A New Way To Look At The Universe.

I love those clear nights when you can look up at the sky and actually see the stars, the aurora borealis, and so much more.  Unfortunately, they built a gas station on the corner near my house and now due to the light pollution, I don't see the sky as clearly as before.  None the less, I can see things.  I found an article about a group of scientists who found a new way to figure out things associated with the skies. 

As you know, they used math to discover at least one planet and several other astronomical objects instead of relying on on observations to expand their knowledge. Recently, researchers used a machine learning algorithm to look one simulated galaxy and from that they were able to predict the makeup of the whole digital universe the galaxy is in.  This is like looking at a grain of sand and using that information to calculate the mass of a moon.

This discovery will someday allow cosmologists to make conclusions about the cosmos based on it's basic building blocks.  What this does is it makes it easier to study the universe because you only have to examine one galaxy to extract enough information to make accurate conclusions about the whole universe.

This developed from a challenge.  The challenge was to create a neural network based on the properties of a galaxy  that could estimate a couple of cosmological attributes.  This challenge was given so the person could become more familiar with machine learning. In the process, it was noticed that the program properly calculated the density of matter.  

The neural network itself analyzed one million simulated galaxies for size, composition, mass, and other characteristics,  but this research only used 2,000 digital universes created by the program. The neural network was able to connect the information to the matter in the parent universe.  Most of these universes were composed of between 10 percent and 50 percent visual matter with the rest being dark matter.  In the simulations, the visible matter and dark matters swirled together forming galaxies.  The simulation even applied events such as supernovas or jets erupting from black holes.  

Furthermore, the neural network was able to predict cosmic density within 10 percent regardless of the galaxy and this result was quite unexpected since galaxy by their nature are quite chaotic. No matter what type of galaxy, they were able to keep up with the overall density of matter. This leads to the idea that both universes and galaxies are simpler than previously thought. 

This then allowed researchers to analyze the neural network to make sure it wasn't getting the data from coding of the simulation and discovered the neural network used things like the property associated with the rotational speed that determines the type of matter found in the central zone.  In addition, the neural network is able to look at multiple properties at once rather than one or two at a time like humans. 

Scientists are not sure which properties the neural networks are able to connect with which galaxies or universes yet since this is just in the beginning stages but the process does open up some interesting lines of investigation.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

 

Warm-up

 

History Of Math Online Journey

There are two questions a math teacher hears on a regular basis.  The first one "When will I ever use this?" is often asked whenever a new topic or concept is being taught. The second  "Who invented this stuff?"  is asked when students get frustrated or don't think it's important enough to learn.  Well in answer to the second question, you can take the students on a field trip to the history of math exhibit.

I think it is great being able to take students on an expedition to checkout the history of math with real articles rather than having them read a book on it.  In fact, some of the exhibits are ones that students can relate to. For instance, there is a clay tablet with a student's practice of multiplication tables from Ancient Sumerian.  The tablet written about 1900 B.C. includes all the errors the student made and it is something students can relate to. 

This exhibit was put together by the National Museum of Mathematics in New York City and Wolfram Research.  There are about 70 artifacts contained in the exhibit showing how math cuts across cultures throughout history.  These artifacts are divided up into 9 areas that show the development of various key topics from counting and prime numbers to geometry, to algebra, and arithmetic.

Each of the 9 areas have a short timeline with artifacts that act as entryways into the topic so students can explore milestones in the history of math in more detail.  One artifact is considered one of the earliest surviving calculating machine from the Greek island of Salamis and dates to 300 B.C. This calculating machine predates the abacus.  It operates by having people move pebbles across a board.  On the other hand, the earliest documented time that zero is used for a place holder, appears in a document dated from 300 A.D. that comes from India. 

For those into algebra, there is a reference the Al-Jabr.  This book established the field of Algebra and written in 820 A.D. by a Persian author. It is said that the term Algebra comes from this title. In addition to having all this historical information, the exhibit takes time to show that modern math developed from a need to keep track of people and supplies due to the growth of cities. 

The exhibit also shows the way some cultures shared the same idea at different times.  We all know the Pythagorean theorem of a^2 + b^2 = c^2.  It is taught in geometry and is used to find the hypotenuse of a right angle triangle but did you know know there is a clay tablet from Mesopotamia created a 1000 years earlier contained the same information.  There is also evidence that both the Chinese and Indians knew about it too.

I took a look at the site to explore the history of Algebra because I have to teach it. I clicked on the page from Al-Jabr which explains how X^2 + bx = c is solved visually. The page shows where the page was found, where it appears in the mini timeline, an interactive point where you play with the visual representation, and an in-depth explanation of this concept and links for additional information.

Check it out, see if you can use it in your class.  Let me know what you think, I'd love to hear.  Have a great day.

Purposeful Play In Math

 

When we think of using play in the classroom, we automatically think about the elementary classrooms since that is where most of it seems to happen.  That may be because much of the research shows that student learn well when using play in the early years to develop literacy and numeracy skills. In addition, students tend to be more involved when they are able to "play" as a way of finding solutions to problems on their own. 

Furthermore, play can help students use the creative part of the brain to come up with solutions to problems and they can discover new concepts and ideas at the same time.  In addition, integrating play into the classroom motivates students to want to learn math because it's fun and they are then engaged.  

One of the all round toys to use in all grades from kindergarten to twelfth is Legos.  Legos can be used to build things while letting students learn more about math skills.  I've seen fractions taught using different sizes to represent fractions such as the block that is 2 by 4 is designated the whole, a 2 by 2 is considered a half, the 1 by 2 represents a fourth, and a one by one is the visual for an eighth but this way tends to restrict the fractions that can actually be used.  

A different way of using Legos is to use different colored blocks to show the fractions.  For instance, if you want to show 3/5, you would have say one blue 2 x 4 block, a other blue 1 by 4 block and one red 2 by 4 block so that you have 5 rows of blocks and 3 are shown in one color to represent the 3/5th's.  This method of representation allows students to find ways of showing more fractions with many different denominators.  It also opens the way for a discussion of 2/5th being needed to make a whole. 

Legos blocks can also be used to help students visualize prime numbers.  If we think of the numbers representing the total area and the factors as the sides, we arrange the factors to create either rectangles or squares.  If the number is prime, there is only one way to arrange the legos into a rectangular shape and that is 1 by the number.  Composite numbers can be arranged in more than one way.  An example would be 8 where it can be arranged as 8 by 1, and 4 by 2.  

These blocks can also be used for questions like how many ways can you get an area of 64 feet squared, or  what is the biggest area you can get from using a certain number of blocks.  In addition, students can figure out the perimeter of visuals such as the perimeter of a rectangle that is 4 by 2. The same exercise can be used with students to work on their multiplication tables since multiplication is represented using the area shape.  

Speaking of multiplication, if you designate the 2 by 1 as X and the single 1 by 1's as ones, you can show binomial multiplication or use the 2 by 2 to represent X^2 and create trinomials for students to factor using a square.  

Finally for geometry, they can use Legos to create three dimensional shapes for geometry so students can work on volume and surface area of rectangular prisms, and cubes. 

I've discovered that middle school and high school students don't mind using Legos to create visuals of the concepts because they don't see these as manipulatives, especially if you allow them to build what they want at the end.  Let me know what you think, I'd love to hear.  Have a great day.

Subtracting Mixed Numbers - Borrow or Use Improper Fractions

 

Due to the Christmas holiday travel, our school has gone red for the second week in a row.  It had to do with the number of positive cases discovered.  As part of sending packets home, I'm including lots of detailed instructions.  One of the topics I am covering this way is subtraction of a fraction from a mixed number such as 5 1/3 - 2/3.  I realize the usual way to teach this topic is by teaching them to borrow before subtracting but it is extremely hard to teach by distance.

In addition, my students automatically rewrite it as 5 2/3 - 1/3 in their heads so they don't have to borrow.  With this in mind, I decided to teach this type of subtraction using improper fractions because there is no misunderstanding about which fraction is larger.  Furthermore,  they find changing mixed numbers into improper fractions much easier than borrowing.

The school sends home packets since most of the students in the village do not have internet access and limited data on their phones.  In fact, most students do not have computers at home so any computer work is done at school or when they are allowed to check out a digital device.  So I've been creating packets of work with lots of in-depth information and examples.  As I wrote out the instructions for borrowing a whole to rewrite the fraction, I realized it was just too hard to assemble a clear explanation so I switched.  I switched to  showing subtraction of mixed numbers using improper fractions.

When I wrote up the explanation, it was so much easier.  It made sense and students could "see" what was happening.  Even if they have to find common denominators first before they convert the mixed number to an improper fraction, they won't be readjusting numbers in their heads so they can subtract without borrowing.

In addition, it gives them practice for when we start multiplying mixed numbers later on.  It won't be the first time they are converting to improper fractions, nor is it the first time they've performed a mathematical operation using improper fractions.  Furthermore, learning to subtract, multiply, and divide using improper fractions shows students the need for learning how to convert between the two forms and for learning to use improper fractions.

In the past, I've never thought about using improper fractions for subtraction because I'd only learned to borrow before subtracting and I was teaching the concept in person but the situation caused by COVID has shown me that perhaps I need to teach both ways but focus on using improper fractions since it makes more sense.

I guess this means that an old dog can learn new tricks.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

 If there are 120 beans in a pound and each costs $0.95, how much is a pound of vanilla beans.

Warm-up

 If a vanilla bean plant costs $22.00 and you can sell a vanilla bean for $0.95, how many do you need to sell to pay off the plant.

Ways To Scaffold In Math

Wednesday, I talked about scaffolding in more general terms but today I'll talk about specific ways of scaffolding.  It is great to have some suggestions to implement this in the classroom since many teachers need the specifics.   

Rather than teaching a full lesson, break it down into smaller mini lessons that build on each other.  This is what I'm doing when I looked at adding and subtracting fractions with unlike denominators.  I looked at everything they would need to find like denominators and decided to have a small lesson and practice on each of the topics.  This provides a safety net and insures they have everything they need to do the assignment. 

Provide an example of what they will be learning.  In addition, talk about the example as you are showing it.  This might include discussing each step so they know what is coming.  In essence it is providing an in-depth preview to the topic. 

When you teach the topic, provide multiple ways of describing the concept so students understand it.  For instance, when defining prime numbers, I give the standard definition of a prime number but I also show a picture of it so they can "see" it.  If you think of the numbers as representing area, the area has to be in a regular shape such as a square or a rectangle.  Prime numbers can only be shown one way as one by the number to have a rectangle while composite numbers can be represented at least two different ways and these representations show the factors for the number.

In addition, after you've shown them how to do it, give them a chance to try it themselves before assigning the work.  Be sure to break all the directions down into small chunks so they can follow the steps one at a time to get to the answer. When you break down the steps, you are providing scaffolding for those who need it. In addition, it is important to slow down, so we give students a chance to process the material.  It is not necessary to race through the book to cover everything.  It is more important to give students a chance to really learn the material. So slow down, pause, give students a chance to process it at different stages throughout the process.

Don't forget to use visuals,  The visuals might be videos, drawings, or an object but it is important for many students to have these.  If you are teaching geometry, you might have cut outs of a concave and convex shape to show the difference.  If you are teaching division of fractions, it might be a short video on how the process appears if we could see it.

Always make sure to cover vocabulary at the beginning so you can use the proper terms when teaching the concept, otherwise the lack of vocabulary might provide a stumbling block.  By front end loading the vocabulary, this stumbling block is removed.  In addition, take time to connect the new concept with previous concepts they have already learned. By activating prior knowledge, they can relate to it better.

This is not all of the suggestions, I'll continue this on Monday, adding several more techniques to help you scaffold your learners.  Let me know what you think, I'd love to hear.  Have a great day.

Scaffolding To Help Students Learn. will

Many of us are dealing with students who are well behind due to lockdowns, quarantines, and COVID cases. The group of students I have now, have not had math in over a year so I'm working on incorporating a lot of  scaffolding to fill in gaps in their foundational knowledge without having to begin at the beginning.  

Scaffolding is a way of providing support to help students bridge gaps and learn better.  In a sense, it is filling in the areas where students are weak so they are better able to do mathematics.  Scaffolding is not just for students who are behind, it is for all students who have any gaps in their knowledge.

There are two types of scaffolding one may use:  hard or soft.  Hard scaffolding refers to using techniques and activities that directly impact student learning and require advanced planning such as games, or the way a lesson is taught.  For instance, I am getting ready to have students review adding and subtracting fractions with unlike denominators.  For my students to do this, I have to teach them about prime factorization and least common denominator.  To take it further, I'm reviewing what a prime number is, how prime factorization works and at least two ways to find the least common denominator including one using their multiplication tables. The other type of scaffolding is the soft scaffolding which is a more indirect method which might use targeted questions.

As far as questioning goes, remember Bloom's Taxonomy when you create questions.  Questions are either remembering, understanding, applying, analyzing, evaluating, or creating.  When you ask a question, the way it is asked will be determined by your end goal which is to help students develop a deeper understanding of the topic.  Furthermore, the use of questioning as a scaffolding tool can actually help students learn faster, not slower as you might think.

Another way to scaffold is to provide two or three simpler problems before assigning the more complex problem of the same type.  For example, if the complex problem asks students to go from one number to another in a specific number of steps by multiplying by one number or subtracting another number,  students might struggle. However, if you have students do two easier versions such as going from the starting number in the complex problem to the next counting number in a smaller number of steps either multiplying or subtracting by the same numbers in the original problem. Then provide another simpler problem starting at the same number but has a larger jump with a couple more steps before assigning the original complex problem. 

So Friday, I'll look at a variety of scaffolding techniques we can easily use in the classroom so students gain in understanding and ability.  Let me know what you think, I'd love to hear.  Have a good day.

Making a Comic Strip Based On A Word Problem.

 

This past Friday, I discussed turning word problems into comic strips to make it easier for students.  I also talked about having students learning to do this as a way of helping them understand the problem.  Today, I'll be discussing how to do this so either the teacher or the students can do it.  

One should decide whether to use a computer based program or one on the tablet.  This article lists 21 different sites that allow students and teachers to make comic strips.  The sites range from fairly simple to more complex.  Try the various ones out because some are made to turn photos into comic strips while others are designed for creating either simple or more complex comics.

For this problem, I used Canva, since I have an account there.  The only real issue I had with it was trying to find free images..

The first thing I did was visit grade 6 in IXL to find a word problem that has student subtracting two fractions.

"Naomi filled a bucket with 9 3/5 gallons of water.  A few minutes later, she realized only 4/5 of a gallon remained. How much water leaked out of the bucket?"

Next, I had  to determine how many frames would be needed to express the problem.  For this one, I could do it in two frames but three is better.  In the first, we have Naomi with a bucket containing 9 3/5 gallons of water.  In the second frame, we have the same scene but the bucket only has 4/5 of a gallon.  The third frame would have Naomi asking the question.

So I know that in frame one, I have a girl holding a bucket that is pretty full.  In the bubble, I'll put something about filling it with 9 3/5 gallons of water.  In the second frame, I'll show the same girl with a mostly empty bucket with a comment about only having 4/5 of a gallon of water.  The final frame shows the question about how much water leaked out of the bucket.

Not too bad.  I made a different one on Pixton.  Pixton can be signed onto using Google, Microsoft, or Facebook.  In addition, it allows both teacher and students to use it.  The reason this one has the bucket as background is that I had to bring the bucket in as a photo.  Pixton does not have "things" like buckets.

Take some time and explore the different comic strip makers.  If you don't like any of the ones on this list, do a search and have fun looking for one you like. Once you've got the hang of using the program and you've provided several problems in comic strip form to your students, let them make their own.   Have them preplan the strip before they make it so they get the info correct.

Have fun, I'm off to make several for next week.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

 If one metric cup (8.5 oz) of coconut water weighs 254 grams, how much does a gallon of coconut water weigh?

Warm-up

 

If the young coconut contains 225 ml of coconut water, how many coconuts are needed to fill a one liter container?

Use Comic Strips Instead Of Word Problems.

 

We are always looking for ways to explain their thinking or understanding of word problems.  We talk about drawing pictures, restating the problem, and other ways to make sure they understand the problem but what if they take the information and translate it into comic strip or are presented with the word problem already turned into a comic strip?  If we eventually have students turn word problems into comic strip, it allows students who have an artistic bent to connect their art with math.

I think one reason students tend not to draw pictures showing the information in the word problem is that so many think they have to be good at art.  They don't understand that it only has to be a quick sketch but if you have them create a comic strip using some program, they are more likely to do it.  Furthermore, too many do not stop and really read the problem.  They see numbers and try to use them in some way to get an answer, even if the answer makes no sense.  They don't see the context in the problem nor do they look for key words, or even slowly read the problem.

Asking them to create a comic strip of the word problems means they have to slow down, read the problem, before figuring out how to show it visually.  Turning the word problem means creating a visual showing what is happening in the problem.  You can see if two people are talking, or there are three pets, or a person and their three friends.  The picture shows so much.

It has been found when students are presented with the word problem in the form of a comic strip, it reduces comprehension problems so even the low readers "see" what is happening. Furthermore, it chunks information making it easier to remember and understand.  In addition, students see comic strips as easier than regular word problems even though they contain the same information.

Another advantage to turning word problems into comic strips is for students who do not know what something is.  For instance, not everyone knows what a landscape is but if there is a nicely done yard in the comic strip, students have a visual reference point.  Furthermore, if these are teacher made, the teacher can make small changes to turn the situation into something students might relate to.  For me, it would be turning those road trip average questions into going to certain local villages.

A good way to introduce students to comic strips showing the word problem is for the teacher to do several so they can see them.  Once students are used to see word problems in this form, it is time to have them make some.  I'll talk about breaking down word problems and turning them into comic strips for both students and teachers on Monday.  Let me know what you think, I'd love to hear.  

Writing Mathematical Poetry.

 

Every so often, the administration gets the idea that we need to do some sort of cross curricular activity.  When I see the edict, I pull out my poetry writing activity and have the students do it once they get the "This isn't English" comments out of their system.  Sometimes, students come up with some really good ones and other times they struggle due to their language abilities.

In addition since April is both National Poetry Month and Mathematics and Statistics Awareness Month, it has been dubbed Math Poetry Month. You can also have students write math poems in April.

A mathematical poem is a poem using vocabulary, definitions, formulas, and concepts to describe mathematical things.  It might be about one of the operations, the math that describes something in nature such as a snail shell, or statistics used for basketball or baseball players.  

Writing a mathematical poem requires the poet follow a few steps and at the end, they will have a poem.  The first thing is to choose the topic and narrow down the topic to something narrow.  For instance, New Year is too broad a subject but if it is narrowed to the dropping of the ball in Times Square, that is doable.

The second thing is to look at mathematical vocabulary associated with the topic.  The idea is to list all the terms you can find on that topic so you have lots of choices.  It might require a trip to the internet, the textbook, or even asking others.  If I did a poem on the Times Square ball, I'd want to look up facts about it before I start looking at mathematical vocabulary so I know what to look for.

There are several different poetry forms that can be used.  A student might want to choose to use a haiku form with three lines, where the first and third lines are composed of five syllables and the middle line has seven.   They might prefer using rhyming couplets, stanzas, sonnets or a longer poem with no set structure.  On the other hand, they might look at mathematically based structures such as diamanté which is a diamond shaped poem composed of seven lines so the lines are one, two, three, four, three, two, one word(s) or a Fib poem whose lines with the number of syllables math the Fibonacci sequence.  Line one has one syllable, line two has one syllable, line three has two syllables, line four has three syllables and so on.

Once the first draft is written, students can go through the process of rewriting to make it better.  They might need to rearrange words, shift out vocabulary for better choices, or check spelling until it is just right. 

Many times when introducing the topic of mathematical poetry, students need to see examples.  The American Mathematical Society (AMS) has a yearly competition and they share the top three winning entries.  Share a few of the winning poems with your students to give them an idea of what it looks like. The Smithsonian Magazine also provides some nice examples written by a variety of people including a former U.S. Poet Laureate.  

The nice thing, is that this activity can be done in elementary, middle school, or high school. So when April comes around or your admin tells you that you must do writing, expose the students to this activity.  Let me know what you think, let me know.  Have a great day.

Games For Secondary Students

There is always a time when students become distracted in class.  They need a chance to give their brains a break so when they return to their work, they are ready to face it.  Students don't need any computers for these games so no need to make sure the internet is up and working.  Most of the games do not require a lot of prep so they are fairly easy to use.

I love using bingo in my class because I can list all the possibilities for answers on the board, pass out bingo cards and let students fill them in.  So instead of calling out I-35, I might say One factor of x^2-3x + 2 is x + 1, find the other. Students generally work hard to be the first one to find the answer so they can call it out.  I've used bingo with both algebraic and numerical factoring, fractions, multiplication or division, order of operations, and just about any other topic.  It just takes a bit of creativity.

I've also used a home grown version of battle ship so students can practice using the coordinate plane. Simply hand out four quadrant graph paper.  Have students draw in their five ships on the paper.  Make sure they have a divider or other way to block their view of each others paper.  Then they proceed by having the first person call out a location using (x,y) coordinate and letting the other person say hit or miss, then call out their choice.  The winner is the one who sinks the other person's ships.

Another variation on this is to prepare a paper with two 5 by 5 grids. Then the teacher creates two separate sheets.  One sheet is for the first player and the other is for the second player.  The top grid has the answers while the bottom grid has the problems but the problems are not the same for both players.  The answers to player A's problems (bottom grid) are on player B's top grid and vice versa.  When the first player calls out the location (x,y) from the bottom grid, both players work the problem to find the answer and if the answer is in the corresponding top grid with the ship it is a hit, otherwise it is a miss. Who ever sinks the others ships first wins.

I've also created cards with problems like I have x = 7, who has the answer to 2x - 1 = 3.  The person then solves the problem and calls out the answer.  The person who has the solution then says, I have x = 4 who has the answer to 3x+2 = 17.  This continues until it arrives back at the first person and everyone has a chance to participate.  I have been known to make some easier and harder problems so all my students can play.

Then there is the variation on basketball.  You provide each student with a worksheet.  They work through it and the when they have all the answers, they check them and if they get the answers correct, they can then throw them in the basket.

So here are a few to start.  I'll provide more later on in another column.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

 If one cup of Nestle's chocolate chips weighs about 6 ounces, how many cups are in 18 ounces of chocolate chips?

Warm-up

 

If a pound of chocolate has 32 pieces, how many pounds of chocolate do you have if there are 145 pieces?

Early Intervention Is Worth It.

I still have to figure out why more schools are unwilling or unable to offer math intervention in the very early grades.  I've seen schools provide reading interventions to the students who need it but when it was suggested these students would benefit from math interventions, I've heard the same response.  Students have to read to do math.  Although the gains from early interventions seem to fade by the upper elementary, a new study has discovered that interventions begun in preschool and kindergarten help students significantly.

It is well known that students who arrive in preschool with low number skills are more likely to not do well in math during elementary school.  In fact, they are more likely to struggle through elementary school, into high school.  This lack of foundational knowledge means they begin behind and stay behind.  Unfortunately,  most schools, if they talk about starting it somewhere between first and third, have a reason to begin interventions much earlier.  

A recent study showed that when students in preschool and kindergarten participate in intervention programs, they do better in math and miss less school by the time they reach the 3rd grade. They looked at students who began preschool with  lower than normal language skills and short attention spans showed significant improvement in math by third grade especially. When they participated in the intervention programs in both preschool and kindergarten, they gained enough ability by 3rd grade to close the gap between low and high income groups.

It was also discovered that these same students had better attendance by 3rd grade.  It is thought that the early intervention changed the view of these students towards math and school so they wanted to go.  It is also speculated that teachers were more likely to build relationships with students since they saw the students enjoyed math more.

These are things for school districts to think about due to the pandemic. Many of these incoming students have not had a chance to build the normal foundations to prepare them for school.  It is commented that many of the students in the lower socio-economic groups do not get the same support at home that others do and this explains why they are already behind when they start. 

To counter this and the effects of the pandemic, school districts should start looking at providing interventions to students who are already behind when they begin preschool and kindergarten rather than waiting until they are far behind.  If we can catch them early enough, the gap can be closed.  Let me know what you think, I'd love to hear.  Have a great day.

Helping Students Catch Up.

 So many of us are struggling to help our students regain at least some of the ground that was lost when schools went virtual due to COVID.  We know they've fallen behind but we also know that we have to begin doing things to help them begin to move forward or at least not lose any more ground.  This semester I have seventh and eighth graders, most of whom do not know their multiplication tables and who are extremely weak in fractions.  So I've been looking for ways to help them do gain some ground in their math skills. 

It is strongly recommended that one assess students to determine what they've learned and where their gaps are because this allows you a chance to figure out how to incorporate missing pieces with the current math material so students don't feel stupid.  As mentioned earlier, many of my students do not know their multiplication tables.  

I am not going to spend time on that.  Instead, I'm handing out multiplication tables so they can refer to it to help them since I began working with fractions, specifically converting mixed numbers to improper fractions and vis versa.  I'm using the first couple of weeks to bring their understanding and skills up to where I need them to be in order to start the material in the math book.  I'll be supplementing their multiplication fluency by using Kahoot so the students can practice their multiplication skills 

In addition, I'm letting students collaborate on work.  I monitor the students to keep them from copying while encouraging them to talk to each other about how to do the work.  The discussing helps them express their ideas and at the same time, I've got peer tutoring going on.  Research indicates that peer tutoring is quite effective so I'm encouraging it. 

It is also recommended that we have a system to support students built in so they don't have to ask for help.  As said earlier, I try to get students to help each other while I rotate around the classroom checking on students, glancing at work, and answering questions. This allows me to catch misunderstandings and clarify things before students get too far along.  If I'm not sure what they are doing, I'll ask them to explain their thinking. This helps them verbalize their thinking and it provides me with another chance to assess their understanding.  I used this to discover that many of my students had weren't sure how to translate the division results into a mixed number.  They'd mix things up so I could help clarify things immediately.

It is also important to review mathematical vocabulary throughout the lesson.  Vocabulary does not have to be anything formal but it must include it so students hear it used in a proper manner so they can learn and use it correctly.  If a student has no idea what the term "cubed" means, they may not be able to do the problem.

Finally, use explicit instruction that is chunked for students who struggle.  This is important since many students who are struggling often struggle trying to discover things on their own.  Furthermore, if the material is done so it is chunked in small amounts with a chance to practice between "lectures", it makes it easier for them to learn the material.

Remember, it is all one step at a time.  Let me know what you think, I'd love to hear.  Have a wonderful day.

Statistical Analysis Shows The Outcome Is More Predictable Than Expected.

 

Over in Europe, the most popular sport is football or soccer as it is known in the United States.  I was in Germany one year during the time of the World Cup and restaurants had those huge televisions placed around outside so people would watch the matches.  Young men and women played with balls in the empty spaces between diners and in any empty space they could find.  People showed their support for their favorite team by flying their flags off their balconies.  So it is as big in Europe as it football is in the United State.

Two researchers at the University of Oxford analyzed over 88,000 matches from the European League professional football matches to discover something quite amazing.  Many watchers have commented that matches now are not as exciting as they have been in the past because the outcomes have become more predictable. Some suggested this is due to the teams with more money being able to attract the best players by paying them more.  In addition, it is suggested that those who are not as good have seen their skill levels have stagnated or fallen.  It is speculated that these two possibilities have lead to people being able to predict the outcome before the game even starts.

The two researchers at the University of Oxford decided to find out if this was true. They analyzed over 88,000 games played by 11 major European Leagues between 1993 and 2019.  They looked at matches played between the teams from Scotland, Greece, Belgium, Spain, Turkey, and the best teams in England. Specifically they looked at the results of the matches as compared with the predictions. 

They first built a model that would use various factors to predict the winner of the match which they used to to predict the winners of the actual matches before comparing the results with the actual results. The model was written to predict whether the home or away team should win based on the results of a certain number of previous matches.  In addition, the model used was less sophisticated than the programs used by modern betting houses but it allows the researcher the ability to go back 20 or 30 years easily.  Even though it was simpler, it still ended up being about 75 percent accurate.

As they analyzed the results, they noticed that as the years passed, the predicted results matched the actual results more accurately. For instance, when the model looked at the results of a German league back in 1993, it managed to predict the correct winner for 60 percent of the matches but by 2019, it was 80 percent accurate. Furthermore, they discovered the points were distributed less evenly as time passed and it appears the stronger teams have grown stronger while the weaker teams get weaker.

This could be due to an inequality of resources meaning the stronger teams tended to win their matches more often possibly due to making more money due to popularity which means they are able to attract the better players creating a stronger team.  One surprising result was that the home team advantage made very little difference as the years passed.  In the early 1990's the home team had about a 30 percent better chance of winning but by late 2019 it had struck to about 15 percent. 

Who would ever have thought of analyzing soccer games to see if the predicted winner actually won?  Let me know what you think, I'd love to hear.  Have a great day. 

Warm-up

 

If the United States shot off 266 million pounds of fireworks each year and only 27 million pounds are used in displays, what percent is shot off by the population?

Happy New Year!