Interesting Buildings part 2

 

Welcome to part two in which I’ll discuss more mathematically interesting buildings.  Buildings that are not the standard cube or rectangular prism.  It is interesting that much of the math is what Ill call hidden math in that it helped shape the building but isn’t always immediately visible. 

We’ll start with Chichen Itza in Mexico.  It was built by the Mayan Civilization which is the same ones credited with “inventing” zero.  The 78 foot tall structure is based on the astrological system.  The 52 panels on each side of the building represent the length of the Mayan cycle and  the 18 tiers on each side represent the months of the Mayan calendar while there are 365 steps that represent the days in a year.

Then there is the Segrada Familia in Spain.  The architect created buildings based on hyperbolic paraboloid shapes so they look like saddles.  In addition, there is a magic square in the passion facade whose numbers add up to 33. 

In Bilboa, Spain, one should visit the Guggenheim museum, Its design imitates a ship with titanium panels that look like fish scales.  The architects used a program to create 3 dimensional renderings which allow them to make buildings that are not traditionally shaped.

Of course, one can pop over to Belgium to visit Philips Pavilion. This pavilion named after an electronics company was built back in the 1950’s to celebrate the technological advances experienced after World War II.  The structure uses several  hyperbolic paraboloid shapes combined with steel cable to create a soaring visual. 

For something different, head to Toronto to check out the $24 million dollar home built by a ravine.   The owner who used to be a calculus professor, wanted a house that incorporated the integral sign into its design and it’s named the “Integral House”. In addition to being a home, it is large enough to accommodate 200 for a concert and the walls and windows are designed to vibrate ensuring beautiful sound.  

Finally, check out the Cube Village, designed by a Dutch architect. These cube shaped buildings are on top of a pedestrian bridge and imitate a forest.  It is composed of three levels, where the top level is made up totally of windows which makes it feel as if it is a completely different structure.  

There are other structures out there that are built on mathematical principles.  These are just a few one can show practical applications of math and the world out there.  Let me know what you think, I’d love to hear. 

Mathematical Interesting Buildings pt 1

I love that mathematics is all around us, especially in the buildings we live in and see everyday.  We see squares and rectangles in the doors, and windows, cubes and rectangular prisms in houses, trapezoids and triangles on the roofs.  Then there are cylinders and ovals in ancient ruins.  In addition, there are several mathematically based buildings that are especially interesting.

The Great Pyramid of Giza is one.  It is the largest of three pyramids found near Giza in Egypt.  It held the record for being the tallest building in the world for close to 3800 years. In addition, it has several interesting math facts associated with it.  If you divide it’s perimeter by  twice it’s height, you end up with the value of pi - 3.1415.... If you measure the perimeter in cubits, you end up with 365.24 which is the same as the number of days in the year.  Lastly,the measurements of the King’s chamber is based on the 3-4-5 pythagorean triangle.

Then there is the Taj Mahal in India.  This beautiful mausoleum was built around 1600 by Shah Jahan for his wife.  If you stand in front of the building, you’ll see a beautiful example of line symmetry with two visible lines.  One line runs down the middle of the building itself while the other one is the reflection of the prayer tower on the water in front.

Next is the Eden Project in Southwest England.  It was built in 2001 and is filled with greenhouses but not the usual ones.  Most of the greenhouses are shaped as geodesic domes with hexagonal and pentagonal cells.  Furthermore, the educational center was designed based on the Fibonacci numbers to reflect nature.  

The Parthenon in Greece is built with a beautiful height to length ratio 4:9 for both vertical and horizontal proportions of the columns, spaces, and other parts of the building.  To overcome a visual illusion of having columns look thinner in the middle if viewed from a distance, they built the column with thicker centers.  

The Gherkin in London is unique in that it is a circular building with a bulge in the middle, tapers at the top with a spiraling design.  The design of the building makes it more stable and creates the illusion that it is shorter than it really is.  In addition, the design makes it easier to heat and cool so it cuts down on costs and it was designed using a CAD program using parametric modeling. 

On Wednesday, I’ll cover  more mathematically interesting buildings so you can share them with students to see additional real examples of how math is used in the real world.  Let me know what you think, I’d love to hear.  Have a good day.

Math and Designing New Cars

 

Most students end up with a car. They might buy their own or their parents might give them one or they might have to share a family vehicle.  If you mention math to anyone in regard to a car, they’ll mention the cost of the vehicle, the cost of insurance, licensing, repairs, etc but there is more math involved than that.

Few students ever think about the math involved in its design and more.  When a company thinks about creating a new vehicle to join it’s line, the first thing they do is carry out a market analysis to determine if the proposed vehicle fits into a certain niche.  The market analysis allows the planner to  determine what is selling where at what cost.  They use this information in addition to things like miles per gallon, horsepower, weight, and size to create a list of specifications for the proposed vehicle. 

The engineers who design a car rely on computer based modeling programs to both physically design the car and “crash” test it before it is actually built.  Furthermore, they use geometry and trigonometry when designing suspension throughout the car.  They also use computers to calculate the aerodynamics of the design.  They tweak the design based on the results of the calculations to make it more efficient and better looking.

Math is also used in the interior to determine how to place every seat, every gage, every window, everything so it is ergonomically placed for the driver and everyone else.  They have to know where to put supports, dashboard, mirrors, and more for maximum space.  Once they start building the car, the programs controlling the robotic elements use a lot of math to make sure the production is done properly. In addition, math is used to determine the best order in which to make parts and put the vehicle together.  

Although the car was built and tested digitally, manufacturers usually test it on the track and do crash testing in real life to obtain real data from how quiet a ride it is to handling to miles per gallon, to so much more before they begin marketing it. Once the cars are built and ready for distribution, math is used to  determine how the cars are being delivered to dealers across the United States.   Furthermore, the company has used math to set a price for each car they sell to the dealers and the dealer adds a percentage to that for his profit.  The price the company sets is enough to cover all the costs involved in designing, producing, and transporting each vehicle.

Thus math is used from start to finish when a company decides to add another vehicle to its line.  Sometimes, manufacturers must adjust things as regulations change, or new ones come in so again the math is used to adjust things in the engine or the car itself.  Let me know what you think, I’d love to hear.

Warm-up

 

What do you notice/ what do you wonder?

Warm-up

 

What do you notice? What do wonder?

Concept Before Shortcut

 This year, I chose to do something different.  Students arrive in high school knowing so many shortcuts but they don’t know why the shortcut works.  For instance, when students divide one fraction by another, they are told not to divide but flip the second fraction and turn it into a multiplication problem, yet I’ve seldom seen students shown why it works this way.  

Unfortunately, shortcuts tend to make it faster to do certain problems but it doesn’t help students learn about the associated topic.  In fact, I would say, shortcuts emphasize procedure rather than understanding the concept they are working with.  I will be the first to admit that I have been teaching shortcuts because I learned math that way and my teachers program didn’t discourage their use.  

In addition, learning a shortcut often contributes to students solving problems by rote rather than seeing the concept that can be applied to a variety of situations.  When students learn shortcuts such as the turn the division of fractions into a multiplication problem where you are multiplying by the reciprocal, they often flip the wrong fraction.  

At the community college, they used the saying “Flip the right one, not the wrong one” to remember it was always the fraction on the right that was flipped.  If you had a problem where one fraction was over the other one, the saying didn’t work and students often set it up incorrectly.  Now I go through the whole process if we are dividing fractions or algebraic fractions so they see why one of the fractions is inverted.

Unfortunately, the use of shortcuts discourages the conceptual understanding of the topic which makes it harder for students to apply their knowledge to the math in later math classes.  When they learn the butterfly method for arithmetic fractions, they don’t know how it relates to algebraic fractions and if they try to apply it, they often end up with a mess.

I have nothing against shortcuts but I do believe students need to master the concept before they are introduced to the shortcut so they know what is going on.  When shortcuts are taught before they’ve learned the concept, most students never get the concept because they are focused on remembering the shortcut itself. Thus when they hit a more complex example of the concept, they may not understand how to solve it because they do not see how to apply the shortcut.

In Algebra I, I hit the chapter that shows the formula for multiplying perfect trinomials squared and the (x +8)(x-8).  I showed students via multiplying the two binomials why the book said (x +c)(x-c) = x^2 - c^2). When they saw the math, it made sense and without the mathematical explanation, I don’t think they would have connected the reason for no middle term.

So it’s important to show the whole process before teaching them the shortcut.  Let me know what you think, I’d love to hear.  Have a great day.

Looking At Mathematical Connections

 Last time,  I spoke about connecting certain equations with different situations.  Today, I’m looking at making mathematical connections.  If you look at most math textbooks, there doesn’t seem to be any connections between sections or chapters in general so students see each topic as unique and unrelated to anything else they’ve studied.  If one looks at the world enough, math is connected and interrelated.  

As teachers, we need to teach students that there is connectivity in math otherwise students will not see it and will continue to struggle.  If you look at everything in math, you will see there are connections between and within strands, subjects, the real world, the past, and the future.  If you were to create a picture of the connections, it might resemble a spider web.

Let’s start by looking at the connections with the strands.  There is a connection among rational numbers, fractions, decimals, ratios, proportions, percentages, and measurement.  Or direct variation, arithmetic sequences, and linear equations.  We need to find the connections and share them with our students.

Then to show connections between math and other subjects, one can explore vocabulary and concentrate on words that mean one thing generally but have a specific meaning in math such as product or look at similar words such as hundreds and hundredths that are similar but are different.  Another thing to look at is the idea that certain things can be spoken of differently such as dividing by four is the same thing as multiplying by one-fourth.

Of course there is connecting between the past, present, and future. For instance, if you show students how to multiply binomials using a vertical set up just like you do for 12 x 36 so instead of place values of 100’s, 10’s, and 1’s you use x^2, x, and ones.  When I’ve introduced it this way, rather than beginning with the foil, students have seen the immediate connection. For students who learned to multiply using the lattice method, students found they could easily use it with binomial multiplication.  

Then if one can take what is being taught right now and connect it with something a student might run into the future, it adds another dimension to the learning process.  This might take the form of sharing when they will use it again in future classes or in the real world.  One example is the use of area.  Once students are familiar with finding area, extend it a bit to show that certain products such as paint are labeled as covering so much area per gallon.  Assign them a task to find the number of cans of paint is needed to cover the walls in a room.  This provides students with a chance to see how they might use it in their lives in the future.

The last example shows the connection between the area formula and its application in a real world situation.  Students need to see how the abstract math they learn in school is applied outside of the classroom.  One visual way to do this is to set a bulletin board up with one section devoted to each topic.  Then as a topic is taught, add it onto the board and connect to other sections till you have what looks like a spider web.  As new things are taught, you might need to change items out each time.  Let me know what you think, I’d love to hear.  Have a great day.

The Context Of Equations

Last Thursday, I covered direct and inverse variations.  As I wrote the equations on the board for each one, I realized the direct variation formula of y = kx is similar to a linear equation once a value for k is calculated while the inverse variation is similar to the 1/x equation.  As I’m writing them on the board, I’m talking about mathematical context helping to determine the proper choice for interpreting the equation.

I recently reported on how being able to read well helps in math.  Although it didn’t address context, good readers understand context and how it is used to define certain words such as product.  

In one context, a product is something that is sold by stores while in another context, it is the result of multiplying two numbers together.  It might also mean a result of a process or situation such as the child is a product of a marriage.

Thus being able to understand context in math is extremely important. As noted earlier if I see the equation y= 1.5x without any context, I don’t know if it is an arithmetic sequence, a linear equation, a direct variation, or anything else.  We tend to teach the equations in isolation rather than referencing having used them before in this situation or that situation.  

In one pre-calculus textbook I use, it does have a nice link from arithmetic sequencing to linear equations.  It took time to show how finding the distance between terms uses the slope formula with the number of the term and it’s value.  It also took it a step further by having students find the linear equation by starting with the nth term = the first term + (n-1)d and rewriting it.

This made me wonder if one reason students have difficulty transferring knowledge is simply that we do not take time to show how the arithmetic sequence relates to a linear equation and how the difference relates to the slope.  If you look at direct variation, you have y = kx.  Normally, they give you the value for X and Y at a certain point and ask you to find the value of k.  When you divide y by x, it is like having the delta y/delta x or the slope.  

If you look up mathematics and context, you’ll find an abundance of articles which talk about relating the math learned in the classroom with real world situations.  This is nice but it is also important to show students how certain equations seem to be related in different situations based on context. 

This is one way to show mathematical connections.  In the next entry, I’ll talk more about mathematical connections.  Let me know what you think, I’d love to hear. Have a good day.

Warm-up

 

If it takes about 540 peanuts to make 12 ounces of peanut butter, how many peanuts will you need to make 32 ounces of peanut butter?

Warmup

If there are about 35 peanuts in an ounce, how many would be in a pound?

Moving Sofas

 It seems like when I move, I’m always ending up in places where I have to haul a sofa around some corner.  You’ve done it.  You and someone else are trying to move it around a corner and inevitably, you start and stop, back up, swear, adjust it and after a while you’ve either done it or you give up.  Well in 1966 mathematician Leo Moser wondered “What is the shape of the largest area in the plane that can be moved around a right-angled corner in a two dimensional hallway of width 1?” Eventually it became called “The Moving Sofa” problem.

When we think of a sofa, we think of something long and straight almost like a rectangle.  A normal sofa usually seats three people and generally can be between 70 and 96 inches long but the average sofa is 84 inches long.  It is extremely rigid and doesn’t curve around a corner easily.  

When this problem was proposed, the idea was to find a shape that allowed the sofa to be as large as possible and still go around corners, so people had to decide the size of the “sofa” and its shape.  If the sofa is a square with a length of one and the hall is also one unit wide, so it slides around the corner easily without any struggle.  It hits the wall and then changes direction without actually rotating.  This has a size of 1 unit.

On the other hand, if the shape is a solid semi-circle, it is able to rotate around the corner so the flat edge is along the interior wall and the curved is along the outside wall.  As it goes around the corner, the curved outside is moving the sofa but it only has an area of around 1.57.

For a semi-circle with a cut out center, it is a bit different.  This version can be longer because the cut out part allows it to rotate around the corner more easily because it has that scoop which provides some leeway as the sofa works its way around the corner.  There is more wiggle room so to speak which makes it easier to move it around a corner. This one has the largest area at about 2.2 times the square one.  This one is composed of 18 curves glued together.

The last shape explored is an unusual one for most sofas.  It looks like the base part of a guitar with two indentations, one on each side.  The rounded edges on both sides make it easier for it to move through the corners and the indentations make it so it has space to move around the corner.  Although this shape slides through the corners better, it only has an area of 1.6.

This site has some wonderful animations of each situation.  I just know that when moving a sofa, one usually ends up swearing and getting frustrated.  Let me know what you think, I’d love to hear.  Have a great day.

Prime Spiral

 I just learned about something called prime spirals or Ulam’s spiral. Ulam’s spiral is named after Stanslaw Ulam who was born in Poland in 1909 but moved to the United States in 1936, becoming an American Citizen in 1943, during World War II.  He worked at Los Alamos from 1943 to 1965 when he began lecturing at various Universities.  Over his lifetime, he published multiple papers on a variety of different mathematical topics. In addition, he contributed to the creation of thermonuclear reactions.

Ulam’s spiral is a grid with prime numbers written in beginning with 1 at the center, and continuing in a spiral around it so one is in the center, two is next to it, three is above the two, four is to the left of three, five is to the left of four, six is right below five and the spiral continues.

The prime spiral is  named after mathematician Stanislaw Ulam.  Back in 1963, while he sat through a boring lecture at a scientific meeting, he doodled.  He created a numerical spiral out of integers beginning with one and spiraling outwards. One he finished creating his spiral, he decided to circle only the prime numbers.

In the end, he noticed a pattern.  All of  the prime numbers appeared on diagonal lines. These diagonal lines are made up of both prime and composite numbers but the primes only appear on the diagonal lines. Since he’d used a small sample, he expanded the grid to 200 by 200 and the same pattern appeared. Mathematicians are not sure why this happens but they do know it is significant. 

During his tenure at Los Alamos, he worked with many of the early computers and he even tried this using a computer.  He is credited with being one of the first people to develop mathematical computer graphics.  Eventually, he produced a short paper with two other mathematicians and published it with computer graphics. 

It is possible to find grids on the internet that you can print out and have the students do the same thing so they see how he made the discovery.   I found the whole topic illuminating and quite interesting.  I’d never realized this before and I think it’s cool.  Let me know what you think, I’d love to hear. Have a great day.

Reading’s Connection To Math.

I just found an article that I found quite interesting. It was published by the University of Buffalo and looked at reading, writing, and arithmetic.  I love it because I will have an explanation to all the students who tell me “This is math, not english.”   They discovered that the ability to read and comprehend well actually affects how we approach tasks and solve problems.

A researcher wanted to know if a student could be identified as dyslexic based on how the brain worked.  In his initial study with 28 students, half with dyslexia and half without, he was able to identify those with dyslexia about 94% of the time but he needed to do additional research to see if he could generalize it.

So he went to a math study which measured the functional connectivity based on a mental math task.  Functional connectivity describes how the brain is wired every moment and changes according to the task being done. The wiring of the brain changes based on what you are looking at, what you hear, and what is happening.  For instance, you read a sign but as you move a bit to catch the end of the sign, you knock something off a shelf with your elbow, so your brain rewires itself to catch the item rather than read the sign.  So as the brain rewires itself to do a different task, your functional network changes.

He then had students look at completing a language activity and a mental math activity and in the process discovered that the connectivity markers were the same. In other words, although the tasks are different, the functional networks are the same.  He could still identify those with dyslexia but he concluded that the way our brain is set for reading actually influences how the brain works when doing math.

This reinforces the idea that your ability to read affects how you problem solve and helps educators understand why children might have difficulties with both reading and math.  He also concluded that learning how to read shapes how our brain is able to do other things.  

I found this article so interesting.  I found the article here if you’d like to check it out. I realize that the sample size is quite small but it is a start.  I can now rebut the “This is math, not english” with information from this article.  Let me know what you think, I’d love to hear.  Have a great day.

Learn more

 

Learn more about the sofa moving problem on Friday.

What Size?

 

What is the biggest sofa you can move around a corner?

History Of Fractions

 

The other day, in my trades math class, I was asked a question I couldn’t answer.  It wasn’t the usual “when will I use it?”, instead it was “Where did fractions come from?”.  My mind just blanked because I’d never thought about it.  As far as fractions go, they didn’t appear in Europe in the form we use until the 17th century but where did they come from?  How did they evolve?

The word fraction comes from the Latin word “fractio” meaning to break.  The Egyptians used their own version of fractions as early as 1800 BC.  Although Egyptians used their version of base 10, they had individual symbols for 10, 100, 1000, etc so when you did a number like 3250, you ended up using the symbol for 1000 three times, the symbol for 100 twice and five symbols for 10. 

Egyptians even had their own version of unit fractions where they had a numerator of 1 which was assumed because they used a mouth to indicate it was a fraction with the denominator illustrated below the mouth.  For instance, ⅕ had the mouth with five sticks below it. This meant they had to express fractions such as ¾ using two different fractions such as ½ + 1/4 .  However, they had a restriction where you couldn’t repeat a fraction.  So if you wanted to represent 2/7, you can’t use 1/7 + 1/7.

On the other hand, the Romans preferred using describing parts of a whole using words rather than symbols for fractions.  The system was based on 12 unica rather than base 10. Unfortunately using words only to describe the fraction made it hard to carry out calculations.  

The Babylonians had a base 60 number system so their fractions were based on sixteenths but they didn’t have a real way to distinguish fractions from regular numbers which made it more difficult to tell what was going on.  Eventually, they added a zero which helped a bit but it was India where fractions began appearing in a form closer to today’s form except they didn’t use that division line to separate the numerator from the denominator. The Arabs were the ones who added that part to make it much closer to what we know.

Along the way, the Chinese established many of the rules for calculating with fractions such as reducing fractions to simplest form and rules on adding, multiplying and dividing them while using a common denominator.  The Chinese didn’t use improper fractions, instead they relied on mixed numbers. In the mid 1500’s one mathematician published the first book relating fractions to their decimal equivalents.  

Now I know a bit more about the history of fractions and I can answer the question the next time it is asked.  Let me know what you think, I’d love to hear.  Have a great day.

Where in The World Are Conic Sections?

 

I am beginning conic sections in the pre-calculus class.  I’m used to teaching it using the basic equations but I seldom take time out from instruction to have students apply what they’ve learned to provide more of a context to their learning. I knew of only two applications of conic sections but I’ve discovered there are more.

1.  The paths of planets circling the sun.  One can use the formula for ellipses to calculate the equation for orbits of all the planets.  All that is required is to find the length of the major and minor axis.  I did this one time in class and the students had fun calculating the equations for each planet’s orbits.

2. Satellite dishes and parabolic mirrors and microphones.  With just a few measurements, students can calculate the equation for each one.  In fact, they could even research these to learn more about the sizes for each one. Since solar ovens and cookers use parabolic mirrors, have students design their own solar oven.

3. For bridges, both suspended and the ones on the ground, are parabolic in nature.  It would be easy to find measurements and using those measurements, students can calculate the equation for the arch part of the bridge.

4. There is a long range navigation system referred to as LORAN uses hyperbolic conic sections.  This particular system was developed during World War II and was especially popular in the 1950’s.

5. Of course, there are all those video games such as Angry Birds where the object that is shot ends up traveling in a parabolic trajectory.  In addition, many rides such as roller coasters rely on a series of parabolic shapes to create a wonderful ride.

6.  Did you know that the sides of a guitar form a hyperbolic shape?  Image having the students calculate the equation of the hyperbola used on the sides of a guitar?  In addition, the Kobe Port Tower gets its hourglass shape from a pair of hyperbolas.

These are just a few ways conic sections are used in real life.  This makes a nice Padlet activity where students can research each shape or only one and find several examples and calculate the equation associated with the object or trajectory.  Remember earlier when I commented that things that are shot such as angry birds, follow a parabolic trajectory?  Introduce students to the National Pumpkin Chunkin contest via youtube so they can see the pumpkin fly.

It is always nice for students to find real life applications for conic sections rather than situations that work out neatly and seem contrived.  Let me know what you think, I’d love to hear. Have a great day.

Logs and Natural Ln

 I’m in the chapter dealing with exponential functions in Algebra II, specifically ln and log.  I have one student who is always questioning its use in real life.  My first example is always the Richter scale but I need more than that.  First of all, logarithmic functions are used to help solve exponential equations such as the Richter scale, calculating how bright a star is, finding pH or the decibel level of sound.

The Richter scale is calculated as the log (the amplitude of the wave/amplitude of the smallest detectable wave)  so if there is an earthquake and it is found to be 392 times greater than the smallest detectable wave, the equation becomes log (392/1) or log 392 = 2.59…. Or 2.6 if rounded.  Although 392 times larger sounds like a lot, it is really a small earthquake.  

The equation to determine the decibel levels for sound is almost the same basic equation.  It is the decibel rating = 10log(power or intensity of sound/the weakest sound heard by the human ear). So if the students in the  classroom are quiet you have a reading of 10^-7 and if the room is empty, the reading is 10^-12 so the equation would be Decibel rating = 10 Log (10^-7/10^-12) which becomes 10Log10^5 or 10*5 = 50 decibels.

As for finding pH, the equation is -log[H^+] or -Log[ the concentration of hydrogen. If you have a concentration of .0025 for the HCL solution.  Then you simply use the .0025 in the equation so it looks like pH = -log[0.0025] and the pH is 2.6.

In addition, ln and log are used to find missing values in both exponential growth and decay, interest, populations and so much more. Unfortunately, most of my students do not think they will every use this even after I shared with them the time I sat down to calculate whether financing via the credit union or the dealership was better. They figure I did that because I am into math.

For all the time, I’ve taught math, I’ve never had a book that took time to look at the different applications of logs other than in a purely theoretical way. I think that the next time I teach logs, I’m going to take time to have students explore the equations used for earthquakes, sound levels, etc and create a small sticky note in the Padlet application.

Students could research the topic so they could explain the basic equation and then provide examples for using the equation to find different things.  For instance, have students find the rating of the earthquake just like I showed in my example and then give them the rating of an earthquake such as 7.2 and have them determine the intensity of the wave.

This type of activity would put  their learning into context and provide real situations where the math is really used rather than being so isolated.  Let me know what you think, I’d love to hear.  Have a great day.

Happy Easter

 

One More Day

One more day!

KFCW Method of Solving Word Problems.

Yesterday, I showed how to solve word problems using the 5 W’s from english.  Today, it is the KFCW or “Kentucky Fried Chicken Wings”  which is a take off on KWL or know, wonder, learn.  I found it was another way to have my students read and really comprehend the information in the word problem.  

The K stands for what do you know.  In other words, what information does the word problem give you to work with.  The F is for find or what are you supposed to be finding.  In other words, the question.  The C stands for consider or what are the things you need to think about in order to solve the problem.  The W is for the work itself.  

Let’s look at this problem.

“Biologists need to know roughly how many fish live in a certain lake, but they don’t want to stress or otherwise harm the fish by draining or dragnetting the lake.  Instead, they let down small nets in a few different spots around the lake, catching, tagging and releasing 96 fish.  A week later, after the tagged fish have a chance to mix thoroughly with the general population, the biologists come back and let down their nets again.  They catch 72 fish, of which 4 are tagged.  Assuming the catch is representative, how many fish live in the lake?

Know - I know they tagged 96 fish.  I also know that one week later they caught 72 fish of which 4 had tags so 4/72 are tagged.

Find - They want to know how many fish are in the lake?

Consider - This looks like I have to use ratios due to the 4 tagged out of 72 fish.

Work - 4/72 = 96/x.  So 72 x 96 = 6912/4 = 1728 total fish.

This is a bit shorter than what I spoke out earlier in the week but it does require students to think in more detail about what they need to do to solve any problem.  Some problems might require them to find the area of a circle but they only have the diameter so they’d have to remember that the A formula needs the radius which is half of the diameter.  This thought would go under consideration.

Some problems will give a decimal as part of the answer but they can only use whole numbers such as with cans of paint.  Most cans of paint tell you how many square feet  they cover and you have to divide the total area of the walls by the square footage to get the number of cans needed but it’s important to include a note about rounding up to the next integer, otherwise you might not have enough paint.  

So this is my “Kentucky Fried Chicken Wings” Method.  Let me know what you think, I’d love to hear.  Have a great day.