Tag: algebra

Maths in Music

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This task aimed to apply some of the things we’ve learned about Percentages, Data and Algebra within the context of music.  Here’s as it was given to the class.

I only wanted to do this for 2 lessons hence I chose the song and gathered the resources, even though I knew the girls could do that. The girls all love Taylor Swift and are learning to play the guitar in Music. I also knew that they’ve used all the ICT tools mentioned except for Wordle which some had Java problems with, WordItOut was a great alternative.  In fact, the girls found the most challenging part of this task is the mathematical component, i.e. finding patterns and expressing patterns using Algebra.

Note: This was not an assessment task, a mere immersion and contextualisation of maths, with some ICT integration. Not everyone submitted their work but pictures from those who did can be viewed on the class website.

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Show me the maths in music!
Maths is everywhere even in music. There are patterns, rules, percentages and fractions. In this activity, we will attempt to capture some maths in music.
Work with your buddy to create a poster. We will vote on the best ones to be displayed for Open Day and uploaded to the class website. Winners will also get merits.
Posters will be judged on (1) visual appeal (2) relevant maths used and (3) completeness.

Here’s my work in progress using a different song, Love Story just to give you an idea.

Your song is also by Taylor Swift, The Best Day
Task Checklist (Lesson 1):

1. Get the lyrics of the song

2. Create a word cloud via Wordle; save as a jpg in Paint or PhotoShop. What are the most common words?

3. Have a copy of the lyrics in Word to count the number of words. Did Wordle get the percentages right?

Option 1

4. Use algebra to express the patterns and rules you see

5. Create a snapshot of your algebra rules; save as a jpg in Paint so your maths symbols look right on your poster

Option 2

4. Write 2 algebraic rules, e.g. 3x + 1, and show as a pattern in pictures

5. Save your rules and patterns as a jpg so your maths symbols look right on your poster

6. Put it all together into a poster using Publisher or PowerPoint with: (1) Word Cloud, (2) Lyrics, (3) Algebra rules, (4) a poster title, (5) 1-5 photos/images

7. Save and Upload to the Posters document library; use your first names as a filename

Task Checklist (Lesson 2):
You need to look at the Music Sheets (PDF) for this task.
  1. Tally the guitar chords used in the song. How many chords are there? Are there chord patterns?
  2. Create a Frequency Distribution Table and a Pie Chart in Excel. What is the mode?
  3. Add your pie chart to your original poster

Option 1

Save a copy of the song (mp3) to your desktop.

4. Import the song into Audacity and play with the Tempo. Create 3 mp3 versions of a 15-second grab: (1) original (2) faster (3) slower; you can choose the percentages but keep track of the numbers. What happens when you change the tempo?

5. Create a snapshot of your changed music and add to your poster. In textboxes, describe the changed music.

Delete the song from your desktop.

Option 2 (especially Music students)

4. Looking at the music sheets, find 3 – 4 mathematical patterns, e.g. Rhythm, Tempo and note values

5. Crop images and describe using both your knowledge of maths and music. Add to your poster.

For example: This bar has many notes, each one is played at 16th of a beat. Play 4 sixteenth notes or semiquavers for the time it takes to play a quarter note or crotchet. All the notes add up to 4 beats because….

Tangential Learning

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I’ve always been fascinated how learning is facilitated through fun, play and games. We have all had personal experiences attesting to this.

Professionally, I first heard about this by reading Marc Prensky’s work. Since then, I’ve always sought ideas on injecting fun activities into my classroom.

More recently, I watched the following video and first heard about Tangential Learning.

This video defines Tangential learning as

what you learn when you’re exposed to things in the context you’re already engaged in…..it’s the idea that some portion of your audience will self-educate if you expose them to concepts in the context that they already find interesting….

I’m not really a video-gamer, much less an author of one, but I lived by these principles and have indeed found that students were most engaged in their learning in such cases. Here are some examples:

1. Decimals

Throughout this topic, the students had to work out how many jelly beans will fit across their desks.  I worked through how they would have solved this problem as they learned more maths from primary school. It started with counting (teacher lines up jelly beans, student lines up jelly beans) to estimating to measuring and calculating. I should add that I had to throw the jelly beans lined up as they ‘carried my germs’. The students were aghast at such waste but they soon figured out that by measuring just one – and learning operations with decimals – they can earn their jelly beans. And so they did!

2. Algebra

I introduced this with a personal experience of wanting to learn how to play Rihanna’s Unfaithful on the piano. Not having played for years, this was a real challenge particularly because the music sheets were several pages long! As if learning Rihanna wasn’t cool enough, the students were particularly impressed that I used YouTube, i.e. this videoshowing me how to do it. The main point was that the video highlighted that the song only really had 3 repeating sections. Learn the 3 and you learn the song. So, not only did I get to learn to play the song, I memorised it really quickly. Patterns are the crux of Algebra. This has set the tone for the students to see that Algebra can be used in something so seemingly far removed from mathematics. Some of them also felt affirmed that teachers use YouTube to learn.

3. Reading Tide Charts and Timetables

The class was told that they were to plan a group’s day out to Manly beach using a wiki.  This YouTube video on wikis was really helpful to enthuse the class. The activity included collecting shells on the beach, watching a movie, catching public transport and be home in time for a particular TV show. I gave them all the web links they needed and they really did not mind reading tide charts at all; they knew it had to be low tide when they go shell-collecting. Some even appreciated learning about Sydney’s Trip Planner. All that besides, they also learned about the challenges of planning and collaborating.

Tangential Learning works and I like it. The real challenge is finding that context that students will be engaged in.

Reflection as part of learning

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Our maths faculty head teacher set a ‘soft’ challenge a while ago to have some subjective assessments.

I took this on board and have devised an assessment task that incorporated self-reflection which was another skill I’d like my students to practice; 2 birds in one stone. Reflection is essential in developing sound mathematical thinking, apart from the fact that it is a good life skill itself. At its core, it necessitates looking at something from a different perspective such as looking back, looking deeper, looking from a different angle. From a Jesuit perspective, it also involves Action. Solving mathematical problems often involve all these.

Here’s what I’ve learned so far:

1. Reflections (or learning journals) are just as good for assessing understanding.

What a number is and what it stands for can be dramatically changed just by putting a tiny, little decimal point in front of it. For example, the numeral ‘1’ is a whole number and stands on its own. But as soon as you make it ‘0.1’ it now stands for 1 over 10.

This shows an understanding of the decimal point as the anchor that defines place values.

2. Aim for conceptual understanding; maths is more than just mechanics and shortcuts

I did not really understand the purpose of it and therefore was confused as to why we were doing it. It made me realize that just because you can do something and get the answers right, it does not mean that you don’t struggle with it.

Do I stop to check for conceptual understanding when the students have demonstrated they can do the mechanics?

3. Passion rubs off – case in point, Algebra

Many of my students expressed struggling with Algebra and yet have found it interesting, even fun. Algebra is my favourite maths strand and know that many are put off by it – often from what they’ve heard adults or older students say. With extra effort as well as creativity – and surely joy – many of my students have admitted to struggling with, but finding Algebra interesting; some even said useful and fun! No amount of objective testing will ever show this.

4. Students can be Teachers

I see myself in the reflections of my students. I can see where they’ve struggled and what they’ve enjoyed. I am learning new ways of perceiving topics in a way that is impossible to capture in ordinary class discussions – it is just impossible to hear everyone out every lesson. I can see that some students do appreciate the variety in delivery – modelling, direct instruction, investigation, problem-solving, questioning.

5. There can be some objectivity in the subjective

This sounds more like a philosophical discourse and, a lifetime ago, it was indeed (Philosophy of Man at uni). For now, all I mean is that there can be guidelines to help in marking subjective answers. I used Depth, Breadth and Details.

While I initially dreaded the prospect of marking these assessments, I was surprised to find myself enjoying reading my students’ comments especially the like the ones quoted above. These are just year 7 students. I look forward to Parts B and C of this assessment task.

Analogies and Algebra

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Analogies

In the course I attended recently, we spent a fair bit of time on discussing Analogies as a teaching-learning strategy.  There is probably not one person who has never used this strategy before, particularly if you ascribe to the thinking that we learn by connections.  Hence, the initial omission.

However, I’ve learned today how good this strategy really is. Hence, the special mention.

Analogies and Algebra

Undoubtedly, the abstract nature of Algebra – it’s very power – is the reason people are generally averse to it. Having overcome the obstacle of why we use x (variables or pronumerals) to begin with, I thought the next problem was balancing equations. My observations proved otherwise. Balancing equations was not an issue (drilled as they were that whatever you do to one side, you do to the other to keep them balanced).  Rather, the problem really was why we needed to isolate x in the first place, to find its value. Pretty obvious, I thought.

The analogy I used today involved the use of a bowl with chocolates and crackers. The question was how many chocolates were in the bowl.  I showed them the bowl. They could see hints of chocolate hidden by a mound of crackers. A quick discussion ensued. Soon enough they realised that the best way was to remove the crackers so the chocolates can be seen. Light-bulb moment. It was wonderful.

For the record, I’ve actually extended this activity to involve revision of the distributive property of multiplication (I had 2 bowls with equal number of chocolates and crackers), substitution of pronumerals (to work out total quantities) and simplification of equations. As soon as I rewarded one insightful comment with chocolate, I got a whole class motivated to contribute to the class discussion, rather Pavlovian. Not enough chocolates for everyone, but I obviously chose the right brand of crackers – everyone had some!

Polya, GGSC and Algebra

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Polya and Algebra

I introduced Polya to my year 7 maths class when we started Algebra. Well actually, all I said was Polya stated that “if the problem is too hard, try a simpler one“. A bit of googling and I learned that Polya said it more elegantly –

If you can ‘t solve a problem, then there is an easier problem you can solve: find it.

For some reason, this has really struck a chord and nary a lesson would pass that Polya is not mentioned, by me or a student. I reckon it’s because the class knows he was real. It’s rare we acknowledge the great minds behind maths as a body of knowledge. I guess we all need a hero and in Algebra, more so. I can’t wait to introduce Descartes!!!!

GGSC and Algebra

GGSC is my concoction, an adaptation of Polya’s steps to solving mathematical problems and the KWHL mentioned in a previous post on learning strategies. One day I will tell my class who the inspiration really was. Anyway, GGSC is supposed to help students deal with word problems, regardless of the strategy used (yes, we did a quick revision of the strategies they’ve learned: guess-check-refine, working backwards, drawing a diagram, solving a simpler problem, etc). The acronym stands for Given, Goal, Solve, Correct, i.e.:

1. What is Given?

2. What is the Goal?

3. How will you Solve it? This is where the chosen strategy is used.

4. How do you know you are Correct?

My process was to unpack the following word problems step-by-step, first using words, and then using Algebra.

1. Five years ago, John’s age was half of the age he will be in 8 years. How old is he now?

2. The sum of the least and greatest of 3 consecutive integers is 60. List the 3 integers.

First mistake, using a fraction and distributive property of multiplication in the first problem. Combined with the algebraic pronumerals/variables and the students forgot everything they already knew!

Second mistake, showing all the steps at once too soon (I should have kept my step-by-step presentation animation used in example 1 for example 2).  As soon as the students saw the algebraic solution, they became bamboozled – seriously…”what is an integer?” let alone differentiate it from a decimal. At least they all agreed  an integer is a number.

In the end, I had to quickly wrap up – abandon ship, in a way – and focus more on GGSC itself. I challenged them to think of a memorable mnemonic. They enjoyed this activity which gives me hope that I can mention GGSC again, even if not with Algebra. The last mentioned before the bell rang was ‘Good Girls Stay Cool’.

Speaking of mnemonics (a useful learning strategy), I once again draw inspiration from Polya

How I need a drink, alcoholic of course, after the heavy chapters involving quantum mechanics

This is his mnemonic for π (pi). The letter count represents the digits 3.14159….. Brilliant!