Ways To Improve Computational Thinking.

 

Last week, we looked at what computational thinking is and today, we'll look at it in general first, then specifically for math since both are important.  This is especially true if you work with teachers of other subjects so that you can work on it across the curriculums.

The first part of computational thinking is decomposition of the task, or breaking it down into smaller tasks.  One way to do this is to ask students to write a set of instructions on how to do a basic task such as mashing potatoes, planting a potato, or putting on your socks.  My English teacher did this to us in high school.  Although we did it in English, she chose a mathematical topic. She had us write the instructions on drawing a square.

We were given the activity in class but she let us work on it at home for a couple of evenings.  On the due date she collected all our instructions and set about following each and every word to determine who had the best set.  We laughed hysterically because people forgot to give a direction on the lines so she'd head off the paper or start in the middle of the page and she had fun.  I admit, I had a great set of directions because I wrote them and had my parents try to follow them so every time the directions messed up, I rewrote them until they were 100 percent correct.  This is an activity I do in my math classes and it is awesome.

Next is pattern recognition.  In foreign languages, it would be discussing the patterns used for the endings of verbs depending on the verb and tense.  It might also be dividing words up into piles based on if they have long or short vowels. It might be sorting words in science or math so the words associated with certain concepts end up in the same group.  Other possibilities for math include dividing equations into slope (positive, negative, horizontal, or vertical), how any steps needed to solve it - such as one, two, or multipole step equations.  When you do a sorting activity, have students explain the criteria they used to sort the items so they are sharing their thinking.

Let's look at ways to learn more about abstraction.  Have the students write down what they know on a topic but give them a final word count because this helps teach them to get rid of the irrelevant details. You'd start with a nice word count like 100 words and then cut it down to 90. then 80 so they learn to work with fewer words.  Another activity is to read a passage, looking for very specific information.  In English, you might have students read a description of a house and then have them find the details a house buyer might want before having them reread the description looking for details a housebreaker needs. For math, give a paragraph on quadrilaterals and ask them to read it looking for the details for squares, or rectangles, or other shape.  

Finally, is the algorithm which is one of those wonderful things we see a lot in math but in other topics such as English, it is asking students to write the instructions to make a peanut butter and jelly sandwich. In math, it might be asking students to explain how to solve a type of problem, prove something, and they could do it via a video or written poster.  

This site has some great lesson plans for algorithms for K to 8.  The lessons come with everything you need but they are more general activities than math specific.  I simply tell my students that they need to communicate using English so they need to do this.  On the other hand, this place has lessons specifically designed to teach computational thinking in all subjects including math however, they do not have a ton of lessons available.  

I have to have some things for the first week that do not start them fully in math, so this topic is great for that time.  Let me know what you think, I'd love to hear.  Have a great day.

Happy Memorial Day.

Happy Memorial Day.

Warm-up

 

If every box of bananas weighs 100 pounds and each banana weighs 4.2 ounces, how many bananas in the box?

Warm-up

 If one stalk of bananas has 185 bananas on it and each banana weighs 4.2 ounces, how many pounds of bananas do you have?

Improving Computational Thinking.

 

I plan to return to the idea of suggested games for students to play over the summer but I got diverted with the idea of helping students improve their computational thinking.  Computational thinking is defined as an interrelated set of skills and practices needed to solve complex problems.  A good definition for students is that computational thinking is learning to think like a computer or breaking your data down into a form a computer can use.

Although we often associate computational thinking with math, it can be applied to any subject. It is just learning to rearrange your thoughts logically.  Computational thinking is composed of four parts, decomposition, pattern recognition, abstraction, and the algorithm. 

Decomposition refers to breaking the problem down into smaller, more manageable parts. This is something we do everyday when we clean house, work on a project, or build something.  If we clean house, we break it down into smaller tasks such as starting with the kitchen.  We might rinse dishes and load the dishwasher, clean the sink, the area around the sink, etc.  We break the whole task into a step by step process.

Pattern recognition is just what it sounds, learning to recognize the patterns we see. In order to break things down, it is helpful to recognize the patterns.  In math, we do it all the time but it happens in other subjects such as in foreign languages, we learn endings based on the type of words and the case or tense.  The world is made up of patterns so it is important for students to learn to identify the patterns.

Abstraction is the ability to cut through all the information available to select only that which is important.  It is like reading a mystery and sorting through clues to separate the red herrings from the actual clues. In life, it would be like going to the store and only buying what is on the grocery list rather than what ever appeals to us.  In math, it would be like reading word problems and taking only the things we need to solve the problem.

Finally is coming up with the algorithms. Algorithms are the step by step instructions to do something.  We follow an algorithm when we get dressed, when we follow a recipe to cook, or follow that plan to make a bookcase.  In math, the algorithm is the process we follow to find the equation of a line from two points, or solve two simultaneous linear equations, and more.

Although we associate computational thinking with math, science, or computers, it is actually found in just about every subject, and life in general.  It is a skill used when we plan research papers, manage projects, figure out which work to delegate to whom to get it done. 

Now that computational thinking has been broken down, the next step is to discuss how to teach it since this is an important enough subject to teach to all grades and all students.  So next week, I'll tackle the topic of discussing how to teach it.  Let me know what you think, I'd love to hear.  Have a great day.

Do Digital Games Help Children's Math Skills?

Digital games are a great way for students to gain fluency of number facts. We know that students who can quickly recall their math facts tend to do much better in higher levels of mathematics.  If a student is not fluent, they find it more difficult to find answers for the more complex problems.  This is because space in the brains is freed up when students fluently know their number facts.

In addition, students are more likely to retain the material and are better able to apply it to future tasks. There is evidence that shows students can achieve fluency in number facts through the use of digital games.

Last time, I suggested sending home a list of digital games for students to play over the summer so they maintain their knowledge for fall.  As stated, one needs to make sure the games recommended will actually help students with their math skills.  The games could be against the computer or others. 

It has been found that games using math strategy to practice the four basic operations, percents, etc tend to have increased achievement rates on regular and state tests. In addition to raising fluency, the right digital games can also increase student enjoyment of math. However, it is important that the games must be engaging so students want to play and are motivated to continue playing. 

The good thing about digital math games is that they allow students to take an active role in their learning, provide immediate and constant feedback, and provide a visual connection between playing the game and the math they are learning. Furthermore, students do not need to spend hours online in order to see results. In fact, students only need a focused session of 5 to 10 minutes, three times a week to see results. 

Unfortunately, most of the information found focuses on its benefits in elementary school but it is known if students know their facts, they do better in the math classes they take in middle school and high school but there is nothing that says students can't work on fluency using digital games if they are in middle or high school.

There are games out there that also help students improve their ability to work their way through open-ended math problems since the right digital games help improve math proficiency, ability to think through problems rather than just learning the fact by rote. Many games focus on speeding up the ability to give answers to standard mathematical facts, not all help students conceptualize problems. In addition, a good digital game helps students learn productive practice which also a students ability to make sense of the problem. 

Next time, I'll suggest a few games that come highly recommended that help students develop both their numerical fluency while providing immediate feedback, and improving ability to work open-ended problems, Let me know what you think, I'd love to hear from you.  Have a great day.

Helping To Counter The Summer Learning Loss

 

We are getting to the time of year again when schools break for the summer.  The kids head out for around three months and when they return, there is a noticeable drop in scores when tested in the fall. It is often much lower than it was in the spring.  As teachers, we know our students do not want to do "Homework" over the summer. 

These nice long breaks up instruction, leads to learning loss, and requires teachers to spend quite a bit of time at the beginning of the next year to review so students end up where they needed to be on the first day of school. In addition, students suffer a 17 to 34 percent decline in learning over the summer without some sort of support.

Although we don't teach over the summer, we can send home suggestions of things students can do to help them reduce the drop they experience during their holidays.  It doesn't matter whether you are talking elementary or secondary students, we can still send home information with the students and to the parents suggesting ways students can keep up their skills.

One way to share information is to send it home via the school newsletter, Facebook pages, or other ways your school communicates with parents.  So let's look at ways to help students enjoy math over the summer.

First, it is suggested that parents take time to point out where math is found everyday.  If everyone is watching a basketball, baseball, or other sporting game, take time to talk about the stats.  How are they calculated, what do they mean, and how can you use them to tell which player is better.  Look at having a student cook regularly.  Let them choose the recipes, figure out the budget, learn to double or cut recipes in half, and more.  Expand this to understand sales tax, discounts, calculating sales prices, etc as the kids are shopping.  Or if someone is planning a project like installing fencing, redoing the paint, etc parents can take time to talk about how you go about figuring out the amount of supplies, shopping for the materials, and more.  

Second, Find books that deal with math such as those on Sir Cumference, or find math mysteries on Amazon, graphic novels such as the Japanese ones that talk about how certain math is done.  Look around, the books are there and parents can read them with their children if they are younger, and let the older students read them by themselves but ask all children to write a book report or book review on the story.

Third, make a list of math games recommended to help students practice various skills.  It is important to send the list rather than make a general suggestion because not all online games are designed to help students practice skills. Some games look as if they have a math flavor but really do nothing.  Don't forget to include a few card games such as cribbage, or sites with games such as Kahoot or Jeopardy.  If you want, prepare a list of possible games and assign them to students for the summer so they can play them.  I enjoy the games which require students to translate word problems into equations, or have them practice a skill.  

Finally, send home problem of the day calendars giving students one problem a day to practice and don't forget to send the answers home so students can check their work.  Don't be afraid to send home links to activities they can do at home involving mathematics.

If students do something math oriented every day for 15 minutes, they will do a lot towards retaining their level of math over the summer. Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

 2.4 million baseballs are used by the MBL during a season.  There are 2430 games each year so how many baseballs are used per game on average?

Warm-up

 

Wilson produces 2,100 footballs everyday.  How many footballs is that per hour, per week, per month, and per year?

Math Games Website.

 

The other day, I looked a nice game called Math Dash Ninjas on Math Games.  This site has a ton of other games one could use for the classroom either as a way to help with scaffolding, giving students who finish first something to do, or provide extra practice.  I wanted to see how the games differed from other sites and games.

I began with Math Agar.  Again, it began with asking me what grade level I wanted and what skill I would work with. I chose 6th grade and the skill of practicing multiplication.  It gave a problem and the idea is that the student solves it while moving your circle around to avoid the things that eat you and trying to find the correct answer in the background.  Without a calculator, students might find it difficult because the problem I was given was like -18.2 x = -200.2.  Not something you can easily do in your head.

Then I tried Number Worms, again began with asking what grade (6) and what skill (multiplication) it gave problems based on that information.  In this one, you are given a problem and you have to steer your worm to the correct answer before one of the bad worms catches up to you and drains you dead.  This one is a bit easier in that it seemed to stick to the more basic types of problems which a person can do without a calculator.

I had to try Cat Wars because of its name.  In this game two cats are playing tug of war. The idea is the more correct answers you get, the better your cat character does.  I chose 6th grade estimation and had problems like 92/8 and I had to come up with the closest answer rounded to the nearest tenth.  This was nice because I wasn't trying to avoid other creatures so there was less stress.

Then I saw Zombie Math.  I had to try it out since I know people who love playing zombie type games.  I again chose 6th grade and ended up practicing reciprocals.  Green zombies came out of the underground bunker. I had a problem such as find the reciprocal of 3/7.  The answer was not listed as 7/3 but as 2 1/3 so it requires more thought.  You have to flip the 3/7 to 7/3 and then make it into a mixed number.  I liked it.

The site listed 27 total games, each a bit different and each can be adjusted to grade and specific skill.  I did not check out every game but I did like the ones I tried.  I strongly suggest you choose a game and play it so you know what it expects and whether you need to choose a lower level for a student who needs additional support. 

Go look at it, pay some of the games so you know what is expected.  I plan to use it next year for some of my students, especially those who are well below grade level.  Let me know what you think, I'd love to hear. Have a great day.

Math Dash Ninjas - Online Game

Last time I looked at ways to help middle school and high school students learn their multiplication facts since knowing them fluently is extremely important.  In the process, I included the link to a suggested game - Math Dash Ninjas - which allows students to find a more focused practice.

When you click on play, you'll see a dialog box asking for the student's grade of Preschool to 8 it works for students who are below grade level or need to practice certain skills.

Each grade level brings up certain choices.  For grade 8, it brings up decimals, fractions, algebra, or mixed equations but 7th grade has these plus addition, subtraction, multiplication, division, ratios, equations, and estimation.  

On the other hand, if you click on skills, you have a choice of counting and cardinality, operations and algebraic thinking, number and operations in base 10, or geometry.  However, there are no live links connected with this page.

Once you choose the skill, the screen comes up showing a ninja getting ready to run the  course.  The student starts with 25 free dodges so if they hit the object in the way, they use up a dodge and for every correct answer, they gain a free dodge.  

A problem is shown with four different choices for answers.  If the answer is correct you get a green light and a yeah its correct but if the answer you select is wrong, it says incorrect, shakes, and you loose a heart.  If you have three wrong answers, your ninja dies and the game tells you how far you went.

This is a nice game because allows students to practice so many different skills based on grade level and topic. If a student needs extra practice in one area, they can work on it because the game structure is the same for all the games. Check it out.  On Friday, I'll be looking at some of the other games at the site.  Let me know what you think, I'd love to hear.  Have a great day.

Multiplication In Middle And High School.

 

Due to Covid, I have students who do not know how to multiply or divide and can barely add or subtract.  It is well known that many students missed out on learning and practicing this due to not being in school.  In addition, we know that students do better when they know their multiplication tables solidly and know how it relates to division.  Students who know their multiplication tables are better able to see patterns, relationships, while supporting more complex mathematical processes.  

Fortunately, there are ways to help students practice their multiplication tables in class without losing too much time from the regular instruction.  Think about using apps or websites that allow students to practice their multiplication.  Do not go for sites that only drill them repetitively since drilling does not always work.

Think about providing a multiplication chart but have it filled only with the facts they have not learned so the ones they know are blank. If they know their facts up to 6x6, then the chart would only have the facts beginning with 6 x 7 and above.  Or think about using factor triangles with the 6, 7, and 42 so they see the relationship between multiplication and division.

Furthermore, take what they know a step further.  If they know 6 x 6 = 36, then take time to help show them understand that 6 x 6 means 6 groups of 6 so 7 x 6 is seven groups of 6 so they know they are adding one more group of 6.  It is also important to help them see that 6 x 7 can be seen as 6 groups of 7 which still gives them 42 objects. When teaching nines, do the finger trick so they can easily find the answers with their fingers.

Start the year with multiplication facts via warmups or bell ringers.  You could use a mad minute worksheet or one that is blank multiplication chart where students write in what they know so you have a better idea of what they don't know.  

Think about an online math station by having students enjoy something like Math Dash Ninjas. This site has you choose a grade level and math topic to practice.  The idea is that for every correct answer, you get dodges but students do start with some so if they make a mistake, they can still avoid being hurt.  It is fun and done well so students have a chance to answer.  

Think about having practice on Friday via Kahoot or Quizz which have premade games.  I've used time on Friday to have students practice writing equations from word problems by playing Kahoot. They like it a lot.  

These are just a few ways to help students practice their multiplication. Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

 If a hummingbird flaps its wings 200 times per second, how many times does its wings beat in one minute, one hour, and one day?

Warm-up

If an anteater eats 30,000 insects every day, how many would they eat in one year?

 

Algebra Balance Scales

Teaching students to solve equations can be difficult, especially when you want to find a visual representation.  Although there are so many different ways to show how to solve equations visually, I like the Algebra balance best.  I think it is able to visually show the operations one uses to solve them and they see the order.

There are several reasons to use the algebra balance either virtually or in reality.  First, as mentioned earlier, it provides a visual representation of the actual equation and why you add or subtract or multiply and divide to both sides.  Without the visual, students end up automatically following the rules without understanding the why. 

In addition, a visual representation helps make the abstract concept more concrete and algebra has some very abstract concepts so these balance scales really help make them easier for students to see.  They also allow students to manipulate the equation so they see what is happening on each side.  They can see when something is not correct because the scales are out of balance. 

Another thing is that using a balance scale helps students develop a deeper understanding of equations because they see that 2x + 1 balances 5 and they are equal.  Often students do not understand that so this way of visualizing the equations allows them to understand equality a bit better. They also connect the idea that the variable represents an unknown and the variable does represent a value that we don't yet know.  

The Algebra balance scale also helps a variety of students learn.  It provides the hands on activity for those who need a kinesthetic element and also helps with differentiation and scaffolding.  Furthermore, having students use an algebraic balance helps improve their problem solving abilities because they have to bread down complex problems into simpler steps while developing strategies to solve each step.  This is important because students will face problems in real life that require these same skills.

If you haven't tried them before, do so.  Many students actually have fun while learning the algebra.  Let me know what you think, I'd love to hear.  Have a great day.

Online Math Games For Middle School/High School

 

In addition to having a few paper based games, I like having some available for students who finish early or need a bit extra practice in certain areas such as algebra or pre-algebra.  I know that many of my students like to play games if they finish early and it is a good incentive  to have them finish.  So we are off to learn about some free games.

Hoodamath has a nice little game called Integer Tilt.  It is a bit different than many of the integer balancing games.  The idea is that you use your left or right arrow keys to move the dropping block to one side or the other.  You want to keep the two sides as close to being in balance as possible.  

In addition, there is a x - 1 square that drops down every so often and it changes the value from positive to negative or vice versa so it is handy at certain times.  If you get too much of a difference or you miss one, the game ends.  It is a game that can be played easily with little time needed to learn it.

Another site, Math Game Time, has several games that are based on algebraic skills.  One that I played that I like was Spider Integer  The idea is that you are a spider and a number is given such as -1. Then you spend the next minute or so finding flies with the correct number on them so you end up with -1.  You look for -8 + 7 or -3 + 2, click on them and you end up winning when you find the most pairs that equal - 1.  There are other games that have you practicing your addition such as 6(3) - 2(3) and you select the correct answer.  Although some of the games labeled as "Algebra" are not really, they do give students the opportunity to practice such skills as multiplication which they may need.

Try "Who wants to be a hundredaire?" based on the Who wants to be a millionaire TV series.  The questions for this game require students to practice using square roots. The questions require everything from a standard addition or subtraction problem to requiring one to simplify sqrt20.  If they can answer ten questions in a row, they earn $100 pretend dollars. I found it fun.

Mathplayground has a lovely game using an algebra balance.  This gives the students a problem and guides them through the process of solving it step by step.  It is not so much a game as it is a way of practicing solving one or two step equations and really cool.  

So now you have a few games and at least one practice manipulative to help students improve some of their basic skills.  Games are always a good thing to have for students when they need something to do when they finish early.  Let me know what you think, I'd love to hear.  In the near future, I'll look at some appropriate geometry games.  Have a great day.

Math Games To Use In Middle School and High School Classrooms.

 

Since I work in a place with 2G cell phone service and questionable internet, I like to keep a few games around in my room just incase I need a filler, or the kids are having an off day.  There is nothing worse than having a fire drill in the middle of math and trying to resume study when we get back so its nice having a few games around.

So I am always on the lookout for games that one can just pull out. There are several I've found that I'll share with you today.  I like making folders that contain several copies of the games so they are ready to go as needed. There are some games that will use online sources and I'll include a link to those.

First is a math tic-tac-toe game.  This one can be used for just about any math topic but for this example, I'll talk about it using decimals.  Begin by creating a worksheet with problems such as addition or with all four operations.  The class is divided into groups of two.  Each person is given a worksheet and assigned a color. The students take turns answering the questions.  If they find the correct answer, they get to color in a box, if they are wrong, they do not color in a cell.  The winner is the one who establishes three in a row first.  This is one that would be good to have laminated tic-tac-toe boards and dry erase markers.

Next is math baseball which is a game to get students up and moving around. First thing is to divide the students into two teams.  Next have each team create a bunch of questions and rank the questions as worth one base, two bases, or three bases based on difficulty. When the game starts, the first team asks the second team a question.  They announce the value of the question, give the question, and the other team tries to solve the question correctly.  If the team gets it, they get that many points, if they miss it they get nothing.  The teacher acts as scorekeeper and the winner is the team who reaches 10 or 15 points first.  The teams alternate back and forth asking questions rather than waiting for three outs.

Think about setting up a trasketball game one day.  First step is to create a supply of questions. Then divide the classroom up into teams of 4 to 5 people. Each team is given a response sheet and one student is designated as team captain who is the only one who can answer questions. Decide the value of a correct answer such as one correct answer gives the team two chances to make a basket.  As the questions are asked, the students work together to correctly solve the problem.  If the answer given by the captain is correct, the team wins the opportunity to shoot a ball into a basket or trash basket.  If the teacher wants to, they could designate the team that is behind, call "Shoot!" so the team has a chance to equalize the score using their basketball playing ability.

Although the next game is set for trigonometry, it could easily be used with families of graphs and similar groups of math items.  For this game, make up two sets of cards with the six trig functions, with their associated graphs, graphs, period, domain and range.  These cards are distributed among all the students.  Then students take time to match up the equation with the graph, period, domain, and range held by other people.  If someone makes an incorrect match, the others get to say "Go Fish." The student with the most matches is the winner.  

Check back Wednesday for more games.  I will finish off this grouping then.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

 

If there are 3 cups of shelled peanuts in a pound and one pound makes 1.5 cups of peanut butter, how many pounds of peanuts do you need to make 64 cups of peanut butter?

Warm-up

 

If 18 percent of a pound of unshelled peanuts is shells, how many ounces of peanuts are there?

Some Misconceptions Associated With Negative Numbers

 

One of my Algebra students had trouble with solving two step equations with a negative coefficient on the variable.  He saw it as subtraction rather than a negative number so he was rather confused.  I spoke with him more and learned that due to Covid, he'd missed out on learning that x - 3 is the same as x + (-3).  This lead me to looking into some of the misconceptions associated with negative numbers.

One misconception has to do with the idea that any negative number is less than zero which they get from number lines that have the zero in the middle with positive numbers increasing in value from zero and negative numbers decreasing in value as you move left on the number line.  Although, most students are great with determining order of positive numbers, they are more shaky when it comes to determining whether -3 or -7 is more.

I don't know if this is a misconception or if it is something students do because they don't want to borrow but I've seen students do one of two things. First, they have 62 - 29 so rather than borrowing, they treat it as if the actual problem is 69 - 22 = 77.  The other one is if they have a problem like 29-62, they just can't do that and want to do it on a calculator.  I think it might be due to the type of problems they work with as they learn the concept.  Most textbooks keep the problems quite simple like 5 - 8 rather than throwing in some like 87-105.

In addition, I get students who hit middle school and high school who still have issues with the idea that subtraction is the same as adding a negative number or is indicating a direction on a number line.  If a student see a problem like 8 - 2, they will tell you that it is 6 but if you write 8 + (-2), they are confused and do not see the two problems as the same. The same applies to 5 - 8 being the same as 5 + (-8).  As far as directions on number lines, I've seen students find the 5 and when they applied the -8, the begin at zero and moved to the -8 on the number line rather than moving 8 in the negative direction to get -3.

Furthermore, they also have trouble with multiplication and division of signed numbers.  They know that a negative times a negative is a positive but they forget that a negative times a negative times a negative will yield a negative number.  The same applies to division.  Add to that, the issue of a slope with a single negative sign, they often try to apply the negative to both the change in y and the change in x rather than seeing it as applied to only one as in -1/2.  Then if that isn't enough, trying to explain that a slope of 2/3 is the same a -2/-3.

I'm sure there are more but these are the ones I've seen repeatedly in my classroom.  I'd love to hear of other misconceptions from you.  Let me know what you think, I'd love to hear.  Have a great weekend.

Misconceptions Associated With Fractions

I teach grades 7 to 12 all in once classroom.  I have a 7th grader who never learned about fractions but could do them with a calculator as long as it allowed him to do fractions. However, he couldn't tell if the answer from the calculator was correct.  I convinced him to try learning fractions by using fraction strips.  

Just this past Friday, I discovered he had misunderstood equality of fractions.  He uses those foam fraction strips and so he could squish some a bit so they seemed to be equal such as 3/4 = 7/10.  I had to take time to explain they could be split up evenly.  It left him a bit confused but I'm glad I saw the misconception.

Today, we'll look at other misconceptions that arise when students study fractions.  Some folks believe that fractions are only used in math, not realizing they are found in cooking, building, the size of tools, and so much more.  Knowing how to work with fractions is to have a real world skill.  In addition, although it seems as if fractions are a small part of mathematics, they are used over and over again in several forms.  When one understands fractions, it makes it easier to understand decimals, and percentages and the relationship among all three.

Students often think that the bigger the denominator, the bigger the fraction rather than understanding that the larger the denominator, the more pieces and each piece is smaller. Thus 1/2 is bigger than 1/4 which is bigger than 1/11. Speaking of denominators, students are taught to use the other denominator to find a common denominator when adding or subtracting.  An example might be 1/2 + 1/4 and they use 8 as the denominator rather than 4 which is the lowest common denominator. They learn this way of finding the denominator because elementary teachers often teach them something called the butterfly method.  I gather you would cross multiply to get the numerators while multiplying the denominators to get a common denominator.

Another misconception is that fractions must equal one or less, they cannot represent something bigger than one.  So when they are asked about 5/3, they have difficulty understanding it is one whole plus two thirds.   Of course, most students learn that fractions represent a part of a whole which is nice but it doesn't help students when it is time to learn about ratios, unit prices, division, or comparing two quantities. 

These are the big ones that I find students have when they get to middle or high school.  I have to go back and clarify the topic.  Let me know what you think, I'd love to hear.  Have a great day.

The Connection Between Math And Card Shuffling!

 

As we all know, math is used to explain the world around us.  We've found that certain mathematical equations can explain more than one thing.  Mathematicians have been studying the math of card shuffling for the past several decades. This has lead to the development of other concepts such as random walks, group theory, and probability theory.

When the cards are shuffled, the order of the cards s randomized. This means the probability of any particular ordering of the cards is the same as any other possible ordering, assuming the shuffle is truly random.  This means the number of possible orderings of a deck of cards is very large, specifically 52! or about 8 x 10^67 possible orderings.

In order to study the probability of certain card shuffling methods, mathematicians used techniques such as Markov chains and permutation groups.  Markov chains are used to model the probabilities of different outcomes based on previous outcomes, while permutation groups are used to study the possible rearrangements of a deck of cards.

In addition, mathematicians developed a variety of different card shuffling techniques designed to produce different possibilities of different outcomes.  Some of the most commonly used shuffling techniques include riffle shuffling, overhand shuffling, and Hindu shuffling.  By studying these different shuffling techniques mathematically, they have been able to gain a deeper understanding of probability and randomness and even have developed new methods for shuffling that are more effective and fair.

So the study of card shuffling has focused on the basic question of how many shuffles does one need to completely mix a deck of cards. This question is one that is great from both a theoretical view and from a practical because the answer to this question has applications to cryptography and computer science.

One theory - "The Seven Shuffle Theorem" which states it takes no more than seven shuffles to thoroughly mix the deck.  This theorem was first proved in the 1980's but since then it has been refined and extended by other mathematicians. 

The interesting thing about the connection between card shuffling and math is the way shuffling techniques are used in cryptography. See, randomness is essential to any cryptographic system and shuffling is one way to generate random numbers used to encrypt messages. 

The connection between math and card shuffling is a fascinating area of study and has lead to many other insights and applications in both mathematics and in other areas. I've always found this topic interesting and remember hearing about the seven shuffle theorem so when I open a new deck of cards, I always shuffle it seven times.  Let me know what you think, I'd love to hear.  Have a great day?