Different Way of Graphing Quadratics.

 I attended the Northwest Math Conference in Tacoma this past weekend and learned so much.  I will be sharing various ideas over the next few days.  Ideas I think are awesome.

One session shared with us a slightly different way of graphing quadratics without completing the square or using a graphing calculator but he started with the idea that a quadratic is composed of two linear equations multiplied together. This is a unique way of looking at quadratics.

In addition he stated any polynomial is made up of a specific number of linear terms multiplied together.  So a 3rd degree is made up of three linear equations multiplied together, etc.

The graphing method is so much simpler and so much clearer than traditional ways while combining finding the zeros and the vertex and line of symmetry.
Lets look at y = x^2 + 3x - 7

Step one: (x^2 + 3x) - 7.  Place the first two terms inside parenthesis.

Step two: x(x+ 3) - 7.  Factor out a common term from the terms inside the parenthesis.

Step three: x = 0 and x+3 = 0.  Set the terms equal to zero and solve.  So we

Step four. x = 0, -3.  These are the two points of the equation that produce the symmetrical points.

Step five: These points give (0, -7) and (-3, -7) when you plug the x values into the equation.

Step six: (0 + -3)/2 = -1.5. Add the two points together and divide by two to find the x value of the vertex.

Step six:  (-1.5)^2 + 3(-1.5) - 7 = 2.25 -4.5 -7 = - 9.25. produces the vertex of (-1.5,-9.25)

Now place dots on (0,-7) (-3, -7) and (-1.5, -9.25)

According to the speaker, this method works on all equations, even the ones with irrational roots.  One does not have to apply the quadratic formula, complete the square, or any other method.  I've not had a chance to investigate these claims but I agree it is much easier to graph the equations this way as long as you are not trying to find the roots.

Give it a try and let me know what you think, I'd love to hear your opinion.  This is just one of the new things I've learned.  Have a great day.

Warm-up

What math questions could you ask about the photo

Warm-up

What math do you see in the picture?

Desmos, Desmos, Desmos

Yesterday,  I attended 2 very good workshop.  Each workshop was 3.5 hours long, packed with awesome information, and we got to learn so much more about using Desmos.  I spent the morning learning more about using various Desmos activities, add pauses, and copy and edit some already done.

I knew the basics but I learned more about using the activities including how to pause it so students had to stop and listen or how to set it so students could only do part of the activity rather than rushing through.  We also learned the Polygraph activities which allow students to chat with each other as they play a type of twenty questions.  The great thing is they are designed so kids cannot ask questions like "Is it the upper left one?" because the arrangement is different on each screen.

In addition, we came out of the workshop with a list of links for list of activities including a list of activities for kindergarten to fifth grades.  There was also a Desmos scavenger hunt with activities to do ranging from beginner to advanced and most have a solution so if you get stuck trying to do it, you can pull up the solution to look at it.  I need the visual examples to help me see how the instructions fit together to make the final product.

In the afternoon, I got to do a workshop with Dan Meyer and it was awesome.  He began with a "What if" activity where he set up a list of numbers and then placed the variable representing the numbers into a coordinate system.  He asked up to make predictions for each of these before trying them on Desmos.  It was so open ended and promoted so much discussion.

Towards the end, he introduced us to the Desmos activity "Graphing Stories".  He had us divide up into groups of two or three people because this arrangement encourages more conversation that having each person doing it individually.  He showed a 15 second video and then asked us what things were qualitative and quantitive.  After some discussion, we were told to create a graph of the distance between his waist and the ground.  He let us know the time was over by pausing everyones screen.

He put a couple graphs up and asked us what was good about the graphs before asking us to identify areas or things that could be looked at to be revised.  We were all given the chance to revise our graphs before he stopped us to compare the original graphs with the revised graphs.  Once we had experience doing this, he asked us to create graphs for two out of four short 15 second videos. AT the end, he went through and shared one or two videos for each one.  We were running out of time, so he had us look at someone's graphs but it was awesome to see that graph against the one we made.

Throughout the hole workshop, Dan provided guidance and leadership without controlling exactly what we learned.  We did the figuring out and it was awesome.  I loved it and I am coming back with new ideas and tools to use in class.  I'll share more with you tomorrow.  Let me know what you think, I'd love to hear.  Have a great evening.

At Math Conference.

I am at a math conference.  Today, I will be learning all about Desmos.  I"ll share tomorrow.

Examples in Math

One of the things I have to do is set a goal to work towards from a list.  I've chosen to work on giving the students ways to help them become more independent learners. Over the past nine weeks, I've realized my students are unwilling to pay careful attention to examples.

Textbook have lots of examples with detailed information explaining how it was solved.  My students prefer being passive as I discuss examples and they really don't want to write things down so beginning next week, I'm going to work with them on learning to "read" examples.

I have one student who takes time to look at his textbook when he does his assignments.  He asks questions when he doesn't understand something and is understanding way more than most who take a passive attitude towards learning.  Most examples include drawings or diagrams to provide the visual key but my students read the problem but won't draw anything so it makes it more difficult for them to understand what is being requested.

I admit, at their age, I didn't want to read my textbook.  I didn't see why I needed to and I'm sure they feel the same.  So it is going to be a bit of a struggle but I think it will be worthwhile.  Over the years of my own education, I've learned how important examples are to learning.  This is something I hope to share with my students.

One way to use examples is to begin with a completed example and having students explain the steps in the example.  This reduces anxiety and memory load so students can focus on understanding the concepts behind the process.  It allows them to explore the equations and process it more deeply because they are not just following a procedure.

There is research to indicate that students learn more accurately from worked examples because they are paying more attention when they concentrate on the examples.  However, students need to be taught to examine examples.  Researchers found there are three components when using worked examples with students.  The first is formulating a problem, the second is examine the steps used to complete the problems.  The third is sharing correct answers with the students. One way to working on the second component is to leave out some of the steps and asking students to supply them.

Research it is not the words the teacher uses to explain the examples but it's getting the students to think about the example as they examined it.  Some of the ways worked examples can be used more effectively in class is to assign a few worked examples to students as part of their homework.  In addition, the teacher should model their thinking as they solve problems.  Finally, create animated worked solutions for students to see solutions develop.  The nice thing about animation is students can rewind and review parts they don't understand.

Another layer to include is to ask students questions like "Why was this strategy used?" or "What principal was applied and why?" because these require deeper thinking.  The question is how do you get students to use worked examples.

One way is to have students explain to themselves as they go through worked examples.  Some of the things they might consider are "where did that number come from?" or "Why did they do that next?"  Another way is to used the faded examples where the first problem is fully worked out, the next example is worked out for the last step which the student fills in.  As the student works through the examples, each one has one less step until they work the complete problem from start to finish.

If you make the problems more complex, its important to show examples with the added complexity so students see how things change.  Let me know what you think, I'd love to hear.  Have a great day.

The Cost of Launching a New Product.

 If you keep your eye on Kickstarter, or Indigogo, you'll see companies trying to raise money for various products they want to bring to fruition.  Yes, I've invested in quite a few things ranging from electronics to clothing.  I bought a pair of shoes that I can wear in water, or in the city, or in the country without a problem.  They are comfortable and I don't need more than one pair when I travel but I've often wondered about the cost involved in bringing the product from conception to the market.

The cost all depends on what the product is being developed.  For instance, the cost of developing a new shampoo from scratch can be upwards of $25,000. Here is a breakdown of the costs.

1. $4000 to $8000 to develop and test the formula for shampoo.

2.  The minimum order for the manufacturer is 400 kg of product which will have a unit cost of $0.80 to $1.20 or a total cost of $8000 to $12,000 for the inventory.

3.  The manufacturer has to make a few samples to be sent to a laboratory to be checked out for stability and testing for microbes for $1100.  It would be more if you want the laboratory to conduct patch testing for allergies.

4. Then the person has to set up an e-commerce or fulfillment site which probably run about $2.40 per order filled only. This cost does not include shipping.

5.  Then add to this the cost of branding and registering your trademark is going to run between $1500 and $2500.

6. There is also the cost of incorporating the company, advertising, sales cost, etc.

What if you prefer to develop something with electronics, how would that change the cost?

The Electronic Development Stage covers the design of the printed board circuits from the preliminary stage to the test and documentation and costs between $15,000 and  $45,000.

The Software Development Stage covers firmware, mobile apps, and P.C. software which will set you back between $15,000 and $100,000.

The Industrial Design where the enclosures are created and runs between $6,000 and $24,000.

The Package Development Stage is where the packaging for the product is created and it can cost between $1,500 and $4,500.

Finally is the Project Management or the person or persons who keep the whole thing together and this costs between $4,000 and $10,000.

So overall you are looking at a cost of between $35,000 to $180,000 for just one products with electronics involved.

So this is another application for real world mathematics.  Let me now what you think, I'd love to hear.  Have a great day.