6 ÷ 2(1+2): BEDMAS vs Distributive Law and final conclusions.

So when I read this problem on the internet the other day, I couldn’t possibly imagine all of the thinking I could do on it! I really went down the rabbit hole here, so I am posting all of my conclusions on this problem.

First I watched a bunch of YouTube videos to see what others thought about the problem.  Essentially, there were two lines of thought.

BEDMAS RULES

On one side were the black/white pure bedmas/pedmas people, who got the answer 9. They saw this problem as this:

Part of the problem with this approach is that if any of the term were to be replaced by a variable, the equation gives you a different answer. Which brings us to those who got the answer of 1.

Distributive Law is #1

Distributive law,  symbolically, is  a(b + c) = ab + ac; saw the relationship between 2(1+2) as a case of distributive law which should be done first (part of the “B” in bedmas – take care of the brackets first).  They viewed the problem as this:

 

Stated symbolically,

 

Conclusions

So which one is right? My personal conclusion is this…

I always tell my students math is a language. Form matters, just like punctuation matters in language (Eat, Grandma vs Eat Grandma). The guilt of ambiguity lies with the author. While mathematically, the (÷) and / are equal, the ÷ is not used in algebra, therefore its usage in a simplified equation leads to confusion. Instead, the problem should be rewritten without the (÷) in multi-term equations because if any term were replaced with a variable you’d get another answer. Write the problem as either of the following.

But more importantly than form, math out of context, like words out of context, means nothing, and represents nothing. Context matters! Otherwise you’re not actually solving anything real! So don’t make math about meaningless numbers and algorithms. Make math meaningful. It shouldn’t just be about the rules. Mathematicians solve problems, and so should students.

 

 

 

 

 

 

 

 

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Math Language: Saying “groups of” instead of “times by” = understanding

Instead, say “groups of”

A small tweak in the language here will make a big difference in building student conceptualization. Without formal instruction, children know what it means to have a certain number of groups of something. Even very young students organize toys into pairs or understand when snacks are evenly distributed, or not. 

“Times” gives them nothing to hang onto, but thinking about groups does. Students may not be able to readily visualize “6 times 10,” but “6 groups of 10” is easy to imagine and even draw.

Maria Howard  on September 19, 2017

To read the rest of the article, click here:

https://www.weareteachers.com/multiplication-vocabulary-mistakes/?utm_content=1516539555&utm_medium=social&utm_source=facebook

done28099t-say-e2809ctimese2809d-when-teaching-multiplication-and-what-to-say-instead-01

Politics & Philosophy / Capitalism vs. Socialism

I teach this as part of a “Shark Tank” learning cycle. It is really good at bringing in politics and philosophy into the classroom.

Note, the videos contain some “classic art” pieces in the background, two of which contain partially clothed people. I say, “pretend you are in a museum.”

Watch the History of Capitalism and make a t-chart of the pros and cons of it:
https://www.youtube.com/watch?v=dIuaW9YWqEU 

    • Efficient as there is a much higher level of specialization, so there can be a much higher level of production
    • Higher specialization means people have a narrow, alienating focus on life
    • People at the bottom are exploited
    • good business is good for business
    • Value based on monetary worth and not necessarily on the things that make us happy.

Watch a video on Marxism which does a good job and looking at the ills of capitalism and shows the ideals and shortcomings/impracticality of socialism :
https://www.youtube.com/watch?v=fSQgCy_iIcc

Problems identified with capitalism include

  • modern work is alienating
  • modern work is insecure
  • big gap between rich and poor
  • capitalism is unstable (lots of peaks and crashes)
  • capitalism is bad for capitalists (wealth doesn’t equal happiness or fulfilling lives)
  • Ends with the quote, “Philosophers, until now, have only interpreted the world in various ways. The point is to change it.”

How can kids change it? Learn about politics and voting. 

List the political parties in North American and have students try to plot them from left to right.

  • Left = high taxes and high services
  • Right = low taxes and low services

Talk about the differences between Canada and the USA. What are the biggest differences? E.g., health care (impact of no health insurance (heart attack or premature babies = huge bills) , cost of education and the impact of higher tuition costs, the distribution of wealth, and opportunity.

The American Dream? The Canadian Dream? What should our dream be?

Tell students to ask their parents who they vote for and why.

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Debate:

What are the potential pros and cons of the minimum wage increase?

Intro to Entrepreneurism

This is a scaled down version of a “Shark Tank” learning cycle I do.  It gives you the outline without getting into the specifics. Having done this several times with students, there are a lot of different ways it could go, depending on your timelines. I will often drift into politics, capitalism vs. socialism – which leads into conversations about current events (increase in minimum wage, housing boom, wants vs. needs).

Intro Project

Key Terms

  • Discuss the difference between “Goods” & “Services”
  • Discuss “Supply & Demand”
  • revenue (gross) vs profit (net),

Brainstorm!

  • With the goal of selling something in the student’s home school, think of a good or service that they could sell for an upcoming event (e.g, Christmas, valentine’s day, graduation).
  • List as many ideas as you can. Split them into goods and services
    • Some samples of what my students have come up with E.g., hot chocolate, open gym, candy canes, valentine’s, school merchandise, movie in library, Stationary Supply in each homeroom available for purchase from teacher. Proceeds go to school.

Warm up activity:

Do this Google Doodle activity. I usually just do the phone as a warm-up activity. What are the problems with current phones (breakable, battery charging, privacy, expensive, disposable, easy to lose) and then design a new phone that fixes the current issues.

google doodle

Next, as a group, brainstorm the problems with grocery shopping. Then show them the video about Amazon Go and how they solved the problem of the long check-out lines. See if they come up with any more ideas to revolutionize shopping.

Finally, watch the video How to be an Entrepreneur by the School of Life and/or “How to Start a Business“.

Evaluate

When you start a business, you need to look at the market in which you are entering. Your task for this element is to analyze your industry (school demographics).

  • Can use SCAMPERto discuss what has been done before, whats worked or hasn’t worked and then tweak to fit the desired event.
  • Supply & Demand: This depends on many factors
    • Demand considerations: E.g., Age, Sex, Local. others?
      • Local (Time and space):  TIME: E.g,. In winter, what types of events would be in higher demand (e.g., indoor events, hot chocolate).  SPACE/Location: what is in scarcer supply in school? Junk food? Computers? Free time?  
      • Sex/Gender: Raffle tickets for a video game might be in higher demand for boys.
      • Age: E.g., Fidget toys might be more popular with kids than teachers.
      • Other?

Project Profitability

How much will each individual item cost? What will you sell it for? How many will you sell? What is your projected revenue? Profit? Costs? 

 

item cost price profit per item sales Profit Overhead Costs Revenue
Bath bomb $2.75 $5.00 $2.25 200 $450.00 $550.00 $1,000.00
candy canes $0.10 $0.25 $0.15 1000 $150.00 $100.00 $250.00
hot chocolate $0.25 $1.00 $0.75 500 $375.00
pencil $1.00 $2.00 $1.00 25 $25.00 $68.75 $50.00
eraser $1.00 $1.50 $0.50 100 $50.00 $275.00 $150.00

What might your Shark ask?

If the Principal is your “Shark” invite them in to discuss all of their considerations: safety, chaos, mess, teacher time, hall congestion, health, permission forms, etc…The principal will likely ask where the money will be going to (field trips, technology, etc..)

Create a Business Plan  

Now that you know what you’re selling, a successful business needs a plan to follow. Develop a business plan that outlines what your business will do, your staffing needs (labor), your sales and marketing approach and how much start-up financing you will need (how much $$ to start everything). Once you have your business plan, you can follow it to create your successful business and use your business plan to interest investors in your company (aka the “Sharks”)

Questions to answer

  1. Company name
  2. Product name
  3. Product: Are you providing goods or a services?
  4. Who is your target audience (don’t say everybody!)
  5. What words would you associate with your brand?
  6. What would you pay for your product?
  7. Where/how would you sell your product?
  8. What would be some of your expenses as a business (what do you need to buy before you can sell? Would you need a loan from the school?)
  9. Who would you need to hire (set up crew, clean up crew, money counters, teacher supervisor)?

In this case, keep your Principal in mind. What would they consider most important (safety, staffing, lack of chaos, not disrupting the school day, etc…)

Marketing  

How will you get people to know your business exists, how will you market your product/service and advertise it to your target audience?  This could be part of your business plan but if it is not you MUST include some marketing and advertising strategies in this project.

  • Signs, announcements, school Twitter or Facebook pages, school website?
  • How will you make your product seem attractive? Research successful sales techniques and try to implement them (catch phrase, make it seem cool, limited time only, bargain, etc…)

Your Business Proposal  (the Pitch)  

You and your group will be creating a business proposal.  In this proposal you will include all of the elements listed above.  Research what makes a successful business proposal (body language, key phrases, being prepared, enthusiasm, etc…). Be creative & good luck!

Co-create rubric with class

Here’s one idea to get you started: https://www.gallup.unm.edu/pdf/shark-tank.pdf

shark tank rubric

Resources (some places to start)

Math – The Fractal Foundation Educator’s Guide

The Fractal Foundation

A fractal is a never-ending pattern. Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop. Driven by recursion, fractals are images of dynamic systems – the pictures of Chaos. Geometrically, they exist in between our familiar dimensions. Fractal patterns are extremely familiar, since nature is full of fractals. For instance: trees, rivers, coastlines, mountains, clouds, seashells, hurricanes, etc. Abstract fractals – such as the Mandelbrot Set – can be generated by a computer calculating a simple equation over and over.

For a simple description of fractals, please download our “One Pager” (380Kb).

For more detailed info, please download our 20 page “Educators’ Guide” (7.5Mb).

Fractal Triangle – Fractal Foundation

Mathigon – Textbooks come to life

mathigon-for-teachersMathigon contains amazing content for the entire mathematics curriculum. Simply pick a chapter and tell your students to work through it – as a homework assignment, or on within a flipped classroom setting. A teacher dashboard shows detailed analytics on their progress and mastery.

Mathigon works on tablets and laptops, and every student will automatically get a highly interactive and personalised experience. Our content is aligned to mathigonthe Common Core (US) and other national curricula.

Every chapter comes with a corresponding lesson plan for teachers, and we have a library of all the interactive games and components to use.

 

About Mathigon

Everything in our world follows mathematical laws: from the motion of stars and galaxies to the transmission of phone signals, bus timetables,

 

weather prediction and online banking. Mathematics lets us describe and explain all of these examples, and can reveal profound truths about their underlying patterns.

Unfortunately the school curriculum often fails to convey the incredible power and great beauty of mathematics. In most cases, school mathematics is simply about memorising abstract concepts: a teacher (or a video, or a mobile app) explains how to solve a specific kind of problem, students have to remember it, and then us

e it to solve homework or exam questions. This has changed very little during the last century, and is one of the reasons why so many students dislike mathematics.

“It is a miracle that curiosity survives formal education.”

– Albert Einstein

In fact, the process of studying mathematics is often much more important than the actual content: it teaches problem solving, logical reasoning, generalising and abstra

ction. Mathematics should be about creativity, curiosity, surprise and imagination – not memorising and rote learning.

Mathigon is part interactive textbook and part virtual personal tutor. Using cutting-edge technology and an innovative new curriculum, we want to make learning mathematics more active, personalised and fun.

Active Learning

Rather than telling students how to solve new kinds of problems, we want them to be able to explore and “discover” solutions on their own. Our content is split into many small sections, and students have to actively participate at every step before the next one is revealed: by solving problems, exploring simulations, finding patterns and drawing conclusions.

We built many new types of interactive components, which go far beyond simple multiple choice questions or textboxes. Students can draw paths across bridges in Königsberg, run large probability simulations, investigate which shapes can be used to create tessellations, and much more.

Personalisation

As users interact with Mathigon, we can slowly build up an internal model of how well they know different related concepts in mathematics: the knowledge graph. This data can then be used to adapt and personalisethe content – we can predict where students might struggle because they haven’t mastered all the prerequisites, or switch between different explanations based on students’ preferred learning style.

virtual personal tutor guides you step-by-step through explanations and gives tailored hints or encouragement in a conversational interface. Students can even ask their own questions.

Storytelling

Using Mathigon requires much more effort and concentration from students, compared to simply watching a video or listening to a teacher. That’s why it is important make the content has fun and engaging as possible.

Mathigon is filled with colourful illustrations, and every chapter has a captivating narrative. Rather than teaching mathematics as a collection of abstract facts and exercises, we use real life applications, puzzles, historic context, inter-disciplinary connections, or even fictional stories to make the content come alive. This gives students a clear reason why what they learn is useful, and makes the content itself much more memorable.

All these goals are difficult to achieve in a classroom, because a single teacher simply can’t offer the individual support required by every student. Of course, we don’t want to replace schools or teachers. Mathigon should be used as a supplement: by students who are struggling and need additional help, students who want to go beyond what they learn at school, or even by teachers in a blended learning environment.