Happy Easter

 

Warm-up

 If people buy 16 million egg dyeing kits to color 180 million eggs, how many eggs does one kit dye on average?

Pigeonhole Theory And Hair

I always find it interesting when someone comes up with a new use for an older theorem.    The pigeonhole theory or dirichlet's principle was applied to the question of "Can two people have  the same number of hairs?" and came up with the answer "yes". This theorem was first proposed back in 1622 and says that when you want to split a certain number of objects among a specific number of drawers, you will end up  with several objects in each drawer.

Although this theorem first appeared in a book published by Jean Leurechon in 1622, it was attributed to Peter Gustav Lejune Dirichlet who lived almost 200 years later.   Although the theorem is rather simple, it is used to explain more complex situations and relationships such as if you have five points arranged randomly on a sphere, four points end up in the same hemisphere. 

A person decided to apply this theorem to the question of "Can two people have the same number of hairs on their heads."  First one has to find out the maximum number of hairs that can be found on a head and the approximate number of people on the earth.  Most people have a total number of hair strands falling between 90,000 and 150,000. The approximate population world wide is around eight million people.  Thus there should be some people who have the exact same number of hairs based on this theorem.

Furthermore, one can assume that if you have a million rooms and  all eight million people have the same amount of hair.  This means that everyone will be in one room and the other 999,999 will be empty.  On the other hand, if people divide themselves up so that the minimum number end up in each room, how many would that be?  I believe that ends up as around 8,000 people per room. 

Another example might be the question of how many people share the same birthday in New York City. We know that some will based on this theory. If you take the population of New York City and divide it by 366 days, you get 8.5 million/366 and end up with 23,000 people who share the same birthday everyday.

Notice the conclusions that people come up using the pigeonhole theory is always based on simple assumptions.  Let me know what you think, I'd love to hear.  Have a great day.

Unpacking Math Standards Part 2.

 

In the last column, we looked at unpacking math standards in general.  Today, we'll look at it in more detail because for many of us, we need to know how to do it so we can prepare lessons for our students . Here in Alaska, the math standards for high school are all together rather than split out for the different types of math and that makes it much harder.  The elementary standards are split up according to grade so it is much easier to unpack but as a high school teacher, I have to find the standard that fits what I'm getting ready to teach.

Let's look at one way to unpack the standards.  First step is to read the standard and any supporting standards completely.  Many times there is something in the supporting standards that help you through the process.

Next, it helps to create a T chart to list the knows and do's. For clarification, the knows refer to the content and the do is described by a verb to talk about the action but it is important to use the same verb as in the standard.  An example of this might be the knowledge is of inverse operations but the do is finding the inverse of a function. Some examples of verbs to show action might be apply, solve, represent, determine, calculate, predict, write, model, or convert.  It is worth taking the time to determine if the standard is a procedural, conceptual, or an application.

Conceptual understanding refers to the way ideas, patterns, and procedures are used to connect new knowledge and use it to solve unfamiliar problems. Procedural knowledge is the ability to use procedures to solve problems, and application is connecting conceptual with procedural knowledge so the student can solve real world problems.

Once you get this far, you should take time to see if this standard aligns with the previous grade and the one above for middle school or younger.  In high school, one can see if it aligns with 8th grade and how it aligns with the next math course in the sequence. At this point, you are ready to write the learning objective, target, or goal.  These are measurable. 

Next comes formulating the big idea or essential question.  The big idea is the idea or concept that flows through the whole unit connecting the material with real life while the essential questions support the big idea and are addressed throughout the unit. Essential questions can be used both at the beginning of the unit and at the end.  

Now you are at the point of deciding what vocabulary is important for students to learn in regard to the material, designating key points of the concept or topic, determining the prerequisite knowledge and skills needed, and how to teach the material.  This is the meat of the unit lesson.  I also realize that most of us do not have time to go into this deep a dive, especially if we are teaching multiple levels of math throughout the day.  Let me know what you think, I'd love to hear.  Have a great day.

Unpacking Math Standards Part 1

 

One of the things I've heard the past few years of my teaching career is to unpack math standards so you know what to teach.  It was a big deal at one school but over the years, especially since COVID, I see it as much more important. 

Unpacking the standards refers to looking at standards in detail so the teacher knows what the student needs to know, what they should understand, and the prerequisites needed to get there. In addition, one should look at various ways to represent the knowledge visually and common misconceptions students are likely to have. 

One thing I see time and time again is when students incorrectly subtract.  They have a problem like 81 - 37.  Instead of borrowing from the 8, they switch the 1 and 7 so they actually subtract 31 from 87 and end up with the wrong number.  I don't know where that comes from but it is one that I see frequently in middle school and high school math.

Now to break things down a bit further.  When deciding what students need to know, it should include vocabulary, facts and rules when doing the math so they are able to complete the concept, topic, or unit successfully.  As far as understanding, one needs to look at the big ideas of the unit, the concepts within it and connections so students learn the relationships among everything.

Teachers also need to determine what students need to be able to do and and how will they do it.  Furthermore, it is important to think about all the prerequisite skills needed to learn the current material.  Know what prerequisites are needed means the teacher knows if certain skills need to be pre-taught or retaught. I have a student, I had to go back to the basics with fractions because he had no idea how 2/6 was the same as 1/3 and couldn't find a common denominator for say 1/2 and 1/3.  He could use a calculator for fractions but since he lacked a knowledge of fractions, he didn't know if the answer was reasonable.

Teachers can find the visual representations or the manipulatives needed to help students learn the concept or material for the lesson.  In addition, when a teacher knows what the common misconceptions are associated with the concept or topic, they can clarify and help students avoid them.  Next time, we'll look at how to actually do all this.  Let me know what you think, I'd love to hear.  Have a great day.   

Warm-up

 If the Bee Hummingbird weighs 1.5 grams and a penny weighs 3 grams, what percent of the weight of the penny is the Bee Hummingbird?

Warm-up

 

If a shark lives for about 25 years and they grow and lose around 30,000 teeth, how many teeth do they grow and lose in an average year?