Geometry Scavenger Hunt.

 

I will have about 5 students this coming year in Geometry. Rather than beginning the year with the standard definitions, I think I'm going to give them a list of shapes to find but it will be done with certain restrictions. 

What I hope to do is to give them a list of shapes such as circle, square, rectangle, trapezoid, and others.  Their job is to find examples of the shapes around the classroom, on the playground, or even around their house.  

They will be required to snap photos of the examples (all my students have a camera) and then transfer the pictures to their computers.  Once the phots are downloaded, they are required to highlight the shape and then write a short description of the shapes they found in each photo.

For instance, if I were to talk about the picture above, I might say that there are three triangles on the front of the house.  The three triangles are actually iscoceles triangles.  There are also two windows that look as if they are square and one of the garage doors is rectangle but the smaller one is almost a square but not quite.  The front door is a rectangle. 

I feel by doing this, I am encouraging them to work on their mathematical vocabulary while I use this to assess their knowledge of geometrical shapes.  If they can only tell me there are three triangles but can't tell me the type of triangles they are, I know I'll have to cover that.  

Another reason for this activity is to show them that they can find geometric shapes all around them.  I am hoping they take photos of cars to show circles, squares, and rectangles.  I already know, they might have trouble finding an example of a right trapezoid and a regular trapezoid.  Those can be hard to find.  I am also hoping they will do a quick search on the internet to figure out what a rhombus is, or an equilateral triangle.  

I've never done anything like this before but I want to try it.  I want to see if I can move the students from passive learners to more active learners. Unfortunately, Covid caused many students to be less independent as far as learning, so I've been working on activities that will have them learning but will help them be more independent.

Let me know if you think this might work.  I'll report back in a couple of week after I've assigned it to them.  This way I can tell you how I'd change it to make it better.  Have a great day.

Warmup

 Douglas Fir burns at 662 degrees F while Western Red Cedar burns at 670 degrees F.  A cord of Douglas Fir costs $415 while a cord of Western Red Cedar costs $500.  You need to buy a cord of wood to burn over the winter.  Would you buy a cord of Douglas Fir or Western Red Cedar?  Justify your choice.

Warm-up

 

The maximum temperature that spruce burns at is 1148 degrees F while oak burns at 1652 degrees F.  How much hotter (in percents) does oak burn than spruce?

Computer Assisted Proof Of Packing Color Problem.

I think this is cool that someone used a computer to help solve the packing color problem.  The packing color problem is also known as rectangular packing problem, or the geometric packing problem.  The problem looks at how many numbers are needed to fill an infinite grid so that no two identical numbers end up either next to each other or too close to each other.

An undergraduate at the University of Chile spotted a question posed at an online forum and he fell in love with it, wanting to find the answer.  The question asked about filling an infinite grid with numbers and in order to solve it, he had to leave Chile to attend Carnegie Mellon University in the United States.

In August, this young man met with a computer scientist who loves using the computer to solve hard problems and they began collaborating to find an answer to the question.  This past January, they finally came up with the solution.  15! A nice beautifully simple answer. 

 Now let's go back in history a bit.  In 2002, some folks were looking at the question about coloring maps with certain constraints applied such as no two contiguous areas could have the same color. To work the problem, they relied on the premise that a grid that goes on infinitely in both directions is a good place to start. They made the constraint that the distance between two numbers could be closer than the number itself.  Distance was defined as adding the vertical and horizontal separation together. 

An example of this would be the number one.  Ones cannot be next to each other because their distance is one apart but they can be placed diagonally from each other since the distance is two ( 1 over + 1 up). When they finally published their results, they'd calculated the numbers needed as 22 in 2008.  One thing about this problem is that they didn't have to do it for the whole infinite grid.  Instead, they could use a small subset such as a 10 by 10 grid to show how many numbers needed because it is thought that the smaller grid repeats itself infinitely.

The young man from Chile, Bernardo Subercaseaux, originally attacked this problem by trying to sketch out various scenarios using pencil and paper.  When he got tired of this he asked a friend to design a web-based application he could use and play with like a game. Eventually, he hit a point where he couldn't move any further because it is harder to prove can't be covered by a certain group of numbers than it is to prove they can.  

Once he began collaborating with the computer person, he was able to expand his search and discovered 15 numbers is the perfect set and they also were able to rule out 12 numbers in the subset.  Shortly after they realized that others had concluded that the answer could not be less than 13 or more than 15.  In fact, there was 20 years of research on this particular problem.  

In order to effectively use the computer, they needed to find ways of limiting the number of combinations the computer had to try.  They didn't want to use brute force but find a way that would be more elegant. They were able to acknowledge that many of the combinations are pretty much the same.  For instance, certain locations are essentially the same such as one up and one over or one down and one the other direction. These two are symmetrical to each other so it's not necessary to check both. 

Using this and a couple of other constraints, they were able to rule out 13 as a possibility in January of 2022. Then redoing a couple of things, they were able to rule out 14 in September of 2022 and thus came to the conclusion that 15 was the proper answer.  If you want to read the article it's here. 

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

Different Types Of Choice Boards

 

For the past several years, there has been a movement to give students the opportunity to take more control over learning.  This can be done in tests, or assignments but not always in what they want to learn. However, they can be given a choice in how they practice the topic or concept.  One way is through the use of choice boards which are also known as menus or tic-tac-toe boards.

Choice boards are usually a subdivided board filled with activities that students choose from.  It might bar 3 by 3, 3 by 4, or even larger and each square contains a suggested activity .  

The choice board could contain the option of creating a rap song, making a visual representation, a poem, play a board game, do a few real world projects but these activities are designed to help the students learn. The board you create will have a specific focus so the choice in it will be different based on the topic or concept. 

When we use a choice board, we offer a powerful learning tool to our students because choice is a powerful motivator.  In addition, it allows students to do activities they want to do and are more comfortable working.  Students have more choice over how they progress through the unit, and it provides more time with teachers to conference with students. 

There are various types of choice boards.  There is the standards aligned choice board with choices that help students meet the standards.  Each column focuses on a specific standard while the choices within the column allow the student to choose an activity which helps them meet the standard. 

Another type of choice board is one that focuses on strategies such as problem solving.  The type of problem solving might be focused on word problems, figuring out the first step, converting words to numbers, or any other strategy.  The strategy board might not focus soley on math, it might focus on reading or vocabulary skills which is just as important in the math classroom as math skills.

Then there is a thematic choice board.  This one might be used more often in elementary school but could be used in the upper grades.  With a thematic choice board, one looks at a topic that is important but is not one found on the standards.  In elementary school, that topic might be looking at a holiday but in middle school, high school, it might be used to provide students with a brain break for students who finish first.  It might also cover real world activities, or just why show work.  

Instead of a study guide, use a study guide, make a traditional review and practice choice board.  This is the one we think of when we think about choice boards.  This board helps prepare students for the upcoming assessment by targeting vocabulary, concepts, and skills.  Encourage them to select one activity from each column so they choose the one that is most interesting to them.

You might create a performance task board that allows students to communicate their learning more effectively.  This is the one where you have students create the larger project that looks more at the overall amount learned.  It might be having a student create a TED talk, Infographic, create a slide show, etc.

Once you've decided which type of choice board you want to use, think about creating three different versions (advanced, regular, and scaffolded) for the different levels of learners in your classroom. This way you provide the correct level of rigor for all students in your class.

Next time, We'll look at the types of things you can use in a choice board.  Let me know what you think, I'd love to hear.  Have a great day.

Manufacturers Suggested Retail Price, Discounts, Markups, etc.

 

I was talking to someone the other day about how the had trouble budgeting because they kept receiving emails offering them things for discounted prices.  You know the ones!  This is your last chance before the price goes up, or hey for the next 24 hours, we'll give you this product for 35% off so instead of $47, you can buy it for $17.

Since COVID, I don't think we've done much to teach discounts, mark-ups, and even sales tax.  In addition, when the topic is taught, it doesn't cover things like MSRP or Manufacturers suggested retail price or the price that manufacturers suggest the price be set at. This suggested price is about 2 to 3 times the wholesale price but stores set their own prices.

This is when stores use it to their advantage by using a little sales psychology.  When I was quite young, I asked my dad why prices ended in 9 rather than 0.  He said when the price was $9.99 rather than $10.00 so the customer would feel as if they were getting a bargain.  The same thing with stores that list the MSRP and the price the item is actually being sold for.  This makes it seem as if you are getting a bargain.  Some stores use the term Rollback to persuade you, this is a deal.

The other way is to tell. there is some sort of sale and what you want to buy is marked down a specific amount so instead of paying $52.00, it is only $39 because they've marked it down 25%.  This is the discount they take when you check out.  I don't know if the price printed on the tag is the MSRP or the price the store set but you feel good when you get the deal.    I love shopping the clearance because the discount is being applied to things that have already been discounted.  This is an area that needs clarification for many students because some think that the discounts should be added together.  In other words, if the item is marked down 50% and then reduced another %25, the kids think the item should be marked down a total of 75% rather than the last discount applied to the new price.  

On the other hand, most students are unaware that most merchandise is sold at a lower price by suppliers referred to as wholesale.  Then suppliers markup the price to the amount they sell it for.  Most of us do not know what the markup price is and that markup is not always the same.  It depends on the product.  Usually gas is only marked up some 10 to 15 cents a gallon but out of the markup they only make about 2 cents net profit. 

It is important to take time to teach students about markups, discounts, even profit and net profit so they understand these real world situations, they will end up in at some point.  It is important to know what these terms mean and their application because it is something they run into regularly.  In fact the applications and such are a bit different when in the bush.

Where I live, the stores often order products from the local Costco, have them shipped out and the markup is on top of that. I usually include information on what the markup covers such as the building, salaries, etc.  I try to include one week of this information so students are exposed to it.  Let me know what ou think, I'd love to hear. 

Warm-up

 A cord of wood is defined as being 128 inches cubed.  What different ways can you arrange the wood to create a stack measures 128 inches cubed?

Warm-up

 If your company makes 7,500,000 toothpicks from 700 pieces of wood, how many toothpicks does one piece of wood make?

Why Is Math Fact Fluency Important.

 

I've had students who asked me why they needed to learn their multiplication tables, learn to do fractions, or some other why question with regard to math.  It falls under the overall umbrella of why is having math fact fluency important.  Since COVID, students are not as fluent with number facts as they had been and today we are exploring all the reasons why it is important to have math fact fluency.  

When students know their fact fluency, they are able to perform calculations more quickly and accurately. In addition, if they are able to add, subtract, multiply, or divide, they can solve the more complex problems. Furthermore, when they have fact fluency, they are able to answer questions more effectively in timed situations such as standardized tests, or solve problems in real life situations.

They are also able to do mental math much faster and do not need to rely on the results of a calculator or needing a piece of paper and pen for physical calculations. Being able to do mental math is something that can be every day either when estimating costs, weighing options to make a quick decision, or solving problems without tools. 

Furthermore, math is a cumulative subject.  Basic math facts serve as the foundation for more advanced topics. When they have a good understanding of these basic math facts, they are more likely to grasp the more advanced concepts and do well.  When students have a good fact fluency, they have more confidence in their mathematical abilities and that increased confidence helps provide the motivation needed to tackle more complex problems and topics. They are more likely to persevere on harder problems and they are more likely to like math.

In addition, when they have fact fluency, they have more brain power available to focus on the underlying clues, understand, and solve problems rather than being stuck on basic calculations. Consequently, it paves the way for both improved problem solving skills and critical thinking ability.  It also makes real life applications in budgeting, balancing a check book, shopping, and other skills, much easier to do. 

Math fact fluency also reduces math anxiety because they feel confident in what they are doing. So when students have fact fluency, they are more likely to do better in math, have confidence, solve problems more easily, and perform operations for accurately.  Let me know what you think, I'd love to hear.  Have a great day.

Benjamin Franklin And Money

 

As we know, money is one of those topics taught to kids in elementary school.  The idea is that by the time they reach a certain age, they can identify the different coins, know their values, total, and make change.  On the other hand, we think of a kite with a key that was struck by lightening when anyone says Benjamin Franklin but did you know he had a connections with money? 

According, to a recent study published, it appears that Benjamin Franklin was involved in printing colonial money using new techniques.  During his career, Franklin was associated with the printing of over 2,500,000 colonial notes using some very innovative techniques.

At this time, the British colonies faced a coin shortage due to the need of the colonies to use the coins to pay for imports and this hindered both trade and economic growth.  To solve this problem, Franklin suggested that the colonies print their own paper monies backed by the assets of each colonial government. Franklin understood that the colonies needed to produce their own monies to gain financial independance.

Benjamin Franklin owned a series of printing shops that were responsible for printing many of the paper notes used in the colonies. One of the biggest problems that came with producing paper notes was the possibility of  counterfeiting. When Franklin opened his first printing shop in 1728, paper money was still a new concept,  Without the backing of gold or silver like the coins, these paper monies could easily depreciate in value.  

In addition, none of the paper notes were standardized so it was much easier for counterfeiters to make fake notes and pass them off to the unsuspecting populous. We know Franklin made decisions designed to make it harder for notes to be counterfeited but the records of his decisions are lost to history but several researchers using cutting edge spectroscopic and imaging instruments to look closely at the inks, papers, and fibers that Franklin used.  What he did, made it harder to have his bills counterfeited.

One thing they discovered was that the fake bills had higher levels of calcium and phosphorus that were only in trace amounts in the original ones.  Although Franklin used and sold a black ink made of burned vegetable oils called "lamp black", he used a special black dye made of graphite found in rocks. This was different than the "bone black" made out of burned bones that was often used by counterfeiters and others.

Furthermore, Franklin appears to have scattered small silk threads in with the paper, much like our modern paper.  In addition, many of the paper notes printed by his network of printing shops also contained muscovite which added a translucent quality to the paper. The researchers proposed that Franklin originally added the muscovite to make the paper more durable but kept using it when he realized it made it harder for counterfeiters to make fake money.

So these researchers proposed that Benjamin Franklin contributed significantly to the original creation of paper money so that it became harder to counterfeit.  I found it quite fascinating.  Let me know what you think, I'd love to hear.  Have a great day.

Encouraging Higher Order Thinking Skills In Math

I think one of the hardest things to get many students to do is to develop higher order thinking skills, especially in math.   This is one area that I'm having to encourage due to COVID and it is hard but I'm working on ways to help students move past rote to doing more thinking and connecting.

I'll be sharing a few ideas with you since we know that not all suggestions work well with every student. One way is to encourage problem solving rather than encouraging rote memorization of formulas. Encourage students to think creatively by presenting open ended questions that require deeper thinking and multiple ways to find a solution.

One way to encourage problem solving by encouraging students to try different ways to solve a problem.  You might even have a whole class discussion on the different ways used by students to solve the problem, take time to compare and contrast the different methods.  This helps students enhance their understanding and critical thinking skills.

Next, work on connecting the math being taught to real world math so students see relevance to what they are learning.  This helps students understand the value of learning math because they see how it is used.  Take time to teach students how to ask thought provoking questions so they can use these types of questions to prompt them to analyze and evaluate a mathematical concept. In addition, ask them to support their reasoning by providing evidence.  

Encourage true collaboration where students share their ideas, explain their thinking, and learn from different perspectives.  Collaboration can encourage higher level thinking.  In addition, when having students collaborate on open ended questions or projects, it requires students to apply their mathematical knowledge in new and creative ways. Look for questions that are real world based, use data analysis, research, or mathematical modeling. 

Furthermore, encourage reflective thinking into the weekly routine.  Have students reflect on their problem-solving strategies, take time to identify misconceptions and errors, and evaluate their understanding.  Self-reflection helps students identify metacognitive skills so they become aware of their thinking process.

Take the time to ask challenging questions that require students to think beyond their basic level of understanding. These questions should encourage students to analyze, evaluate, and synthesize. One way to help this is to integrate technology to help encourage exploration and visualization.  Look at using dynamic geometric software, graphing calculators, and spreadsheets.  When students use technology, it can support higher level thinking skills by having students manipulate data and create visualization. 

Work on encouraging persistence.  Math can be challenging and when students learn persistence, they can work their way through the more difficult problems that require a lot of work.  It is also important to help students understand that mistakes are good because it helps tell them what doesn't work.  This is not something that can be done in one day, or one month, it will take a while.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

 You just bought a 72 ounce container of gummy bears for $36.90.  How much do you pay per ounce?

Warm-up

 If there are about 195 gummy bears in a pound, and you buy a 72 ounce package, about how many gummy bears did you get?

Why Use Board Games

 I realize that children do not play boardgames as much now as they did years ago but I love board games for so many reasons. I feel physical boardgames encourage more interaction than when the same game is done online.  It is also a great teaching tool, and it can be so much fun.  I remember we played Monopoly for days.  My dad was a cribbage man (I call it a board game) so I learned to play it and it helped with addition and focus.  Although it is the digital age, it is still good to have students interact with boardgames.

Boardgames are a good way to reinforce basic skills such as addition, subtraction, and other basic operations. A boardgames allows students to practice these skills in a fun setting so they are more likely to be engaged and willing to interact.  In addition, boardgames help students develop strategic planning and decision making.  Students have to decide what the best move is based on the information they have. When students play checkers, chess, and strategic card games encourage logical thinking, problem-solving, and critical reasoning skills.  All of these skills are needed in Math.

Furthermore, boardgames offer students the opportunity to develop a strong number sense. In many games they have to estimate quantities, compare numbers, while understanding the relationships between numbers. Some games that help encourage these skills include Snakes and Ladders, Monopoly, or Yahtzee. Many boardgames demand that students perform mental math when one has to add the total of the dice, or cards, subtract points, or totaling scores. It helps them become better at mental math.

If you have students play math specific games, boardgames can be used to introduce or reinforce mathematical concepts in an enjoyable situation.  Rather than assigning a worksheet, let the students play a game so they can learn while having fun.  In addition, boardgames are great for encouraging collaboration since students have to work together to play the game.  Students have to communicate effectively because they are working towards a common goal. In the process, they learn to discuss strategies, communicate, and collaborate which are all necessary skills to do well in math.

Certain boardgames such as Tangram help improve spatial reasoning because they learn to visualize and manipulate shapes, identify symmetry, etc which improves geometric understanding and mathematical thinking. One of the biggest pluses of using boardgames is that they encourage persistence.  Many games have setbacks, challenges, and often requires multiple attempts to be successful.  In the process of developing persistence, they often acquire a growth mindset, all of which help students attacking complex math problems while overcoming obstacles faced during learning.

It is important to know what skills the boardgames are targeting so one can reinforce learning.  In addition, boardgames can be used as supplemental activities and make math more enjoyable for the student.  Let me know what you think, I'd love to hear.  Have a great day.

Board Games Can Improve A Child's Ability To Do Math.

 

Who would have thought it.  Board games are awesome for helping young children learn mathematics as long as the game is based on numbers, like Monopoly, Othello, or Chutes and Ladders.  

There was a comprehensive review done of all the studies addressing this issue over the last 23 years. It is well known that having children play board games improves reading and literacy but the study published in the journal Early Years focused on children 3 to 9 found that number based board games help improve counting, addition, and identifying if numbers are more or less.

In addition, it appears that children benefit from playing these types of board games two to three times a week under the supervision of a teacher or trained adult.  Board games are a fun way to help build skills in younger children.  It can also be considered a strategy to help young children improve their math skills.  Furthermore, it is easy to adapt learning goals to the games so one knows the specific skills a child is working on.

Most games require students to take turns and the rules limits what can and cannot be done.  In a game like Monopoly, students get an opportunity to practice using money while getting some fiscal exposure. Unfortunately, most preschools never use board games as part of their curriculum so they don't have as much exposure as others.

The study looked at 19 studies beginning around the year 2000 and later and the studies looked at children between the ages of 3 and 9.  In addition, all but one study looked at the connection between board games and mathematical skills.  The studies had children playing board games twice a week for twenty minutes at a time over a period of one and a half month period.  All sessions were monitored and lead by teachers, therapists, or parents. 

In some of the studies children just played board games while in other studies they played number based board games. Or students might be playing number based board games but were allocated different types such as dominoes etc. Furthermore, children were assessed before they began and after these sessions. The assessment might involve counting out loud to see where they were before starting and how much they gained over the time of the intervention.

The authors of the study rated success based on four categories such as basic numeracy, deepened number comprehension,  can they accurately add or subtract numbers, and do they have an interest in math. The results indicted that children's math skills significantly improved (52%) over the time of the study, while one third (32%) playing board games improved more than those who didn't participate at all.

It is felt that additional scientifically based studies addressing this questions specifically should be done.  If you want to check this out, this link takes you to the article.  So I'd like to know how students would do if they played specific board games designed to help students learn specific skills.  The ones you can buy from educational companies.  Let me know what you think, I'd love to hear. Have a great day.

Teaching Algebraic Fractions

 I always find it challenging to teach algebraic fractions in high school, especially now.  Too many students have struggled with the past.  They had difficulty finding common denominators, remembering the rules to add, subtract, multiplication, or division so trying to transfer their knowledge to algebraic fractions can be problematic. 

One of the first things one should do is to review the concept of a fraction beginning with the type of fractions most people are used to. It is important they know how to work with fractions, to compare, to simplify, add, subtract, multiply, or divide.

The next step is to introduce students to algebraic fractions with variables in the denominator, numerator, or both. It is important to show how the rules concerning regular fractions also apply to algebraic fractions.

Once students have some comfort to algebraic fractions, it is time to how to simplify algebraic equations by canceling common factors.  Then take it a step farther by showing how to factor both numerators and denominators to find common factors so one can simplify.  This is an important step.

Then one needs to introduce students to adding and subtracting algebraic fractions with the same denominator, just like one does with numerical fractions.  Have them practice this so they become comfortable with the process, find  and don't forget to show how to simplify fractions so the answer is in the simplest form.

The next step in the process is to instruct students in adding or subtracting fractions that do not have the same denominator.  This requires showing them how to find a common denominator through the use of factoring, using the factors to determine the common denominator, and the process of changing the fractions so they both have the same common denominator. The first examples should be fractions with two different denominators and then move on to fractions that require factoring before finding the common denominator.  This is often where it gets harder for students because there are no real numbers to work with.

Once they learn to find common denominators and learn to change the fractions to have the same denominator, it is time to have them practice adding and subtracting fractions with different denominators.   One should keep reinforcing the idea to check the answer to see if it can be factored and terms crossed out to simplify the answer.

From adding and subtracting algebraic fractions, it is time to teach students to learn to multiply fractions. Before teaching this step, review the process of multiplying binomials, trinomials, and monomial terms so when they actually multiply, they can do it.  I admit, I tend to teach students to factor all the terms so they can eliminate common terms before they multiply.  Usually the books tell students to multiply first, then reduce but sometimes the final product can be hard to factor so if they factor first, then reduce, it becomes much easier.  Once everything is crossed out, students can multiply for a final answer or they can leave it in that form.

The final step is to help students learn to divide algebraic fractions.  Explain how division is actually multiplying by the terms reciprocal.  Again, I like having students factor the terms after they rewrite the equation so they can eliminate the common terms.  This leaves fewer terms to work with and a smaller, reduced answer at the end.

Finally, always have students practice the process at each step so they become proficient. I realize this can take time to do it right but it is important to give students time to learn.  Don't forget visual aids when possible and remind them to simplify, simplify, simplify.  Let me know what you think, I'd love to hear. 

Warm-up

 If there are 39 skittles in a two ounce package, how many skittles are in a two pound bag?

Warm-up

If one recipe of clam chowder requires 2.5 pounds of clams and there are 14 clams in a pound, how many clams will you need?

The Importance Of Math To Brain Development.

It is well known that math has a lot to do with encouraging specific aspects of brain development.  In addition, math and brain development are closely related so when students learn math, their cognitive processes are being engaged and brain development is stimulated in a variety of ways.

In addition, many of these skills they learn that develop their brains, can also be applied in real life.  Skills from problem solving skills, recognizing patterns, and so much more. 

First, when students learn and practicer mathematics, they are strengthening neural connections within the brain. As students engage in mathematical activities such as solving problems, manipulating numbers, new connections are formed between the brain cells thus enhancing brain development. This also helps with the neuroplasticity of the brain causing it to strengthen its ability to learn and problem solve.

In addition, mathematics requires students to practice critical thinking, logical reasoning, and problem solving skills.  When students regularly participate certain types of mathematical activities, they develop these cognitive abilities which benefit them in other parts of their lives.

Furthermore, math often has students using spatial relationships, practicing visualization of objects and manipulating geometric shapes in both 2D and 3D. Using these skills in math helps student develop the brain functions associated with visualization and spacial perception and their use in other areas.

Mathematics also helps students develop their memory and recall because students are expected to memorize multiplication tables, memorizing and recalling formulas, concepts, and processes.  This helps students strengthen their memories while improving the brains ability to retrieve information to be used in both in and out of school. 

Problem solving in mathematics has students breaking down complex situations into smaller easy to do steps and finding Stratagies to solve these situations. The analytical skills learned in this situation contributes to higher order cognitive skills.  The same skills used in math to identify patterns and relationships help the brains ability to recognize patterns in the real world.

Finally, math has students planning tasks, organize information, and manage time, all skills that are needed in other jobs and in life.  The brain practices using these tasks and is able to transfer the same skills to projects in real life.  So anytime your students ask why learn math, you can answer this by talking about how it benefits them and their brain.  Let me know what you think, I'd love to hear.  Have a great day.

Using Sticky Notes In Math Classes.

 

I am a strong proponent of using sticky notes in math classes because they are a great way to give students the ability a physical way to work with both math problems, and concepts.  I usually prefer the physical ones but one could also use digital sticky notes for problems being collaborated on. 

One can write the equations being used on a sticky note and placed in the corner of the desk for quick reference.  With it right on the corner, the student doesn't have to have their notebook out.  There are graph based sticky notes that could be used for graphing equations in notebooks or placed on the desk for reference but these graphs can be created on regular sticky notes.  One can also place coordinate graphs or number lines on sticky notes.  

One can also use multiple sticky notes to create equations using different colored notes for variables, constants, or even operations so students can see what is going on.  Then they can use sticky notes to show the steps and using different colors allows students to see which one can be combined as they are the same color.  Another way is to write the equation and each step needed to solve the equation on a separate note so students can see the order.   In addition, the steps can be listed on the notes so the student can write solve the problems using the steps to the side.  They can also mix these notes up and then place them in order as a way of practicing.

Furthermore, sticky notes allow students to move things around, place like terms together, same sign terms , match up opposite signs for terms that equal zero, solving systems of equations, finding points on a coordinate plane, and so much more.  Sticky notes also help students organize various problem solving strategies. The notes also allow students to break complex problems down into smaller more manageable steps.  Sticky notes are perfect for doing substitution or substitution in calculus or trig. 

Sticky notes also allow you to find the key information in word problems and separate it out so they are not distracted the problem itself. It makes it easier for many students if they only have the important information available to solve word problems.   

In addition, sticky notes can be used as reminders.  These reminders might be about where common errors occur, not to forget certain things, or the exception such as when you divide by zero, you get undefined. Finally, sticky notes can be used to define mathematical vocabulary, create flashcards for concepts, formulas, or vocabulary.  So many ways to use sticky notes.  Let me know what you think, I'd love to hear.  Have a great day.

Music + Math = Higher Scores

 

You have the kid who tells you they can't work unless they are listening to music.  I have a few who insist on listening to music but it seems as if they often spend more time looking for the "right" tune than they spend doing work. Well that might be true. A scientist looked through 55 studies involving 78,000 students conducted over the last 50 years to see what combining math with music does. She also searched various academic databases on the same topic.

There were three type of musical intervention mentioned in this analysis.  The first were standard music lessons where students listen to, compose, or sing.  The second is instrumental where students learn to play one or more instruments either as part of a band or individually. The last is when music is integrated into the math lesson.

As far as general methodologies, tests were given to students at the beginning of the intervention and again at the end and then the gain in test scores were compared to students who did not participate in the intervention. When music was included as part of the math lesson or as separate lessons, scores increased over time.  In fact, 73% of the students who had lessons that integrated music and math improved significantly as compared with those who had none.  

In addition, 55% of the students who had music lessons improved while 68% of the students who learned to play a musical instrument improved over the students who didn't do either. Furthermore, the results indicate that music has a bigger impact on elementary students who are learning basic concepts or learning arithmetic. This may be because certain core concepts such as fractions and ratios are common to both music and math.

Math and music also have much in common such as abstract thought, quantitive reasoning, and using symbols symmetrically. Integrating music into math may be so effective because this type of lesson allows students to establish a connection between math and music while providing opportunities to explore, interpret, and understand math.  Since these lessons may be more fun,. students experience less math anxiety. 

Unfortunately, the author of this analysis was unable to look at how factors such as gender, socio-economic status, and length that music was studied effected everything. So learning to play an instrument and having general music lessons helped improve overall test scores, integrating music into math lessons had the biggest effect.  Let me know what you think, I'd love to hear.  Have a great day.

warm-up

 If there are 6 mangosteen in one pound of fruit and it a pound costs $25.00, how much did you spend to buy 182 pieces of fruit?

Warmup

 If an average umbrella has a diameter of 40 inches, approximately how wide is the street in feet?