The 5 W’s Word Problem Solving Method

If you’ve taught math for any length of time, you keep hearing that math needs to be taught as if it is a foreign language.  That requires teaching vocabulary, structure, and interpretation.  Unfortunately, when I went to school for my teaching credentials, they never discussed that or even the idea of teaching students to do word problems as if you are teaching a form of reading comprehension.

One year, I decided to teach students to solve word problems as if they were magazine or newspaper articles.  For each problem, I had students identify who was involved, what did they do, how did they do it, why did they do it, where were they, when did they do it, before they looked at the question.  This actually had students slow down and really read the problem before they tried to solve it.

Take this word problem - 

“You are installing rain gutters across the back of your house.  The directions say that the gutters should decline ¼ inch every four feet of the lateral run. The gutters will be spanning thirty-seven feet.  How much lower than the starting point should the lower end of the gutter be?

So students would answer this way.

1.  Who - you are doing it.

2. What - you are installing rain gutters.

3. Why - It doesn’t say but it could be because you need to replace the current ones.

4. Where - The back of the house.

5. When - not given.

6. How - by hanging them so they decline ¼ inch every four feet.

7. How Long - for 37 feet.

After identifying the who, what, where, why, and how, I have students define words they might not know such as decline, lateral run.  

Question Asked?  How much lower than the starting point should the lower end of the gutter be?

Finally - what mathematical concept should I use to solve this problem?  In this case it, I’d use proportion.  ¼ / 4 = x/37.

When students have a procedure like this, they have to take time to slow down, really read the problem, identify important information before they even try to solve it.  Tomorrow, I’ll be sharing my “Kentucky Fried Chicken Wings” Method of doing word problems. It is similar and the students did pretty well with it.  Let me know what you think, I’d love to hear.  Have a great day. 

What Happened When I Used Numberless Word Problems.

 

The other week, I wrote about numberless word problems.  I spent one week giving my students numberless word problems and a simple question.  Such as the first day, I asked students what they noticed and what they wondered about the problem.  Almost every student noticed I didn’t include any numbers and they promptly provided their own numbers and solved it.

The second day, I selected a different problem, again without any numbers and asked students what operation they needed to solve the problem.  I didn’t ask anything other than what operation is needed to solve the problem and most students answered by substituting numbers into the problem to get a numerical answer.

The third day, I provided a different problem and asked why I chose not to use numbers in the problem and again too many students put their own numbers in the problem to solve it.  The same thing happened on the 4th and 5th days.  

So based on this short trial, I can make some reasonable conclusions:

1.  Students do not actually read the problem.

2. Students have had it drilled into their thinking that all word problems must be solved, even if no numbers are provided.  

3. The solution seems to be more important than even deciding what operation is needed to solve the problem.

So I”m wondering if we spend too much time teaching students how to solve word problems rather than teaching them how problems are structured, how to “read” the actual problem, and how to look at a numberless word problem without the urge to solve it.  

Unfortunately, by high school, students are so into finding numbers and just solving the problems that I don’t think they really take time to read the problems.  I have one class that has a lot of word problems.  When they ask me how to solve each problem, I ask them to read it out loud to me before they even try to write out the problem.

I think it is important to move students past the feeling that they must have a numerical answer from any word problem immediately, rather than taking the time to explore the problem itself. It is almost like chugging a glass of wine rather than taking time to look at the color, checking out the aroma, and enjoying the flavors.  

I don’t know what it would take to get students to savor solving word problems.  Let me know what you think, I”d love to hear.  Have a great day.

Warm-up

 

Write a word problem based on the photo

Warm-up

 

Write a word problem based on the photo.

Electric Bills and Math

 

Sometimes it is nice to take a few minutes out of your regularly scheduled pacing to introduce students to the connection between electric bills and energy consumption of appliances and devices. I'm not sure students know how much power is used by the refrigerator or stove.  Yes, it is usually a topic covered in life skills, consumer math, or personal finance but it doesn't hurt to have students take time out of an algebra class to learn this.

I've seen it suggested that students use information on appliances and devices to calculate how much electricity each one consumes to give students an idea of their families monthly electric bill.  To calculate this, students need three pieces of information. They will need the wattage used by the appliance or digital device, the number of hours it is used each day, and the price per kilowatt hour the electric company charges.  

The kilowatt hour cost is easily found on the electric bill or it can be gotten by calling the local electric company.  I know that in many places in Alaska, the cost if about $0.34 per kWh.  The number of hours the item uses electricity will vary depending on what it is.  For instance, the refrigerator is on 24 hours a day while the dishwasher may only be on a couple times a week.  

To find the amount of power it uses, it will be necessary to go one of three places.  First try the back or bottom of the appliance.  You are looking for a small metal tag with the information on it.  If that doesn't work, check for the book that came with it and lastly, look for the information on the internet by searching for the year and model.  The information can usually be found in the technical specs.

Once students have all three pieces of information for all their appliances and electronic devices, they can get a good estimate by multiplying the wattage times the number of hours used in a 24 hour period, before dividing the product by 1000 to convert from watts to kilowatts. The next step is to multiply the result by the price per kWh to get the final amount per day.  The last step is to multiply the daily amount by 30 to get a monthly amount which is a part of the total monthly bill.

If this is done for all appliances and electronic devices used by the family, students can see how much of the total bill each accounts for.  Then take the results of this exercise to create a pie chart or other visualization to see which ones use the most and which use the least.

This exercise can be extended quite easily by having students calculate annual usage based on taking the daily amount and multiplying it by 365 days.  If they do this for each appliance or electronic device, they can use the results to determine how much they spend every year for power.  Another extension would be to get a year's worth of electric bills to see when usage spikes and drops.  They should notice there is an increase in the middle of winter and summer due to the additional heating and cooling demands made by the season.  

Sometimes the yearly amount is more than most students think it is and they are shocked.  It's usually enough to have a nice visit somewhere or even buy that new electronic device they've been wanting.  Let me know what you think, I'd love to hear.  Have a great day.

Beats Per Minute into Miles Per Hour!

 During the winter, when it is horrible outside, I use exercise DVD's to get my workout in.  Many of my DVD's state if you do the whole video, you'll have covered 4, 5, or 6 miles and if you only do one mile, it takes somewhere in the range of 12 to 15 minutes.  I've often wondered how the person determined that you'd done a miles since many of the moves involved going backwards or sideways or even in place such as knee lifts.

Did they used the 10,000 steps a day idea where 2,000 steps approximately equals a miles but if the person is trained, they'd know that originally started out as a marketing slogan and stayed around. It took a bit but then  I discovered  a mathematical formula involved in helping the instructor determine when they've reached a mile.  If you've every worked out in some sort of group exercise such as aerobics, the instructors always have music on.  If it's a cool down, the music is slower, if its at the point of highest activity, it is faster and we love moving with it.

The background music inspires us to move faster or slower and it is the music that helps instructors calculate when they've incorporated enough movement into the session to reach one mile or more. The calculation begins with the number of beats per minute the music is set at.  The slower the music the fewer beats per minute and the faster, the more beats per minute.  It is the number of beats per minute that is the starting point to calculate miles per hour.

The beats per minute is either comes with the music if it is specifically designed to be used with an exercise routine or it can be found by counting the number of downbeats during one minute of music.  The next step is to determine how long your stride  or the distance between one step and another.  An easy way to do this is to place a marker on the floor as your starting point, then walk or stride for 10 steps and mark where your foot is.  Take a measuring tape and measure the distance between starting and finishing marks.  The last step is to take this distance and divide it by 10 to determine the distance you cover in one step.

So now we are ready for the actual formula used to convert beats per minute into miles per hour.

(Beats per minute x stride length in feet x 60)/ 5280 feet

This means the instructor on the DVD used this formula to determine how far they might travel in one hour. Then using a bit more math, they'd figure out how long it takes them to cover one mile and you end up with the time it takes to complete the mile based on the pacing of the music.  This is why they separate the warm-up and cool downs from the actual mile so the calculations can be done 

Based on this, the miles done on the DVD's are based on the instructor's stride but I tend to look at time only when I record my daily exercise but it is still cool to see the math behind this.  Let me know what you think, I'd love to hear.  Have a great day.

Step Functions Are All Around Us!

I've been teaching for many years and I have taught about step functions but it hasn't been until fairly recently that I've been able to find real life situations that illustrate step functions.  To begin with, step functions are defined as a function whose graph looks like a series of steps because the values jump.

If you go by the math textbooks, step functions seem to be completely theoretical because there are never any real examples to show us how step functions operate in real life.  However, with a bit of research, it is easy to find real life examples.

Let's start with postage stamps which are priced for mail weighing up to one ounce.  Back in 1926, it cost 2 cents to mail a letter but the price increased to 3 cents in 1932 where it remained at that price until 1958 when it jumped to a whopping 4 cents for the first ounce.  The next jump came in 1963 when it moved to 5 cents and five years later in 1968, the price changed to 6 cents. The price jumped every few years until 2019 when it cost 55 cents to mail a letter.  The USPS website has a lovely chart with the cost of a stamp from 1885 to 2019.  The jumps are generally from one to three cents except in 2019 when it jumped from 50 cents to 55 cents.

Again, looking at postage stamps, we have a step function for just mailing something first class.  As mentioned earlier, the first ounce costs 55 cents which means that as long as the letter weighs between 0 and .99 of an ounce, it will cost you 55 cents to mail but the minute you go over 1 ounce, the letter will cost an additional 20 cents for the next full ounce and 20 cents each additional ounce.  The cost is not decreased if you are mailing a letter that is 1.5 ounces. It will cost 75 cents if it is 1.1 ounce or 1.99 ounce as the cost if for anything up to the full ounce weight.

Another example of step functions in real life is the minimum wage.  The first minimum wage was set at 25 cents per hour back in 1938 and one year later in October of 1939, the government raised the minimum wage to 30 cents per hour where it remained till 1945 when the minimum wage jumped to 40 cents per hour.  Over the years, the wage was raised periodically until 2009, when it was set to $7.25 per hour. This governmental website has a lovely chart showing the raises over the years.  This lists the federal minimum wage because some states such as Alaska have a higher minimum wage.

Then there is the wonderful Hershey's candy bar.  In 1908, this candy bar cost 2 cents where it stayed till 1930 when they raised the price to 5 cents per bar. The chocolate bar continued to cost 5 cents until 1969 when they raised the price to 10 cents and the price jumped to 15 cents in 1974.  Three years later, in 1977, the price was 20 cents and one year later the cost was 25 cents.  The price continued to rise till today when you can find the standard bar for around $1.00. This site has a nice summary of the price of Hershey's candy bar and the change in weight as the prices changed. 

Here are four examples of step functions because when graphed as a step function, the graphs reflect the data more accurately and give a better representation of the situation.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

If a tiger can run at 52 km per hour but only for 100 meters? How long will it take them to cover 100 meters?

Warm-up

If a male tiger weighs 98 pounds at birth and 572 pounds when an adult, what is the percent increase of weight between birth and adulthood?

Problems With PEMDAS Or Please Excuse My Dear Aunt Sally!

By the time my students get to high school, they've memorized the order of operations but still do not understand that it can be used fluidly.  Students are taught the pneumonic PEMDAS or, Please Excuse My Dear Aunt Sally or something similar as a way to help them remember the order of operations but too many of the students have a disconnect between the pneumonic and performing the order of operations.

I've found there are situations when I can ignore the rules and others where I have to adhere to them in order to get the right answer. For instance, if the equation uses the distributive property, it is sometimes better to divide rather than distribute because it makes it easier and more likely to get the correct answer because students often forget the second term inside the parenthesis.

On the other hand, students love multiplying the coefficient and the value of x before squaring it in a problem like 2(3)^2.  I admit that I remind them that the 3^2 has to be carried out first before they can multiply the answer by two to get 18 but they could easily write the problem as 2*3*3 and get the correct answer.  

In addition, the P in PEMDAS can limit students because it stands for parentheses so when they run into any other type of grouping symbol, they don't know what to do because they aren't the proper shape.  Since multiplication and division are right next to each other, students often perform all the multiplication first and then all the division rather than doing them as you come across them in the equation.  The same applies to addition and subtraction.

One solution I've seen suggested is the GEMA one where G stands for grouping symbols because it includes square brackets, parentheses, and even implied grouping symbols such as division in fractions, or even absolute value.  E stands for exponential operations, M for multiplicative operations, and A for additive operations.  

The A for additive operations is really more appropriate since subtraction can be written as adding a negative number but unfortunately, many students do not understand that when they enter high school.  I've talked about subtracting the number or adding a negative and my students ask me what I'm talking about.  They don't see the two as being interchangeable.  The same applies to the idea of M being used for multiplicative operations.  They don't see that dividing by 12 is the same as multiplying by 1/12.

I think this is one reason students have difficulty applying the order of operations to any problem.  They don't fully understand the relationships between multiplication and division and addition and subtraction.  This might be because in elementary school, they are taught that multiplication is repeated addition and division is repeated subtraction.  Then once they get to signed numbers they see subtraction as subtraction and not as adding a negative number.

Perhaps when we teach anything in high school math, we should take time to talk about how we use order of operations to solve problems as a way of reinforcing it's use.  Let me know what you think, I'd love to hear.  Have a great day.

Numberless Word Problems

The other day, I ran across the idea of using numberless word problems in class.  Most students when they see a number problem look for the numbers, apply one or more operations to it in the hopes they arrive at the correct answer.  A few understand that the material taught in the section is to be applied to the word problem but most haven't gotten the hang of it.  

Numberless word problems have a every thing a regular word problem has except for the amounts.  We know students automatically look for numbers rather than reading through the whole problem.  An example of a numberless word problem might be "I travelled ______ miles on a tank of gas.  I started with ______ gallons of gas, how many miles per gallon did I get?"

When the numbers are removed, students are forced to look at the meaning and context behind the problem. It helps develop mathematical reasoning.  They are forced to move away from the numbers and finding a solution to understanding exactly the problem itself.  It allows them to focus on context and the underlying structure of word problems.

In the past,   I've given students numbers and asked them to create their own word problem using the values and over half of the incoming freshmen wrote a one sentence problem such as two plus half of twelve is 15.  This indicates they had not yet learned how a word problem is put together instead they'd learned to focus on looking to create an equation to find the solution.

The great thing about giving students numberless word problems, is that you've removed all numbers so students can't "solve" the problem.  They have to slow down, read the problem itself to determine the context of the situation.  It gives them time to figure out how to go about finding a solution or a plan of action. 

Furthermore, by removing the numbers, students have an opportunity to notice that certain problems are similar to each other such as addition problems have them combining things  rather than treating each word problem as something you look for key words to tell them what to do with the numbers written down.  They have to really think about how they would solve it by creating a plan.  

In addition, many students hesitate to solve word problems that have fractions or large numbers or scientific numbers because they are uncomfortable working with them.  In situations like this, many students shut down and refuse to try but by removing the numbers, a limitation is removed and students are more willing to look at word problems because they cannot get a "wrong" answer.  

One way to utilize numberless word problems is to create a set of numberless word problems so students can sort them by operation.  This can easily be done in pairs or small groups where students talk about the phrasing to determine if the problem is asking for addition, multiplication, subtraction or division.  

A second way is to provide one problem to a small group of students and have them do a "What do you know about this problem?", "What do you need to know?" before answering questions such as "What is a possible answer to this problem?"  Then pass out a second problem which is the first problem with one piece of information added so students can answer "What changed in this problem?" "Does this piece of information change the problem?" if so "How?"  Then ask them the "What do you know and what do you need to know "questions before asking for possible solutions.  Finally, provide a third problem with all the information for students to solve independently before they work with a partner to prove their solution.

This way of teaching word problems can be used from Kindergarten to seniors in high school.  Perhaps by using numberless word problems, students will unlearn the idea they have to grab all the numbers, combine them in some manner to get the "right" answer.  Let me know what you think, I'd love to hear.  Have a great day.

Daylight Savings Activities.

This past weekend, clocks moved forward one hour as part of the daylight savings time ritual.  This is the one where we wake up and our bodies tell us the clock is wrong.  It is the time when we wake up to a bright day, instead of early morning.  

There are several activities students can do to find out more about the mathematics involved with daylight savings time.  First, assume that daylight savings time lasts for 34 weeks of the year.  Ask students to calculate what percent of the year daylight savings time uses.  Then take this a step further and have students calculate the percentage of the year in days before comparing and contrasting the two figures.

Another aspect involved with daylight savings is to compare sunrise and sunset times a week before daylight savings time begins with both sunrise and sunset times a week after the change for Colorado and Arizona.  Colorado participates while Arizona doesn’t so you’ll want to have students determine the length of daylight in the two states so students can see how the change actually affects things 

The next activity requires a map of the four corner area with the Navajo nation, Arizona, Utah, New Mexico, and Colorado. You will have students create a trip which has them changing clocks five times without repeating any part of the trip.  It is important for students to know that although the Navajo nation is in Arizona, it goes on daylight saving time as well as Utah, New Mexico, and Colorado.

Another activity is to have students research how much going on to daylight savings time actually saves for New Mexico, Montana, and Alaska as compared to before or after being one hour advanced.  Then students can prepare two graphs showing the way expenses are divided up between the two.

The final project is to have students research claims that going on daylight saving time causes more deaths, more health issues, more accidents at work, decreases productivity on the job, and an increased number of strokes.  Students can research the claims and provide evidence to support or prove them incorrect and put it all in a report with graphs etc.

Most times, students tend to live through the process without knowing how the change affects our lives and the world.  This gives them a chance to explore it.  Let me know what you think, I’d love to hear.  Have a great day. 

Did You?

 

Did you remember to change the time?  Tomorrow, we will look at the cost of daylight savings time.

Daylight Savings Time

 

Clocks spring ahead one hour tonight.

The New Assessment.

 

Since Covid hit a year ago, we’ve had to readjust our thinking on how to assess students, especially if having to teach by distance.  There are ways to create assessments for these times.  One of the first things recommended is to quit assessing everything students turn in.  I know I had to get my students over the “Will this count for my grade?”.  They seemed to think that they should receive credit if they turned it in but I only grade some of it but I don’t tell them which ones I’ll be grading.

Performance tasks and projects are wonderful for this situation but they should be designed so that students are required to apply their knowledge to new and different situations.  When performance tasks are applied to new and different situations, the teacher is creating an potentially engaging opportunity for them to apply multi-standard thinking. 

If performance tasks and projects are designed to be large, multi week assignments, one might consider breaking it down into smaller units, especially if it requires students to research a topic.  When a big task or project is broken down into smaller units, students are less inclined to feel overwhelmed.  

Rather than assigning tons of problems, cut down on the number of problems to one or two and have them create a google slide showing their work and then they can record an explanation of their thinking as they solved the problem.  They could discuss what they had trouble with on the problem, or where they got stuck, or where their thinking was wrong. Or students could create videos to share how to do a problem with the others.  They become the teacher and help other students.  

Take advantage of technology tools such as google classroom to administer tests and quizzes.  If one chooses the short answer possibility, students can provide a written explanation of their work or show their work for the problem.  It is also possible to create  multiple choice questions but I’ve discovered one has to check student answers because if their answer does not match the solution exactly, it might be marked wrong. 

I’ve been known to assign a Kahoot or Quizzes to students and disabled  the timer so students didn’t feel rushed when completing it. I analyze the answers and assign a grade based on their work.  I’ve found that students usually like the games so much, they want to complete the assignment rather than trying to “find” the answer on the internet.

Hopefully, everyone will be back at school for the 2021-2022 school year but if not, these are some things to think about.  Let me know what you think, I’d love to hear.  Have a great day.

R.E.A.L. Criteria

Here it is, March of 2021, almost one year after Covid struck.  My school was one of those that shut totally down for the final quarter of school but once the new year arrived, school began in person after a delay and several times of changing from staying distance to hybrid, to full green with masks, etc.  Overall, we’ve been pretty much green but we have a certain number out everyday due to quarantine, especially with the one case found last week.

Unfortunately, many schools are still closed and having to teach by distance which can make assessment even more difficult. Schools still want to have some sort of assessment but under the circumstances, it can be harder than normal.  One recommendation is that teachers focus on what is important in the curriculum.   

Unfortunately, it has been noted that a teacher  left with no guidance, they tend to  revert to teaching what they are most comfortable with rather than looking at the whole picture. For this situation it has been recommended that one look at the  R.E.A.L. criteria.  R.E.A.L stands for readiness, endurance, assessed, and leverage.

Readiness refers to the skill or piece of knowledge that is necessary for students to have so they do well in the next level.  For algebra I that might be being able to manipulate or rewrite formulas to use in Geometry or Algebra II.  Endurance refers to knowing the material for longer than the unit so when it is revisited, they will still know it or at least be able to recall it.  Students will know it in future classes all the way to college.

Assessed means that the material may not be essential to future learning but it is tested on nationally standardized tests such as ACT or SAT.  In other words, it is a skill they need to do well on future state and national tests but is not necessary to do well in college. Finally, is this skill or piece of knowledge that could be used across the curriculum or in other classes.  

This is a nice guideline for schools who are trying to focus on what should be taught during the year.  The guideline focuses on information that is important for future classes and for standardized tests and unfortunately, those are not always the same.  Using the R.E.A.L criteria means teachers have to sit down to determine which standards meet this criteria and are the most important.  This step is a time consuming collaborative step but it is well worth it.

On Friday, I’ll look more at ways to assess during the Covid.  Your district and mine might be face to face but I know of several that are still distance.  In addition, many of these suggestions work in a face to face situation just as well as distance.  Let me know what you think, I’d love to hear.  Have a great day. 

Crossword Puzzles

 

We’ve had times during the school year when we’ve had to send packets home.  For the most part, my students do not do the work. In fact, many of them “forget” it at home or “lose” it. For the last packet, I sent home a crossword puzzle to help students become more familiar with the vocabulary associated with the current section.  My principal said I needed to send home real work because crossword puzzles are not real work.

It turns out that the use of crossword puzzles can help students learn vocabulary because the clues makes the student think about what they’ve learned in the past and recently. It helps them go through their own vocabulary to see if they know the word.  In addition, it helps students work independently while actively engaged.  

There has been research conducted on the use of educational crossword puzzles in schools.  They are good for introducing and reinforcing mathematical concepts.  Puzzles can be used to informally identify what students are missing from their knowledge base and provides students with a way to self-assess their understanding.   It allows students to study together and work collaboratively. Many students felt they learned more using this form of review than standard methods.    

Fortunately, there are so many premade puzzles online or sites where teachers can make their own puzzles but there are things to keep in mind when looking for a or creating a puzzle. 

It is suggested that the crossword puzzle have single word answers rather than answers composed of multiple words.  Unfortunately, math often has vocabulary with multiple words such as greatest common factor, or mixed numbers or improper fractions.  The hints should also be short if possible.  

Try to group words that are related in the same area of the crossword because it makes it easier for students to come up with related words.  For instance, you might use  hypotenuse and right angle in the same part as the Pythagorean Theorem.It is important to make sure the clues are extremely specific.  If the clues are too vague, students might think of the wrong word.  

In addition,  a good way to help students with their vocabulary is to have them create their own crossword puzzles and have other students try them out.  This comes highly recommended by several sources.  Graph paper is wonderful for making crosswords by hand or students can go to a site specifically designed to create crosswords from a list of clues and words.

I love using crossword puzzles.  I have given students the opportunity to use the computer to look up the word associated with the clues.  I find it’s another way for them to learn vocabulary.  Let me know what you think, I”d love to hear.  Have a great day.

Warm-up

 

If 2 ounces of dried rose petals equals 2/3rd of a cup, how many cups of dried rose petals will you need for 1.5 pounds.

Warm-up

 

If 2 cups of rose petals weigh about 2 ounces, how many cups are needed for a kilo of rose petals.

Math Solving Apps And Distance Learning.

 

In these times of uncertainty, it is hard for students to get the personalized help they need when learning mathematics.  Although my school is mostly green, a certain number of students are in quarantine each and every day.  Just this past few days, we’ve had COVID hit, so all students are on distance learning and we’ve had teachers out for various reasons including quarantine.  

The science teacher is off long term, so he sends his sub regular work packets but the students have discovered that with a bit of searching they can find the answer keys to those work packets and simply copy everything down so they don’t have to do much.  I suspect several of my students are doing the same thing for their math assignments or are using a app that will show them the steps and the answers.

I suspect this because one of my students who struggled, suddenly began turning in work that was 100 percent correct.  Unfortunately, when I gave the test, he failed it completely.  He also made a comment about “Photo Math” so I think he was using that program or a similar one.  This is a very real issue both in distance and in person learning.

I am not opposed to using technology to find the answer but most students just copy without trying to figure it out.  I’ve looked at one app where they have a few steps written down but the steps are not written out in typical math language.  

For instance, I gave it the problem 2x + 3 =7.  For the first step it wrote 2x = 7-3 and for the explanation it said to move the 3 to the other side and change the sign rather than telling students to add -3 to both sides.  So it tells the user the short cuts without the mathematical reasoning.  

Sometimes, I assign just a few problems but I require students to show their work and explain what they did for each step.  The reason for the justification is that students who use the apps have to slow down and include everything.  If they use an answer key, the answer keys often do not list the explanations for each step so the student either doesn’t include that part or they have to figure it out.

In addition, when students have to write out justifications or explanations for steps, it lets the teacher know if they really understand the process and the concepts involved in finding a solution.  Furthermore, this is good  because it helps students develop the mathematical vocabulary necessary to communicate their thinking to others since many have trouble with this.  

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

Traffic Signal Optimization

 

Have you ever gone somewhere and it seemed like you hit every single red light between your house and your destination?  You could swear the traffic gods are against you because it doesn’t seem to matter what speed you travel at, you still get stopped?  I have felt that way quite often.  In terms of mathematics, it is referred to as traffic light optimization.  

Traffic light optimization is defined as the process of optimizing the timing of traffic lights to as to make it so people do not have to idle too long and cars do not have to speed to make it to the next traffic light which causes a decrease in the release of carbon dioxide and cars consume less fuel.  It is also used to help increase overall traffic flow.  

We know that traffic modeling looks at unidirectional flow where the density of cars is based only on a single spatial dimension.  Recently, certain traffic models have begun looking at intersections because they form a large part of the whole system.  The models do one of two things.  They either look at minimizing travel time or maximize the flow of traffic.

We know that traffic models look at traffic as a fluid flowing including traffic jams and traffic lights are used as a tool to redirect traffic so as to mitigate issues even on roads that have a high traffic flow.  So mathematicians look at the ratio of traffic volume to capacity among other things. Another thing that traffic light optimization needs to include  how long it takes for people to cross any street at a traffic light.

There are two types of traffic signal optimization.  The first is time-of-day scheduling where traffic signals are set to have exactly the same cycle length which can change based on the demands of traffic throughout the day.  This schedule is the same day after day.  This schedule is good for roads with over 400 cars per hour, downtown areas, areas with bicycles and buses, and special events.

The second is set to respond to traffic. The signal has several pre-programmed timing plans which are implemented based on the demands of the traffic. This works best in situations where there is only one traffic light here and there.

When looking at traffic signal optimization, people have to consider how long the whole light cycle is for red, yellow, and green, the time the amount of time it takes for traffic to get from one light to the next based on the posted speed, 

In addition, traffic signal optimization is not done just once but is redone multiple times to account for population growth in the form of new developments, and increased traffic, to cut down on delays or stops so drivers are less frustrated, to improve traffic flow so vehicles use less fuel and produce less emissions, and finally to allow current infrastructures to carry higher numbers of vehicles thus delaying the cost of adding infrastructures.  If utilized properly, traffic signal optimization can produce a benefit to cost ratio of 40 to 1 which is quite good.  

This is the basics of traffic signal optimization and why it is important.  There is a lot of math involved in setting everything up.  Let me know what you think, I’d love to hear.  Have a great day.

Traffic Modeling

 

Have you ever been frustrated driving to work because you ended up sitting in a line of stopped traffic and the route you chose was the same one the local radio station suggested as being faster?  Or instead of the trip taking 20 minutes, it ended up taking 55 minutes?  The study of traffic is called traffic modeling and is a valid application of math.

Modeling traffic is done to help pinpoint why traffic slows down, or becomes jammed so cities and states can create better road and highway systems.  It is also used to find the best routes for traveling from point A to point B. The thing about traffic is that it varies from day to day based on signal lights, number of cars, accidents, behavior of the drives and the way the roads are set up and connected to each other.  

There are several methods used to model traffic.  One method is to use fluid dynamics to treat traffic as if it is a flow.  Another uses cellular automata to model traffic flow.  In other words, the roadways are divided up into cells which cars occupy and follow certain rules to explain how cars change location and speed.  Furthermore, traffic modeling uses a certain amount of randomness so they have to apply a type of Monte Carlo simulation to the trials.

Modelers have also used differential equations to model a dynamic system where they calculate how one car changes position constantly in relation to surrounding vehicles and based on the speed limit.  One of the results of traffic modeling is that it is possible to have a traffic jam artificially created by the behavior of drivers and a certain amount of congestion.  If you are on the freeway and someone suddenly hits their breaks, others behind will hit breaks and it can move back a mile or two and you have a parking lot. 

One thing about traffic modeling is that you can read two different reports with two different results.  Unfortunately, traffic modeling is extremely complex because although all roads are connected in some way, what happens in one part can affect the whole.  There are so many different things that can affect traffic.  For instance, one would think that if a person had one more possible route added to choose from to get to work, it might speed things up but in reality, it can actually slow things down without increasing the number of trips someone makes. This means that one small change can have an impact on the whole system.  

Tomorrow, I’ll look at the study of traffic light optimization.  Let me know what you think, I’d love to hear.  Have a great day.