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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A Poem: Nature Eternal

I found a poem I wrote, presumably in high school.

I bask beneath the twilight night.
Everything so serenely right.
And all is loose that once was tight,
As stardust steals away the fright.

I feel their magic fall on me,
And coat me with their mystery;
Telling tales of chivalry,
Of times behind and times to be.

The shameless trees stand tall and proud,
Taking comfort in the crowd,
Protecting life as they’ve always vowed,
While cries of the fallen still echo loud.

A velvet mist hovers o’er the lawn,
Muffling grass’ joyous sound.
To keep the night quiet until the dawn,
When father time completes his yawn.

A soothing breeze comes floating in,
And playfully dances with the wind,
Changing direction at any whim,
It embraces me and holds me within.

I smile up at the man on the moon,
And listen for his distant tune.
But it’s locked within a chainless tomb,
And left lamenting is the call of the loon.

The waves of the lake search for the land,
And try to crawl up shell-less sand.
But they slip from the earth from which the’re banned,
Because no one will grab their outstretched hand.

The air drowns out their cries for help.
Their mouths are gagged by chains of kelp;
Unable to breathe one final yelp,
And never discover how dry land felt.

And as I stare into the soul of the dark,
An ugly stain has left its mark.
But I can’t trace it to its start.
It once was subtle, but now grows stark.

Now the temperature starts to fall.
The earth is cloaked in a ratty shawl.
And traces of nature’s eternal brawl,
Taints the beauty of it all.

By Tammy Gaudun

 

 

Lesson Plan: Bullying & Entropy – Scientific Analogies

Question 1: Think of a time when you’ve been angry or mean or both!  What did you do?

You answer is likely that you were filled with powerful, negative emotions and you either took them out on yourself or others.

In simplified terms, entropy is nature’s need to dissipate energy.  For example, heat transfer is dissipative because it is a transfer of internal energy from a hotter body to a colder one.

The second law of thermodynamics states that the total entropy can never decrease over time for an isolated system, that is, a system in which neither energy nor matter can enter nor leave.

Entropy Comic:

056c026d-1c66-4d42-9fae-a8e96df290c5-1020x817

Hopefully, you can already recognize where this is going and how it relates to bullying. Basically, when negative (or positive) emotions build up, they do not disappear, they dissipate or spread from areas of high concentration to low concentration.

Bullying and anger can result from a build up of negative energy, which the person spreads or dissipates to those around them. Since the person wants to get rid of that energy, and does not want those feelings returned, they often choose targets who are unlikely to return fire, which may be why some people, “kick the dog”.

Another conclusion from this is that NO HAPPY PERSON BULLIES. If someone is not filled with negative energy, they are not going to spread it around. No happy, accepted, loved person bullies.

 

Question 2A: Think of instances in nature where energy accumulates? How does it dissipate?

E.g., electrical energy in the sky becomes lightening.

lightningstrike

Other examples include all natural disasters: earthquakes, volcanoes, hurricanes.

Question 2B: Can you think of any positive examples?

E.g., a seed becomes a tree, an egg becomes an animal, food becomes energy dissipated by the eater, etc….

Task 1: WATCH: “To This Day” by Shane Korzan

Task 2: Mind map a Negative scenario

  • In a group: think of an example when someone was mean to you. Often, we think of how other’s actions affect us. What we don’t realize is, is that we also spread those around. How might those around us react to our negative energy? Create a mind map of how far those emotions will spread or draw a picture.

bully

 

Task 3: Think of ways to dissipate the negative feelings without passing them around

For example:

  • Listen to music
    • Play “Thunder” by Imagine Dragons
      • The metaphor works. Thunder is the effect after bullying.
  • Journal
  • Talk to someone
  • Exercise
  • Play a video game
  • Here’s a list

Task 4: Mind Map a positive scenario

  • In a group: Think of a time where someone was kind to you? How did you feel? How might you have spread the joy to others? Mind map or draw a picture.

Task 5: Share a smile

  • The simplest way to spread joy is to smile at someone. Give your best or silliest smile to a classmate and pass it around until the whole class has been smiled at.

b9aa5a9057fa0d1fac6b64b93d8d4551-monkey-business-happy-weekend

YOU ARE CONTAGIOUS!  YOU CAN CHOOSE TO SPREAD LOVE OR HATE. CHOOSE WISELY!

 

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.

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 $2750,00 $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!

Resources (some places to start)