Financial Algebra

 Last year, several teachers from my district attended a mathematics conference and one of the vendors offered something that I’d never seen before.  As you know, most high schools offer the traditional Algebra I, Geometry, and Algebra II but this vendor offered a financial algebra course which could be used to replace the traditional class.

This is a class that offers many of the same topics but shows how they are applied in finance.  It might show students how advanced algebra is used in things such as discretionary spending, banking, home or auto ownership, business, or retirement.  Furthermore, financial algebra also offers a better opportunity to integrate modeling with the topics.

I looked at one textbook from National Geographic which had chapters on the stock market, learning to model a business, banking and services associated with them, consumer types of credit, employment and taxes, and planning for retirement along with several other topics.   In other words, the class gives students a more in depth understanding of financial topics for personal and business settings.   It takes things a step further than the usual consumer math classes and gives them a better mathematical exposure to finance.

This particular class is a good one to add to the current offerings because it offers a more relatable context for most students who see Algebra as totally unrelated to their lives.  A good financial algebra class will cover topics like linear equations, fractions, decimals, percents, moving averages, exponential functions and exponential growth and decay, limits, recursive thinking, piecewise functions, expected value, parabolas and quadratic formulas, scatter plots and correlation, natural logs and lns, rational expressions, square roots and so many more topics found in a traditional Algebra II class.

If done properly, students will only need to complete the Algebra I class in order to be prepared and a good financial algebra class will reinforce what they’ve learned while introducing them to more complex topics.  

I like the idea of financial algebra because it will answer the question “When will I ever use this?”  I think it will help students to see when and where the material is used in the real world and in their lives.  This is important because it gives context to the math they are learning in school. 

In addition, such a class provides students who struggle with traditional algebra 2 classes a nice alternative and is good for students who do not plan to major in one of the hard sciences.  It offers more advanced algebra than the traditional consumer math classes which are more arithmetic based.  

So schools should offer financial algebra in addition to the more traditional algebra II classes so more students have their needs met while preparing them for the future.  Let me know what you think, I’d love to hear.  Have a great day.

Data Science In High School

 

Things are changing. When I went to college many years ago, one wanted to have enough math in high school so you could go straight into calculus if you were majoring in the hard sciences.  In fact, if you could manage to take calculus in high school, even better but the requirements for what is needed in the real world has changed.

People are now thinking that a different approach is necessary for advancing in the hard science fields because data and statistics are being used more and more rather than calculus.  In fact, the collection and interpretation of data is being used to make predictions, interpret the events of the world, or explain the world.  

In fact, most data science has become a major player in the world and many are predicting that all high schools should offer a class in it.  Although sciences traditionally have used data and statistics over the years, other professions such as economics, politics, and education are relying more heavily on it.  Jo Boaler is recommending that schools provide instruction to students to help them develop data literacy.

Unfortunately, traditional pathways in modern education still have students taking a sequence leading to calculus. Furthermore, most schools still do not offer stand alone data science or statistics courses.  Statistics is often incorporated into Algebra I and II classes such as in my district.  This means students only get a surface knowledge of the material rather than a more in depth understanding.

It is hoped by offering additional classes in data analysis and statistics in high school, students will be better prepared for college and the workforce. If this cannot be done, it has been recommended that classes offer students the opportunity to work with large sets of data.  It is something students need to do before they graduate from high school.

In fact, if students take data science classes in high school, it opens up a chance to participate in the up and coming career of data scientist.  A data scientist sorts through data and provides the information necessary for various industries to make informed decisions. At the moment, there are more jobs available than candidates to fill them.

So if we want our students to understand more of the world around them while preparing them for a career or at least offer the tools needed to help them succeed, we need to begin offering data science classes in high school.  Unfortunately, education tends to be a bit slow in changing to meet the needs of a rapidly changing world.  Let me know what you think, I’d love to hear.  Have a great day.

Warm-up

If there are 33 dried apricots per pound, approximately how many apricots are there when you have 258 pounds?

Warm-up

If it takes four pounds of grapes to produce one pound of raisins, how many pounds of raisins will you have if you start with 2688 pounds of grapes? 

Japanese Multiplication

One of my students showed me something new.  She’d seen it on Youtube and chose to share it with me.  It was interesting. It’s called the Japanese multiplication method also known as Chinese multiplication, or the stick method.  Apparently, it's been around a while. I've used it and it reminds me a bit of the box method without the box.

It uses parallel lines that run diagonally and when the line cross, they form intersections and it is the intersections that provide the multiplication. I've done a problem showing each step so you can see how it works. I'm using the problem 24 x 13.

For 24, I've got two parallel lines in green with a space and then four more that are dotted. 13 is represented by one red line and then 3 more a bit a way. It is easy to see the 24 and 13.

The line are supposed to cross at 90 degree angles or be perpendicular. Usually the lines are running along the y = x or y = -x pathways.

The next step is to place dots on the intersections so they are so much easier to see.

These dots are what provides us with the product for each step of the multiplication.

The intersections at the middle left represent the hundreds, the group at the top and bottom represent the tens, and the group to the right represent the ones.

So based on the last photo, we have 200 from the left, 60 plus 40 from the middle and 12 from the bottom. Each of these represents one number from the standard multiplication algorithm.  When you follow the standard multiplication algorithm you would go 3 x 4 = 12, 3 x 20 = 60, 10 x 4 = 40 and 10 x 20 = 200 so when you add all the numbers up, you get 312 using either method.

There are problems when trying to use this method with larger numbers.  For instance, if you multiply 8 x 9 you'll have 8 lines crossing 9 which gives you 72 intersections to count.  Or if you so something like 599 x 798, you end up with lots of lines which can be quite confusing but when children are learning to multiply, they tend to use small numbers so this system works well.

I am glad to know about this as it gives me one more method I can use with students who struggle with multiplication.  In fact, I have a couple of students who do not know their multiplication tables so I want to show them this method.  It might help them so they are not as reliant on a calculator. Let me know what you think, I'd love to hear.  Have a great day.

Is John Hattie Incorrect?

I don't remember where I stumbled across this but it was about John Hattie and his work. As you known, he's written several books which contained conclusions based on synthesized data on how to improve learning. My district uses his information out of his "Visible Learning books to talk to us about what we should be doing. In fact, I set my personal goal based off of something I found in his "Visible Learning in Mathematics.

There are quite a few articles out there questioning the methodology used. One claim is that Hattie conducted a meta-meta-analysis of over 1200 meta-analysis studies that synthesized over 50,000 studies which covered all sorts of topics from interventions to pay performance teachers. These studies had sample sizes from one to several hundred, or carried out with everything from great parameters to bad. It was also claimed that any biases contained in these meta-analysis studies into his final work.

In addition, several people argued that since many of the original studies had sample sizes of one or lab studies using paid volunteers and other issues, the results may not be valid. Supposedly, Hattie synthesized all the results and applied a metric of confidence so they could utilize the date.

At least one statistician claimed the method of calculating averages and statistical deviation used by Hattie and his people shows a lack of sophistication. Many questioned whether this method of calculating averages and statistical deviation can provide enough data for accurate conclusions. It has also been said that he does not use proper baseline comparisons and has also used comparisons of other factors in correction. Furthermore, they allege that Hattie didn't really understand most of the normal methodology used which lead to incorrect conclusions.

Another person raised the issue about using a multitude of studies that do not share the same methodologies, implementations, or having students who are in the same socio-economic group. These differences can make comparing results even more problematic. Others have asked if conducting a meta-meta-analysis of meta-analysis studies is the best way to obtain information since one is synthesizing information that has already been synthesized.

I don't know if his conclusions are correct or incorrect. For me, my district follows the results published in "Visible Learning", and I have to accept that they want me to use the information. Let me know what you think about this topic. I'd love to hear. Have a great day.

The Problem With Remainders

I am returning to the topic of division, specifically remainders because so many of my students do not seem to understand what it represents.  I’ve observed many students who took a remainder and made it the decimal portion of the final answer.  For instance, if they have 24/5, they do the division and instead of 4 4/5 or 4.8, they put 4.4 because the remainder is four. I have no idea where that misunderstanding comes from.

I’ve tried to research why this happens but have not been successful as most every article only discusses the process of division and how difficult it is.  It is hard finding articles discussing why students have issues with writing down the correct form of the remainder. I wonder if they do not connect the remainder with a fraction and  a decimal equivalent. I'm not sure they even understand the concept of a remainder . 

Most articles state that the process of long division is extremely difficult and students struggle to learn the multi-step process but few address misunderstandings associated with remainders. I found one author who indicated that higher order thinking skills are needed to interpret the type of remainder. The remainder requires students to determine its context in order to figure out the type of remainder that is needed.

There are four types of remainders that students will run into when doing division.  The first type of remainder is the one you leave alone as a fraction such as in 25/4 gives 6 and 1 left over as the answer.  The numerator tells you how many were left over out of the groups.  In other words, the part of a whole.  This is the case where my students tend to write 6.1.  The second type of problem looks at only the remainder so the problem might tell you that you have 25 quarters and wants to know how many you have left over once you turn the quarters into dollars.  In this case, the answer is one quarter. 

The third type of remainder is one that is either rounded up or down depending on the circumstances.  For instance, if your can of paint will cover 300 square feet with two layers and you need to cover 1100 square feet with a two coats, how many cans should you buy.  If you do it with a calculator, you’ll end up with 3.666666667 or 3 ⅔ cans.  In this case, we need to round up because we cannot buy ⅔ of a can.

The final type of remainder is the sharing remainder where you want a fractional answer such as you are sharing 25 cookies among 4 people, how many cookies will each person have?  It is 6 ¼.  The answer tells us that each person is going to get 6 ¼ cookies.

Based on my own experiences teaching, students have issues with remainders because they have not fully learned how to use long division in addition to not knowing how to interpret remainders.  Furthermore, they are also weak on connecting remainders with fractions and changing those fractions into decimals.  Let me know what you think, I’d love to hear.  Have a great day.