Why Is It Important To Teach Long Division For Dividing Polynomials.

 

This is one of those topics I really hate teaching, especially right now because most students have difficulty doing regular long division.  This has become even more apparent due to COVID and I have so many students who cannot divide without a calculator.  Although there is a wonderful shorter method referred to as synthetic division but it only works in specific situations.  

What I've learned over the years is hat if a student struggles with regular long division, they struggle with polynomial long division due to the process being the same. So when teaching this topic, it may be necessary to backtrack all the way to regular long division.

There are many reasons for students to learn long division with polynomials.  First, since division is a fundamental operation, teaching them to divide polynomials is helping them to apply the algorithm to algebraic expressions. Understanding this is essential for higher mathematics classes such as calculus and Complete factorization is important for solving equations, finding roots, and simplifying expressions. 

Next, being able to divide polynomials by using long division is essential to  factoring polynomials because it is a systematic approach that can be used in multiple situations.  The long division algorithm provides a step by step method for dividing polynomials while reinforcing algorithmic thinking.  It also helps reinforce problem solving skills for the more complex problems.  

As far as problem solving goes, long division requires careful organization and attention to details. It encourages logical thinking,  The process has students analyze the problem, break it down into smaller steps, and apply the appropriate strategies to find the quotient and remainder.  These skills are transferable and can be used in other math courses and in real life.

It also provides a foundation for higher level mathematics. In addition, it is the foundation of other mathematical topics such as synthetic division.  Of course, students will ask "When are we going to use this?" Or "When is it used in real life."  This is fairly easy to answer. 

Polynomial long division is used in circuit analysis when they are analyzing electrical circuits with complex transfer points or calculating the stability of a system by diving the input by the output. Another place is in control systems to determine the stability of feedback systems. In economics, polynomial division to determine roots and critical point used to understand market equilibrium and economic behavior.  In data interpolation, one use is to determine missing points and values within a data set. Furthermore, it is used in error correction codes such as the Reed Solomon codes which help detect and correct data transmission or storage systems. 

Although students will argue why learn since there are calculators out there that will do it for them, it is still important for them to learn the process so they understand how it works.  Let me know what you think, I'd love to hear.  Have a great day.

Kirigami And Shape Shifting Materials.

 

Recently, people have applied the math behind Kirigami to shape shifting materials so that more can be done.  Kirigami is the Japanese art of paper folding and cutting to create cool three dimensional designs. Scientists have taken the concept of Kirigami and applied it to other areas such as shape shifting materials. 

 Now shape shifting materials are also known as smart materials or programmable materials. In other words, these materials can change their shape or properties as a direct response to external stimuli such as heat, electricity, light, or a mechanical force.The change these materials undergo is reversible and the transformations can be controlled so they can adapt to different forms or configurations.

When scientists combine Kirigami techniques with the shape shifting materials, so many possibilities open  for creating complex and adaptable shapes. With careful cutting and shaping of the shape shifting materials, it opens the way to designing objects that can change their shape, size, or functionality.  One use of Kirigami is to apply it to shape-memory polymers. These particular polymers are able to "remember" a specific shape and with an application of stimuli such as heat, are able to return to that shape. When Kirigami is combined with these shape-memory polymers, the result something that can undergo specific transformations.

On the other hand, kirigami can be used with meta materials which are engineered materials whose properties are not found in nature. certain structures are created.  These structures change their mechanical properties such as stiffness or flexibility by cutting or folding certain areas selectively. These are just two examples of how Kirigami is combined with shape shifting materials.

Now the question of how is this information used in real life.  It can be used in robotics, aerospace engineering, biomedical devices, flexible electronics, and other fields.  In robotics, this process creates a material that can be used to design adaptive and self reconfigurable robots who can change their shape so they can navigate in different environments. On the other hand, in the biomedical field, this technique can produce materials used in pacemakers that can adapt to the shape of the organs.

Combining Kirigami with shape shifting materials offer a future of new materials that can be used across a variety of multiple industries. This is a field that will continue developing due to its potential.  Let me know what you think, I'd love to hear.  Have a great day.

Skip Counting, Multiplication, And How To Get From One To Another.

 

As I have been doing more research on number lines, I realized that many of my students get to high school with the ability to skip count but didn't know their multiplication tables.  I've accepted that skip counting might be the only way they know but I've often wondered why they didn't make the jump from one to the other.

Skip counting is often the first thing students are taught in their journey into multiplication.  From there , they move on to learning multiplication and multiplication facts but some students never make the jump.

It is never too late to do things to help students make the jump from skip counting to multiplication.  These steps can be done in middle school or high school as part of the scaffolding activities. First, think about using visual representations such as number lines, counters, or arrays to help make students see how skip counting and multiplication is related since both are a form of repeated addition.

Another activity using counters, cubes, or blocks is to have students arrange these manipulatives to represent skip counting. So if a student is skip counting by three's, arrange the manipulatives in groups of three so they count by three's, then they can ease over to the idea of counting the number of groups to arrive at the total. Both ways get the student to the total but one is strictly addition and the other ways shows the idea of groups times number in group gives total.

Look at having students explore patterns and extensions associated with skip counting.  For instance, list the pattern for four's skip counting and leave blanks such as 4, 8, ____, 16, _____, etc.  Extend this to the idea that one 4 is 4, two 4's is eight, so three four's is 12 which is the missing number.  This helps students make the jump from skip counting to multiplication.

Throw in some questions which require students to use skip counting in a real life situation such as buying 5 packages of 12 screws so a student might do 12, 24, 36, 48, 60 or 5 x 12 is 60. This provides a real life connection between skip counting and multiplication.  

Finally, give students a chance to practice, practice practice.  Provide students the opportunity to play games, make posters, and other methods to give them a chance to practice. Start with easier problems and move to problems that are more complex.

These are just a few ways to help students move from skip counting to multiplication.  Once students have multiplication down, it is important to include continued practice.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

If there are 480 skittles in a one pond bag, how much does one skittle weigh? 

Warm-up.

 

If there were 108,000 new registrations of the French bulldog in 2022 and they were the most popular breed. The number two breed is the Labrador with 21,000 less. What percentage more bulldogs were registered than labs?

Combining Like Terms

 

Combining like terms is one of those topics that students either get or struggle with. The standard way to teach it is to have students identify like terms, rearrange the terms so like terms are next to each other, simplify by combining like terms, and the remaining terms constitute your answers.  Sounds quite straight forward by it doesn't always work that way.

It doesn't always work that easily because some students have difficulty understanding the concept of like terms.  One way to help students identify like terms is to have them use circles, squares, triangles, diamonds, and other shapes. They would circle all terms that are constants, terms with x's are inside a square and x^2 are inside triangles.  This way they students can rewrite the terms so those in the circles are next to each other, squares, next to each other, and triangles gathered together.  

Another way is to use either physical or digital sticky notes in different colors.  Each color represents a type of terms such as constants, x's, or x^2's so write each term on the appropriate color so that when it is time to rearrange the terms, students can put terms on the same color notes together to make it easier to distinguish among terms.  

Then there is using base 10 blocks with the individual squares representing ones, the 10's become the x's, and the 100's represent the x^2's, so 2x^2 + 3x + 4 would be represented by 2 - 100 squares, 3 - 10 strips, and 4 - individual squares.  Once they've translated the terms into the base 10 blocks, the student can physically see which terms are alike and it makes it easier for them to combine like terms.  

Once students have been instructed in the concept of combining like terms, it is time to include a variety of activities to reinforce and practice this.  One activity is to whip out a game of combining like terms bingo.  Prepare a variety of cards with answers and pass the cards out to the students.  The other possibility is to like the answers on the board, hand out blank bingo cards, and have the students fill out their cards with the answers they've chosen from the board.  Always have more answers than squares so that students will not have all the same answers.  To start the game, have a container full of problems and draw one out.  Write it on the board and have students come the like terms.  When they have the answer, they check their cards to see if they have the answer.  When they are beginning, I always go over the problems so struggling students can see how to do it.  As they gain skill, I no longer of it or only do it for problems where many students struggle.

When students are more proficient at combining like terms, it is time to enjoy either jeopardy or Kahoot games.  For Jeopardy games, I like having students work in pairs and write the answer on a whiteboard so all the students who get the right answer will receive a score otherwise students who take a bit longer do not feel penalized.  

This is also the perfect item for students to do a scavenger hunt activity.  On a piece of paper, write down a problem and an answer but not the answer to the problem on this paper.  Write the answer down to another  problem.  On the next sheet, record the problem to the answer on the other paper, and write down the answer to a different paper.  Continue till you have 10 to 15 papers completed and post around the room.  Give each student an answer sheet so they can start at any paper.  They work the problem, and then search for the answer.  Once they find the correct answer, they do the problem on that page and search for the matching answer.  Continue until each students has worked all the way through the problems.

These are some ways for students to practice combining like terms.  Let me now what you think, I'd love to hear.  Have a great day.

Need To Make A Decision, Use A Scoring System.

 

When it comes to making a decision, most people look at it as a yes or no proposition.  You do it or you don't do it but others look at a decision being a choice of several possibilities, much as businesses approach a decision. When the decision is difficult, we often list and rank our alternatives to determine which is better.

The process of using a scoring system involves identifying the outcome, determining the criteria, assigning a weight to each based on importance, developing the actual scoring system, evaluating, totaling the score, analyzing and making a decision.  Although this is the normal system used, it is sometimes flawed.

Often decisions come out of a limited number of choices. If the decision is based on one criteria such as cost, then it becomes much easier but if there are more possibilities, then it involves the pros and cons for each one. This means there is a multi-criteria choice involved.

Most of the weighted systems used to make the decision. This process requires the decision maker to eliminate those options which are a no go before ranking the remaining ones according to preference and assigned a score based on each criteria. The scores usually range from 1 to 5 or some similar ranking for each possibility, then multiplied by a weight, and the scores are totaled to find an overall score.  

Unfortunately, the weakness with this the when it comes to assigning the value because the value is based upon the human evaluation. The better approach. is to consider using a scoring system that contains negative numbers with an adjustment to keep the values between 0 and 10. The actual formula is 

                                    weighted score = (score – offset) × weight + scale_shift.

Offset refers to the midpoint of the score range and scale shift is the smallest number needed to make all values positive. Thus if the values are 0 to 10, then the offset and scale shifts are 5 and 50. 

This method still sees those that have the lowest numbers are not the best choice but what this alternate system does is that a low score does not immediately put it at a disadvantage whereas the normal system does. It was originally developed to use in engineering.

In the normal selection process, it is possible to get have several possibilities end up with a zero regardless of their importance and depending on whether they are along the left side or the bottom, determines whether they are zero regardless of importance or the choices are penalized against unimportant criteria.  In the alternative system, unimportant scores are neither good nor bad. 

This is just another way of looking at weighting choices. Although it originated in engineering, it is a methods that could easily be using in other places such as businesses.  Let me know what you think, I'd love to hear.  Have a great day.