Happy New Years Eve

 Happy New Years Eve.

Math For New Years

It is time for the New Year and most schools are not in session but these activities would be good as a way to let the students ease into the school year after being off for a couple of weeks.  So I found a couple activities that focus on New Year's itself.  

First off is one from Yummy Math on the Times Square ball, the one that is dropped at midnight in New York City.  The ball is actually a geodesic icosahedron covered in Waterford crystal triangles.  The activity allows students to examine the ball in more detail. It includes a three page worksheet to do and has a video students can watch to see how a truncated polygon happens. 

There is also an activity from Yummy Math that looks at the year of the rat from January 2022.  Although it is a lunar new year, it can be used to show that different groups may have their year begin at a different point and explains more about Chinese culture.  In this activity, students learn more about the Chinese calendar and how it works.  Students have the opportunity to figure out which New Year it is for 2023 to 2028 or so. They learn the patterns used and so much more.  At the end, they get to answer questions about the calendar and the New Year based on their work.

Another site has several worksheets which are great for the New Year.  Some of the worksheets listed on the page are free and some you need to be a member for.  I'm only reviewing the free ones. For instance, one asks students which discount is better when you buy fireworks or party supplies.  This worksheet is made for younger students so I would ask students to show via mathematics that their answer is correct.  This would add an element of communications. 

The other activity for this site has students practicing identifying reflection by placing the dot in a new location after it was reflected over the x-axis or y-axis.  Both worksheets have answer sheets available so students can check their own work. 

Finally is a short video on the the math of time zones and how they work in regard to celebrating the New Year.  If you watch television, you'll start celebrating with Times Square and continue each hour until you reach your region's celebration. So you have several options to explore various aspects of the New Year in your math class.  Let me know what you think, I'd love to hear.  Have a great day.

Winning At The Gift Stealing Game.

 

I am going to apologize for not having my usual entry this past Monday but I had been traveling and didn't have the ability to access the internet.  I took a semester long job in a beautiful place in Alaska and I'm finally getting back to normal.  Now, on to today.  We are going to look at the best way to win at the gift stealing games.  

One mathematician, decided to use mathematical modeling to determine how one can best "win" the game. The game is played where everyone brings a wrapped gift, puts it in a pile and each person in order has the choice of selecting a gift from the pile and opening it or selecting a gift that someone else opened so you know what you are getting.  If a person has their gift taken they may choose a new one or take an already opened one. Now if you are the first person, you can only choose an unopened gift and if you are the last person, you know what everyone has gotten so you can make a better guess.

The mathematician used agent based modeling which is used for everything from companies bidding in the electricity market to the way the human immune system works.  The person decided to have 16 people with 16 gifts for the model. In addition, all opened gifts were assigned a value of 1 to 10 based on how desirable they were.  The likely hood of the opened gift being taken increased if it was rated at above a 5.  In my family, if the gift was some sort of alcohol, it was automatically at the top of the list for being taken.  

Furthermore, it is known that there are multiple variations in which people play depending on the family.  The fairest rules are the ones that allow a gift to be taken multiple times during each turn, if a person holds the same gift  three times, it cannot be taken, and after the last person has chosen their gift, the first person is allowed to take a gift.  

It is possible to utilize the three rounds and the gift becomes locked rule if you utilize two co-conspirators.  Say you take friend ones gift, friend one takes friend two's gift, and friend two takes your gift  so each of your friends have the gift three times and can keep you.  You can go after another gift so they get what they want.

It is important to steal gifts in this game because the model indicated that participants are 75% less happy with their gift if they don't steal any gifts and steal something even if you don't want anything.  Think about stealing a gift that someone wants so that when it is stolen from you later in the game, you have a wider choice of games to choose from.  So now you know how to mathematically use the rules to help you get the gift you want.  Let me know what you think  

Happy Boxing Day

Happy Boxing Day to all. 

Seasons Greetings

 

Happy Christmas Eve

Happy Christmas Eve  

Bad Weather

Weather and internet issues. Should be back later.

 

Chat Stations

 

When I was researching Hexagonal Thinking, I came across a discussion protocol called "Chat Stations". This is a type of learning center but one with strict protocols for having students talk to each other about a topic.It enhances small group discussions.  In addition, this is a type of cooperative learning tool and it is one I'd never heard of before. 

Chat stations require very little prep but are designed to encourage conversation among students rather than just having them sit around a table and it encourages movement between discussions.

A chat station usually has a discussion prompt that goes with a photo, picture, or object associated with the concept or topic being taught. In math, the prompt might be asking them to analyze the problem to determine what possible equations of with the picture, talking about everything they know on a topic, deciding which answer is wrong and explaining why, list the sequence of steps needed to solve a problem but they don't have to actually solve it, which equation is needed from the reference card to solve the equation, etc.   Each group needs a piece of paper that is divided up so it has enough individual cells to match the number of stations. This is where students will record their thoughts and answers associated with the prompt for the station.  

As far as groups go, each group should have no more than between four and five students but the number depends on how many students are in the class and the number of stations available. In addition, each group should only be at a station for three to five minutes because any longer and the conversation will turn to something else.  You want them engaged so this short time is perfect. Once the time is up, have the students move to the next station, chat, and move one.  

When students have rotated through all the stations, it is time to call everyone back together to discuss their findings from each station.  When asking the groups, choose a different group to ask them for their results  for each station.  This group starts it off but you can call on all the groups to contribute.  The teacher writes down any insightful thoughts or comments from the students. In fact, these chat stations encourage whole class discussions because all the students have explored the topic in smaller less threatening situations.  

Often times, chat stations are used when students are starting a new concept or topic.  It is a way to trigger prior knowledge so they  have something to refer to. Chat stations can also be used as a lesson opener, a way to summarize the lesson, or as a way of integrating writing into the math classroom, and as a way to communicate mathematically. 

When students are at the chat stations, the teacher can go around, monitor the discussions, do a quick assessment to see what they do or don't understand, help if a group needs a bit of a nudge, while encouraging student independence.  So if you have never used chat stations before, give them a try, especially since they break up the normal day, let students move around, and have a chance to talk.  Let me know what you think, I'd love to hear.  

Hexagonal Thinking For Project Based Learning And More

It has been suggested that when we have students work on project based learning activities, we help them by introducing them to something called hexagonal thinking.  Hexagonal thinking is an activity specifically designed to help students learn to think critically, make novel connections, increase discussion while providing evidence to support their reasoning.  This is done by visually connecting ideas that have been written on paper or using a hexagon either digitally or on paper.
Picture this if you will.  You have a bunch of empty six sided figures or hexagons.  You write a name or idea on each hexagon and then assemble the hexagons so ones that are related or connected in some way touch upon each other.  The nice thing about this is that every person and every group will assemble the cards in a different way.  Hopefully students will question some of the connections and discuss which connections work better.  

In addition hexagonal thinking helps promote a more rigorous project in project based learning activities.  There really are three levels of rigor when thinking about PBL’s.  The first is composed of simple I know skills or ideas which is the lowest level.  The second encompasses a more deeper level where the student is able  to relate the skills or ideas.  The last level is the transfer level, so students are able to apply the skills or ideas to other contexts and this is where we want our students to end up.
Within the six steps of hexagonal thinking, these three levels are applied and students move deeper in their understanding.  Each step is designed to help students go from the entry event which helps build the need for surface and deeper learning.  The first step is the entry event or launch.  This is where the context or contexts are created for the students.  The entry event might be listening to a podcast, watching a situational video but it needs to initiate the intellectual engagement.

The second step is for surface exploration.  This is where hexagonal notes are passed out and students are asked to write down key terms, the context and content which shows relationships.  Context refers to the situations where the content applies and content is the specific knowledge or skills to be applied.  If students are not sure where to start, supply them with a word bank to use.
The third step is where students move on to a deep level connections.  This step is done using smaller steps or parts.  Step one is to have students discuss the connections between the words they have on the hexagonal notes.  They can also write down the connections but they need to talk about connections between content, between context, and between both.  Once they’ve done this, they are then asked to compare and contrast their results with the results of another group.  

Step four is designed to help encourage transferring knowledge by asking students to think of other contents and contexts associated with this project.  This is where they might create analogies, compare contexts across groups, speculate on how their configurations might change over time, and draft any questions they have yet to solve.  In step five, students share their thinking from the previous step to help identify the driving questions, learning goals, and success criteria.  One way to do this is by using structured protocols such as chat stations.  

In the final step, the class decides which questions they will work on to answer. It is here that students can write down what they know and what they need to know, what steps they will take to answer the driving questions, and how they know they are successful. Although this is designed to help students get started doing their projects, I can see where it could be used to help students learn how math is used in the everyday world.  For instance, slope is found on road signs, roofs, ramps, and so much more.  This is a way for students to make the connections on their own.  Let me know what you think, I’d love to hear.  Have a great day.

Warm-up

 You and your family harvested 2789 kumquats. If there are 35 kumquats per pound, how many pounds did you harvest?

Warm-up

 If a kumquat tree grows 18 inches a year and reaches a height of 15 feet, how long does it take the tree to reach its full height.

Writing Better Math Journaling Prompts,

 I have tried using writing prompts in the past with less than spectacular results. It might be that I have never learned to write a proper prompt and have had to rely on prompts I found in books or on the internet. A lot of prompts I found didn’t seem right but I used them anyway because they were available. The other day, I came across something that explains how to write better prompts. Ones that will help students think more deeply and not just regurgitate the lesson.

It has been suggested that teachers move towards prompts that encourage multiple solution paths and away from ones that have students simply retell everything contained in the lesson, or just rewrite the steps needed to solve a problem. Prompts should appear in a variety of formats that encourage more writing, more reasoning, and even debates or arguments. The prompts need to be carefully constructed. 

Unfortunately, this not happen in the math classroom as much as it should due to state testing and mandated curriculum and pacing requirements. What is nice, is that with a bit of adjustment, it is possible to create opportunities to engage students in higher order thinking, writing, and conversation. One can make small changes to already existing material to make them more effective. 

Instead of writing the prompt to tell students how to do it, the language needs to be changed so the student has to do more work.  For instance, rather than writing "Use a drawing to show how to add two fractions together", change the wording so it is more like "Describe an efficient way to add two fractions." This makes it more open ended which requires students to do more thinking.  

They have to decide what way to use to show how to add fractions.  This could be done by using a number line, a drawing, or even photos of manipulatives.  This leads to higher level thinking and it provides more opportunities for multiple solution paths. If there is a fear students won't write much, it is easy to rewrite the prompt so the student is asked to provide two to three different ways of solving the problem and have them include their solutions with an explanation of why all the ways are accurate.

Another problem with many writing prompts is they ask students to explain how they solved it.  This is often interpreted by students to mean they need to write down the steps they used to solve the problem.  Instead of asking students to calculate the volume of a box of cereal and explain their answer, ask the student to explain the meaning of each term in the volume formula to a friend.  Think of using phrases like "Explain to your friend how you solved......" or "Describe the meaning of........" or "Explain the patterns you found...........".

These types of prompts allow the students to communicate their understanding of a mathematical. concept in the way that they answer the question. For instance, if they use more mathematical terminology correctly rather than general English vocabulary, it shows a higher level of understanding.  If the written answers are constantly  vague, then one should talk to the individual to determine how much they understand.

Finally, look at prompts that ask students to defend the validity of their answer or friends answers to a specific problem.  This allows students to see there are multiple pathways to get to a solution, see the common errors that occur, and help them become mathematical writers and thinkers.  If you have a prompt which asks if the work of a student is correct, rephrase it as two students are debating their answers.  One has one answer, the other another answer so students have to write a text, an email, or answer to both to explain who has the correct answer and why it is correct.  

When prompts include the phrase "How do you know" offers the opportunity for students to explain how students approached solving a problem, provides an opportunity to practice mathematical writing, and argument while practicing higher levels of critical thinking.  Let me know what you think, I'd love to hear. Have a great day.  

One Way To Encourage Note Taking Via Assessment.

 

In math, we are always trying to encourage students to take notes and them use them.  Many teachers even have students copy down notes so they have them but then need a way to have them look at the notes later.  One teacher found a way to do this. Although she teaches government classes, the technique works as well in Math.

The way this works is that students take the test twice.  First they take the test without using their notes and then they take the test a second time with access to their notes.  The two scores are averaged to provide their final score.

When notes are taken by hand, it improves retention of material and improves understanding. I realize students can have a difficult time staying organized if they take notes on sheets of paper, I usually have students take notes in either a composition or spiral bound notebook.  They write in the notebooks and their notes are there.  The other advantage to this is it allows the instructor to spend time teaching students how to take notes since few learn how to do it.  It is an important skill for college or training.

If you use a system like Schoology or Power School, you can modify it so it allows students the chance to take the test twice and average the scores. When they take the test first without using the notes, they must rely on what they have learned through classwork, notes, and studying.  In order to get a higher grade, they need to do well on the first test due to averaging the test scores. When they retake the test, they retake the exact same test using their notes.  

One thing to remember is that students need to be taught how to take notes.  We need to show them a method to use so they have a way of organizing their thoughts.  I've use Cornel notes in the past because it allows for additional comments and learning. One way to teach students note taking is to give a lecture with examples of good note taking technique. Take time to grade student notes while providing feedback on their notes.  In addition, give open note quizzes so students get practice looking at and using their notes.

As the teacher, you decide when and how often to do this type of assessment.  This could be used on the major tests rather than every assignment, quiz, or assessment.  It is acceptable to tell students they will be able to use notes on tests but not on which tests.  You would never use the double testing method for any pretests.  It works best on material that is much more complex such as algebraic fractions.

You can include short answer questions which ask what the next step is, or why would you do this at this point, or something similar. If students answer these correctly on the first test, let them skip them on the retest because they know them but they do need to retake any calculations problems. No matter what type of questions you use, you need to look at the length of time you have in class.  When you have them take the test twice in a row, that takes time so you may end up with two days being taken up.  In addition, the length of the test needs to be set so students have time to really work the problems.  I had a professor in college who felt that if he could do it in 10 minutes, we could do it in the regular class period.

You need to always grade both tests because some assessment software does not always read the way the answer is inputed.  Once students have taken both tests and you've graded them, you can let them know what their final score is.  On Friday, I'll explore the question of how to get your students more independent so you are not doing all the thinking for them.  Let me know what you think, I'd love to hear. 

7 Ways To Differentiate Math Instruction.

 In today's semi-post Covid landscape, we have students of differing abilities and we need a way to meet the needs of all students.  This means we need to think more about differentiating instruction so none of our students feel left out. Differentiating math means we think about changing up the way we do things, think outside the box, and not get into the habit of doing the same thing every single day.

Differentiated math instruction is when an instructor uses a collection of techniques, strategies, or adaptions used to reach a diverse group of students so math is made accessible to everyone. When one differentiates a math lesson, one is providing a variety of entry points or exit points designed to support student thinking.  This makes math accessible to all students and no one feels left out.

One way is to set up a series of math centers or stations that students can work their way through.  Centers might include watching a video, reading an article, solving a word problem, or doing an activity. Once the teacher has given the whole class lesson, students break up and work their way through the centers, spending 10 to 15 minutes at each stop.  Math centers are great because they help facilitate independence, small group learning, and gives the teacher some time to provide additional support to struggling learners. It is important for the teacher to customize groups and centers to math the needs of students better.

Next, look at using activity or task cards that allow students to decide what they do and the choice gives students more power.  Activity cards might be math problems, tasks, or questions and the material should span several lessons while offering students the opportunity to work individually, in small groups, or with a partner.  

Another possibility is through the use of choice or menu boards since they provide students with a way of them making decisions about their learning.  What ever type of choice board you choose to use, they have to focus on specific learning needs, interests, and skills.  Use of choice or menu boards tends to increase student ownership because they can pace themselves and decide how they will engage with the information, and show what they've learned.

In addition, look at having students fill out math journals.  When students write about math, they can reflect about their learning while having the chance to practice English, especially in written form.  This is a great way for ELL students to practice their language skills while giving all students the opportunity to practice communication.  Students can summarize key points, answer open ended questions, connect math with everyday life, or write about what they find most challenging.  To make the entry point good for all students, do not set a minimum amount they must write.  Give them the choice to write as little or as much as they want or even let them draw their ideas.  

A slightly different idea here is to set up learning contracts.  One way to do this is to ask students to reflect on their learning, set learning which includes what skills they need to learn, or which skills they want to improve, or the areas they want to explore.  This is one way to set up personalized learning plans and these can be done at the beginning of the year and have students revisit these on a regular basis throughout the year.

Don't forget to use math games in class because they are fun, motivational and help students to deepen their mathematical thinking and reasoning.  When using a game in class, make sure the games learning objective matches up with the mathematical objective.  Always have a variety of games to use so you can change them out. Some games might involve the whole class while others allow each student to play individually.  It all depends on what is needed.

Finally include digital practice for math. Look for websites and apps that are specifically designed to reinforce the current material being studied.  Make sure the apps or website is not timed so students do not develop anxiety and are more likely to learn the material.  Using these type of digital materials can increase the level of fun and participation without discouraging them as much.  

When it comes time to implement these strategies, you don't have to do them all at once.  Think about introducing one at a time so they get used to doing them but these are all ways to differentiate math instruction. Let me know what you think, I'd love to hear. 

Warm-up

 Once the flow reaches the Saddle Road, it will cover it.  So if the lava continues flowing at 21 feet per hour and the road is 10 feet wide including the entire paved road, what percent of an hour will it take to cover the road?

Warm-up

 In Hawaii, lava is 1.9 miles away from the Saddle road.  It is flowing at 21 feet per hour, how long will it be before the lava meets the road?

Using Math To Make Christmas Easier..

Most of us used the age old method for placing decorations on your Christmas tree.  Look at the tree and put it where you see some space.  Make sure you don't clump the tinsel and hope the top can hold your angel or star.  

A professor of Information Theory at the University of Bristol did some interesting research on the statistics associated with Christmas.  Professor Johnson spent most of the pandemic helping explain the various Covid statistics to the population.  As things have slowed down, he decided to explore the statistics associated with Christmas.

He looked at the statistics on decorating your tree, stacking tree decorations, wrapping presents, to selecting favorite chocolates, and of course Santa. He explored the math of everything from what happens if a person miscalculates the amount of time needed to defrost or cook a turkey to figuring out how to seat everyone at the table for Christmas dinner.  

In regard to decorating the tree, most people try to create a random pattern of decorations so there are no two of the same color right next to each other.  Humans are not really good at randomness so we cannot create truly random patterns.  So, let's say you have 100 ornaments to hang on 100 branches, then if you "randomly" place the ornaments on the branches, you'll end up with all the decorations placed on about a two thirds of the branches. This means about 37 branches will be bare and other branches might have up to four ornaments on them. 

Using Maclaurins inequality, they've found that the best shaped box to use to save money and wrapping paper is a cube because cubes have the smallest volume. In addition, the most popular flavor of chocolate at Christmas time is chocolate orange.  If you want to save money on wrapping it, don't buy it in the box with the individually wrapped orange, buy a regular shaped chocolate bar in that flavor.

According to this same mathematician, the 12 days of Christmas song represents the numbers in Pascals triangle.  On the first day, you get one partridge in a pear tree.  On the second day, you get two turtle doves and the partridge in a pear tree.  This means you got 1 on the first day, 3 on the second day because 1 + 2 =3.  On the third day, it would be a total of 6 since 1 + 2 + 3 = 6.  If you do this for every day, you see Pascals triangle with 1, 3, 6, 10, etc.  At the end of the 12 days, you will have received 364 presents in total.

It is well known that those glass ornaments break so easily.  About 400 years ago, someone decided that the best way to store them was in a hexagonal shapes in layers so each bauble touches 6 others in one layer.  The next layer is set so the baubles are over the openings of the lower layer.  It wasn't until 1998 that someone was able to prove this is correct. 

Finally, let's look at those boxes of chocolate where there are a few flavors that are not that popular.  If you say the box has 30 total pieces, where 24 have the preferred flavors and 6 do not, then you can calculate the possibility of getting a less desirable flavor is 6/30 = 1/5 = 20 percent.  This is assuming people eat the flavor they chose but what if they return the nasty flavor and exchange it for a preferred flavor. That changes the statistics and raises the numbers so you are more likely to get a nasty one toward the end of the chocolates left in the box.  Let me know what you think, I'd love to hear.  Have a great day.

Implementing Project Based Learning In The Classroom

 

Last time, we looked at Project Based Learning in general and today, we'll be looking at how to implement it in the classroom. This can be difficult, especially if the students have never done anything like this before.  One cannot just start doing it full blast because we need to teach students how to do projects.  We can create the best learning experience but if they don't know how to do it, they will get frustrated and no one learns  anything.

1. Begin with small focused steps.  Select a few goals you want the students to work on throughout the year.  Focus on helping students work towards mastering the goals while concentrating on growth.  One might only look at minimal scope and sequence, revisit a previous project, and take time to get feedback from students.

2. Look at the project as if you are a student.  Think about the questions they might have in regard to the project.  Have some easy to understand and use resources available to get them started and to help them make sense of the project. At the beginning, help get them going by providing the steps they will need to navigate their way through the project.  All projects require that students practice a variety of skills such as researching, summarizing, problem solving, working in groups, learning to determine if the source is valid, and so much more.  

3. Plan to introduce students to the project and process over several days.  If there is someone at school who is experienced with doing project based learning, get their help, otherwise look for help from an outside source.  Introduce students to the project using some sort of entry or launch event.  This sets the tone of excitement and a need to get it done. Once you have their attention, clarify by setting clear expectations, clarifying the purpose of the project and explaining why they are doing it, taking time to talk about how they will do it (interviewing, research, etc), and what options do they have for the final project.

4. Generate a list of possible project ideas or head to places that already have projects available such as PBLWorks. If you want to look at possible topics, think about climate change, meeting a design challenge, exploring a question like "Is violence ever justified?", conducting an investigation, or taking a position on a controversial topic.  In math, topics might include having students working for the NSA as code breakers in which they have to break a code in order to determine the who, when, and where of a terrorist attack, working on making connections between geometry and real life such as roof pitch being the same as slope, or which house shape has the most room, or building a house for a spider where they have to determine the size of the house, the size and positions of doors, etc.

5. Think about how this experience will be assessed and make sure students understand what criteria they need to meet to be successful.  In addition, do not put too many checkpoints in the time line but let them know that a final product is expected as part of the final assessment. 

6.  Sometimes trying to do everything including the PBL within the 60 minute class period can be difficult.  It is suggested the instructor focus on one or two objectives and arrange them to model inquiry and and design thinking.  Offer instruction on process or concept for a day or two and then give students a day or two to work on the project after the project launch. Do not have PBL time separate from the rest of the subjects so students see a connection. 

This is a framework for actually implementing Project Based Learning in your classroom.  Let me know what you think, I'd love to hear.  Have a great day. 

Project Based Learning Basics.

One way to help students learn the material better is to integrate projects into their learning but it can be hard, especially when the administration wants everyone to follow the pacing guide or at least cover the same material over the same period of time.  

Project based learning is a teaching method that has students using the math they learning doing projects based on real world situations.  Usually, students are expected to work on these projects over a specific period of time from a week to the whole semester.  It depends on the needs of the students.

If you've never done any project based learning, it is recommended that the initial units be short so everyone gets used to doing them.  A period of no more than three weeks is suggested for a successful project.  One important thing is that there are several different types of projects in project based learning.  One type of project is the one used at the end of a unit so students can practice applying what they learned while another project is the unit itself.  

When designing the project, there are seven things a person should keep in mind.  First, there should be an essential question or problem that needs to be solved with just the right amount of challenge.  Second, all students should be engaged in a process of asking questions, looking for resources to use, and putting together all the information. Third, it needs to be real world task, context, and speaks to the students about something in their lives. Fourth, students need to have choice so their voices are heard. This allows them to work in their way, express themselves, and be creative.  Fifth, there needs to be a reflection component where they discuss their learning, how effective they managed the research and the whole project, and obstacles they encountered. Sixth, students learn to take criticism and apply it to their projects to make them better.  Finally, they need to share their finished work with the public.  These are the most important things to keep in mind when planning the project.

Then there are seven teaching practices to apply when planning the project.  First is to either plan or adapt a project based on student context, plan it from start to finish, and include some student choice. Second, know the standards the project meets and make sure it key knowledge and understanding from the math class. Third, promote student independence, growth, team spirit, learning to produce quality work, and include open ended inquiry. Fourth, work with students to organize all tasks and schedules, set deadlines, find legitimate resources, use those resources, create the products, and learn to share them with the public. Fifth, use a variety of lessons, tools, and instructional strategies to scaffold student learning. Sixth, use both formative and summative assessments for knowledge, skills, and understanding.  Assessments should  include both peer and self reviews of individual and team work.  Finally, it is important for teachers to monitor the class to decide when students need skill building, redirection, encouragement, and when to celebrate.

This can be a lot to think about when starting to do your first project with your students.  The Buck Institute for Education has a site MyPBLWorks which has so many resources. They offer a free account so it's easy to sign up for. There are articles on how to build a PBL culture from the start, a template for a letter home to parents, and projects one can use.  At this time, they have 19 math based projects available for grades K to 12. These projects cover everything from creating financial plans, to voting, to reducing impact on the environment, and so many more.  If you are interested in exploring this site and lessons, click on the MyPBLWorks and explore.  Let me know what you think, I'd love to hear.  In the next column, I'm hoping to share more on project based learning. 

Warm-up

 If a single hazelnut tree produces an average 24 pounds of nuts each year and there are 108 trees per acre, how many pounds of nuts are harvested if you have 9 acres of hazelnut trees you are growing?

Warm-up

 

If a hazelnut tree is 12 inches tall when it is planted in your yard and it grows 15 inches every year, how long till it reaches its height of 17 feet?

5 Ways To Improve Math Instruction.

 I suspect many of you are like me in that you are always looking out for ways to improve your instruction technique since not everything works well with your students and what works seems to change from year to year.  These suggestions are made by teachers who have used them rather than the administration trying to find another magic program to boost test results.

1.  It is important to teach vocabulary in context.  Many elementary teachers teach students that less than means subtraction but depending on context, it could mean an inequality rather than subtraction.  Even then it might mean subtracting the first number from the second such as 3 less than x where it is x - 3, not 3 - x. 

It is suggested that when working word problems, divide students into groups of two. Have the stronger reader, read the word problem and then have the other child summarize the problem. Next both students use the summary to decide what operations and steps are needed to solve the word problem. The summaries help students understand the problem and helps them avoid trying to use all the numbers in some way.

2. Use the Concrete Representational Abstract method of teaching which has teachers making sure students have the concepts that are prerequisites to the new concept. The teacher begins teaching students the concept using some form of concrete representation such as manipulatives. Then the teacher moves from the manipulatives to a representational form such as a drawing to represent the concept.  Finally, the teacher moves to the symbolic stage using symbols such as numbers, operations, etc to show the same concept so the teacher takes the student from manipulatives to mathematical symbols.

3. Don't avoid some sort of project based learning because the project helps students deepen their conceptual knowledge of various mathematical topics.  Furthermore, it gives students a chance to see how the math is actually applied in a real life situation.  Up until I attended a presentation on piecewise functions, I'd never seen a real life situation so I didn't know that a piecewise could be used to track the price of a postage stamp.  

4. Find a way to make the math culturally relevant. This might mean helping students see a connection with their community so they develop a personal connection.   I work in Alaska and there was a program done up at the University of Fairbanks where they took Native math knowledge and created lessons from that.  They worked with elders and others to create the lessons and I use them in my classroom since many of my students can relate to the material. 

5. Always break the material down into smaller chunks to make it easier for students to learn.  I know that some of those pacing charts just don't work with most of the students I teach because they shut down if I try to present too much at once.  Research shows, students learn the material better if if it chunked rather than done all at once.

These are just five easy ways to improve your math instruction.  They don't take much but are easy.  On Monday, I'll talk about project based learning ideas to integrate into your math class.  Let me know what you think, I'd love to hear.  Have a great day.

Neuroscience And Fire Flies Flickering In Union.

About once a week, I check out the latest on math news just because I love seeing what is happening and I came across this article on why some fireflies flicker together in union.  I admit that wasn't a topic I ever thought about but the researchers from Pittsburg used neuroscience and math to come up with an explanation.   

When you see it happen, it looks like a swarm of lights that flicker on and off synchronized like a string of Christmas lights. If you've ever had the pleasure of watching fireflies, you'd have seen a series of quick flashes followed by a pause.before the flashing starts again.  What that actually is is male fireflies producing a glow from their abdomen indicating they are looking for a mate.  Now, the fireflies in the species Photinus Carolinas found in certain places in North America and they have managed to coordinate synchronization through the whole group so they blink at once. Only a few species of fireflies are able to do this and it only occurs when males are mating.  This has peaked the interest of mathematicians who wonder how this was accomplished.

In order to determine how the fireflies managed to synchronize, mathematicians relied on a complex mathematical model called "elliptic burster" which is the same model scientists have used to describe the behavior of brain cells.  They began with simulating the flickers of one firefly, then expanded it to two fireflies to determine how the two insects coordinated their flashes. Then they expanded to larger swarms to see how speed, distance, and number of fireflies effected the blinks.  

As scientists varied distance, they noticed the fireflies could "see" each other and were able to respond to each other.  With a few tweeks, researchers could get the fireflies to blink in spirals or ripples and their results aligned with real world observations. It was also noticed that individual fireflies flashed in a more inconsistent pattern while groups produced a more regular pattern.  In addition, when new fireflies joined the group, they were able to join perfectly in time.

The mathematicians indicated that their research will help scientists make amount of light pollution and the time of day effected the ability of fireflies to coordinate with each other since both factors could interfere with who well they "see" each other.

It was interesting to see how the individuals began acting in unison when they came together in a swarm.  It is also interesting to see how the mathematical model applied to model the behavior of brain cells can also be used to explain firefly behavior.  Let me know what you think, I'd love to hear.  Have a great day.  

Using Social Media In The Math Classroom.

 

Yes, one of the social media platforms was recently sold and the first thing the new owner did was lay off a bunch of people.  No one is sure how this purchase will effect Twitter but you can still use a Twitter type message in math.  Let's look at the the main social platforms and how they can be used in the math classroom.  

Let's start with Twitter.  I know you used to be able to set up a classroom account that students could use but I don't know if that has changed since it was sold.  If you do decide to go this way, you can set up the account so it is private and only students in the group can see what is shared.  Twitter is good for promoting writing in Math to practice using mathematical vocabulary, as an exit slip, assessing student understanding of the concept, and collaboration. 

Specifically, students can tweet a short summary of what they learned in class about a specific topic or concept. Since tweets are limited to 140 characters, students can't just copy everything down, they need to learn to be concise and this helps them really think about the topic.  This activity provides closure to the lesson and helps the teacher determine what needs to be retaught.  It can also be used to have students define a vocabulary word, add a photo in, show what it is and what it's not.  For this use, give each student a different word so they are not all defining the same word.  

Twitter can used for each unit so students submit the rules, the formulas, examples, etc so that students can use this material to study and review.  In regard to assessment, students can share what they understand, what they still have questions about, and share any resources they found on their own.  If you have an actual twitter account, you can share assignments, deadlines, and students can use it to work together on projects and group assignments.  

If you do not have access to Twitter or Twitter changes the rules, you can use a template for google slides or power point.  If you do, make sure you keep the group private if you can so students are not exposed to those who love to scam others.  

Another platform one can use in math class is instagram. Instagram also provides the opportunity to set up a classroom account.  The first thing is that the teacher needs to set up a private account with the name of your class. If you want to use pictures of students, get permission so they can appear on the page.  Instagram is a great place to post information on how to solve certain problems, drop hints, post vocabulary definitions, etc.  You can do much of what you do on twitter but with photos which is great because a student can post their work to the point they get stuck and ask for the next step, or the teacher can post a "what is wrong with this?" math.  If you don't want to get an actual account, check out this template using google slides. 

One platform students love using is Tik Tok.  I don't know of any student who doesn't have an account and tries to post some sort of dance.  Tik Tok is great for having students create their own raps about mathematical topics, share how they solved a problem, even turn vocabulary into a production rather than just being cut and dried.  It is possible to set up a classroom sized group in Tik Tok but you can also use this template to create a Tik Tok type entry using google slides. 

These are the three main ones to post on but this isn't the only way to use social media.  There are activities available such as this one for middle school students where they learn to calculate the percent of engagement on a social media platform.  The site provides everything needed for students to decide which of the three influencers they want to work with.  Another site which has activities to create and interpret a scatter plot for YouTube videos and a different one to calculate instagram success.  Thanks to Matt Miller for the templates for social media if you don't want to use an actual account.  He has some great freebies and lots of cool info.   Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

 If 2.5 pounds of real pumpkin gives enough pumpkin for one pie, how many pies can you bake with a 17 pound pumpkin?

Warmup

 If it takes 2 3/4 hours to cook a turkey that weighs between 8 and 12 pounds without stuffing and 3 hours with stuffing, by what percent does the stuffing increase the cooking time?

Happy Black Friday

 

Happy Thanksgiving

 

Making Worksheets More Student Friendly.

 

Over the years, I've tried getting rid of worksheets but I end up using them when I have a sub, when I know the class is going to be shorter due to an assembly or because it is the only way available to find the type of problems I want to use.  Unfortunately most worksheets end up requiring the students to fill them out and turn them in but that isn't necessarily the most student friendly way of doing things. Just recently, I sat down to really think about how to make worksheets more student friendly and more interactive so they can complete them without getting frustrated.  

The other day, a few things came together in my mind and I think I figured out a way to make digital copies of worksheets that can be posted with help, hints, places for answers, or for showing how to do problems. It relies on using interactive images at genially.  I have several ways of making the worksheets more interactive and helpful.  

The first step is to create a jpeg of the worksheet.  This is easily done by finding a digital copy of the worksheet and bringing it up in a PDF reader like preview on the Mac.  Export the pdf as a jpeg so it is in image form. Once it is in this form, on your desktop, you can then import it into the interactive image frame. 

I chose to do a worksheet on domain and range with mapping so students get to decide if it is a function.  It is a nice visual worksheet so students can easily see the arrows.  Once you've uploaded the worksheet, you can put buttons in it, like the one you see in the corner.  The button is an eye and I set it up as a link to a short YouTube video on the subject and I've got it opening in another tab so the student can still see the work.

As a clue I can link the problems to a second page with a picture and a bit of text showing a function visually with a couple of notes so they can compare.  I can also add a page with text saying the same thing if I want.  I can also add a page with a link to the answers if I want or to a google form so students can post answers with descriptions of why they are a function or not a function.

Another way to make the work more student friendly is to take a screenshot of one of the problems off a worksheet and import it into the interactive image window.  One can link to a short video that shows how to solve this type of problem.  The nice thing about videos is they can rewatch it as needed.

For this worksheet, I would give the students the actual worksheet but I would have several problems posted individually online so they can use the interactivity to help learn to solve the problem. 

The button with the eye has a link to a short video showing how to solve one step equations and the other is page showing how to solve it step by step with explanation.  On another problem, later on, I would only list the first two steps and for the third, I would only do the instructions for the third step.  A couple problems, I'd only list the first step fully, and have the written instructions for the second two steps.  For the final few problems, I'd just list the starting step as a prompt.

The window or page provides support for those who need the extra bit but it is also a great way of providing guided practice at the same time.  If you prefer, you can put all the buttons on the digital copy of the whole worksheet but you also have the option to break it down.  If you desire, you can add a link to a google form with a few problems that students do on their own just to check their understanding.

It is so easy to add video, assessments, explanations, and examples so students get immediate help rather than struggling through the assignment.  You can also have them do the actual work on a physical worksheet and turn that in or have them place the work on a google form.  It is all up to you.  Let me know what you think, I'd love to hear. Have a great day.

Math For Thanksgiving

It's that time of year again when you only have a couple of days to teach this week because it is the week of Thanksgiving.  I love having the students do Thanksgiving themed math activities this week because they are in no mood to do regular work.  Unfortunately, most of the things labeled Thanksgiving math are the standard worksheets that substitute turkeys for chocolate or add cute Thanksgiving pictures to the worksheet.  This is where Yummy math comes to the rescue.

The link will take you to enough activities to last the whole week and give you lots of choice.  The activities span grades from upper elementary through all of middle school but some are quite easy to adjust to use in high school. There is one for planning the turkey for Thanksgiving Dinner and deciding which method - bake, fry, or smoke - is the best based on cost.  The worksheet takes students step by step through it including having them read the chart that suggests how long it takes to cook different sized turkeys. 

Another activity helps students learn to adjust the recipe for mashed potatoes for 4, 8, 10, and 12 people at first and then up to 32 servings.  It also asks students to explain how they found their answers. There is also an activity that has students calculating the amount of ingredients needed for like 23 people.  A very real life activity unless you have family members bring dishes.  Yummy math didn't forget the cranberry sauce in that they have another activity for this. 

There are three activities dealing with football since for many people, those games are extremely important. One has students reading various graphs to make predictions on the next move if it is the fourth down.  Another one has students use the pythagorean theorem to determine how far someone ran and the final has students interpreting data on an infographic in regard to NFL home advantage versus away games. 

For those not into football, there are two activities that have students analyze Macy's Thanksgiving Day Parade.  One activity has students analyzing the route from 5 years ago to determine if it is the best one and how long it will take folks to cover the route.  The other looks at two new balloons and looking at their dimensions in comparison to humans, bicycles, etc to give a better idea of size.  

The activities are not only focused on Thanksgiving itself but there are two dealing with Black Friday deals. One worksheet asks students things like how much do they save if they buy the item on sale, translating it into percent off, and completing a chart, similar to data charts in science. The other is the same type of activity but it uses different items including large screen tv sets and games.  The final financial activity has students interpreting data off of graphs in regard to consumer spending over a 5 year period.

There are enough activities that you can have different groups work on different activities so everyone's interest is met.  You could even cluster the turkey, mashed potatoes, and pumpkin pie ones as planning a meal, the football and Macy's ones as entertainment over the weekend and the financial ones for the traditional Black Friday sales.  Let me know what you think, I'd love to hear.  Have a great day.  On Wednesday, there is information on creating interactive worksheets that help the student.

Warm-up

 If you are planning breakfast for 892 people and you decide each person should be served half a fruit but you know only 33% will want grapefruit, how many grapefruit should you buy for the breakfast?

Warm-up

 If there one grapefruit tree produces 1400 pounds of fruit in a year, and there are two grapefruit per pound, how many grapefruit does the tree produce each year?

Making Videos Active.

 

I try not to assign videos to my students unless there is some sort of interactive element.  If students have to interact with the video, they are more likely to pay attention and really watch it rather than just being distracted and not getting anything from the material. 

One of the places I've used in the past is Edupuzzle because it allowed me to add questions to the video so it would automatically pause.  In addition, they offer a free plan with up to 20 videos in ones account.  I do not know if you can delete the ones you've already used so you can designate more to your account.  

The nice thing about Edupuzzle is that you can use premade videos or you can make one yourself.  When you look on a premade video you can check it out and it tells you where the activities are in the video and what type of activity are inserted.  In addition, they list the other videos based on the one you are looking at.  If you don't like one, you can make one of your own.  You choose the video from one of many sources from Youtube to National Geographic to TED talks and more.  Then you insert the type of activity which could be multiple choice, open ended, or a note and you can use all of the video, or only a small part of it so you have a ton of versatility.  Finally, you can use your google ID to log in and you can set up classes so you can assign the videos to each student but there are permissions involved so Edupuzzle offers letters you can send home explaining it all.

If this site is not for you, you can use google suite to create interactive videos.  This site has a lovely hour long video showing how to create interactive video lessons using G suite with a focus on distance learning but the material is still valid for creating interactive videos and you can select just a specific topic rather than planning for a full class period. I love videos because I can watch, stop it, and try each step myself.  I've found when I can see what is supposed to happen is much more helpful than just reading the steps.  The real information on the video starts around 13 minutes in since this is actually a presentation.  It takes you from start to finish and begins with google hangout which will be google chat after January 1, 2023. I mention this because the person has you begin with meeting yourself on google hangout but google chats will still allow you to have video meetings.

When you meet with yourself, you record your whole presentation just like you would do if this were a real meeting.  She talks about how one would include handwritten examples which is awesome in math. She walks you through the process step by step to make your own document camera so you can do it on a paper if you don't want to use google jam board. 

I like that the instructor takes you step by step through planning a lesson with all the things you need to think about and shows you everything in detail. I like the information and directions included so I can make the videos with examples and assessments.  She talks about including assessments by using google forms or a rubric via google classroom. If you want a set of interactive videos available to help students, check the site out and have fun.

So you have Edpuzzle which uses regular videos from other content creators with questions and notes and gsuite that will allow you to make your own lessons.   Let me know what you think, I'd love to hear.  Have a great weekend.

The Geometry Of Gerrymandering.

 

The midterm elections are mostly over and a couple of cases on voting seem to head to the state court systems or the Supreme Court.  Often, the cases that make it to court deal with the way the area is divided up for voting.  States often want to redraw districts to favor the party in charge but did you realize how much of a role geometry plays in this whole situation?

It turns out that geometry provides a powerful tool for establishing the best division of area into voting districts for places that have at least two parties.  According to the constitution, the candidate who gets the most electoral votes wins but the way the districts are designed can have a huge impact on how the votes are distributed.

When governments rearrange the shapes of each district, they can encourage a favored candidate to win to keep them in power and to provide representation that may not be really representative of the district.  We'll use a small example of 50 voters who live in a 5 by 10 area, one person per square.  The voter distribution is 20 are red and 30 are blue, so if you divide the area into 5 districts, you have several possible results.  If you have two rows of red and 3 rows of blue and divide it so you have two districts of red and three districts of blue, the blue will provide more votes for their candidate so if this were the house, you'd have two red and three blue but what if you want the blue to win all the seats?  You would redesign the 5 districts so all 5 districts are set up so there are four red voters to every 6 blue voters and the blue would take all the seats but what if you wanted red to win the majority of seats.  You would have to redraw the districts so three of the districts had more red than blue and the red would get three votes.  The first few scenarios can be accomplished using regular districts but the last situation requires districts to be done in irregular shapes rather than standard shapes.

These examples are not all theoretical.  Back in 2012 in New York State when 58 percent of the voters voted democratic but the democrats got 21 out of the 27 available seats. On the other side, 51 percent of the voters in Pennsylvania only won 5 of the 18 available seats in the same year.  The practice of redrawing districts to give a certain party the advantage is referred to as gerrymandering which comes from Elbridge Gerry who was governor of Massachusetts in the early 1800's.  His party was good at redoing districts into strange shapes to give his partner the advantage.  A cartoonist noted that the districting looked like a salamander so the term gerrymandering was born.

It appears local and state governments change the lines of districts about every 10 years when a new census comes out.  Over the 10 year period people move and the number of red and blue voters change so the lines have to be redrawn to help a political party stay in power. Although intentional gerrymandering is illegal, most people find it difficult to establish rules for fair districting, and this includes mathematicians.  Every state has their own rules which makes it even more difficult.

The most fair criteria for districting is that the number of red and blue voters should be about equal, each district should contain about the same number of total voters, should not discriminate against any ethnic group, not cross county lines, and follow natural boundaries but in reality, it is hard to do this.  In addition, if one wants to keep compact districts, it can't always be done and doesn't always produce fair voting results.

One study looked at the presidential election in Florida in 2000 because the number of republican voters equaled the number of democratic voters but the republicans had the edge in results.  The scientists redrew district lines to meet the definition of compact districts while meeting Florida state rules but the results showed the republicans had the advantage. The skewed results came from the fact that most of the democrats lived in urban areas while republicans lived in rural areas. This leads to the republicans winning more. Others did more research and discovered you cannot prove the districts were drawn to give one party the advantage by just looking at the lines.

One set of scientists discovered that one has to look at the efficiency gap. The efficiency gap is calculated by subtracting the "wasted" votes of the two parties from each other and then the answer is divided by the total number of votes.  A wasted vote is defined as a vote cast in the district where the other party won or is above the margin needed to win. The smaller the efficiency gap, the more impartial the results.  Sometimes the efficiency gap isn't the best choice when all the voters of a party live in the same place.  So another group wrote a computer program to determine possible ways to set the districts based on state requirements.  They applied this to the state of Maryland in 2011 and the computer came up with 250 million possibilities.  They found the results gave the democrats an advantage.  In fact, 99.79 percent of the 250 million results favored the democrats. 

Many states have moved to using independent commissions to set up the districts and these commissions use computer programs to find the fairest way to set up the voting districts so the results are more fair to all but some states are still trying to adjust districts to give them an advantage.  So this is where math comes into play.  Have a great day and let me know what you think, I'd love to hear.   

Creating Interactive Image

 

Every to often, I want to create interactive photos so that students can click on things to bring up information.  The two programs I've used in the past cost money and I don't always have the money available so I went looking for something free.  I found a program that is free and that allows the creation of interaction photos and other things.

Genially is a site that allows you to create a variety of interactive activities from presentations to gamification to so much more.  One can sign in using any google ID you already have and it allows educators to say they want to use it for class.  

After I set up an account, I then clicked on new project and discovered I could create a presentation, infographic, gamification, interactive image, video presentation, guide, etc.  You have a choice.

So I chose interactive image since that is what I wanted to focus on. As soon as I clicked on interactive image, it went to the page for creating an interactive image and there was a video right there I could play that took me step by step through the process.   That helped so much.  I would strongly recommend watching the video a couple of times to make sure you have an idea of how to do thing.  

Once I imported a photo, I had my choice of these types of interactivity to include with it.  I could do a tooltip which provides a short title or description when you run your curser over it.  A window which is a box that has text, images, video, or other content.  The ability to link a button to a page, audio, or adding a reveal.  So many possibilities.

When you actually get to the editing page, it is much like canva and other similar programs.  It gives you a ton of different buttons you can choose from and it is easy to resize the button.  It is also easy to create the interactivity to make the photo much more interesting.  When you are done, you can download the interactive photo or use it with google classroom.

 In addition, Genially also offers courses to help you learn to use the site for teaching.  At the moment, there is a course on gamification, a micro course on assessments, and a micro course on using it as an add-on for google classroom.  This site has lots of templates one can use for teaching and the site allows you to easily separate the educational templates from the rest.  The educational subtopics are educational presentations, unit plans, graphic organizers, timelines, informational organizers, flashcards, quizzes, planners, and assessments.  

I went looking for a place to create an interactive image and ended up with a place that will allow me to do so much.  Let me know what you think, I'd love to hear.  Have a great day.