Leap Day

Happy Leap Day - The Extra Day To Keep Things Right.

She Will Be Missed.

 We got to know a bit about her from a movie on three women whose contributions to the space program were not really praised until the attitude of the country changed to accept both women and African American women as able to do the same work as men.

Her name is Katherine Johnson who just died at the age of 101.  She began her career with the National Advisory Committee for Space in 1953 at the Langley Labs in Virginia before it became NASA.  We learned about her in the movie "Hidden Figures".

She provided calculations that helped sync Apollo's Lunar Lander with the Command module that circled the moon.  She helped put men on the moon in 1969 with her mathematics.  She also became the first to have her name listed as a coauthor on a research report containing her calculations for space flight.

The scene where she helps calculate Alan Shepards space flight and where she is asked to double check the computers calculation for John Glen's flight before he was willing to go on the trip are all true.  She worked for NASA until she retired in 1986.

Katherine Johnson was born in White Sulpher Springs, West Virginia in 1918 but she skipped several grades due to her ability with numbers before beginning high school at the age of 13.  She attended the high school sharing the same campus as West Virginia State College for African Americans and when she was 18, she began an undergraduate program there.

She managed to find a mentor in W.W. Schieffelin Claytor who was the first African American to publish his material in a mathematics journal and the third person to obtain a pHd in Mathematics.  Katherine Johnson graduated in 1937 with the highest honors she could in math and French before finding a job as a high school teacher in one of the segregated school in Virginia.

Although she was offered the opportunity to enroll in a graduate mathematics program at West Virginia University in 1939, she turned it down to marry and start a family.  She was honored by the invitation because she was the only woman of three African Americans who'd been invited to the predominantly white college.

She returned to teaching when her children were older and in 1952 she heard about jobs opening up at the West Computing section of the National Advisory Committee for Space.  She, her husband and family moved there where she secured a position as a temporary that turned into a permanent position.  Her husband, James Gobel died of cancer in 1956 and she remarried Lieutenant Colonel James Johnson in 1959.

She continued working for NASA until she retired in 1986 but she wasn't forgotten.  She received the medal of Freedom in 2015 and just last year she cut the ribbon on a new facility named after her.  Monday, I heard the news she had passed on.  I didn't realize she was still alive when I saw hidden figures and so this took me by surprise.

I actually have a degree in mathematics and I am proud of it because I am one of two women who were in the program. I didn't know of any women who contributed to the world as we know it. It is only as time passes I learn more about women who contributed to the field of mathematics because even now they are not as well known as men but they are slowly becoming more visible.

I thank all these women who made it possible for me to go into mathematics. They opened the doors for me and all the other women who chose to enter this field.  Let me know what you think, I'd love to hear.  Have a great day.

New Twist On Population Growth.

 The other day, I read an article on some math students who took the time to figure out how long it would take for Tribbles to completely fill the Enterprise.

For those of you who have never watched the original Star Trek, people ended up visiting a place where a trader was selling these cute little purring creatures but a problem arose when Uhura brought one back to the ship.  The creature began reproducing so fast, it was born pregnant.  Soon, they were finding Tribbles all over the place including in the grain they were transporting to Sherman's Planet.  The only good thing about the situation was the tribbles died letting the captain know the grain had been poisoned and it helped them find the undercover Klingon agent who infiltrated the area.

These three undergraduates from the University of Leicester published the paper back in November 2018 in the Journal of Special Topics.  They used information from the television series which was given by Spock.  He stated that tribbles had liters of 10 offspring every 12 hours so they students used the equation -  NT = 11t/tr. to describe the growth of the tribbles where t represents the time spent breeding and tr is the time it takes to breed. After three days, there were 1,771,561 tribbles running all over the ship and within 4.5 days before the tribbles filled the ship from top to bottom, stem to stern.

The reason the tribbles grew so rapidly was they had been removed from their natural environment so they no longer had any predators to curb their growth.  Consequently, they had nothing to slow down their growth.  This is similar to at least two other situations people has experienced in real life.

1.  Australia did not have any rabbits until they were brought over in 1788 by the First Fleet.  The First Fleet released 24 wild European rabbits into the countryside in Victoria so they could hunt rabbits just like they did in Europe.  These 24 rabbits reproduces so well that by 1900, rabbits had made it to pretty much every place in Australia, including Western Australia.

In fact, the Australian government built three rabbit proof fences around the country but they were too late to stop the animals expansion.  It is estimated the rabbit population in the 1920's had reached 10 billion creatures.  Consequently, the government began using other methods from poisoning, to hunting, to releasing a virus to eliminate rabbits but it was not fully successful. The rabbit population is estimated at 200 million in 2018.

Currently, rabbits compete with livestock for pastures, with farmers for produce, and these wild rabbits are damaging the environment.  Another species showing a huge increase is the White Tailed Deer in the United States.

Back in the 1930's the White Tailed deer had an estimated population of 300,000 but since then it is estimated the population is estimated to be around 30 million.  This has occurred for several reasons but one major one is their enemies such as wolves, grizzlies, and cougars have disappeared so the young, sick, and old are no longer dying and live out to a ripe old age.  Then it has been found that as forests are cut down, this helps the deer because they do well in edge type areas.  These are the places between freeways, newly planted lawns, areas around airports, etc.

In addition, the number of animals being killed by hunters is decreasing due to the decrease of hunters.  It is estimated that 6 million deer were killed in 2014 but in ideal situations, the deer population will double every year.  They also compete with farmers for produce, destroy landscaped yards, and cause accidents when they step into the road.

These are three examples of real life population growth where the components that keep the numbers under control go away and allow for explosive growth.  There is enough information to have students determine rates of growth for all three groups.  Let me know what you think, I'd love to hear.  Have a great day.

Programming Video Games

I have at least one student every year who wants to get a job in the video game industry working on games but he insists he doesn't need any math.  If he's going to help write the programming for the game, he is going to need to know math.  He's going to need to know quite a variety of math.

In fact, math is the basis of all games because the math tells which direction and where characters move, how long it takes before they can regenerate, what they need to move to the next level, or anything else.

Most games are built in virtual worlds filled with rules that keep it running.  These rules are based on mathematics because math explains and defines the world.  Some of the math may be based on physics such as when you shoot a banana at the top of a building, it has to follow a parabolic trajectory.  If your character has to climb a mountain, the mountain has to be defined by slope.

Video games require mathematics from Algebra, Trigonometry, Calculus, Linear Algebra, Discrete and Applied Mathematics, and so many more.  They don't necessarily use everything but they do use a lot of math.  Programmers especially need to know how to use matrices, delta time, unit and scaling vectors, dot and cross products, and scalar manipulation among other things.

Mathematics is what is used to make everything in the virtual worlds so they look real and everything  reacts correctly.  If you shoot a bullet in Call of Duty, it is math that provides the proper trajectory.  If you jump over something, the math helps provide the proper movement.  We also need to know how much damage someone will sustain if they fall off a cliff.

Some of the specific types of math needed include:

1.  Probability and statistics.  Probability is used to calculate the chances of something happening and the degree to which it occurs such as if you swing your sword at the ogre, what are the odds you will actually kill him vs just damage him and the extend of the damage.  Calculating odds is needed for every event that happens in the whole game.

2. Linear Algebra provides matrix theory and the more practical uses of Algebra.  This is what programmers use to understand and move things through three dimensional space.

3. Geometry provide an understanding of how shapes are formed.

4. Algebra and calculus help find limits, functions, derivatives, and maximum and minimums along with helping to explain the physics that is happening.  Then there is parabolic trajectories, rate of change, and so much more.

5. Mathematical modeling takes the basic mathematics such as geometric shapes and figures out how to put them together to make the more complex shapes.  For three dimensional space, one creates a cylinder by rotating a curve around it's central axis.

6. Game Theory helps people learn how everything works together so programmers know how it all fits.

These are just a few examples of how math is used in creating video games.  The next time a student talks about video games, let them know how useful math is in this field.

Warm-up

If the manatee swims at 5 miles per hour, how long will it take to cover a distance of 482 miles?

Warm-up

If a sloth travels at a rate of 0.003 mph, how long will it take the sloth to cover 1 mile of terrain?

Making Board Games in Math

 In the day of technology and digital devices, students and teachers no long consider using board games in math.  Teachers tend to automatically look for digital apps to provide the practice students need to learn new material.  What is wrong with reverting to having students create and play board games.  There are apps that replicate many of the most famous board games such as backgammon and others.

There are several reasons to look at having students make their own games.  First, it can make practicing mathematical skills fun, second they can be used to develop mathematical reasoning, third it can be used to teach new concepts, and lastly, it can be used as a method of hands-on learning.  Furthermore, it can increase student involvement when they make their own learning activities or games.

Before they begin the process of actually making the game, students need to figure out what math concept they want to incorporate in the game and how will they do that.  These games are based on math so they need to figure out how to do that.  I've actually had kids play the 5 dice game where they rolled the dice and used the operations to make a certain number.

If you have students create their own board games, there are certain steps to follow to ensure a good game.

1.  Select a game style for the game.  Will it resemble something like Candyland, Scrabble, Backgammon, or will it be something totally new, or a hybrid of both?   Will it require dice, a spinner, a bunch of cards?  What type of board will there be?  Will it have shortcuts, places to send you back?  Do you need the exact number to go out?  These are some questions to think about when selecting the game style.

2.  Next break down all the rules, the movement, everything, so students know exactly how the game is played.  This is the point where the design is worked completely out so they can build it and share it with others to play.  This is where students make a list of what they need for the game from pieces, to cards to dice.  This is also where they have worked out all the rules from starting the game, to movement, to playing it, to ending it.

3. Then the student is ready to build the board.  The board can be made of a manilla folder, piece of cardboard, thick construction paper glued to cardboard or thick card stock.  Students can use permanent markers to create squares or mark the area between colored construction paper squares or painted squares.

4.  If the game requires pieces to move around the board, then the students make them.  If the game does not require pieces, the student can move on.

5.  This last step has students putting everything together, trying the game to make sure it all works, and then playing it with other students to ensure it works exactly the way they thought it should.  This is the "Beta" testing stage, where students discover any adjustments they need to make.  Do the rules need clarifying?  Did they goof and forget to put some sort of obstacles in the way?  What can they do to make it better.

Now it is ready to play with other students to see how it goes.  Try it with other classes, or other grades. Let me know what you think, I'd love to hear.  Have a great day.

Math Based Game Faires.

 One year, I had my students create their own board games.  The idea was for them to choose a topic and create a board game complete with pieces and rules so younger students could play the games and learn something.

The hardest thing was to get students past trying to use the board games they were familiar with but without changing them.  With encouragement, they eventually began modifying some of the games to work with a topic such as adding or subtracting fractions, finding area, or other such topic.

Some of the areas students struggles with included creating a set of rules that made sense and explained everything the players needed to know.  I often read the rules and asked questions so students could revise them until one could play the game.  Once students decided they'd finished their game, I had another group of students try to play it, evaluate the weak spots while finding some good things about it.

Eventually, the students all had a playable game with rules that made sense and anyone could play.  Once they were ready, I made arrangements with several elementary teachers to have my students come in with their games for children to play.  I explained to my students that this is the Beta testing part of the processes to make sure the younger ones could play the games as designed.

The elementary students were asked to provide feedback.  If they were old enough, they provided it in written form and if they were younger, the teacher recorded their comments and students listened to their suggestions.  This often lead to revisions and when the games were as finished as possible, we invited parents in one night to play the student created games.  This event turned out to be quite popular with families.

Another year, I taught one class of high school students who'd missed so much school they were well below where they needed to be.  I happened across a huge book of math games for grades 1 to 5.  I had the students break up into groups of two and they had to look through the book to find a game they thought elementary kids would enjoy.

They had to make sure they had everything they needed for students to play the game.  They had to make sure they themselves understood all the rules and had to be able to explain the rules to the children.  I made them play their games several times through so they'd be ready.  I invited all the classes in grades 1 to 5 to the school library to try out the Math game faire.

It was successful.  The kids loved it and hated to go back to their classrooms.  Even the principal dropped by and joined in along with many of the paras and many hated to leave.  It was great because the older 4th and 5th grade students didn't mind trying some of the games geared for younger students.  They had fun.

Now, I admit, I don't know how successful this would be in today's world with all the mobile devices but I would still give it a try because most modern device based games have lost the human touch.  In another column, I'll talk about creating math based board games in more detail.  Let me know what you think, I'd love to hear.  Have a great day.

Teaching Math In The Middle Of Regionals

I work for a school district that covers quite a bit of territory and must fly students from one school to another so they can play.  Most of the students in this district are 1A while the school I teach at is 2A.  Every year 1A regionals move from school to school and this year, it is being held at my school.

This means for one solid week, there will be games running from early in the morning till fairly late in the evening.  Everyone will be helping out in some way and I'll end up volunteering to do books at various points throughout the tournament.

I struggled in the past on how to teach math to students while huge tournaments such as this were happening.  Most basketball players want to watch the games so they can observe those they will have to play against.  They'll want to look at plays, who throws the most 3 pointers, defensive formations etc.  So if I don't take them, they will take extremely long bathroom breaks or not even come to class at all.

I ended up developing a project for students to do during situations like this.  Before I take the students to the gym, I pass out a worksheet designed for them to select one player in the game they will be watching.  They have to mark down two point and three point attempts and completed ones.  They also have to keep track of steals, rebounds, etc so at the end of the week, they will go through and analyze all the data.  They will turn the data into a graphical representation.

Then they will select their favorite player in the NBA and compare the stats of the players they chose to the stats of the NBA player and determine if the student has the potential to be in the NBA.  They will have to select at least one player for the next part of the project.  Students will pretend they are offering potential players to be drafted by the NBA.

The student needs to take the stats, graphical representation of the stats, and create a sales pitch for their player.  They want the NBA to draft their player so the presentations must be good and they must meet certain criterial. The three main stats students must complete are field goal percentage for two and three pointers, effective field goal percentage and true shooting percentage because these do require math and are important.

Furthermore, they can compare the number of completed two and three point shots to the averages to see if they match up.  The average for shooting a 2 point basket is at 46 percent while the 3 point basket is around 37 percent. If the player is within 3 feet of the basket, the percent of baskets can be up around 74 percent.

I plan to sneak in some of the stats I'm required to do in this project because most of the stats do not fit with anything else in the course and this is a real life application of them.  It will make it much more interesting for the kids and they may not realize they are meeting standards.  I will let you know how it goes.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

If it takes 22 pounds of olives to make one quart of olive oil and each tree produces 36 pounds of olives, how many gallons of olive oil will your orchard of 150 trees produce?

Warm-up

If one olive tree produces 20 kg of olives each year, how many trees do you need to produce a metric ton of olives?

The Importance of Spiral Review.

We've all heard of serial review.  Spiral review is where you have students practice key concepts and skill throughout the year on a regular basis rather than just when it is taught.

Spiral reviews have several advantages such as giving students multiple opportunities to maintain the skills they've already learned or strengthen any skills they are not yet proficient with.

Spiral reviews also give teachers a chance to assess where the student stands in his or her mastery of the concept or skill and can see how much progress the student has made.  Furthermore using spiral review helps increase student confidence and reduces time spent in prepping for state tests.

There are quite a few ways to use spiral reviews in class and many do not take much time to create.

1.  Use problems from previous sections during the warm-up so students regularly get to practice and review this material.  Look at previous skills and rotate through them on a regular basis adding new topics as needed and spreading the practice of mastered matured materials so they appear less frequently.

2.  Use 5 question quizzes at the end of class.  Make sure each week's worth of quiz problems are in the same format for each question.  For instance is the first question asks students to rewrite the standard form of a linear equation into the slope - intercept form, then every first question for the week should be that type of question.

3.  If you have stations in your classroom, one station should have review problems on previous material.  Let students work in pairs so they can help each other because the increases their learning when they have to explain it to others.  This is where you could use previous worksheets, problems, or activities you had to skip over earlier.

4.  Use math games to provide the spiral review.  Students love to play games and are willing to play them, even if they aren't fluent in the topic.  I've found students who are unwilling to try during the instructional part class will make an effort during games.  I often use games as the warm-up with material from previous classes.  My students love, love, love it.

5.  Create task cards that cover various standards and have broken down the standards into different levels.  The task cards should be different types rather than all the same kind. For instance, they might have to find an equation on one card, while they have to explain how to identify what the line looks like when it graphed to looking for patterns.  Get creative.  Add challenges to make each task card differentiated so the more advanced student is not bored.

6.  Throw in a review question in with the exit ticket at the end of the lesson. The review question focuses on previous material and is included with the question on the current material.

The nice thing about using spiral reviews is that this spaces practice out across time and this is a best practice.  It helps move the learning from short term memory to long term memory.  So if you are have not incorporated the spiral review in your classroom, give it a try.  Let me know what you think, I'd love to hear.  Have a great day.

One Important Reason For Students To Explain or Justify Their Work!

 I finally tripped over the answer to "Why students need to explain or justify their work?"  My students see no reason to do any explanations because they got an answer.  Unfortunately, it is not always the correct answer and my students cannot always explain how they got that answer.

What I read, makes so much sense because it is put into a context I can share with them.  In the future, many students will be working at jobs where they are given a problem to solve.

The solution may not be numerical as we get from solving mathematical equations but it will be a solution to the problem.  Usually two or three people will each work towards finding a solution, talking to each other, explaining their thinking, and eventually combining ideas into a solution.  Then one group or many groups will present their solutions to the boss and they have to convince him that their solution is the best.

When we have students draw pictures, or diagrams, it is preparing them to create multiple representations of their solution to a problem at work.  When people prepare a presentation for work, they usually include graphs, diagrams, illustrations, words,  mathematical equations, or tables to convey the information to an audience.  Furthermore, the idea behind these presentations is to convince the audience that the position of the presenter is the best one.  This is done through explanations and justifications of the method chosen with supporting data.

As we all know, more and more of the mathematical calculations are being done by machines via computer programs but people still need to interpret the results to know if the results indicate this choice is the best or if the person is arguing the other way, why it shouldn't be chosen.  Learning to justify or explain prepares people to support their position on the topic.

When the explanations are based on the mathematics, students who have gone through the process of developing mathematical communications are at an advantage over those who have not because they know how to formulate their ideas based on interpreting the results of mathematical calculations so they can argue one way or the other.

This is an important skill to develop.  Many industries rely on people who have excellent communications skills to develop new mines, new products, new ways of doing things.  For instance, when a mining company is looking at developing a new gold mine, they take samples, they do surveys, they determine if the amount of gold will be enough to build the mine, hire the people, pay taxes, and still make a profit.

In addition, many of these same businesses look at amortization tables to determine when a business will break even on the new project so they can account for inflation as part of the process and know if they can break even.  These are all important parts of communication. This is why we need to have students learn to explain and justify their work.  It teaches them an important skill for the future.  Let me know what you think, I'd love to hear.  Have a great day.

Suggestions For Improving Algebraic Knowledge

 Most of the time, we teach in ways that are not suited to help students improve their algebraic knowledge.  The current system is digitally based but it still has the examples set up so you click the screen and each step is shown in the proper order.  The students work their problems and are expected to learn but this method is not necessarily the best way to do it.

In 2015, The U.S. Department of Education made several suggestions designed to help students improve their algebraic knowledge.  At least one of these is contrary to what I learned when I was in my teachers education program but it makes a lot of sense.

1.  Start with a problem that has already been solved and shows all the steps.  You want students to analyze the problem, the process used to solve it and let them make connections among all the strategies and reasons they've learned.  You need to make sure the problem is one that focuses on the lesson's instructional goal.  You should include problems that show common errors so students become proficient at recognizing them.  Be sure to use a variety of methods such as small groups, whole groups, pairs, etc to discuss this.  This example could be pulled from the book, student work, or could be made by the teacher.

It is important to help students learn to analyze problems by having them describe the steps used to solve the problem, asking questions such as "Could it be solved in fewer steps?" or "Are there other ways to solve this problem?" or "Will these strategies work for other problems?  If so which ones?".  Furthermore, it is important to use more than one example and the examples should have different levels of complexity so they can see they cover the same concept.  By using several problems, students are able to see the steps are the same for all problems.

When having students analyze problems that were done incorrectly, let them verbalize why the error lead to a correct answer.  Another way is to have both the correct and incorrect problem next to each other so students can compare and contrast the steps to find the incorrect step.  It is important to ask probing questions such as "What advice would you give the student to help them understand why they did it incorrectly?"

2.  Encourage students to use algebraic representations by using language that promotes said mathematical structure. Teach students to use a type of self reflection questioning as they solve problems to help them see structure,  and help them see that the different types of algebraic representation can help them see different types of information.

Structure refers to the type and number of variables, operations, equality or inequality signs, and relationships among all of these.  It is important to use precise mathematical language so students develop the vocabulary and the connections between words and structure.  Furthermore, it is important to take time to teach students some questions they can use every time they solve a problem so they recognize structure.  The questions might be something like "What can I say about the form of the expression" or "What has happened in similar problems before".

It is important to teach students to compare and contrast different forms to what they focus on.  For instance the "Slope intercept form" makes it easy to graph starting with the y-intercept while the "Point slope form" begins at a specific point on the line.  Both forms have their uses depending on what you need to do and what you have to work with.

Furthermore, it is necessary to teach students to represent problems, especially word problems, using different methods.  Sometimes it helps to teach them to translate a word problem into a specific visual representation so they see what is going on.

3.  Take time to suggest students look for alternative ways to solve problems.  To do this, the teacher needs to help students learn to generate a list of possible ways to solve problems, look at the pros and cons for each method, and explain their reasoning for choosing a particular method.

One way to accomplish this is to show students different ways to solve the same problem including the standard algorithm that is usually taught.  By showing different ways of solving the same problem, students have the opportunity to see which ones might be more effective than others.  It is also important to let students come up with strategies on their own to try.  Furthermore, students should see how the same strategy can be applied to different problems so they see a connection.

Do not use all the alternate strategies at once or you might overwhelm the students. Introduce one or two at a time so they can practice them.  Teaching students a variety of strategies can help them become better at approaching new types of problems they have not seen before because they have tools to work with.

It is important to question students throughout the whole process to help them develop their self reflection and ability to try new problems.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

If the male zebra weighs around 700 pounds and a female weighs about 400 pounds full grown, how much more does the male weigh by percent?

Warm - up

A new born zebra weighs around 65 pounds yet by the age of 2 the male weighs 700 pounds. What is the average weight gain per week over the two year period?

The Price of Super Bowl Commercials.

 The Super Bowl just happened this past weekend.  I understand the Kansas City Chiefs won and yes I'm a bit late on it but I discovered something about the cost of Super Bowl commercials after I read an article of the best and the worst ones broadcast.

Since the first Super Bowl the cost of a 30 second commercial has increased significantly since 1967,  The first broadcast ran advertisers about $37,500  or when calculated for inflation is equal to $292,903 today.  The latest Super Bowl is estimated to run an advertiser about $5,600,000.

Since the first Super Bowl charged $37,500 but by 1973, less than 10 years later, the cost hit $100,000 and by 1995, the cost was over $1,000,000.  Within 5 years, the cost jumped to over $2,000,000.  Then by 2011, the cost went to $3,000,000, and by 2015, it was over $4,000,000.  Now it is heading towards $6,000,000.

This Sports Illustrated site has a list of costs of of 30 second commercials beginning in 1967, all the way to 2020.  It does not have every year but it has enough years to create a nice graph using original costs or the costs adjusted for inflation.  Students could calculate the percent increase based on costs again adjusted for inflation and graph it.

On the other hand, this site has the cost of add for every year from 1967 to 2020, rather than just a select number.  It said the first Super Bowl cost $47,000 rather than the $37,500 and I've seen it both ways.  This data could be used to create a general graph showing the increase from the first Super Bowl to the current one.

Students could also calculate the percent increase or decrease each year from the first bowl to the current one so they can see which years were steeper than others.  They could break the cost down per minute and compare those.

On the other hand, if they used the number of viewers listed here for each Super Bowl, they could investigate the number of viewers who saw the advertisements and calculate the cost per viewer for the commercial or the increase in cost vs the number of viewers.

If you want to give students a bit more choice, let them explore these sites and then suggest a project they would like to do using the data they have.  There are always going to be foot ball fans in the class who would love to explore this topic because they can relate to it.

I admit, I'm not a football fan and only know when the Super Bowl happens but I enjoy analyzing data so you might get some kids who are interested in that.  Let me know what you think, I'd love to hear.  Have a great day.

Justifying Your Work in Math

We are always telling students to justify their work, explain their work, and show their work but how often do we take time to teach students to do this.  I've been focusing on having students "show" their work because it is a way to communicate their thinking.  I do need to work on having them explain or justify their work and that is a bit different.

First, you need to inform students of your expectations on what constitutes justifying your answer.  Do you want them to label everything?  Do you want them to show only calculations or do you want everything including the drawings or models included.  Do you want them to explain how they arrived at their answers in words discussing each step and their thoughts?  Do they need to explain how they made sure their answer was correct?  Put the answers to all these questions on a chart so they can refer to it every they have to justify their answers.

Second, provide each student with a smaller sized chart of their own containing everything they need to include in their answers.  These are the same things you have on the large chart in your room but every student has an individual copy to use.

Once students have these tools, they need multiple opportunities to practice justifying their answers.  This might be taking a word problem and breaking it down into chunks so students have a chance to answer each part and explain how they arrived at the answer.  Furthermore, it is possible to write the questions so they are differentiated to meet all the abilities of the students.

Regularly incorporate opportunities for students to justify their answer when using word problems and regular problems. Students need practice justifying their answers in both situations because they often need to do this for all sorts of tests.  In addition, they need to develop the vocabulary necessary to communicate their thoughts so others can see their thinking.

In my opinion, justifying your answer is being able to write more than just a phrase or identify the property that allows you to do what you did.  It is being able to show understanding of when and why certain things. are used to solve problems.

Furthermore, this process needs to be started down in elementary school rather than waiting till students reach middle or high school.  If they start early, it will become second nature to students and they won't fight it in the upper levels.  This also helps them develop the ability to ask focused rather than general questions.

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

Changes in Prices of Nuts.

 Most people I know absolutely love eating nuts.  All sorts of nuts like almonds, hazelnuts, pistachios, walnuts, and pecans.  They eat them straight, add them to desserts, quick breads, butters, and make non-dairy milks.  Often, I complain about the costs with my friends because they seem so expensive but how much do growers get?

This opens the doors to a real life project using information on how much growers receive versus how much nuts are sold for.  For instance, the average price for a pound of nuts increased from $4.63 per pound in 2011 to $6.07 per pound in 2016.  This information comes from this graph.

If one wants to look at the prices for individual nuts, that can be done.  This site allows one to see how the prices of in shell almonds changed  in California from 2014 to 2020.  It provides prices at least once a month or sometimes twice and includes the percent increase or decrease.  The site also includes a graph showing the same information so students can see it visually too.

On the other hand, one can look at the change in prices for walnuts in Spain between 2013 to 2018 here.  Furthermore this graph published the prices for both shelled and unshelled walnuts per kilo.  This site focuses on walnut production in California.  It has great graphics on the price per pound farmers receive for the years 2000 to 2017.  In addition, it has graphs on California walnut production, walnut yield, acreage used to grow walnuts, farm value of the nuts, value of exports per unit, and projected values for 2018.  Great information to use on a compare and contrast presentations.

Ycharts has graphs for pecans, walnuts, almonds, pistachios, and hazelnuts.  Each graph covers average yearly prices beginning in 1996 to the present.  Not all charts use the same unit of measure.  The information on hazelnut production is based on metric ton while pistachios are per pound.  Not all the data listed on each page appears in graph form so it is easy to have students graph the complete data while calculating the yearly increase or decrease.

The last site gives students a change to perform more calculations in the spreadsheet, have additional data to analyze for trends and patterns while offering an opportunity to see what happened in the real world to create the changes in price.  If students can discover what happened to create the cost to rise or fall, they are on the way to understanding the economics of supply and demand.

In addition, since the unit measurements are different, it means students have to practice converting crops into the same unit to be able to compare and contrast the cost, increases or decreases in percentages.

This is something that uses real life data to see changes over time.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

If one pound of pods produces 1.25 cups of peas, how many pounds do you need for 15 cups of peas?

Warm-up

If a pea pod has 6 peas per pod, how many pods will it take to get 1524 peas?