Warm-up

 

If there are 1.4 billion insects per person on this planet and the population as of 2021 is 7.88 billion people, how many insects total are on this planet?

Warm-up

 

Just learned today that penguins can shoot their poop, four feet out to keep the nest clean.  If they moved 4 feet every day in a straight line, how many days would it take to go one mile?

Math Doctors

 

I was looking for an explanation on why when two negatives are multiplied together, you get a positive and I came across a page of math doctors! Math doctors is a page that is run by lovers of math and has replaced the old ask dr. math.  The archives for the dr math questions are now housed by NCTM and is free for the moment but that could change.

The people who answered questions at the ask dr. math part of the math forum are responsible for this site. Apparently, the question part of the site closed down in 2017 and rather than letting it all go away, they banded together to form this new site. This is a place where question can be posted and answered by reputable math people so the answer is correct.  Consequently, it is not a discussion board but a true place to get help with homework, general questions, etc. 

The site recommends you check out the old archives to see if that particular question has been answered before you submit it to the current math doctors.  In addition, there is a blog which has been active since January 2018.  They use the blog to share answers with the population, including questions and answers from the archives.  

The first entry for April explored how can the determinants of a 3 by 3 give both area and volume. It explores the question when applied to finding the triangle or the volume of a pyramid and one other set of shapes.  The answer has both diagrams and a written answer combining an entry from the archives with the actual answer.

I also checked the January 2018 blog with its multiple entries.  One entry explored the actual meaning of median since the definitions often vary according to the book, the country, and interpretation.  By interpretation, I mean the person who answered the question might explore what the median is when you have an even number of items.  It is quite fascinating reading.  I kind of wish I had time to binge read on the blog.

Anyway, head over to themathdoctors.org and check it out.  Let me know what you think, I'd love to hear.  Have a great day.

Interactive - Math

I was all set to talk about math and how its used in cards but I came across this interactive web site when I was looking for an interactive activity on the Pythagorean Theorem for one of my students. It lead me to Interactive - Maths, a cool site. This is a site, I've not seen before so I was pleased to find it.

The owner of the site, offers several choices such as topic resources, mixed topic generators, starters puzzles, and enrichment, crypto center, and a blog.  I'll take you through everything.

First the topic resources. This page looks at all of the activities available along with an explanation of the type each activity is, and a few generators.  The topics range from arithmetic, to Algebra, to geometry, trig, transformations, and even some college level activities. Some topics have randomly generated questions, one at a time, or ten at a time, or done in relay form so they have to answer one question before it allows them to go onto the next, bingo, worksheets, or a geogebra activity.  Quite a variety.

The mixed topic generator allows you to create either worksheets or questions that are totally random or topic specific.  There is also a treasure hunt creator, and a bingo creator so you can make what you need for the activity you need.  The starters, puzzles, and enrichment has a nice long list of activities that can be used to start the class, add enrichment, or are just fun to do.  One of the puzzles is a generator that creates a puzzle based on the day's date.  Sound like fun.  I looked at one that gave 2, 5, 4, 2, 3 as the digits one can use and by using the four operations, you have to come up with the answers of 1 to 30.  Rather a cool, challenging puzzle.

Crypto corner is dedicated to codes and cyphers so students get a better idea of how cryptography work. This comes with all sorts of codes, resources and such that one could create a fun unit towards the end of school.  Finally, there is a blog that runs between December 2012 and August 2021 with all sorts of how to do it topics.  Some of the topics are really cool.  

Please go check this site out if you need some cool things for your classroom.  Let me know what you think, I'd love to hear.  Have a great day.

Grades Based On Rubrics In The Math Classroom?

 

This past Thursday, our school had a one hour professional development on using rubrics in the classroom by a gentleman from British Columbia, Canada.  It was interesting since he indicated that grades K to 9 will be graded according to a standard type rubric using emerging, developing, proficient, or advanced but not the upper grades.  

Since I've spent most of my teaching career working as a high school math teacher, I'm used to giving grades based on a percent. Some schools I've worked at have regulations like you can't give anything less than 50 percent or if a student is getting too close to the maximum number of absences, you have to allow them to go to study hall to make up the time and other things but I'm not used to thinking in terms of using rubrics for a grade. This morning, I realized that in general, we grade elementary students using some sort of rubric but usually based on satisfactory or needs improvement or something similar.  

We learned to develop rubrics based on the verbs used in the state standards for our grades while using a list of similar words to help write what one is looking for.  He suggested using no more than three for each assignment and to get it down to one word for each cell to make it easier for students.

I semi like the idea of using this type of rubric for grading but it would take a while to implement for so many reasons.  One has to get both the parents and students to accept that this is a valid way of assessing and might even be more accurate than the percentage scale.  In addition, it is hard to determine valedictorian and salutatorian based on this type of rubric. I don't think colleges and other post secondary schools are set up to use a rubric since most still operate using the standard grades with percentages for admission, for scholarships, and so many other things.

State testing has been using this type of rubric but it used well below, below, proficient, and advanced when reporting results back to the school.  I don't think I've taught at a school that has used the overall results to change instruction or to scaffold instruction for students.  I do know one teacher who analyzed the type of questions used, what standards they addressed, and focused her teaching the following year using that. Only problem with that - the state changed out questions and standards so it didn't work well.

I will be honest with you. I teach at a small school with 27 students who are about evenly divided into two groups.  The elementary room has grades K to 5 while my room has 6 to 12.  Both teachers teach all the subjects. It can be difficult to make rubrics for each grade in each subject since that takes a lot of time.  Yes, we had training but who knows if they will give us time to practice or even decide to implement it.  

It seems education gets these ideas that last a couple of years before they are supplemented by some new idea.  I'd love to hear what you think about this.  Please let me know.  Have a great day.

Warm-up

 If there is one dog for every 16 people, what is the population of dogs world wide?

Warm-up

A flea is able to jump 350 times it's length.  If a flea is 3 mm, how far can it jump in inches?

Using Base 10 Blocks In Algebra.

 

When I got my teaching credentials, we didn't use manipulatives in high school.  It was never a topic and I didn't learn to use them but over time it's become more acceptable to use manipulatives in Algebra, especially for students who need the visual component.  So today, I thought I would take time to share some ways I've researched or discovered for using manipulatives in Algebra.

For working with signed numbers one can use those two colored round disks.  The ones I have are yellow and red.  I designate the red as negative and yellow as positive. This way if they have -4 + 5, they have four red disks and five yellow disks.  When they match up one red and one yellow to make zero, they are left with one yellow or -4 + 5 = 1.  One can use algebra tiles, small rods in different colors, or even a number line.  You can also use two different colored singles from the base 10 manipulatives so they have the correct number of singles for -4 and 5.  Visually they can see that there is a difference of 1.

For combining like terms, you can use those base 10 manipulatives.  The singles can make up the constant, the strips of 10 represent x and the 100 squares represent x^2.  Using two colors, one for positive, one for negative, they can combine terms and see what the total is of each term.  

Those base 10 manipulatives can be used to show binomial multiplication using the singles as ones, the 10's strips as x and the 100's as x^2.  The problem is (x +2)(x+1) like in the picture.  

The x + 2 is one 10 strip with two singles and the x + 1 is with one 10 strip and one single.  You multiply the x times x, x time one, x times one.  then you multiply 1 times x, one times one and one times one. If you count all of the parts, you end up with x^2 + 3x + 2. Side note: this visualization also works well with two digit times two digit multiplication.  You can also use these as the reverse so you set up the answer and figure out the solution.  Now if you have say (2x + 3)(3x - 2) you would have two -10 strips for one side and three - ten strips on the other with a different color for the negatives.  You could use algebra tiles but the base 10's work and you might have some of those around.

In addition, these base 10 manipulatives can be used to solve one and two step equations by using the single squares and the 10's strips.  Say you have 2x - 3 = 5.  you have a rectangle with a line about half way through.  On the left side you have two - 10 strips and three squares in the color representing a negative number.  On the right side you have five squares in the color representing positive numbers.  Using positive colored squares you add three to each side which shows how the -3 goes to zero and the other side becomes 8.  Then you can divide the 10 strips representing the x in half and divide the 8 in half so you now have x = 4

If you don't have base 10 blocks, this site has some virtual ones your students could use. I will revisit this topic in the future.  I hope this gives you some ideas. Let me know what you think, I'd love to hear.  Have a great day and a wonderful weekend.

April Is Math Poetry Month

 

I just discovered that April is Math Poetry Month in addition to being National Poetry Month and Math and Statistics Awareness Week.  The poetry used in class could be either sharing mathematically based poems or having students write their own poetry.

Although it is about half way through April, you still have time to engage in exposing your math students to poetry.  Let's start with some sites that have poetry you can share with your classes. 

One is through the Mathematics Association of America with a lovely page of poetry. Part of the page has poems with a mathematical theme such as Geometry by Rita Dove or types of mathematical poems based on structure with examples.  They discuss the visual poem, the concrete triangular poem, square poems, haiku, and other variations of syllable poems.

The acorn naturalists have a lovely page that discusses the idea that by knowing the mathematical structure of a poem, it should make writing poetry so much easier. They talk about how many of the poems with the rhythmic sing song sound is due to the math of the number of lines and number of syllables in each line.  Then there are the constraints or rules for certain types of poems such as Haiku where you are limited by the 17 syllable frame.

This National Geographic blog has a lovely discussion on how math and poetry intersect written especially for teachers. They provide discussion ideas such as comparing poetic structure with mathematical formulas, types of mathematical poems with examples, and even links.  Everything you need for a lesson on the subject.

If you do a quick search for math poetry, you'll find lots of sites and suggested youtube videos to help you do this in class.  This can show how similar math and poetry are.  Let me know what you think, I'd love to hear.  Have a great day.

Encouraging Persistence In Math Class

 

This year, I ended up working with students who ended up on an online class because their original instructor left early and although certified in math, wasn't comfortable teaching it.  I arrived in time for the second semester and I've had to work with most of them to develop persistence.  They've been learning to complete their work without shutting down and giving up.

In addition, many students are hesitant to try possible solutions if they are not sure they are correct. For some reason, many students believe they should find the correct answer immediately without making any mistakes.  Sometimes, that is a hard attitude to overcome. 

One way is to use low floor and high ceiling problems so that weaker students are still able to do the problem as well as those who are much stronger in math. This type of problem has few skills needed to attack but contains multiple levels of complexity to provide challenge for those who need it.  

Another possibility is to use open ended problems that have more than one correct answer. This tends to confuse students who have always been taught that there is only one right answer and it takes a while to figure out how to answer it.  In addition, this type of problem allows them the opportunity to work through the frustration of finding the right path.  This helps build the persistence needed to work this and other types of problems.

Furthermore, if the problem allows for multiple solutions, it shows students that there is not necessarily a single way to solve the problem.  Students need to know that there maybe more than one way to solve a problem because it encourages students to use their creativity while allowing every student to exercise their problem solving abilities.

In addition, try to set things up so that students feel as if they are not being judged.  It sometimes pays to remove any and all factors that might inhibit students from thinking outside the box.  It makes them more willing to try things they might not otherwise try.  One way to make it easier is to assume students will not solve it and rather than grading the final product, grade the process they used to try to arrive at an answer. 

If you grade it using participation, it means weaker students are more likely to do well because they tried and it levels the playing field for them.  Furthermore, the instructor needs to learn to answer requests of helps from students so they don't give the answer but encourage all students to keep trying.  

One piece of advice is to throw in the occasional easy problem so students don't burn out and being able to solve a problem here and there helps student persistence.  Don't worry if the problem doesn't have anything to do with the topic being studied in class.  The problem doesn't have to.  

Don't be afraid to integrate problems regularly into class time so students get used to doing them.  Once a week is enough but be prepared to have students complain the first few times.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

 If the worlds smallest bat, the bumblebee bat, weighs about 1.76 grams, how many bats would you need to weigh 500 grams?

Warm-up

 

If the smallest monkey is the same height as a toothbrush, and a toothbrush is about 7 inches long, would the monkey fit in the palm of your hand?  Explain your answer.

New Math Game At The New York Times

 

Apparently, there is a new daily math puzzle being printed by the New York Times.  It's called "Digits" a daily numbers puzzle. It is available for free right now in the beta version here.

It takes a few minutes to figure out how it works but it is fun once you work it out.  Basically, it gives you a target goal with several numbers that you apply operations to so you get as close to your target goal.

One of my puzzles had a target goal of 195 with the numbers 2, 3,5, 10, 15, 25.  The idea is that I take two numbers and perform an operation.  I might start with 3 x 10 = 30 and the window will show the equation.  Then I might do 2 x 15 = 30 and again the equation will appear in the side window.  I can then take the sums and add them together to make 60. Then I might multiply 25 x 5 to get 125 and add the 60 to 125 to get 185.  I am aiming for 195 so I am 10 off and I can try again.

I decided to try 25 x 5 = 125 and then multiplied the 125 by 2 to get 250.  I multiplied 15 by 3 to get 45 and then did 250 - 45 = 205.  The last equation was 205 - 10 = 195.  Just after I finished the wrong attempt, I realized that I needed to go higher and work down rather than trying to work up. 

For the daily puzzles, there are 5 puzzles you work your way through.  I had 94, 251, 382, and 456, so it starts with the smallest number and works through the numbers up to the highest.  You can earn up to 3 stars for each solution that you make and you do get stars for getting as close as possible.  

I noticed that as long as I do not submit my attempt, it doesn't seem to count so I can use the undo button as many times as needed to go back to the beginning and try again.  I find this helps me eliminate possibilities in my brain. 

This is something you could use in your classroom for older students to help them develop a better number sense.  Give it a try and let me know what you think, I'd love to hear.  Have a great weekend..

 

Examples Of Chat GPT Responses.

 

After writing the entry on Chat GPT, I thought I would share some of them with you so you could see what type of responses you are likely to see. Some have explanations, some don't.  I chose a vocabulary type question, an algebra one, and a calculus one just so you have a better idea of what it can do. I will tell you that I sometimes have issues with it due to how lousy my internet is.

Question - "What is domain and range?"

Answer - In mathematics, the domain of a function is the set of all possible values of the independent variable (usually denoted as "x") for which the function is defined. The range of a function is the set of all possible values of the dependent variable (usually denoted as "y") that the function can output for all the possible values of the independent variable in its domain.

To put it simply, the domain is the set of inputs for a function, and the range is the set of corresponding outputs. For example, if we have the function f(x) = x^2, the domain is all real numbers (because we can square any real number), and the range is all non-negative real numbers (because the square of any real number is non-negative).

Question - Evaluate 5x^2 - 2x + 3 if x = -3

To evaluate 5x^2 - 2x + 3 when x = -3, we simply substitute -3 for every instance of x in the expression and then simplify:

5x^2 - 2x + 3, when x = -3

= 5(-3)^2 - 2(-3) + 3 (substitute -3 for x) = 5(9) + 6 + 3 (simplify exponent and multiplication) = 45 + 6 + 3 (simplify multiplication) = 54 (addition)

Therefore, 5x^2 - 2x + 3, when x = -3, equals 54.

Question - "Find the fourth derivative of sec^2 x.

Answer - To find the fourth derivative of sec^2 x, we need to differentiate the function four times.

Let's start by finding the first derivative:

y = sec^2 x

dy/dx = d/dx (sec^2 x) = 2 sec x tan x

Now, let's find the second derivative:

d^2y/dx^2 = d/dx (2 sec x tan x) = 2 (sec x tan^2 x + sec^3 x)

Next, let's find the third derivative:

d^3y/dx^3 = d/dx [2 (sec x tan^2 x + sec^3 x)] = 2 [sec x (2 tan^3 x + 3 tan x)]

Finally, let's find the fourth derivative:

d^4y/dx^4 = d/dx [2 sec x (2 tan^3 x + 3 tan x)] = 2 [2 sec x (6 tan^2 x + 3 sec^2 x tan^2 x + 3 sec^4 x + 2 sec^2 x) + sec x (6 tan^2 x + 6 sec^2 x tan^2 x)] = 2 [12 sec^3 x tan^2 x + 24 sec^5 x + 18 sec^3 x tan^4 x]

Therefore, the fourth derivative of sec^2 x is 2 [12 sec^3 x tan^2 x + 24 sec^5 x + 18 sec^3 x tan^4 x].

You should check it out and play with it. Head over to chat GPT and explore. Let me know what you think, I'd love to hear.

Chat GPT In The Math Classroom?

 I am sure you've heard of Chat GPT.  It is one of the hottest things out on the internet right now. There are so many ways to use it, at least according to all the e-mails I'm getting from people wanting to sell me training on it.  Yes, I've checked it out for writing things but not really for math. About 80 percent of the time I try it, I can't get a real response.  

Chat GPT is a language model created by Open AI that generates a response to any question typed into the "chat" window because it's been trained to answer questions.  According to what I've read, you can type in a math problem and will receive a response explaining how to solve it. 

One article stated that chat GPT is quite good at performing basic operations such as addition, subtraction, multiplication, and division along with solving systems of linear equations and quadratic equations because mathematical expressions and equations are the central components of mathematics.  It can even solve more complex subjects such as differentiation and trig, depending on the problem, it may not solve it correctly.

I brought up a chat window and asked it a few different mathematical questions from a "what is" to get a definition of a mathematical term, to solving a linear equation to taking the derivative of x^2 and it provided answers for all of them.  It even gave written instructions for each step so if you asked for the steps, all students have to do is copy it down.

According to one article I read, chat GPT may not be able to solve every problem, especially if it is a complex problem that requires a specific method or formula to solve. In addition, it does not always provide the most efficient way of solving the problem, much like a GPS that doesn't always provide the most direct route home.  It also stated that the chat GPT has difficulty solving problems that require a certain level of real world knowledge.

Another article indicated that chat GPT is good for teachers who want to generate new problems that include explanations of how to solve them and additional practice problems of the same type. Since chat GPT is a language model, teachers should check any problems and examples formulated by this AI to make sure they are correct. In addition, it has been suggested that students read explanations formulated by chat GPT to check for logic and any fallacies. 

I know there is a fear that students will use this digital product to do homework but if you as a teacher use multiple and frequent assessments, it becomes quite obvious they are copying rather than trying to learn the material based on their answers.  Let me know what you think, I'd love to hear.  Have a great day.

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?