Building An Algebraic Foundation - Grade By Grade.

A strong foundation in algebra is essential for students' success in higher mathematics. To achieve this, algebraic concepts should be gradually introduced and reinforced throughout elementary school. Here's a grade-by-grade overview of potential algebraic concepts and their connections to future algebra learning.

Kindergarten and First Grade: At this age, the focus is on building number sense and developing problem-solving skills. Children can begin to explore basic algebraic concepts through patterns and relationships. Activities like sorting objects by size, color, or shape can help students recognize patterns and classify objects based on attributes. Additionally,simple number patterns, such as counting by twos or fives, introduce the idea of sequences. These early experiences lay the groundwork for understanding variables and functions in later grades.

Second and Third Grade: In these grades, students should continue to explore patterns and relationships, but with a greater emphasis on using mathematical language to describe them. They can start to use symbols to represent unknown quantities. For example, in a problem like "I have 5 apples, and I eat some. Now I have 3 apples. How many apples did I eat?", students can use a symbol (e.g., ?) to represent the unknown number of apples eaten. This introduces the concept of variables. Additionally, students can begin to explore basic equations through simple addition and subtraction problems.

Fourth and Fifth Grade: As students progress, their understanding of algebraic concepts should deepen. They can explore more complex patterns and relationships, including those involving multiplication and division. Introducing the concept of order of operations (PEMDAS) is crucial at this stage. Students can also begin to solve simple equations involving one variable. For instance, problems like "3x + 2 = 11" can be introduced and solved using guess-and-check or inverse operations. These experiences help students develop algebraic reasoning and problem-solving skills.

Sixth and Seventh Grade: In these grades, the focus shifts towards formal algebra. Students should become proficient in solving multi-step equations and inequalities. They can also explore concepts such as graphing linear equations,understanding slope and intercepts, and working with systems of equations. Additionally, introducing the concept of functions and their representations (tables, graphs, equations) is essential. These experiences prepare students for more advanced algebraic topics in high school.

By systematically introducing algebraic concepts throughout elementary school, students develop a strong foundation for success in algebra. This approach emphasizes problem-solving, critical thinking, and mathematical reasoning, which are essential skills for all students. It is important to note that this is just one possible framework, and teachers should adapt their instruction to meet the needs of their students.

Ultimately, the goal is to create a seamless transition from arithmetic to algebra, making the learning process engaging and meaningful for all students.  Let me know what you think, I'd love to hear.  Have a great day.

When Should We Begin Teaching Algebra?

While the allure of accelerating students through math curricula is tempting, introducing formal algebra before eighth grade can be counterproductive. Algebra requires a solid foundation in arithmetic, logical reasoning, and abstract thinking. Premature exposure can lead to superficial understanding and a negative association with math.

Instead of rushing through the curriculum, educators should focus on building a strong algebraic foundation in the earlier grades. This involves infusing algebraic thinking into existing math concepts. For instance, young students can explore patterns in number sequences, developing a sense of generalization. Creating input-output tables for simple operations like addition and multiplication can introduce the concept of functions.

Real-world problem-solving is another effective approach. Questions like, "If you have five apples and give two to a friend, how many do you have left?" can be represented algebraically as 5 - 2 = x. These early experiences familiarize students with the language of algebra without overwhelming them with complex equations. Games and puzzles can also be used to develop algebraic reasoning. Sudoku, for example, enhances logical thinking and problem-solving skills. Card games involving number combinations can promote algebraic patterns recognition.

By the time students reach eighth grade, they should have a solid grasp of arithmetic, a familiarity with algebraic concepts, and the cognitive maturity to handle the abstract nature of formal algebra. This approach ensures a smoother transition and a deeper understanding of the subject. Remember, the goal is not to rush students through the curriculum but to equip them with the necessary tools for future mathematical success. A strong foundation in algebraic thinking, built gradually over several years, will serve students well in their academic journey.

Unraveling The Physics Of Knitting

 

Knitting, often seen as a leisurely craft, holds a surprising secret – a world governed by the principles of physics. While needles manipulate yarn into beautiful garments, the magic lies in the intricate interplay between yarn properties, stitch patterns, and the resulting fabric's behavior.

Imagine a single strand of yarn: relatively inelastic on its own. Yet, when knitted, it transforms into a fabric boasting remarkable elasticity, capable of stretching well beyond its original length. This seemingly paradoxical behavior stems from the way stitches interlock. Each loop interacts with its neighbors, creating a web-like structure that allows the fabric to deform and then spring back.

Physicists like Elisabetta Matsumoto are delving into the fascinating realm of stitch geometry. By analyzing how loops intertwine, they're creating a "dictionary of knits" that predicts fabric properties based on stitch patterns. This knowledge has far-reaching implications. Imagine altering a single loop configuration – a seemingly minor tweak – yet it could drastically change how the fabric drapes, breathes, or resists wrinkles.

Furthermore, researchers are using computer simulations to model the physics behind knitting. These simulations mimic the manipulation of yarn and predict how different stitch combinations affect fabric behavior. Imagine a software program that allows designers to virtually test their stitch patterns, understanding the drape and stretch of the final garment before a single loop is even cast on.

Unraveling the physics of knitting extends beyond aesthetics. Knitted fabrics, due to their inherent flexibility and adaptability, are ideal for cutting-edge technologies. Imagine wearable electronics seamlessly woven into clothing, or soft robotics that mimic human movement – all thanks to the unique properties of knitted structures. By understanding the physics behind these fabrics, engineers can design smarter textiles that integrate functionality with comfort and style.

The journey to unravel the physics of knitting is as fascinating as the craft itself. It's a testament to the hidden complexity within seemingly simple activities. As research delves deeper, the future promises exciting possibilities – from creating fabrics with tailored properties to pushing the boundaries of wearable technology. In this pursuit, knitting transcends needles and yarn, becoming a powerful tool for scientific exploration and innovation.  Let me know what you think, I'd love to hear.

Why Multiplication Facts Trumps Calculators

Apologies for not having any column this past Monday.  I was off doing a show and had no time to get it together and I just got home with a bug that has me sleeping a lot.  So today, we'll look at multiplication.

In today's tech-driven world, calculators are ubiquitous. Yet, while these tools have their place, relying solely on them for basic multiplication comes at a cost. Mastering multiplication facts offers a range of benefits that go beyond simply getting the right answer.

Firstly, knowing multiplication facts fosters a deeper understanding of numbers. By memorizing these relationships, we internalize patterns and connections. We see multiplication not just as button-pressing, but as repeated addition, doubling,or halving, depending on the context. This number sense allows us to tackle more complex math problems with greater ease. Imagine multiplying a two-digit number by a single digit. With memorized facts, the process becomes a breeze.Without them, it requires cumbersome steps that can lead to errors.

Secondly, fluency in multiplication facts translates to mental math prowess. Imagine being at the grocery store, quickly calculating the total cost of your items without needing a calculator. This mental agility spills over into other areas of life,making budgeting, estimating, and problem-solving more efficient. It's like having a built-in calculator readily available,always at your fingertips.

Thirdly, memorizing multiplication facts builds cognitive skills. The process of learning and retaining these facts strengthens memory and reinforces critical thinking. It requires focus, repetition, and the ability to identify patterns. These skills are essential for success in all academic disciplines, not just math.

Of course, calculators are powerful tools. They can handle complex calculations and save time when dealing with large numbers. However, relying on them for basic multiplication can be detrimental in situations where a calculator isn't available. Imagine being caught in a power outage and needing to calculate a tip at a restaurant. Or, picture a standardized test that prohibits calculators. Without a foundation in multiplication facts, these situations can become stressful and hinder performance.

Ultimately, the goal is not to eliminate calculators entirely, but to develop a healthy balance. Mastering multiplication facts empowers us to solve problems efficiently, both with and without technological assistance. It fosters a deeper understanding of numbers, hones cognitive skills, and equips us for various real-life scenarios. In a world that demands adaptability and critical thinking, the power of knowing one's multiplication facts remains an invaluable asset.  Let me know what you think, I'd love to hear.

Asking Probing Questions

 

Transitioning from providing direct answers or steps to asking probing questions requires a deliberate shift in teaching. Math teachers can cultivate this skill by focusing on understanding their students' thinking processes and creating a classroom environment that encourages exploration and discovery.

One effective strategy is to develop a repertoire of open-ended questions that can be adapted to various mathematical concepts. Instead of asking, "What is the answer?", teachers can inquire, "How did you arrive at that solution?" or "Can you explain your reasoning to a classmate?" These questions encourage students to articulate their thought processes,justifying their answers and building confidence in their abilities.

Active listening is equally crucial. By paying close attention to students' responses, teachers can identify misconceptions and areas where further exploration is needed. Follow-up questions can then be tailored to guide students toward deeper understanding without providing explicit answers. For instance, if a student struggles with a problem, instead of offering a solution, the teacher might ask, "What part of the problem are you finding challenging?" or "Can you try a different approach?"

To enhance questioning skills, teachers can engage in professional development opportunities and collaborate with colleagues. Sharing effective questioning strategies and providing feedback can foster a supportive learning community.Additionally, reflecting on classroom interactions and analyzing student work can help teachers refine their questioning techniques over time.

Creating a classroom culture that values exploration and perseverance is essential for the success of probing questions. By encouraging students to take risks, make mistakes, and learn from them, teachers can foster a growth mindset. When students feel safe to experiment and share their ideas, they are more likely to engage in independent problem-solving.

Ultimately, the goal is to empower students to become independent thinkers and problem solvers. By asking probing questions and creating a supportive learning environment, math teachers can play a pivotal role in developing students' mathematical confidence and competence.  Let me know what you think, I'd love to hear.  Have a great day.

Learning To Listen.

Traditionally, math classrooms have been dominated by teacher-led instruction, with students primarily taking notes and absorbing information. However, in order to encourage independent problem-solving and critical thinking, it is best to  shift towards increased student voice and decreased teacher intervention is necessary.

Active listening is a cornerstone of this transformation. By attentively observing and responding to students' mathematical thinking, teachers can gain invaluable insights into their understanding and misconceptions. This information can then be used to tailor instruction and provide targeted support. For instance, when a student struggles, instead of immediately providing the correct answer, the teacher can ask probing questions to guide the student towards the solution. This approach encourages students to persevere and develop their own problem-solving strategies.

Furthermore, creating a classroom culture where students feel comfortable sharing their ideas and reasoning is crucial.Teachers can establish a safe and supportive environment by valuing all contributions, regardless of correctness. This encourages students to take risks, experiment with different approaches, and learn from their mistakes.

To facilitate increased student independence, teachers can employ various strategies. One effective approach is to incorporate more open-ended problems and real-world applications into the curriculum. These types of tasks require students to analyze, interpret, and evaluate information, promoting deeper understanding and critical thinking.Additionally, cooperative learning groups can empower students to collaborate, share ideas, and build on each other's knowledge.

By shifting the focus from teacher-centered to student-centered learning, teachers can create a more engaging and effective math classroom. Listening attentively to students, encouraging their mathematical thinking, and providing appropriate support will ultimately lead to greater student independence and a deeper understanding of mathematical concepts.

Modeling How Cells Move Like Trains.

The intricate dance of cellular movement, particularly the coordinated progression of cell groups, has long puzzled scientists. How do cells, often likened to tiny trains navigating a complex landscape, manage to move in unison through challenging environments? Recent advancements in computer modeling are shedding light on this cellular choreography.

Imagine a single cell as a locomotive, pulling its cargo behind it. But unlike steel rails, the cellular environment is a labyrinth of proteins, other cells, and extracellular matrix. To understand this complex system, researchers have turned to computer simulations. By creating virtual cells and their surroundings, they can manipulate variables, test hypotheses, and observe patterns that would be impossible to study under a microscope.

A breakthrough approach has revealed a novel perspective on cell-cell interactions. These models suggest that cells don’t simply follow each other blindly. Instead, they engage in a sophisticated communication network. Leading cells send out signals that guide the following cells, much like a train's engineer directing the cars behind. These signals, often chemical or mechanical, influence the behavior of trailing cells, ensuring coordinated movement.

Furthermore, the models highlight the importance of the cell's microenvironment. The surrounding matrix, the stiffness of the terrain, and even the presence of other cell types can dramatically impact the group's movement. By simulating various environmental conditions, researchers can identify factors that facilitate or hinder collective cell migration.

This computational approach is not just an academic exercise. Understanding how cells move in groups has profound implications for fields such as developmental biology, cancer metastasis, and wound healing. By deciphering the rules governing cellular locomotion, scientists hope to develop strategies to control or manipulate cell behavior for therapeutic purposes. As computational power and modeling techniques continue to advance, we can anticipate even more detailed and accurate simulations, bringing us closer to unlocking the mysteries of cellular movement. Let me know what you think.  Have a great day.

Stress Testing Pension Funds

 Pension funds, as long-term investment vehicles, face significant risks arising from market volatility and economic fluctuations. Stress testing, a critical risk management tool, assesses a fund's resilience to adverse scenarios. To enhance the sophistication and accuracy of stress testing, employing a hidden Markov regime switching model (HMM-RSM) presents a promising approach.

HMM-RSM acknowledges that financial markets exhibit distinct patterns or regimes, such as bull, bear, or normal markets. These regimes are often unobservable (hidden) but influence asset returns. By incorporating this dynamic behavior, HMM-RSM offers a more realistic representation of market conditions compared to traditional models assuming constant parameters.

The stress testing process begins by estimating the HMM-RSM based on historical asset returns. This involves identifying the optimal number of regimes, estimating transition probabilities between regimes, and determining the distribution of asset returns within each regime. Once the model is calibrated, various stress scenarios can be simulated by altering the model's parameters. For instance, increasing the probability of transitioning to a severe recession regime or modifying the distribution of asset returns within a crisis regime can create adverse conditions.

By exposing the pension fund to these stressed scenarios, its performance under extreme circumstances can be evaluated.Key metrics such as funding ratio, asset-liability management (ALM) gaps, and risk measures (e.g., Value-at-Risk) can be calculated to assess the fund's resilience. Furthermore, sensitivity analysis can be conducted to identify critical risk factors and vulnerabilities.

The HMM-RSM approach offers several advantages. Firstly, it captures the non-linear and time-varying nature of financial markets, leading to more accurate stress test results. Secondly, it allows for the generation of a wide range of plausible scenarios, enhancing the comprehensiveness of the stress testing exercise. Thirdly, it provides insights into the fund's behavior under different market conditions, aiding in the development of robust risk management strategies.

However, challenges remain. Accurate estimation of the HMM-RSM requires high-quality data and sophisticated statistical techniques. Moreover, determining the appropriate level of stress and selecting relevant stress scenarios can be subjective.

In conclusion, stress testing pension funds using a hidden Markov regime switching model is a valuable tool for assessing risk and building resilience. By incorporating the dynamic nature of financial markets, this approach provides a more realistic and informative perspective on potential future outcomes. As the financial landscape continues to evolve, refining and expanding the use of HMM-RSM in stress testing will be crucial for ensuring the long-term sustainability of pension funds.  Let me know what you think, I'd love to hear. Have a great day.

Math On The Move.

Summer vacations offer a wonderful opportunity to explore new places and create lasting memories. Surprisingly, planning, taking, and reviewing a trip involve a significant amount of mathematics. From budgeting to navigation, math is an integral part of the travel experience.

The planning stage is a mathematician's paradise. Determining the optimal travel dates often involves considering factors like flight costs, hotel rates, and weather patterns, which require data analysis and comparison. Budgeting for the trip is another mathematical exercise, encompassing calculations for transportation, accommodation, food, activities, and souvenirs. Additionally, optimizing the itinerary demands spatial reasoning and time management, as travelers must calculate distances, travel times, and activity durations to create an efficient schedule.

In addition, one can plan tours before leaving home and math helps you decide which tour is the better choice.  Once considers the cost, what you get for that amount of money, how long does the trip take, where does it go, etc.  This is a wonderful way for students to decide which trip is the best for the amount of money paid.

Once the trip begins, math continues to play a role. Navigating unfamiliar places often involves calculating distances, estimating travel times, and converting currencies. For those who enjoy outdoor activities, math can be used to measure hiking distances, calculate calorie intake, or determine the best angle for a perfect photograph. Even leisure activities like gambling or playing games at a casino involve probability and statistics.

Upon returning home, the mathematical analysis doesn't end. Travelers may calculate the total cost of the trip to compare it to the budget, analyze spending patterns, and identify areas for potential savings. For those with a penchant for data,tracking travel statistics such as distance traveled, number of photos taken, or places visited can be a rewarding exercise.Furthermore, comparing travel experiences across different destinations can involve statistical analysis to identify trends and preferences.

Incorporating math into travel not only enhances the planning and execution of the trip but also provides a unique perspective on the experience. By applying mathematical concepts to real-world situations, travelers can develop critical thinking and problem-solving skills while enjoying their adventures.

Keeping Math Skills Sharp Over The Summer

Summer break offers a much-needed respite from the academic grind, but it's essential to prevent the dreaded "summer slide" when it comes to math. While relaxation is crucial, maintaining math skills can lay a strong foundation for the upcoming school year. Fortunately, there are numerous engaging ways to incorporate math into daily life.

One effective strategy is to turn everyday activities into math problems. Cooking or baking can be a fun way to practice fractions, measurements, and ratios. Shopping trips offer opportunities to calculate discounts, compare prices, and manage budgets. Even gardening can involve math, whether it's determining plant spacing, measuring garden plots, or calculating fertilizer amounts.

Incorporating math-based games and puzzles into leisure time is another enjoyable approach. Classic board games like Monopoly, Chess, and Sudoku challenge strategic thinking and problem-solving abilities. Online math games and apps provide interactive and engaging practice. Additionally, card games can be adapted to enhance math skills. For instance,players can calculate sums, differences, or products of card values.

Reading can also contribute to math retention. Many books incorporate mathematical concepts, from geometry in architecture to probability in storytelling. Discussing these mathematical elements with peers or adults can deepen understanding.

For those seeking a more structured approach, online math resources and tutoring programs are readily available. These platforms offer a variety of practice exercises, interactive lessons, and personalized support. Even short daily practice sessions can make a significant difference.

Ultimately, the key to preventing math skill decline over summer is to make it enjoyable. By integrating math into everyday activities and turning it into a game, students can maintain their proficiency while having fun. With a little effort and creativity, the summer break can be a time for both relaxation and mathematical growth.  Let me know what you think, I'd love to hear.  Have a nice day. 

Real World Examples Of Piecewise Functions.

Math often gets a bad rap for being abstract and irrelevant. However, many seemingly complex mathematical concepts find surprising applications in our daily lives. Piecewise functions, with their segmented rules, are a prime example. These functions, though seemingly theoretical, govern a surprising number of real-world scenarios.

One ubiquitous example of a piecewise function is cellular phone pricing or wifi plans. Most plans offer tiered data allowances,where the cost per gigabyte changes based on the amount of data used. Imagine a plan with 5GB of data included for a base price. Any usage exceeding 5GB is charged an additional fee per gigabyte. This scenario perfectly illustrates a piecewise function. Up to 5GB, the cost is governed by one rule (the base price). After that threshold, a different rule (the additional fee) applies.

Another familiar example are store discounts and coupons. Many stores offer tiered discounts based on purchase amount. For example, you might get 10% off for spending between $50 and $100, and then 15% off for anything above $100. This scenario uses a piecewise function with spending amount as the independent variable and the discount percentage determined by the applicable spending tier.

Progressive tax systems are another fascinating example. Taxpayers fall into different income brackets, each with its own tax rate. Someone earning below a certain threshold might pay no income tax, while someone earning more falls into a higher tax bracket with a different rate. This tiered system is a classic example of a piecewise function, with income acting as the independent variable and the tax owed being determined by the applicable bracket and its corresponding rate.

Even the world of shipping utilizes piecewise functions. Many shipping companies offer tiered rates based on weight or distance. For example, a package weighing less than a pound might have a flat rate, while heavier packages are charged progressively more based on weight. This scenario perfectly aligns with the concept of piecewise functions, with different cost rules applying to different weight categories.

Look at employee pay with overtime: Many jobs offer a base hourly wage, but pay a higher rate for hours worked beyond a certain threshold (usually 40 hours per week). This scenario is a perfect example of a piecewise function, with the total pay depending on the number of hours worked and the applicable pay rate (base wage for regular hours, higher rate for overtime).

These are just a few examples of how piecewise functions permeate our everyday lives. From the seemingly mundane tasks of making phone calls or sending packages to the intricacies of tax systems, these functions play a crucial role in defining costs and outcomes. If you want more examples, just do a search. By recognizing these connections, we can bridge the gap between theoretical math and the practical world, demonstrating the surprising relevance of piecewise functions in our daily lives. Let me know what you think, I'd love to know.

Happy July 4th.

 

Demystifying Piecewise Functions

Piecewise functions, with their multiple rules and sudden shifts, can be a stumbling block for students. However, with engaging teaching methods and relatable real-life examples, educators can transform these functions from cryptic equations to comprehensible concepts.  It is important to break down the pieces.

Start with a clear visual representation of a piecewise function. Graphing each piece of the function separately helps students understand the different rules governing each interval. Overlaying these individual graphs creates the complete picture, showcasing the transitions between the pieces.

Bridge the gap between abstract math and the real world by using relatable examples of piecewise functions. Discuss long-distance phone call rates, where the cost per minute changes depending on the duration of the call. Cell phone plans with tiered data allowances can also be used to illustrate piecewise functions in action.

Next build understanding by breaking down the function into its component parts. Explain how to identify the different intervals by analyzing the inequalities that define them. Then, guide students through applying the appropriate rule for each interval to solve problems. Incorporate interactive activities to solidify understanding. Use online applets or physical manipulatives to allow students to explore how changes in the function's parameters affect the graph. This hands-on approach allows them to visualize the relationship between the equations and the visual representation. Don't just focus on solving for outputs; delve into real-world applications of piecewise functions. Discuss how they are used in engineering to model temperature changes during a manufacturing process,or how they can be used to calculate shipping costs based on weight and distance.

Consider going beyond the basics. Create a "piecewise function zoo" where students encounter various examples with unique applications. This could include functions modeling taxes based on income brackets or membership fees with discounts for longer commitment periods. For advanced students, explore piecewise functions with more complex rules or multiple variables. Offer opportunities for independent projects where they can research real-world applications of piecewise functions and present their findings to the class.

By employing visual aids, relatable examples, and interactive activities, educators can transform piecewise functions from daunting equations to engaging concepts. Highlighting the real-world applications of these functions further fuels student interest and demonstrates the practical value of mathematics. With the right approach, piecewise functions can become a stepping stone towards a deeper understanding of advanced mathematical concepts.

Next time, we'll look at some situations that use piecewise functions. let me know what you think, I'd love to hear.  Have a great day.

Are Self-checking Reviews A Good Thing?

 

Traditionally, tests and assessments have been seen as one-way streets, measuring student learning but offering little guidance for improvement. However, a powerful tool called self-checking reviews is transforming this belief. These reviews involve students actively checking their own understanding of the material before a test, fostering self-awareness and pinpointing areas needing reinforcement.

Self-checking reviews offer a multitude of benefits for students. It improves metacognition. By actively checking their understanding, students develop metacognition – the ability to think about their own thinking. This allows them to identify areas of confusion and formulate strategies to address them.  It provides enhanced retention of materials. The act of reviewing key concepts and checking answers reinforces learning,solidifying information in students' minds and promoting long-term retention. It also increases confidence.  When students complete a self-review and identify areas they understand, their confidence in tackling the actual test is boosted. This can lead to improved test performance by reducing anxiety.

Self-checking reviews aren't just beneficial for students; they provide valuable insights for teachers as well. They help teachers identify learning gaps.  Analyzing common mistakes on self-reviews helps teachers pinpoint areas where the majority of students are struggling. This allows them to reteach concepts before the actual test, ensuring everyone has a chance to grasp the material. Based on the review results, teachers can adjust their instructional approach. This might involve providing additional resources, simplifying explanations, or offering differentiated instruction for students who need more support. It also promotes active learning.  Self-checking reviews shift the focus from passive learning to active engagement.Students become active participants in their learning process, taking ownership of their understanding.

There are various ways to implement self-checking reviews in the classroom.  Provide students with short, low-stakes quizzes that include answer explanations. This allows them to not only check their answers but also understand the reasoning behind the correct ones. Pair students up to review each other's work on practice problems or assignments. This fosters collaboration, communication, and a deeper understanding of the material.At the end of a lesson, ask students to answer a quick question or complete a short task that reflects the key concepts covered. This helps teachers gauge understanding and identify areas needing clarification before moving on.

Self-checking reviews are a powerful tool that empowers students to take ownership of their learning and provides valuable insights for teachers. By actively engaging students in the assessment process, we can create a more dynamic and effective learning environment. This focus on self-evaluation fosters a growth mindset, where challenges are seen as opportunities to learn and improve. So, the next time you're preparing students for a test, consider incorporating self-checking reviews. The benefits for both students and teachers are undeniable.  Let me know what you think, I'd love to hear.  Have a great day.