Warm-up

 

If there were 40.3 million flights before Covid and only 19.6 million flights after Covid arrived, what is the decrease in flights between the pre and post numbers.

The Mathematics Of Tear Free Hair Brushing.

I ran across an article on this topic and was totally intrigued.  I have hair down to the bottom of my rear and sometimes it gets nicely tangled and over the years I've learned enough tricks to make it almost manageable to brush my hair without pulling tons out. It requires a good detangled which doesn't always do much, tons of conditioner, and some swearing on occasion. As we all know, you start combing from the ends up. So I had to read this article and share it with everyone based on the title.  It's nice to learn how math explains the way to make brushing hair easier.

A professor of applied math who used to try to detangle and comb his daughters hair noted that the topology, geometry, and mechanics involved in detangling posed some very interesting mathematical questions. The answers to these questions can also be applied to textile manufacturing, and certain chemical processes such as polymer processing.

This gentleman and two others recently published a paper explaining why the standard brushing method is the most effective way to detangle fibers. In order to look at the problem, they simplified it by simulating two helically entwined filaments rather than looking a whole head of hair. Using this model, they examined how the double helix is detangled via a single stiff tine that leaves two separate filaments and they measured the forces involved and the deformations connected with combing hair before simulating the results numerically.

They assigned a mathematical value to the short strokes that begin at the free end and move up towards the clamped end and end up removing the tangles as link density.  Link density refers to the amount of hair strands that are braided with each other. They also identified the optimal minimal length of the stroke, or the minimal length needed to detangle the hair.  Any shorter and the hair would never get detangled. Other people took this information and used it to create an algorithm that allows a robot to comb hair. 

The next step for these researchers is to look at the mechanics involved in brushing curlier hair and how it reacts to humidity and temperature. So they know have the mathematics to explain this method and it's interesting that the information could be utilized to teach a robot to comb hair effectively.  Let me know what you think, I'd love to hear.  Have a great day.

Fact Fluency Games Without Technology

 

I love having the students play games on line but I don't always have that option.  The internet at my school is so-so at best and if we have school wide testing, no one other than the testers can be on, otherwise they get kicked off quickly and frequently.  So I have several games I keep handy to have students play.  

One of my favorites is Bingo.  I use it to help students practice their multiplication and division facts, order of operations, fractions, and everything else but today I'm focusing on fact fluency.

I hand out blank bingo cards and have students write on either products if they practice multiplication or possible answers for division depending what I"m doing.  I let them chose the numbers they wanted.  Then I call out the problem like 7 x 8 and they have to come up with the answer of 56 and then cross out the answer.  I might go with 6 divided by 6 = and they have to come up with the answer of 1 and cross it out.  Yes some problems may have the same answers such as 6 x 6 or 9 x 4 so they might find the number is already crossed out but that has never been a problem.

Then there is checkers fact fluency where you place mathematical problems on all the white or all the black squares.  All pieces follow the normal rules for movement except the student must give the correct answer for the problem to land on that square. The winner is the one who has captured all the other pieces.

What about a match game with factors and answers on separate squares and you have to match them up.  For instance one card might have 8 and another card would have 4 x 2 and a third card would have 8 x 1. All three would match up because 8 equals the other two.  This could be done for any of the operations. A variation of this is to create a puzzle with answers and problems with squares.  The squares have a problem or answer around the sides so students match the problem with the answer on a different square. 

The photo shows how it might be set up for two squares. One is an edge piece and the other is a middle piece.  Students have to match the equation with the answer till it done.  It's like a jigsaw puzzle and requires some thought.

The last suggestion is for a fact search.  In this you create a grid filled with the numbers that make up factors so that students fill in the appropriate signs to make it true.  They might see 7. 8.  5. 6 and rewrite it as 7 x 8 = 56.  These are easy to find online such as this web site.  This is a nice way to have students practice it.  

This is just a few suggestions for things to do.  Let me know what you think, I'd love to hear.  Have a good day.

Practicing Fact Fluency Using Technology

 

In many school districts, students have not had proper math instruction over the past two years.  The students I work with have not had much math since the pandemic started due to being red.  They do not have access to the internet and their reading levels are quite low so they have trouble following any written instruction I include in take home packets.  So now that they are back in class, I'm slowly discovering their weaknesses.

I gave my students an assignment the other day that required them to read a verbal description of money and translate it into a written amount.  The description might be three dollar bills, six quarters, three dimes, and four nickels.  I discovered that half of them didn't know a nickel was worth five cents and about the same number couldn't multiply six times 25 or three times ten.  One of my students did have a strategy to account for that so she could do the answers.  She'd draw the number of coins and then skip count to get the answer.

Since my students do not know their multiplication tables, I have to include time in class for them to practice.  Usually, I incorporate it into the assignments but sometimes I have to look at ways for students to get a focused practice session.

One way is to have them use apps or web based programs for practice. The assignment should not be one in which they fill in the blank in ordered tables such as 2 x ___. = 2, 2 x ____ = 4 etc or 2 x 1 = ____, 2 x 2 = _____ etc because in the first they understand they have to write in the numbers 1,2,3..... in order.  In the second one, they can skip count to the end without really learning their facts.  Therefore, one should look for a program or app that has lets them solve problems that are not in order so they can't use a pattern to complete the assignment.

 One iPad app I've had students use is something called Sushi Monster which has them building fact fluency for both addition and multiplication.  It has seven levels for addition, five for multiplication and each level is made up of four rounds.  Each level targets 14 different numbers and if a student redoes a round because they didn't do well, or they need additional practice, the app uses new numbers so they can't write down the answers to use again. It also increases speed so they get faster at using their fact fluency.  The app is made by Houghton-Mifflin and best of all, it is free.  I have used it with both middle school and 9th graders effectively before.

Another free to download app is called Monster Math. This one is geared for grades one to five but it can be used for higher grades.  It allows students to practice all four operations in a game format and it is set up so students can practice multiple skills at once while meeting the common core standards.  The program uses stories and adaptive technology to set the expectations for where they are and not where they should be. In addition, it is set up with voice over narration so students can follow along. Another nice thing about this app is that it allows students to play with others if you want a more competitive game.   Although it is listed as free and can be used it does have a subscription option.

Then there is Prodigy games which is a web based online game site that is free. It is easy to sign up for and can be done at home or in school.  In addition, students can access it with or without a class code but if the student does it at home, they need permission from their parent to access it for more than one month.   The game site has students answer math questions to complete quests and earn in-game rewards and it is good for students up to the 8th grade but could easily be used in high school.

I'll be back next time with games that can be done in the classroom without technology.  I'm including nanotechnology based games because many schools do not have great internet including mine.  We have to quit using the internet if there are students are doing online testing otherwise people get kicked off.  Let me know what you think, I'd love to hear from you.  Have a great week.

Warm-up

 

If you need three pounds of tomatoes to make one quart of tomato sauce, how many quarts will you end up with if you harvest 327 pounds of tomatoes?

Warm-up

 If there are 3 globe sized tomatoes in a pound, how many tomatoes will be in 50 pounds?

Can You Get More Of The Pizza?

 

Is it possible to share a pizza so one person gets more of the pizza?  Let's see how it all works out.  We know that most pizza's are cut with an even number of pieces so if two people share a pizza, they will both end up with an equal number of slices. If the slices are of equal size, no problem.  If as in reality, the slices are not equal, then the person will go after the largest piece available to end up with the most.

Now to look at this from a more mathematical point of view since mathematics can explain the world.  When taking slices from a pizza, you either take the next one over - a shift - or you skip the empty spot - a jump.

If the person's strategy includes a number of jumps, then it is referred to as a j-jump strategy.  In addition, the pizza can be represented by a circular sequence with each piece being given a certain weight or the weight of P.  Now a player can have gain g if the strategy they use results in a certain number of pieces whose sum of weights is equal to g.

If a person does not have a jump strategy, there is only a 1/3 possibility of making it but if they have a one jump strategy, it jumps to a 7/16 possibility. If the person has a two jump strategy, it moves up to a 4/9 possibility.  Each jump increases the chances of coming out ahead. This is all based on an even number of pieces.

If there is an odd number of pieces, then it is a characteristic cycle instead of a circular cycle.  The characteristic cycle has an arc defined as a certain number of consecutive pieces with a specific length and weight. Furthermore, the arc of length defined as (n + 1)/2 is called a half circle. In the zero jump strategy, it has a potential of 1/3 as the lower bound.  This means it covers three half circles since the potential of v is the minimum number of weights of half circles covering v.  The upper bound is 100100100.

For a one jump strategy, the results depend on how and when the jump is made and could be a 1 if n is one,  4/9 if n is an odd number beginning at 15 or 1/2 for everything else,  a nice piecewise function.   It is best to use a one jump strategy when n is even or 9, 11, 13 or a two jump strategy for the odd numbers beginning at 15. 

The best two numbers of pieces the pizza should be cut into is 15 or 21 so one person gets more pieces than the other. I admit, it's been a while since I've had to read anything written in formal mathematical language so it took me a bit.  As a teacher, I don't spend much time reading things like this because I'm just trying to fill in missing foundational knowledge.  Let me know what you think, I'd love to hear.  Have a great day.