The Different Types Of Curriculum?

 

The curriculum is something all teachers have to face.  It is the list of what we are supposed to teach in a year.  It used to be a huge book filled with pages of scope and sequence and material that seemed to get added every time it was redesigned but nothing ever seemed to be removed.  Eventually, they looked like door stops.

Recently, it has been replaced with "Common Core Standards".  The ones used in Alaska are broken down for grades K to 8 but for high school, all the math standards are mixed together and teachers have to separate everything out.  

This year, the math curriculum department went through the textbook series, deciding what should be taught, what can be skipped, and what can be lightly touched upon.  Although it is cut back, there are still areas, I need to include several suggested omitted topics because they are often the foundation of the skills I need to teach.

In addition, about one third of the students never finished Algebra I and they are lacking certain skills needed for Algebra II so I have to teach them.  For those who completed Algebra I, the year before, I have to review certain skills so students are able to do the work. I also have to provide more scaffolding to help students get through the class.

When I talk with teachers who graduated in the past few years, they always refer to curriculum as the textbook the school uses rather than what should be taught. I looked up curriculum to find out when it became associated with textbook series rather than a guide on what should be taught.  It appears there are multiple types of curriculum which may explain the confusion.  

First there is the recommended curriculum which covers the topics that experts feel should be covered.  Then there is the written curriculum found in most state documents as "standards" or in the school website, specifying what should be taught. In addition, there is a supported curriculum done via textbook series, software, and multimedia materials.  Then there is the tested curriculum or what the state, companies, schools, and teachers create and test students on.  Of course, there is the taught curriculum which is what teachers actually manage to teach regardless of the written curriculum or pacing guides.  Finally, is the learned curriculum which is what the students have learned and this is really the most important. 

The question then becomes how do these all relate to what and how topics are taught in the classroom.  Although the recommended curriculum is what experts believe should be taught, it has less influence on the written curriculum and even less on the classroom teacher because they have to take into account their students, what has worked in the past, and what appears on district and state tests. In fact, the material on state and district tests seems to have the most influence on what is taught in the classroom.  

The supported curriculum has more influence on elementary teachers because they have to teach multiple subjects and the textbook forms the basis of their content knowledge.  Furthermore, there is always a gab between the taught curriculum and the learned curriculum due to short attention spans, a lack of motivation, a failure to monitor student progress, and a failure to make the topic meaningful and challenging.  

To have a high quality curriculum it should focus on a smaller number of topics in greater detail which is the opposite to the commonly accepted pacing guide which covers a bunch of topics with just a glance.  It should also have students using various learning strategies to solve problems. Students need to acquire both essential skills and knowledge of a topic so they understand it better.  In addition, the curriculum has to be set up so it meets students individual differences. Make sure the classes are multi- year, multi-level sequential courses so as to build upon previous knowledge.  

Take time to focus on learning a smaller number of essential curriculum objectives while maintaining an emphasis on what has been learned.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

If it takes 14 ounces of cranberries to make a small 10 ounce jar of cranberry jam, how many jars of jam will 150 pounds of cranberries make?

Warm-up

If the average serving of turkey for each person is .25 pounds, how many people will a 18 pound of turkey feed?

Love The Book "Reach Them All"

I seldom get around to reviewing books but this is one I think could easily solve some of the issues we face as teachers.  One big issue is how to work with students who are missing certain fundamental  skills which make it harder for them to do the expected work.

Most of the time, the recommendation is to work on filling the gap while teaching the current unit rather than giving them the work at the level they test at. Unfortunately, I've never seen a plan which provided a reasonable way to accomplish it. Thi s is where this book by Chris Skierski comes in.

He suggests teachers, including high school teachers use Learning Stations to accomplish this.  The idea is the teacher determines which basic skills are needed to do well with the current topic.  For instance, if you are teaching students how to solve one and two step equations, they should be able to add, subtract, multiply, or divide integers both positive and negative before the lesson begins.

The process involves a pretest, learning stations, and a path of specific requirements to move on.  Students take a pretest to determine if they have all the necessary skills to begin solving one and two step equations or if they are missing some skills.  The results of the pretest determine which learning station the student begins with.  Each learning station is set up with a video on the topic and students are expected to at least write down the examples given on the videos.  Then they practice on-line at IXL, Khan Academy, or other location.  This is followed by a work sheet and then they finish the station by taking a quiz.

If the student does not pass the quiz, they are expected to redo the material with different videos, practice, worksheets and quizzes.  If they pass, they move on to the next learning station until they've completed every skill needed for solving one and two step equations and the lessons on the topic.  In addition, as students complete the skills, there is a small celebration held acknowledging their success.

The book includes all the forms, details, and information needed to create this in your room. Furthermore, students are not expected to work at learning stations for the whole period so the process is broken down into smaller chunks to be completed each day. For instance, the student should do the video and notes on the first day, the online practice on the second day, the worksheet on the third day and on the final day, the student takes the quiz.  In addition, there are some enrichment or extensions included for students who have most of the skills but just need a brush up.

I have been looking for how to do learning centers effectively in high school and this is the first book that offered solid information.  Usually the information I've found on learning centers is geared for elementary or just tells you what should be there in a vague way.  This book provides concrete examples so I can set up learning stations for all the topics I need to cover.

My next step is to sit down and figure out how I'm going to incorporate learning stations into my classroom. I've realized that I can combine topics in the book so they line up with basic skills such as combining like terms when I address adding and subtracting polynomials and binomial or trinomial multiplication for multiplying polynomials.

In addition to the book, Chris has a great website with some really helpful resources and classes. At least one class is free and I've signed up for it and need to get started. Let me know what you think, I'd love to hear.  Have a great day.

Happy Thanksgiving

Pacing Guides Are Unrealistic!

 

I am trying to figure out why districts have the idea that the pacing guides provided by math textbooks are realistic.  I have a pacing guide for every textbook I teach from and I can tell you the one I have is totally unrealistic and do not meet the needs of my students.  

It seldom allows students more than 2.5 days on any particular section but most are between a day and a day and a half. The lessons begin with an exploration, vocabulary, notes, practice and then the assignment.  Most of the explorations take my students at least 20 to 30 minutes instead of the allotted 10 minutes.

Unfortunately, the author's idea of differentiation is to assign different problems depending on a person's level of performance.  Every lesson is the same and none of the lessons are designed to really help students who are behind.  The year before I arrived, the district insisted teachers follow the pacing guide as written. If  a teacher got behind, they had to meet with the principal to determine what could be done so they got back on pace. Last year, they didn't push as much and when the pandemic hit, everything fled from their minds.  

I've known for a very long time that when we follow any pacing guide, we are just forcing them through a class with no regard to their actual learning.  The lessons are not designed to take time out to help students work on weaknesses, and really learn.  It doesn't really follow the practice of breaking information down into small, understandable chunks.  If you have students who are classified as English Language Learners (ELL), the pacing doesn't give them a chance to work on vocabulary or learning the language of mathematics.

How can we expect students to learn and progress if we do not include time for meaningful exploration, a chance to develop visual representations for various concepts so we can take students from concrete to abstract.  It's like the pacing is designed to get students through the whole book rather than learning the concepts.  It makes students feel as if they just need to make it through lessons, put the "right" answer down without transference of knowledge.  This may be why students often see each problem as something new.

I think it is also responsible for students not seeing the connections between basic topics and their applications to various situations.  That may be due to the pacing because it doesn't give students a chance to develop relations between one topic to another.  In addition, most lessons still rely on material and methods from when I was in school. It's almost as if everyone is brainwashed into believing one has to teach students everything in the book. It just seems to me that if we expect students to learn, we need to either cut significant amounts of material , or we have to throw out the pacing guide.

Rather than following a pacing guide, I would rather have a list of the most important material students should learn, and the time to present it in small chunks with additional time to provide differentiation, scaffolding, and a chance to let students work on any missing skills at the same time.  I've been at schools where the mantra "We are here for the kids" resonated around the school but we still followed the pacing guides rather than looking at where the students were and deciding ways to help them gain their missing skills.

On Wednesday, I want to share a few ideas from a book I am reading that makes so much sense.  It is totally in opposition to following a pacing guide yet provides a proper way to help students fill gaps and move forward.  Let me know what you think, I'd love to hear.  Have a great day.

Thanksgiving Math Activities

 

It is that time of year again, the Wednesday right before students are off to enjoy Thanksgiving and a four day weekend.  We know they aren't interested in doing a regular lesson so why not plan an activity that uses math to address some aspect of Thanksgiving.

One site that can be counted on to have Thanksgiving themed math activities is Yummy Math.  If you check the site, they have an activity on cranberries.  In Alaska, the cranberries we grow are quite low and close to the ground since they are classified as low bush cranberries.

The cranberries the activity cover those grown commercially for juice, sauce, jelly, and the fresh ones sold at the grocery store.  The activity looks at why they float, how much each weighs and how many make a pound, etc after watching a video on harvesting cranberries.  Students are expected to estimate, predict, calculate, and use problem solving skills to complete the worksheet. 

In addition, to looking at cranberries, there is an activity on cooking your turkey, making mashed potatoes from so many pounds of potatoes and making pumpkin pie for what people eat.  There are also a couple activities on football.  One activities have students interpret information about NFL teams on an infographic while the other activity has students creating a graph from data on fourth quarter decisions  whether to punt or try a field goal.

There are also several activities designed to look at consumer spending at this time of year and the savings people find during the Black Friday sales.  In addition, this site has two activities relating to Macy's Thanksgiving day parade but they have the cravat that these are older and do not take into account the changes Macy made for this year.  The final couple of activities look at the distance someone ran and building structures out of cans.

On the other hand, Math-aids has some four quadrant coordinate plane pictures available so students can practice plotting coordinate points and end up with Thanksgiving themed pictures. There are two turkey pictures, one pilgrim hat, and a pumpkin with a pilgrim hat.  Once students finish graphing the picture, they can color in the final product. 

Finally, Wallethub has a great Thanksgiving infographic that includes information such as the average cost for a 10 person dinner, or how many total calories consumed by Americans on Thanksgiving day. It lists people favorite dishes, the number of hours a male has to work to work off a meal, the number of total turkeys killed, the number of questions answered by the Butterball hot line and do many other interesting facts.

It makes an awesome what do you notice?, what do you wonder? and what type of mathy questions could be asked about this infographics.  In addition, it would be easy to have students take some of the information and turn it into pie charts, bar graphs, and other graphical representations.  Let me know what you think,  I'd love to hear.  Have a great day.

Warm-up

If  2.75 cups of dried rose petals are in one ounce, how many cups are needed to make a pound of rose petals?

Warm-up

If there are 14 cups of dried lavender buds in one pound, how many cups of lavender buds are in an ounce?

Sunrise, Sunset Data = Sinusoidal Waves.

 

Want to get students involved in a project which will help them model real life data using a trigonometric regression?  Set them up to learn more about the increase and decrease of sunlight throughout the year in various locations around the world.  Arrange things so they do the whole project from the beginning by collecting data, to using a sin regression to find the equation to identifying all the parts of the transformational sin formula.

The first thing to do is to assign each student a city such as Honolulu, Hawaii,  Reykjavik, Iceland, Oslo Norway, Brownsville, Texas, Rome, Italy from all over the world. The choices should range from far north, to around the equator, northern hemisphere and Southern Hemisphere so students have a chance to see how the sin wave changes according to it's location on the earth.  This site has sunrise and sunset information from most major cities around the world. 

Students should write down the times of sunrise and sunset for same day every month.  When I tried out this activity, I chose the 15th of each month because it was the middle which I felt was an average.  After recording the data, students should subtract sunrise from sunset to get the total length of daylight.  The next step is to create a state plot of the data which can be done by hand or on a calculator.  

I've seen the data entered in two different ways.  The first way is to count the months so January is 1, February is 2, all the way to December being 12 and the second piece or L2 is the day length, or you can count the days themselves so Janury 15 is day 15 which February 15 is day 45, all the way to December. This is a great opportunity to discuss which way would be better.

This is a great point to have groups of students place several plots onto one graph so students can see the similarities and differences based on the different locations.  The cities should be grouped so one is located fairly close to the equator, another as far north as possible and two others somewhere in-between.  It is interesting to see how the data curves if it is from say Oslo, Norway compared to Honolulu, Hawaii. The one from Honolulu will have a smaller stretch than the one from Oslo.

Once all the data is entered and the stat plot is completed, students will be able to see that the sine waves will provide the best fit so this is a great opportunity to teach students how to carry out a sine regression to find the an equation for the data so it its the f(x) = asin(bx + c) + d general formula.  In addition, one can show how the a, b, c, and d are calculated individually so students see how each part is calculated.  At the end, they can create a written report comparing and contrasting their city with others.

One of the easiest things to find first is D because one just has to add the maximum and minimum hours of daylight together and divide by 2.  That would be the vertical shift and center line.  To find A which is the distance form the maximum to the center line one just has to subtract the maximum -  D to find A. In order to find B it's just 2 pi/the period such as 365.25 for the year.  Once students have A, B, and D, they can calculate C.

This exercise provides a wonderful real life application of sin waves where students learn more about how amplitude is found, how b is pretty much standard with 2pi/period, and the horizontal and vertical shifts.  Real data, real calculations and real modeling.  Let me know what you think, I'd love to hear.  Have a great day. 

Real Life Data = Piecewise Functions

Another talk I attended discussed the use of real life data and then figuring out an equation using one of the regressions for line of best fit.  Unfortunately most text books focus only on using linear regression to find an equation for the line of best fit for a scatter plot.  

This talk was so interesting because the presenter used more than linear regression to find an equation for the data points.  This person provided real life data from the CDC on the number of opioid deaths over the years.  

Once he had the data inputed and the stat plot up on the screen, he began using various regressions to see which produced a line that fit the data as well as possible.  It was obvious the standard y = ax + b created a line that had an r^2 indicating it wasn't that accurate.  Furthermore, the end of the data showed a steeper curve indicating the equation needed might be an exponential one. 

I loved the way he showed us how equations created by quadratic, cubic, quintic regression matched the data.  This lead us to seeing how different parts of each equation matched up with the curve of the data.  It showed a real visual reason for using piecewise functions to end up with a series of functions to match the data.  In the past I have taught piecewise functions but I never had a context for why one would use it but now I understand a good application of it.

After watching the presentation, I realized that I need to include this type of information when teaching either linear regression or piecewise functions.  My pre-calc students are taught linear regression in the first chapter without a proper context so I think I'll expand that lesson the include other types of regression applied to the same data so students understand the line of best fit might be made up of sections of a variety of equations.

The presenter used the TI-84 to show how to do each regression and the piecewise.  He used a newer one that had the piecewise function choice.  The ones I have do not have that choice and as much as I like the TI-84, Desmos is easier to read for discontinuities, find the values for intersections, and other pieces of information.  I actually have students use both so they have the ability to use both applications. 

This gives students the experience to use a variety of tools depending on the circumstances.  They have to be able to use both because they may not have access to a handheld calculator but they do to the other.  It is important they exposed to a variety of tools.  It appears that Geogebra can also be used to create a variety of regressions. 

The gentleman also had data for the coronavirus and a few other things which makes it easier to start teaching the topic with something more relatable for my students. Furthermore, I don't think I've done any regression other than linear regression since college and even then it was by hand because they wanted us to know how it worked.

So next time I teach linear regression I am going to use this information and the different programs to create a piece wise equations. I'll be looking for some nice reliable data for other activities.  Let me know what you think, I'd love to hear.  Have a great day.

Circle Packing = Social Distances

I just finished attending the Virtual NCTM meeting this year and I loved being able to attend talks without leaving home.  In addition, it is great I'll have access to recordings because I can see the talks that I missed due to all being at the same time.  One of the talks, a short 30 minute one, looked at circle and sphere packing and it's applications to the modern world.

The one thing the speaker said throughout the talk is that once people figure out the math, they never know all of the idea's possible applications at the time.  He began with the circle inscribed inside the square.  The square was say one unit by one unit while the square had a radius of 1/2. 

He showed how the area of a circle is about 78% of the area of the square.  I gather this is an old problem whose new application only became apparent during this pandemic with social distancing.

With today's 6 foot distancing mandates, offices often have to figure out how to arrange workers so companies can have as many workers as possible while still maintaining social distancing.  I used the basic idea when I arranged my classroom for my biggest class while trying to maintain the appropriate distances.  

The specific idea being used is called circle packing which is where circles are arranged within a certain area so that many of the sides are tangent to each other but do not overlap at all.  Most people think of packing circles within a square area by doing rows upon rows, so everything lines up such as in the picture of flowers.  

This is not the best arrangement for circles because there is still quite a lot of space available between the circles.  This is one of the usual arrangement teachers choose for their classroom, rows of seats all neat and orderly.  As stated earlier, this is not the most efficient arrangement.

After a lot of exploration, mathematicians discovered the best arrangement is actually hexagonal with a density of almost 0.91.  I didn't realize the hexagonal arrangement was the most efficient when I arranged my classroom. I just eyed things and put the chairs into this orientation because it was the only way I could figure how to arrange all the chairs to maintain distancing.  Although it seems that it might be a good idea to pack circles in side a circle, it really is not the most efficient.  

In addition, the idea of circle packing can be extended to sphere packing which is the idea that one can arrange spheres in the most effective arrangement.  This problem was first proposed in 1611 and Kepler came up with a solution but it was until around 1998 that it was proven the best arrangement is just the way they stack oranges in a pyramidal shape at the grocery store. Others researched the best arrangement up to 24 dimensions and found one that works.  Sphere packing is used currently for data transmission and error correcting code. So we have something first proposed in 1611 whose applications are extremely important in today's society.

I found this extremely interesting.  Math first proposed centuries ago that is now playing a huge part of our daily lives.  Let me now what you think, I'd love to hear.  Have a great day.

Warm-up

If one candidate is leading by 0.26% and that equals 27,000 votes, how many votes would he be leading by if he were ahead by 1% of the votes?

Warm-up

If 130 secret service people have tested positive or are in quarantine out of a force of 1300, what percentage cannot work?

Manipulatives Can Still Be Used In A Virtual Classroom.

 

Teachers often use manipulatives in class to help students move from concrete to abstract or vice versa but with many school districts moving to a virtual classroom, one has to have virtual manipulatives to use in class. Fortunately, there are quite a few sites out there that are available and many of them are free.

My experience is that some sites do not work as well with iPads because they require flash but others work well.  I'll let you know which ones seem to work on mac's and iPads so you'll know ahead of time.

Didax has a page of virtual manipulatives including algebra tiles that work on the Mac and iPad.  The nice thing about the algebra tiles aside from adding or subtraction, multiplying or dividing is that they can be used to show binomial multiplication and trinomial factoring.  It offers so many possibilities.  In addition, there are fraction strips with a number line, two color counters for basic math and other possibilities.

Math Playground has quite a few virtual manipulatives including a geoboard complete with rubber bands and a choice of grid lines or no gridlines for students to work on area, perimeter, or transformation.  It's a good way to visualize an open ended question like what are the measurements of a rectangle with an area of 32 square units.  In addition, there are apps designed for ratios, fractions, decimals, and measuring angles.  A few of their virtual manipulatives require flash but most don't so you can use them on your iPad or Mac.

I have an activity I use in Geometry which uses tangrams to create certain shapes based on specific pieces.  Some shapes are concave while others are convex and this site has tangrams one can use in whatever activity you want rather than requiring people to try to reproduce a specific picture.  This also allows a person to use pieces of the tangram to work on transformations, or rotations.  This is nice and works on Macs. 

The National Council of Mathematics has some nice virtual manipulatives in Illuminations. Their Algebra Tiles manipulative allows students to solve simple equations, substitute values into equations, practice binomial multiplication and factoring trinomials or binomials.  This one includes instructions for using the virtual manipulatives where other Algebra Tiles ones do not.

Hooda Math has a nice algebra balance activity which allows students to solve problems step by step once they've set the balance up with the original equation.  This set of virtual manipulatives allow students to set up the equation they want to solve or they can use one the program selects but it does not do anything more than a simple equation with a variable on both sides.  They also offer students the chance to work with Algebra Tiles which is actually a nice one to work with as it allows y^2 and a few other choices I've not seen before.

Although the National Library of Virtual Manipulatives (NLVM) is still out there and has so many manipulatives, I've never had it work on a Mac or iPad due to the language it is written in. I've never been able to download the app they offer but I have had it work on a windows based machine so it might work on chrome books.  This site, if you can get access to it has some great manipulatives that cannot be found anywhere else.

I hope these help with teaching during the pandemic.  Let me know what you think, I'd love to hear.  Have a great day.

Designing A Lesson For Any Situation.

I was so happy to stumble across the outline of a lesson plan that is designed for in class, hybrid, or remote and it doesn't have to be changed up should your school move from one to the other without hesitation. 

Although my school is face to face and has been since opening in September, we face the possibility of going hybrid or totally online at any time.  In addition, I have students who are out on a regular basis due to having to go out of town for medical or someone in their family travelled so they end up on quarantine for one to two weeks.

The general lesson plan covers everything from start to finish.  One should always open class with some sort of bell work such as a joke, a picture or, a cute meme.  For online lessons, this is the section where students are logging on or getting set up in class. 

Next it is important to go over rules and norms.  This is where you'd include information on expectations for behavior or expected interactions.  I'm taking two distance classes and both instructors include this information just after they welcome us to class and have given us something to work on to warm-up.  

This is followed by sharing time or a chance to connect with everyone. This might be a time to share a birthday, recognize someone who might have done something cool, or answer questions about work.  Then it is important to look at learning goals both as the learning goal and with the success criteria so students have a way to determine if they've learned what they need to learn.  It is important to address the learning goals every single class. I take time to post the learning goals on google classroom and have my students comment on the goals either with a question, or how they feel they are doing on learning the material.  

Once everyone is settled down, had time to connect, check to the learning goals, and looked at things students have done, it is time for a warm-up where you might have an open ended question, vocabulary, a previous problem, a problem that works on a skill, or even a three act task spread out over three days.

At this point, you get to the lesson and in math it is more important to let the students do the problem first so as to encourage productive struggle rather than demonstrating the problem first, having them do it with you and then letting them do it by themselves.  It has been shown that by having students try the problem first before doing it with the teacher, they receive more benefit from the lesson, and the teacher part of the lesson should be last.  

When letting students try a problem by themselves, it is good to encourage discussion either in small groups in class or via breakout rooms or via chat.  For the we do, where the teacher works with the students, it can be done with the teacher having students suggest steps, or if online, let them share it via jam board, or in chat. The last step should be the teacher lecturing on the topic but it should include interactive elements otherwise students won't pay enough attention.

The final step should be some sort of exit ticket where teachers share a joke, ask students to complete something to show what they learned or include a question showing that students listened to the lesson if it is online or if they were absent.

The framework is here with all the information you need to create the lesson.  I love it because I can use it to create a lesson for any circumstances without making a lot of changes.  That is important for all of us, especially since the numbers of infections are increasing across the nation. I am happy to have the information because it is stressful to create lessons with an audience that can go from in person to distance at the drop of a hat.  Let me know what you think, I'd love to hear.  Have a great day.  

Paying For Education During the Coronavirus.

 

I stumbled across an article on the news about a hospitality college in Indonesia who is working hard to stay open during the coronavirus.  The Venus One Tourism Academy in Bali, Indonesia has taken the step of accepting alternative payment for tuition since people do not have as much money.

The school will accept coconuts, moringa leaves, or gotu kola plants.  The school uses the coconuts to produce virgin coconut oil which can be sold as a way of making money.  The moringa leaves and gotu kola plants are used in herbal soaps which are sold.  In addition, both moringa leaves and gotu plants are said to have medicinal properties.

The school changed it's tuition policy back in March when the virus hit by allowing students to split tuition into three payments but as the coronavirus continued, they moved to accept produce so they can stay in business and provide other skills for these hospitality graduates.  In addition, the owners of the school claims this helps students learn to entrepreneurs so one day they might start their own businesses.  

The way the scheme works is the student brings in fresh picked coconuts and the school accepts them at the current market price which is around 35 cents.  The amount is totaled up and applied against their tuition.  In addition, they do the same thing for the moringa leaves and gotu plants so a student has an alternative way to pay for education even if their parents have lost their jobs.

In the United States there are at least four colleges who prior to the pandemic allowed students to exchange work for their college education.  Two of the colleges are in Kentucky, one in Missouri, and one in Nevada.  Most of these colleges require their students to work 10 to 20 hours per week to pay off tuition, they are expected to still pay for the other costs such as books, room, and board.  When students do not have to pay tuition, it makes the cost of going to college much easier.

Unfortunately due to the coronavirus, many of these opportunities may not be available but as a math teacher, one can still use these examples to help students determine the amount actually saved by going to one of these colleges.  I found a list of 16 places of learning but many of these restrict their offers to residents of the state or a certain area, to people who want to learn a certain trade, or of certain religious beliefs.  

Many of these colleges do pay a minimum wage to help students pay for the rest of their costs.  Many schools ask seniors to compare the costs of colleges so students know how much they need to find scholarships and loans for.  If you have students do a comparison on if the work programs will pay for all the additional costs or if they'll have to take out a loan to cover the extra amount.

This makes for a wonderful compare and contrast activity for college or trade school. I've learned that many of my students pick out a college based on where their family went but they never think about how to pay for the experience.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

The smaller elephant is about five and a half feet tall. Estimate the height of the tree in inches.

Warm-up

How high do you think the arch is in dolphins?  How wide is the base?

Place Based Math

 

The other day, I attended the third webinar in a series on teaching math via distance or hybrid because there is always the possibility that our school district could end up going red. The class addressed turning math problems into open ended problems but at the very end, the instructor mentioned place based math.

By definition, place based math activities are activities that use local history, culture, economy, arts, or literature to help students learn more about math.

The instructor indicated that one needs to create situations that are based on local knowledge and understanding.  For instance, here in Alaska, I might post a picture of someone standing under a pair of bones that form a parabolic shape.  Based on what they see, you might ask students to estimate the height of the bones, or ask them to create a quadratic equation for the parabolic shape.  

In another place such as in Colorado, it might be looking at pictures of signs showing the grade of a mountain road with the numbers blacked out and asking students to come up with the grade or perhaps asking them to explain what a 6 percent grade represents in terms of the mountain.  Take it a step further by asking them to decide what grade is too steep for a fully loaded truck going uphill.

Something like that wouldn't work well where I live because the only local roads go a few miles inland and there are barely any speed limit signs and no grade signs for certain.  But if I asked my students to tell me how long it takes to get to the next town by boat or by snow machine which opens up the possibility of  having students create their own word problems.  

Another possibility down in the area of the Gulf would be showing a shrimping boat and asking students to estimate the amount of shrimp the boat could hold and what the load could sell for.  Up here, it would be taking a picture of a local fishing boat and asking students about the number of salmon it could hold in pounds.

One of my favorite activities for locally based math is to have students recreate a cross section of the local landforms from a topographic map so they can see an outline of something they are familiar with.  Once they have the cross-section completed, I ask them to calculate the slope or grade of a few different points on the cross-section.  

There are also wonderful mathematical exercises one can do with the differences in daily highs or daily lows over the period of a month for the area.  If it's in a colder place, some of the math would involve negatives from start to finish. There is also the possibility of having students calculate the daily average temperature for the month based on data collected by the students every day.

A couple of my freshmen argued they didn't need math at all.  I said they did and they should name something that didn't require any math.  They promptly suggested they didn't need to know math for when they went out hunting with their guns.  This lead to pointing out that they use an intuitive math they've developed over time based on the experience of learning to shoot.  I also took time to explain that the calculational math we do in class is what we use to explain how the bullet moves through the air, it's speed, it's trajectory.  In other words, we can use mathematical equations to explain the actual firing of a gun. 

Again it was something the boys could relate to and introduced the topic of mathematical modeling to explain how things work.   I'll take this a step further later but in the next week, I've got some ideas for locally based math to use as warm-ups. Let me know what you think, I'd love to hear.  Have a great day.

The Importance of Number Sense.

We talk about number sense in regard to our students but I can tell you with absolute certainty, that too many students arrive in high school with little to no number sense.  Unfortunately, the lack of number sense creates some problems of its own.  

Students who have a well developed number sense tend to do better in math because a good number sense encourages them to think flexibly while feeling more confident. 

In addition, if students do not develop a strong number sense early on, they will have difficulty developing the foundational knowledge to do even simple arithmetic and certainly will struggle with performing more complex mathematics. A study of seventh graders was conducted back in 2013 lead researchers to conclude that the students who struggled at this age were the same ones who had the least amount of number sense in first grade.

Students can develop number sense by visualizing numbers in a variety of different contexts, exploring numbers, and relating to numbers in different ways but it takes time.  One way to help students develop a better number sense is to show there are multiple ways to complete a problem.  Ask students for suggestions on how to solve it and write the suggestions on the board.  This gives students a chance to see different strategies.

Another suggestion is to ask students to calculate certain problems in their head because when students perform mental math it helps them build their knowledge of numbers and their relationships.  When the numbers get a bit too large or does not work with the memorized steps they need to learn to be flexible with ways to solve these problems.

In addition, teachers should have students participate in classroom discussions where they talk about different strategies that could be used to solve various problems.  This helps students clarify their thinking while developing the ability to critical think about other peoples strategies.  It is important to note the strategies on the board so students can see a relationship between their mathematical thinking and symbolic representation.

Take a moment to embed estimation and rounding activities into various situations as it is something that textbooks do not cover as well.  Furthermore, the teacher should ask them about their numerical thinking for both correct and incorrect answers because it shows you value their thinking and it has to make sense to them.  Finally, take time to pose questions which have more than one answer such as two numbers added together make 35.  What two numbers might they be? or Suggest three sets of numbers that add up to 35.

So it is important to work with students on developing number sense, even if they are in high school.  Let me know what you think, I'd love to hear.  I do plan to revisit this. Have a great day. 

Estimation and High School Students.

I've discovered that most of my students have trouble with estimation.  In fact, they often equate estimation with making guesses rather than using certain strategies to come up with a reasonable answer.  I'm not sure how much instruction they get in the earlier grades but I do know that estimation is an important skill to have.

First of all, estimation is a higher level skill requiring  students to understands the concepts and carry out mental math manipulation.  They have to look at the numbers in a problem, decide if they should round up or down while keeping a running total.  

Basically, it is important to help students develop the ability to estimate because it shows how well a student understands the concept and the reasoning behind it when they are not relying on the standard step by step procedures.  In addition, it helps students develop strategies which in turn strengthens their problem solving skills and logic and as stated before, it helps develop their number sense.  It is also a great skill to have when planning for things such as a Thanksgiving dinner.

Furthermore, the ability to estimate helps students develop number sense so they know if their answer is reasonable.  Without a developed number sense or ability to know if their answers are reasonable, it is much easier for students to make computational errors.  The other day, I discovered my highest level students in Math didn't seem to grasp that when they are due to get a raise every year, their salaries should be going up.  Over half the class had a decreasing amount earned.  This tells me they lack certain skills.

Second, estimation allows students to carry out mental math faster because they can round numbers to come up with ball park figures. This is an important skill in business where people look at the estimated costs for upcoming projects.  Someone who is good at estimation can look at the numbers, do some quick calculations in their head and come up with an estimated cost for the whole thing.  They rely on spread sheets to do the actual calculations.  

In addition, to estimating using addition, subtraction, multiplication, and division, students also need to be able to estimate how long it might take to do something, or how long it might take to travel from point A to point B or even how many calories in the cookie they are eating.  

It is worth taking time to discuss why estimations are important in real life and the type of estimations one runs into.  There are general estimations such as why the local football field might want to estimate the number of people who are planning to attend the next football game?  Aside from an approximation of the amount of take at the gate, the concession stands have to know so they have enough food or souvenirs available to sell.

Or if you need to have something fixed, most people get estimations before they agree to have the work done so they can plan for the output.  Although the estimation is usually a specific amount such as $3967.28 for redoing the plumbing in the bathroom, or $256.98 to have the breaks redone, we rely on that as being the largest amount we pay.  

Unfortunately, the math books I use at school do not really address estimation.  That is a shame because estimation is such an important skill.  I'm working on figuring out how to integrate it into my classes so students get better at it.  I'll talk more about it another time. Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

 

If an African Elephant weighs between 5000 and 14,000 lbs depending on it's size, what do you think would be an average weight for one?  Explain your thinking.