Warm-up

If you need 7 1/2 pounds of tomatoes to make one quart of ketchup,  How many pounds will you need to make 24 gallons of ketchup?

Warm-up

If you need two pounds of grapes to make one quart of jam, how many gallons of jam can you make from 156 pounds of grapes.

Reflections On Dividing Fractions

 I find it difficult to teach students to divide fractions using diagrams.  I realized that if I had difficulty picturing the process, it means I don't fully understand the concept.  I can do the process swiftly and come up with the answer.  That is because I learned the process only.

When I was growing up, they didn't worry about students being able to visualize concepts.  They were concerned with the process.  They wanted to know that you could follow the algorithm correctly and they used lots of short cuts and sayings to help you remember the steps

Over the years, there has been a move to providing visualization for the concept as it is being taught.  After repeatedly looking at pictures on line, I think I finally figured out how to read the pictures showing dividing fractions. I also finally connected the idea of part to whole when dealing with dividing fractions.  

For instance, I have 3/4th of a pizza I want to divide into 2 pieces.  So the whole instead of four pieces making up the whole, I have 8 pieces of pizza if I had a whole pizza but only 6 pieces are relevant.  Now we want 3 parts of the 6 so we each part of the pizza is 3/8th.  On the other hand, if I have the same 3/4th of a pizza and I divide it into quarters, I am asking myself, how many fourths are there, so Im looking at the parts which means there are 3 one quarter sections.

So if I have 3/4 divided by 1/8, it means I start with 3/4th of a pizza and I want 8 pieces instead of four which means each piece is divided in half and my pizza now has 6 total pieces which is what I needed to find.  On the other hand if I have two pizzas and I divide them into quarters, I end up with 8 pieces because each pizza is divided into four and four times two is eight.

As far as improper fractions go, I figured out that one needs to convert the improper fraction into a mixed number.  The student would then draw the mixed number using circles or rectangles with the fractional part in it's own circle or square.  After it's all set up, then apply the division.  It's important to see it all in context and as a whole.  This makes it so much easier to "see" what is happening.

Honestly every time I've tried to figure it out with pictures, I'd stumble around until it made sense and then promptly forgot how it worked.  I think that is because I never took time to verbalize what was happening.  The verbalization allowed me to focus on the concept while moving it from short term to long term memory.  

The next thing I want to figure out is how to apply visualization to algebraic fractions.  I'm not sure how to draw 1/(x-1) or 1/(x-1)/1/(x+2).  My next step is to see if I can figure out how to express algebraic fractions while showing addition, subtraction, multiplication, and division of those same algebraic fractions.

Let me know what you think, I'd love to hear.  Have a great day.

The Exploratorium and Math

 

Unfortunately, the Exploratorium in San Francisco is currently closed but they still have online activities available for teachers to use in their classroom.  Many of the lessons labelled as math tend to also be science which is great because it shows students how the two are interrelated

The activities range from applying the inverse square law, to calculating the weight of a car based on tire pressure, to packaging, to the radioactive decay model.  So many choices for so much fun.Each lesson has a short description of the concept, list of materials, along with everything needed to conduct the lesson.

The activity on the radioactive decay model uses pennies to show how it works.  The instructions ask that someone toss the pennies onto a flat surface and then take all the pennies that land tail side up and line them up to form a column.  The remaining pennies are picked up and tossed on the flat surface where all the pennies tail side up are picked up and placed in a second column right next to the first column.  This is continued until there are no pennies left. At this point the columns of pennies are used to explain half life so students get a good visual picture of what half life is and how it works.  This provides a wonderful visual for when it is time to teach decay.

Another activity involves paint and fun to create fractal patterns which are found in nature.  Basically a small dot of paint is placed between two plastic sheets and pressure is applied.  Then the two sheets are carefully pulled apart so they come straight off with no smearing and voila you can observe a beautiful fractal.  There is a great explanation of fractals along with information on perfect mathematical fractals such as Sierpinski triangle.

One activity that is not straight mathematics but answers the question of can square wheels roll.  Yes they can with a scalloped road.  The activity uses toilet roll tubes, glue, and a few other things to create a road that a square wheel can travel across.  The mathematics comes at the end when explaining how it all works.

In addition to activities, the Exploratorium has videos on some fascinating topics such as one on cameras and the math of vision.  It looks at the history of photography and how our understanding of vision grew as cameras developed.  The 20 minute video is short enough to put into class yet long enough to show the connection.

Another video shows an artists evolution from math to an in-depth study of polyhedrons to creating them using a variety of materials.  The 5 minute video explains the whole process and something more about how she played with variations and adjustments.

There are a few activities which are connected with videos such as using a handy measuring tool to figure out the height or distance of an object.  Students learn to use their hand as the measuring tool from the video and then the activity has them practicing this in person.  It includes a mathematical explanation and even goes so far as to offer ratios using parts of the hand.

So if you are looking for something a bit different, check this out.  Let me know what you think, I'd love to hear.  Have a great day.

Fashion and Geometry

 

Most clothing is based on geometric shapes, some are left alone while others are adjusted to fit better. In fact, many countries designed clothing out of rectangular shapes sewn together. Other times, the material is printed with geometric shapes such as the hexagonal pattern on the dress worn by Mrs Trump when she arrived in Florida. 

As far as popular fashion, one automatically thinks of the 1920's when a revolution occurred, freeing women and allowing Art Deco to influence life.

Art Deco evolved from cubism by breaking down the visual shapes into the basic geometric shapes. From art, it trickled into fashion and jewelry.  As far as dresses went, fashion designers eliminated darts and extra seams so the rectangular pieces were sewn together to form a cylinder.  

This many of the dresses of the time were created with rectangular pieces to create a dress that hung loosely on the body so women had more freedom.  In addition, they often used geometric patterns on outfits such as checkered pockets, square necklines, or even material with geometric shapes.

In Alaska, native clothing referred to as Kusp'iks are often based on rectangular shapes.  I was taught to take a long piece of cloth that was the same width as my shoulders and twice the length from neck to just below the hips.  The neck is cut in the middle with an elliptical shape.  The arms are two smaller rectangular pieces of material that are sewn to the edges of the shoulders and then the seams are sewn shut.  Then cuffs are added to the sleeves to make them fit and a hood is added to the neckline.  The final step is to make a skirt out of shorter rectangular piece of material that is two times the circumference of the bottom.  It is pleated and sewn to the bottom of the outfit.  Add trim and you are done.

As far as countries, look at how clothing is constructed.  The shape and way geometric pieces are put together depends on the width of the material.  For instance, one type of Japanese kimono, the Yukata or unlined kimono, is made out of material that is about 14 inches wide and 12 to 13 yards long.  It is designed to use the 14 inch wide pieces so they are sewn together to form the complete piece of clothing.  I'm using numbers for a standard sized Yukata.  Think of two sleeves that are 14 by 42 inches, the actual body panels are 14 inches by 2 times the body height plus one inch. The front strips are 7 by 55 inches, the collar is 7 by 76 inches and a collar reinforcement that is 7 by 35 inches. 

If you look at photos of traditional clothing you will see that many are designed and made out of rectangular pieces of material as shown above.  If you can get your hands on a book by Max Tilke, his drawings show every seam, so you can actually see how things fit together. 

Other traditional clothing uses long pieces of cloth such as the Indian sari. The sari is a piece of material about 6 feet long by 36 inches worn by women.  The piece of material wrapped around their body once or twice and then a bunch is pleated and stuck in the waist band and the last bit is thrown over the shoulder.  In that vein, the Scottish kilt is also a long piece of cloth that is pleated, sewn, and it's ready to go.

Take this all a step farther and have students create their own fashion out of squares or rectangles like a project Runway.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

A young male tiger weighs 90 pounds at 6 months old and reaches the weight of 225 pounds at 2.5 years of age.  What is the weekly average weight gain from 6 months to 2.5 years old.

Warm-up

If a male tiger weighs 3.2 pounds at birth and by 6 months old weighs 90 pounds, what was it's average weekly weight gain.

Baseball Hall of Fame and Math.

As winter slowly turns to spring, it will be time for baseball to make it’s annual appearance.  It is a sport played in school, on playgrounds, and by professionals.  As baseball is extremely important in this country, there is the National Baseball Hall of Fame one can find lesson plans for all sorts of subjects including Mathematics. 

Although the number of lessons available are limited, they are well designed to cover three levels from third grade up to high school.  For instance, they have a nice unit on baseball statistics with 4 to 5 lessons included.  The lessons do include one requiring either an in person or a virtual visit. It is easy to fill out a request for a virtual visit and I didn't find anything about a cost. it also looks as if it can be done without the actual visit.

Each one, rookie, intermediate, or advanced has students working on using a math skill appropriate for the age group.  Grades three to five learn about batting averages in both fractional and decimal form, and other stats that are kept for basketball players. Grades 6 to 8 focuses more on batting averages, averages and percents, and how the stats are used to show a successful player. The lesson plans for grades 9 to 12 goes into batting averages in even more details, slugging percentage, the use of mean, mode, median while learning to calculate standard deviation, and ends with a long term project where students have their own fantasy baseball league.

The second lesson is on geometry and circling bases.  Unfortunately, they only have two lessons for grades 2 to 5 and 6 to 8 but the second one can easily be used in a high school geometry class as one lesson has students calculate the perimeter and area of a rectangle, the circumference of a circle, and an application of the Pythagorean theorem to the baseball diamond.

The final lesson is on the economics of baseball. The lessons for grades three to five focus on introducing students to the basics of supply and demand and the use of decimals in regard to ticket prices. Students in grades 6 to 8 work on learn how the Industrial revolution helped create a market for professional baseball, ways in which professional teams make money, the amount it costs to be a fan, and the the amount of money baseball stars can make. The lessons designed for high school learn about utility and marginal utility, factors that effect supply and demand, what goes into creating a price index, and what factors set the price of a sports production.

These lessons have everything needed to teach several one hour classes. The virtual visits are designed to reinforce the material students studied in previous lessons but if the visit cannot be arranged, it is possible to still do things with your students. Let me know what you think, I'd love to hear. have a great day.

The Super Bowl Is Coming!

 

The Super Bowl. is just around the corner, on February 7, 2021, in sunny Tampa Bay Florida.  Most students will look at you strangely when you comment there is lots of math one can do on the Super Bowl.  One place that has tons of possibilities is at Yummy Math with at least 9 different activities.  

Two of the activities focus on the amount companies pay for a 30 second commercial.  In 2019, the average 30 second commercial cost $5.35 million which could be expressed in scientific notation.  One of the activities has students use scientific notation for the cost while a second activity looks at the increasing cost of a 30 second commercial over the years.

Then there is an activity which looks at the typical Super Bowl scores over the years.  The activity requires students to use the mean, mode, median and range to compare scores and see how things change when one switches from mean to median to mode.  It has students look at 10 years worth of data during the comparison.

Another activity looks at how much grease Philadelphia had to use in order to grease city light poles so they could keep fans from climbing to the tops in 2018. Students are required to estimate, calculate, read maps, and use several other skills to come  up with answers.  Think about this, if one quart of paint covers a 100 foot area while 1 tblsp of Crisco covers 9 inches square.

Other activities include having students estimate and then figure out how many chicken wings, sauce, and carrots the Buffalo Bills sent to the Bengals after the game, calculating the amount of cheese needed to make cheesy poppers for superbowl sunday or students have to calculate the best buy for a large screen television so they can watch the game.  There is also an activity which focuses on helping students become more fluent with Roman numerals since all games include Roman numerals.

There are enough exercises to do a full week's worth of activities leading into Super Bowl Sunday. Let me know what you think, I'd love to hear. Have a great day.

Martin Luther King Day

Today is Martin Luther King Day.

Warm-up

What if you made $100,000 per hour and then lost $1000 per second?  In one hour, would you make or lose more money?  Justify your answer?

Warm-up

If you had a choice of being paid $100,000 per hour or $1000 per second? Which would pay you more?  Justify your choice.

Why Is Long Division So Hard?

 

Over the years, I've noticed that high school students struggle with performing long division in classes such as Algebra II.  One of the elementary teachers I know, told me her students struggle with dividing two digit into four digit numbers.  This comment was confirmed when I ran across an article published back in 1942 in which the author discussed why long division is so difficult.

I realized if students struggled with it in elementary school, it makes sense they will struggle with it in high school because they haven't mastered it. If students are unable to divide a four digit number by a two digit, how are they going to divide a polynomial by another polynomial?

If you take time to look at the steps used, long division is quite complex.  Yes, I said steps because that is usually how it is taught.  Long division uses estimation, multiplication, subtraction repeated multiple times until the answer is found.  If any mistakes occur, the answer will be incorrect so each step has to be performed perfectly.

One study indicated that the more steps a division problem requires, the higher the chances are that the student will get the wrong answer due to the increased possibility of making a mistake.  In addition, a student synthesizes all the mathematical knowledge in order to solve long division so if a student is weak in multiplication, they will struggle with the process.

Since most students learn to solve long division using an algorithm, they acquire little knowledge of what is actually happening and lack a conceptual understanding.  I admit, I could do long division in elementary but I never understood I was splitting the dividend into smaller groups of so many.  I couldn't have explained it other than taking you through the algorithm.

Another issue in long division of numbers or polynomials is that students do not understand that zero's perform as place holders indicating that the divisor cannot go into the dividend and another digit must be brought down so it becomes big enough.  It is something that occurs in both numerical and polynomial division.  Furthermore, many of the "hints" they learned such as the big number goes into the house does not apply when discussing polynomials

I will say that long division using polynomials is a bit easier than using only numbers because it's easier for students to make the first term of the divisor "match" the first term of the dividend rather than looking at the whole "number" and estimating. Unfortunately, the basic algorithm is still the same so if students did not get long division in elementary school, they often struggle with it in high school. 

No matter whether you are using numbers or polynomials, long division is a struggle for most students.  I'm still trying to figure out how to make a visual so students can "see" what is happening at each step.  Let me now what you think, I'd love to hear.  Have a great day.

Coding and the 8 Mathematical Principles

 

I really wish I could incorporate coding into my math class but the system is not set up to do that.  One reason I want to do that is because it gives students a real life opportunity to use the mathematical knowledge they’ve learned in class.  It provides an opportunity for students to apply and deepen their mathematical understanding while having them apply mathematical principles in a real life situation.

The first principal asks students to make sense of the problem and persevere to solve them.  When students code, they are more likely to continue trying different things so they work through and solve the problem.  In fact, being able to problem solve is one of the most important skills for coders.  Many companies look for someone who has excellent problem solving abilities even if they are a bit weak in the actual programming.

The second principle is to have students develop abstract reasoning which is also important in coding.  Often times, a student has to look at what they are working on to figure out what is missing so they can get their program to work.  They have to make sure when they want a counter for loops, the counter is working correctly while keeping proper count.  

Furthermore, coding often requires the programmer to model mathematical situations.  They have to identify the important quantities, and use coding to place them into a usable form or to apply to a modeling situation.  It also allows them to create tools or apps that others can use.  In addition, students are expected to be precise in math and they have apply the same precision to coding or their final product won’t work right.

Another mathematical practice requires students to make use of structure and all programming languages have a set structure that students have to follow or things won’t work out.  If students follow the structure of the language properly, they can produce something that works but if they don’t, the app won’t work.

Finally, mathematical principles have students express regularity in a repeated situation.  In coding, students have to write loops so something will be done multiple times.  It would be like having your dinosaur walk far enough forward to stop at the river without falling in.  Sometimes, a program has to have the printer print out a set of results multiple times.

If we want students to develop better understanding and application of the eight mathematical principles, we need to integrate more programming or coding into schools.  Let me know what you think, I’d love to hear.  Have a great day.

Shortcuts vs Process

 

The other day, I read something on twitter asking whether one should teach only guess and check rather than shortcuts or both.  I was surprised to find out that much of what I teach to students to help them factor trinomials and some polynomials are classified as shortcuts.  I seldom actually teach guess and check because so many students find that method frustrating and they shut down.  Even I get frustrated if I have to rely on the guess and check method for factoring.

Once I learned to do the diamond method, I found it so much easier.  If you are factoring a trinomial with a leading coefficient of one, you figure out what two numbers produce a product of the third term and a sum of the second term.  The two numbers you come up with, makes up the second term of the binomials.

If you look at the equation x^2 + 2x - 15, you look at the factors that make -15 but when added together gives 2 so x^2 + 2x - 15 factors into (x -3)(x+5) On the other hand, if you have something like 2x^2 + 5x + 3, the method I use is more like the reverse of multiplying two binomials but use the diamond.

I still have students multiply the leading coefficient by the third term, the constant so you’d so 2 x3 to get 6 so what two numbers produce 6 when multiplied but add up to 5 which are 2 and 3.  The next step is to replace the 5x with 2x + 3x so you have four terms.  From there you use the grouping method to place the four terms into two groups with two terms.  Factor things out and go from there.  I have a complete step by step photo of it.  

So is this a process or is it a shortcut?  Personally I see it as a process because there are steps you follow to go from the trinomial to two binomials which are factors.  These methods also appear in many textbooks so does that validate them or are they still considered shortcuts?  Honestly, this process works if a trinomial can be factored otherwise, you end up using the quadratic formula.

I’ve also been able to use the group four terms into two groups with certain polynomials but not all.  In fact, it is one of the things I try with polynomials to the 4th degree since some of them can be factored this way.  Again, if I can apply it to  multiple situations and it’s a process that can be followed each time, it seems to me it becomes a method rather than a shortcut.  Most things, I think of as a shortcut is something like you can’t divide by a fraction so you flip the second term and multiply.  Let me know what you think, I’d love to hear.

Warm---up

If this sphere has a radius of 3/4 inch, what is it's volume?

Warm-up

 

If snow is falling at 2.5 inches per hour, how much snow will you get after 15 hours?

Prime versus Composite

 Visualizing prime and composite numbers can be very difficult for students. They are usually taught that prime numbers only have factors of one and themselves while composite numbers can have more than those two but what does that mean visually.  I didn’t know a way to provide a visual representation that shows the difference between prime and composite numbers.  

The idea is that if a number is prime, you cannot arrange it in any rectangular or square shape other than in a 1 by the number.  It doesn’t matter if it is vertical or horizontal, it’s still in a 1 x whatever arrangement.

If you look at the picture to the left, you’ll notice I chose 11 which is a prime number and can only be expressed as 1 by 11,  If you try to arrange it any other way, you will not get a proper shape such as in the photo to the right.

I tried it in threes but no matter how you arrange the blocks.

On the other hand composite numbers can be arranged in more than one way.  Take the number 20.  It is a composite number and can be arranged in multiple ways.  It can be done as 1 x 20, 2 x 10, 5 x 4, or 4 x 5, 10 x 2, or 20 x 1, so the factors are 1,2,4,5,10,20.  You can see the results are perfect shapes.

So as far as visualizing, composite once can arrange representations in complete shapes while those for prime numbers only work in a 1 x the number.  

When I looked at the visual, I was surprised to see the current definitions make sense when you look at the photos.  Since I read the definition, I’ve used it and many of my students had light bulbs go off. They went “oh!”  

I’ve even used this as a hands on activity where I gave students some single blocks and 10 blocks to work on finding different ways of making say 20 or 43. They play around and end up concluding prime numbers can only be done in one shape while composite do others.

So the next time you have to revisit prime versus composite numbers, try this to help them see the difference.

Composite function memes.

Over the holidays, I ran into a really awesome use of memes in math.  As most teachers are aware, some students find it difficult to visualize what is happening when they learn composition of functions.  I usually end up using lots of parenthesis, white boards, and sticky notes to physically move one function into the other so students have a kinesthetic way to do it but some still struggle.  

I check Twitter for ideas I can use in the classroom to help students learn math better.  One beautiful meme I saw absolutely capsulized the composite functions of f(g(x)) and g(f(x)).  The idea is to set up two pictures, one as f(x) and the other as g(x) as I’ve done at the top of the meme.  Then one creates a picture by inserting the g(x) photo into the f(x)photo and vice versa to show the f(g(x)) and g(f(x)).

I saw several and tried it myself.  Other than taking a bit of time to find the photos, find a meme maker that was easy to follow and then put it all together, it wasn’t that hard. I’ll share how I did it and which programs I used to do it.  There are probably easier ways to do memes but I wanted four different pictures to use in one meme.

First, I went to Pixabay which is a website with tons of free photos that anyone can use.   I selected two photos, one of a frozen bubble and the other of a can of soda with a circled pattern with it. Using Preview on my Mac, I selected the parts of the photos I wanted to use for the composition and saved those separately.

Second, I used Kapwing to create four different memes because it allowed me to create four different scenes.  Normally, it would allow me to string them together with voice over to make an animated gif but I just wanted the individual memes.  This web based free program allowed me to I took screenshots of each scene so I’d have a photo of each one. 

For the final step, I created the whole meme using imgflip which allowed me to combine all four scenes into one meme with titles and everything. I might have been able to do it all in one or the other but the Kapwing allowed me to create the individual parts and imgflip let me put the four together.  

Both programs were fairly intuitive so I could easily manage it without much difficulty.  I did have to undo things a few times but overall, it was easy and it took about one period to make the whole thing from start to finish and learn new programs. I  hope this helps you figure out how to do things.  Let me know what you think, I’d love to hear.  Have a great day.

Math Games That Are Not Computer Based.

 

This year, I've been trying to incorporate games into my Algebra I class to help students practice the skills they are learning while making the learning more fun.  I use a couple of online games but I also try to incorporate some that are not computer based so my students get a chance to move around the room. 

One that my students enjoy is a mathematical scavenger hunt.  I post 10 to 12 problems around the room on separate paper.  Each paper has a number in the corner telling the student which problem it is so they can write the problem down on their answer sheets, a problem and an answer but the answer is not to the problem on this paper, it is to a different problem. 

 So once they've worked the problem, they have to search for the answer on another sheet of paper.  when they find the answer, they do the new problem on that sheet and continue until they've completed all the problems.  This activity usually takes most of the class period.  I allow students to help each other, check their work.  For a grade, I check for completeness rather than correctness because they have the answers.

Another activity many of my students enjoy, involves paper airplanes. This doesn't require a lot of preparation, only finding or making a worksheet with 5 to 8 problems of the type you want students to practice.  Give each student a copy of the worksheet and they need to work the first problem on the page.  When everyone has had a chance to complete the problem, allow them to fold it into a paper airplane.  Then students stand in a circle around the room and fly the planes towards the other side of the room to another student.  Everyone opens up their plane, works a second problem.  When they have an answer, they fold their papers back into planes, stand in a circle,  fly the plane to someone else and continue doing problems until all of the problems have been solved.  I collect the papers at the end and scan them looking for areas students need a bit extra help with. 

The final game for today, is baseball.  It does require a bit more prep but once everything is made, it is easy to reuse materials.  Step one is to create a set of cards with questions on them.  The cards should be set up as single, double, triple, or a home run questions.  A single should be the simplest form of a questions such as Solve 2x + 1 = 7 while the home run would be the most complex form such as solve 2(x + 3) - 8 = 4(x-8) + 3(x+1)-4.  Once the questions are completed, designate the four bases around the room, set up the score board with it's 9 innings, and divide the students into two teams.  Allow a bit of time for students to select a team name and set the line up.  The last step before playing is to designate which team goes first.

Yell "batter up" and let the first student choose the type of question they want such as single, double, triple or home run.  The teacher asks the student the question and if the student gets the problem correct, they move to the appropriate base but if they miss it, they are out.  Three outs for the side and students have to switch out. At the end of the game, the team with the most points wins.  If you want to make it a bit more exciting, have the team that is not up to bat, stand at the bases.  As the batter chooses a problem to work, the student manning the base also has to work the problem to get the answer.  If the student on base gets the correct answer first, the batter is out but if the batter gets it first, they go to the base.  It all depends upon how competitive you want the game to get.

Sometimes, it is nice to include a few games that allow students to get up, move around, while working on problems.  It makes the work more fun while doing some serious work.  Let me know what you think, I'd love to hear.  have a great day.

Warm-

If it takes 70 people working 16 hours per day for 12 days to finish the float for the Rose Bowl parade, and each person is paid $15.75 per hour, how much will it cost to build the float.

Warm-up

The rose bowl parade has 45 floats which use a total of 3.2 million flowers.  How many flowers are used on each float?

Happy New Year