long it would take them to evacuate the same number of people if they could work 24 hours a day.  They had to include the time for the return trip so they knew how long it would take for a full round trip.

The unit finished with a dance held in the cafeteria.  We dug up some 40's music.  One of the teachers knew how to do that type of dancing and held dance lessons right after school.  We dressed the cafeteria up a bit so it sort of resembled a U.S.O. canteen and had fun.  Just before the end of the day, the dance contest was held and there was a small cash prize for the winners who turned out to be two guys, one with a broken arm in a cast, who decided to do it together as a lark.

Although this is a technology blog, I thought I would share something different.  To add more technology to the above unit, I would provide a list of websites for students to find the information they need to do the calculations.  This activity was done my first year there back in 2006 when there was much less technology available.

Final Note:  Thank you to everyone who has served to keep this country free.

Algebalance

Algebalance is a cute little app by David Collins designed to help students learn to balance equations using little aliens and weights rather than using variables and constants.  The idea behind the app is that you balance the creature with the weight to send the alien home.

The aliens are always small but in different shapes and colors.  As the student hits a new situation, the professor gives hints on how to solve the problem.  Then you remove weight from either side or move the creature to the other side so you have a balanced equation.


This is an app that helps students practice the process of solving one, two and multi-step equations in a fun way.  The pictures you see here are from level 10 out of 25 levels.

The first level actually sets up the scenario and introduces you to the Vumble (the alien) and the concept of balancing by providing balanced equations.  As the student progresses through the levels, the student begins to move things around to balance the equation and the requirements get more complex.

This is an app, I plan to utilize with my pre-algebra and algebra students next year.  It will introduce the process to my pre-algebra students and help scaffold my lower performing algebra one students.

New from Center for Algebraic Thinking

While looking up more information on Flabido and Action Grapher apps, I stumbled across a pdf filled with teaching guides and student worksheets for many of their apps.  I know that the Center for Algebraic Thinking is selling the information for $5.99 but I ran across the pdf file using a regular search engine.


They also have the student worksheets with the information  for  doing the activity and it tells  which app to use.

Each activity explains what to do, some require the student to record data and form conclusions based on the data.  While others ask the student to record their results and then asks what did they have trouble with on the level.  I like that these activities require the students to do analysis on what they are having trouble with and learn to verbalize their thoughts and problems. 

Since it is in pdf format, I can take what I need, upload it and the students can use the pdf file and then return it so I can grade their work.  This is a good way for students to develop literacy in mathematics.  I plan to check back regularly to see what new things they offer.  This pdf is new since I last checked their website.

Flabido vs Action Grapher

Flabido

Action Grapher is put out by the Center for Algebraic Thinking while Flabido is put out by both Shodor and the Center for Algebraic Thinking.  From what I can tell they are both the same app with the same three games.  The big difference for me, is simply that Flabido would only show one orientation on my ipad and I couldn't get it changed.  I could not see the buttons so I could not do the activity.

Action Grapher

Both have three games in the app.  The first is where there is a flask and the student chooses the correct graph that would show the rate at which the flask is filled.  If you get it right it shows as green, otherwise it is red.  In addition, you can select animate so you can see the graph actually being drawn as the flask is filled.

The second game is the bike rider.  If you animate the rider, three graphs (speed, elevation and distance) are created.  The student can watch the graphs to see how the three correlate to rider's actions.
I find it rather cool.

The final section is called Doodle.  The idea behind the part of game that I played was to watch the movement and then draw the line for the x and for the y.  Once you've drawn the lines, you can check your work to see how close you were.

I really like the app because it shows some very real applications of linear graphing. It allows the students to work at their own pace and get to work on transferring the theory into practical.  Tomorrow I'll share something fantastic I found to help use these apps and others from the Center for Algebraic Thinking.

Common Core Nuggets: Building funcitions

This morning, I ran across several apps designed to meet common core standards.  The one I looked at today is Common Core Nuggets: Building Functions (F-BF/3)by Saltire Software.  The app covers vertical and horizontal shifts, vertical and horizontal dilation for linear, quadratic, absolute value, exponential, logarithmic, and polynomial functions.

When you click on the transformation you want, it goes to a screen with the appropriate general equation such as f(x) + k.  Underneath the graph is a slider bar with a small button at the right hand side that says go.  If you click the go button, the linear graph will move up and down until you hit the stop button. 

If you want to see how a different function moves with the same
general equation, you click the change function button at the lower left of the graph. The picture to the left shows both the change function button and the slider bar for the horizontal shift. The small red button at the end of the slider bar is the go/stop button.

This is nice seeing so many functions covered in the app. The only problem is that the instructions are a bit thin and I spent quite a bit of time playing with it until I figured out how to work it.  I also discovered the software company has this on their website.  The best thing is that this is a free app.

smart pirates or fractions into decimals

This app is done by the same people who made the pirates app I reviewed earlier.  This one focuses on students learning to equate fractions with percents.  As with the earlier one, there are two free apps, one if the free and the other is the lite.

This entry looks at the lite one.  As before there are four areas of practice.   The first is fractions as a percent, followed by equivalent fractions and percents, then comparing fractions to percents and finally finding the whole when given a percent.

There are three levels to choose from. The easy level has students changing basic  fractions into decimals such as 1/2 = 50%.  A student is required to get 5 questions correct before being allowed to move on to the next area.

Once a student has completed a level, they start fractions as a percent at the next level. 

If they miss a problem, something happens so they cannot go back to the problem to change their answer and a new problem appears so they can try again.  I believe my struggling students will enjoy this one as much as the young man enjoyed the other app.

Solve for X

 The other night I stumbled across a nice little app called Solve for X.  This app is different from most of the apps I've run across in that it requires the student to tell the app the next step.

The app has six levels.  The first level has students solving one step equations.  You have to complete a certain number of problems correctly before you can unlock the next level.

The second level has students solving 2 step equations using addition or subtraction first, followed by the multiplication or division. It isn't until you get to the third level that fractions are involved. The interesting thing about this app is if you put in the wrong numbers, add 3 instead of subtracting 3, it will do exactly what you said and you can see what the results.

This app will allow my students to practice solving equations and I can use it to help reinforce and differentiate instruction.

Flipping the classroom

I've often thought of flipping my classroom but I was not sure how to handle the students who did not have access to the internet at home or after school hours.  Today, thanks to Josh Fisher, I read a lovely blog post by Scott Haselwood on EdSurge  what to do for students who do not have internet at home.  I finally have direction, a way to flip my classroom so my students can have access to any video's I create or assign. 

At school, each teacher has a wiki to post anything relating to classes.  I do not believe any one has taken advantage of this due to the labor required to post everything. After reading this article, I am having second thoughts because this would be the perfect place to post the narrated stop motion videos I plan to create.


I don't have to record myself presenting a 15 to 40 min video giving the whole lesson.  I could create multiple short videos that introduce or  review material as necessary.  I could even create hollywood style epics with sound effects that last maybe 3 or 4 mins.

Then I could simply place the video on the wiki to download if they desire and have the capability to do so, otherwise, they can stay after school, grab an ipad and view it before heading home to study.

The idea sounds really great. The blog entry did state it would be some work to do it so it looks like I"ll be doing a lot of prep this summer.  

Math pirates- fractions

I found a couple apps called Math Pirates that have a free and a lite version.  There is a minor difference between these two free versions.

The free version allows access to all four areas but you are limited to no more than 5 questions per day.  The lite version requires the player to do simple fractions first.
 Once the student has correctly answered 5 questions, they are moved to equivalent fractions to work on answering another 5 questions.  Then its on to comparing fractions and finally adding fractions.

As each section is finished, the pirate ship sails to the next one.  Each activity uses the pirates either on land or in a boat and you must complete the activity properly for the pirate to complete the action.  For instance, in equivalent fractions, you must select the equivalent fraction so the bridge works and the pirate can walk across.

What I like is that it is very active using pirates and pizza or weights.   The picture to the left from the  comparing fractions section, medium level.  Notice they have two pirates sitting on a teeter - totter.  One has a melon and you select the watermelon to help the other pirate weigh more.

The section on simple fractions uses a pizza and dividing it between pirates on land and pirates in a boat.  I actually find it kind of cute.

I would say these apps would be good for upper elementary all the way to high school for students who need additional scaffolding in fractions.  You can get the full version of this app if you pay for it but I think the free and lite versions are fun if you want students to do some practice every day.

I chose a sophomore from my geometry class to try both apps to get her opinion. She spent about 20 min on it and said it was a good app.  It was because of her playing with the app that I figured out you had to complete a certain number of problems before you could "unlock" the next activity.

I also had a pre-algebra student work with it and he gave a thumbs up on  it.  He wants to use it next year.

Real life uses of math

This school year is almost over.  The time to start planning for next year is now.  I got the bright idea while watching MythBuster of watching an episode and have the kids research the necessary math and science needed to do what the hosts of the show do.

They show the hosts making things but the actual math is done off screen.  What if they had to calculate the thrust of fire works, or a air gun.  What if they had to find out how many watts a mirror needs to make to set fire to a ship?

What about finding out the density between a frozen chicken and a thawed chicken?  How much thrust does a cannon use when you put a 6 oz charge of black powder in?

Then there is the idea of starting to show an episode and stop it just after they introduce the myth.  The students could make a prediction of will it prove true, plausible, or busted?  They could work out what factors or variables do they have to think about.

We watched one today in which the mythbusters checked out something from the show CSI to see if it worked.  The idea was the spark from a stun gun could cause pepper spray to ignite and burn someone up.

This episode is great for showing the steps in isolating variables and following scientific method. I want to use this to show students how mathematicians can also follow this process to help us find the math.

I want to plan activities to do with that episode over the summer.  It would show the process everyone goes through and it would encourage them to do independent research. 

The first week of school is very short and this would be a great activity to include before the actual work for the classes start and get them familiar with how to do mathematical research.

Cool!

HUP Algebra part 2

The other day, I took another look at HUP Algebra and discovered a part of it that just jumped out at me.  It is a free app and covers solving one and two step equations. In addition, it gives students instruction in solving multi-step equations and equations with distribution.

Each topic shows an written example and an example using manipulatives. At the bottom of the page you can choose try it, random questions or watch a video showing how to solve the problem.

If you choose next, you will get a screen similar to the picture on the left.  This is a nice way to show how we solve one step equations.  I think this way almost makes more sense than the traditional way.  I plan to show solving this way in addition to the traditional method so students are exposed to more than one method of solving.

I like the way this second screen also a try it and a video choice. Once you get to solving multi-step equations they assume the student has practiced the individual steps and does not need to practice the opposites the way they needed to in the one step equations.

I know that I have some low performing Algebra I students coming up from 8th grade next year and this will help them learn to solve one, two, and multi-step equations.  This will be good to help scaffold students in learning this material.