Examples of Linear Relationships: Proportion, Modelling, Correlation, Regression

Started a list for linear relationships. Not all linear across an infinite domain but that adds in discussion on piece-wise functions. Can anyone add more to the list?

Purely mathematical:

1) Diameter vs Circumference of a Circle

2) Length of Side vs Perimeter of a Square

3) Number of Sides of Polygon vs Sum of Interior Angles

Connected to Real Life:

1) Temperature of Coffee vs Time

2) Amount of Fuel in Tank vs Miles driven

3) Amount of Ink in Pen vs Pages Written

4) Profit made in first weekend of Film in Cinema vs Total Profit (yummymath.com)

5) Altitude vs Temperature (data can be recorded on some airlines during descent)

6) Taxi Cost vs Time

7) Gym Membership Cost vs Time (Initial Payment to begin member)

8) Internet Cost vs Time (Initial Payment for Installation)

9) Distance-Time Graphs and Velocity-Time Graphs

10) Hooke’s Law and Newton’s Second Law

 – Physically Measure:

1) Length of Foot vs Height of Person

2) Pull back distance of Model Car vs Total Distance travelled of Model Car

3) Number of Bites of Liquorice vs Length of Liquorice (Discussion on the type of bites you would need to keep linear)

4) Queueing time vs Number of Items per Customer (Dan Meyer)

5) Number of Sips vs Amount of Water in Glass (Discussion of the types of sips you would need to keep linear)

6) Time Jogging on Treadmill vs Calories Burned  

- Service Related:

1) Number of Cigarettes Smoked per Day vs Life Expectancy

2) Number of Questions Practiced vs Grade in Test (in class data taken)

3) Percentage of Population in Extreme Poverty vs Time (ATM: MT245)

4) Income per Person vs CO2 Emissions (www.gapminder.org)

5) Math Achievement 8th Grade vs Income per Person (www.gapminder.org)

Things I’d like to know how strong the correlation would be:

1) Weight of Person vs 1 Rep Max on Benchpress in First Gym Session

2) Salary of Striker vs Number of Goals Scored

I did recently ask Grade 10 to look into Number of Tweets vs Number of Followers on Twitter (restricting the domain so as not to include anyone famous). Whilst I knew it wouldn’t be linear, I think it’s important to discuss examples of why relationships are not linear. Often there is a correlation between how many variables are involved and how linear the relationship is. For example, the start up capital of a business vs profit in first five years. There are simply too many variables at play in this, just as there are for number of followers on twitter. I’d love to add more to this list…

1. Charging an iPhone (Michael Fenton): Linear at first and then tails off.

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Starting with what Students know: Solving Quadratic Equations

I thought I’d share a first lesson on solving quadratic equations that I developed for year 10 recently. I think many would agree that starting with what the students already know about solving linear equations is an aspect of this learning journey which often gets side-stepped. I therefore tried to make this integral to the learning experience. Looking at the component parts of this lesson, each one is completely obvious to any experienced maths teacher. The reason therefore that I chose to blog about it is because I felt that the learning sequence and journey in this particular case was a quality way to promote real understanding of solving equations. More technical aspects, solving in different ways and real life application will all come later in the unit.

Starter:

quad 1

Independent, quick and doable. All three aspects in this case were connected to the key parts of the lesson. Even though it pained me to do it, I taught double bracket factorisation before introducing solving equations this year. Usually I start by asking students to try to solve a quadratic and then introduce the factorisation as a means of necessity. This time however, I felt that to really hit home the concept behind why factorisation is used when solving quadratics, the students needed to be skilled at factorising before understanding why it is important (would be interested to know what other people do in this case).

Starter Q1 was important to remind them of balancing an equation and also to remind them of what the word solve actually means. Normally I would say that solving an equation gives you a value/s which, when substituted into the equation, make it work (e.g. satisfy it). In this lesson, I made the students discuss in more detailed terms what this means and we rested on, “The value/s that, when substituted into the equation, make the left hand side of the equation equal to the right hand side.” We plugged in a few numbers to make this clear, one of them being a decimal and the other being a negative whole number.

It’s obvious that starter Q2 helped to remind them of double bracket factorisation. Starter Q3 however is interesting in that I wanted this concept to be completely obvious and simple to the students before I introduced it as integral to the strategy of solving quadratics. Students said it was easy and I was glad that it was explicitly mentioned at the start.

Main (part 1):

quad 2

[The second question animated to appear after discussion of the first question]

The first question was easily solved by balancing although students forgot that there would also be a negative solution. It was very important to the lesson to show how balancing equations to solve them is a very general and wide-reaching tool for a mathematician.

The second question is then designed to show that balancing would not work; something the students were quickly able to identify. At that point I took it back to our definition of solving to ask how we might solve this particular quadratic. That is the reason behind the form of the quadratic; I wanted the student to be able to substitute a number into both sides. Students recommended trial and error to which I agreed but only if they use technology to maximise efficiency of the process. One student mentioned at this point that he’d already found the solution (x=2) to which I said that he could move straight onto the next task (I’ll come back to this). Luckily, Year 10 have TI-Nspire Graphic Display Calculators and so we utilised the spread sheet view. Excel would be just as good if you have quick and easy access to it.

I took the students through the use of the calculator with this example. Essentially we wanted to try lots of different input values for x and check whether the LHS of the equation would equal the RHS. (i.e. when is x^2+10 equal to 7x)

quad 4

Of course the interesting moment was when the student who had found the first solution (x=2) was amazed to see that this equation has another solution (x=5). I responded to his amazement with: “Just because you know a lot about linear equations it does not mean that you know a lot about quadratics. Alarms bells should have been raised when you saw that x^2+8=24 has two solutions. Mathematics requires both intuition and rigour.” His response was both perfectly timed and very ironic: “So all quadratic equations have two solutions?” I told him that he’d be able to answer that questions himself as time went on when he could apply more rigour to his conjecture. No one at this point picked up on the fact that we had only used positive integers as inputs but I was happy to leave unsaid for now.

Students had a go themselves with the following questions and then I wrote their solutions on the board:

quad 6

Of course some students were confused as to why Q2 didn’t fit the mould and only appeared to have one solution. Fortunately I got a “Ahhhh” moment when one student realised that the second solution may not be one of the input values we’d used in the calculator. That’s when I told them that there was a second solution, and in the interest of the pace of the lesson, I told them it (x=-1).

No one managed to connect the starter questions to the bottom question on this slide and so when we realised that the spread sheet method was unreliable, I went back to the starter questions to highlight Q2 and Q3 before stating that these integral techniques would provide us with a completely new way to look at solving equations. So I put up the next slide and facilitated the class through how factorising the left hand side allowed us to know with certainty that one of the brackets, or both of the brackets must be equal to zero.

[The second question on the slide was animated to show up after we’d discussed the first].

quad 7

It was completely obvious to them that the only way this method would work is if one side of the equation was equal to zero. I put this completely down to the fact that they had internalised both the skill of factorisation and the concept of multiplying two numbers to make zero, before I introduced the strategy. Therefore it was much easier for them to connect the dots and fully understand why factorisation works.

Here’s the objectives that I showed at the end of the lesson:

quad 8

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Help Needed: Collaboration on a Rubric for Presentations about Pi

Quick one: Can anyone provide some advice on developing a rubric for student presentations on: “Understanding what pi actually is and why it is significant.”

It makes me cringe when people walk around reciting the number with no understanding of what pi is and so I’m determined this year to have my students educate the school community. I’ve got no problem at all with people memorising pi for fun and this can provide a motivation for certain students but I simply feel that we should aim to help students understand what pi is and why it is historically significant.

This is what I have so far but I feel like it could definitely be improved. Would hugely appreciate some help with this.

pi rubric

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Exploring and Creating Mathematics Part 3: Investigating Powers using Pascal’s Triangle

Before reading this post, you might want to see the other two posts in the series:

Post 1 – Creating and Exploring Mathematics: How many different directions can we take a topic in 15 minutes?

Post 2 – Discovering and Creating Mathematics Part 2: Does investigating problems make it easier to pose interesting problems?

Whenever we get a chance as mathematics educators, I think it’s great if we can explore and discover our own mathematics. Of course it’s almost always been created before but I don’t see how can we ask students to investigate and explore if we’re not prepared to do so?

So today I spent a few hours investigating powers using a binomial theorem/Pascal’s triangle approach. Would love to see how other maths teachers would investigate this given the chance. Here’s some interesting stuff I discovered today:

1. The sum of row n of Pascal’s triangle is equal to 2^n.

2. Odd powers of 4 always have final digit ending in 4; Even powers of 4 always have final digit ending in 6.

3. The first four powers of 11 follow exactly the same digits in Pascal’s triangle. E.g.

11, 121, 1331, 14641, …

After that, the pattern doesn’t work. This is a great example to show why proof and reasoning are important in mathematics.

4. The sum of the digits of powers of 11 are always even.

Continue reading

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Exponential Functions: What’s been missing from my teaching?

To get straight into the meat of the post, I believe that one fundamental aspect has been missing from my teaching of exponential functions in the past.

To help students understand exponential functions, constant comparisons must be made to linear functions.

In the past, I’ve started discussions on exponential functions with a shocking statement (taken from Albert Barlett’s youtube lecture) – “The greatest shortcoming of the human race is our inability to understand the exponential function.” I’ve then tried to show, by getting students to do problems in which they struggle to predict the rapid, exponential growth, that this statement is on-the-whole correct. I still believe that this approach is valuable. However, when I reflect on how well students’ really understand the exponential function, there’s been something important missing from the learning equation. i.e. the connections and differences between exponential and linear growth.

The key difference between linear growth and exponential growth is that the rate of increase of linear growth is always the same, but the rate of increase of exponential growth always increases.  It sounds obvious, but I believe that in making this direct comparison, students’ are more likely to understand why exponential growth is so rapid.

I saw the WTF problem below  on Dan Meyer’s blog  (The basic principle behind “What the F” problems is that they display a concept in a surprising way), and so started the lesson with what the students’ already know about growth.

Continue reading

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A Geometrical Approach to Factors and Primes: Understanding an Algorithm

Reasons for Post:

  1. To introduce factors and primes through a geometric approach.
  2. To introduce an algorithm to quickly find how many factors a number has.
  3. To discuss why the algorithm works based on combinations of factors.

Recently I gave my Year 8 (Grade 7) students a mini-investigation based on factors and primes – without saying it was about factors and primes.

“For this investigation, consider a square to be a special type of rectangle. If the number 10 represents the area of a rectangle, how many different rectangles can you make with a whole number length and width? Investigate this for different numbers.”

Before discussing two examples as a class, I told them that if anyone figured out what this investigation was actually about, then they should keep it to themselves. Of course some realised quite quickly that the width and length of a rectangle or square represented a factor pair of the area. Those students generally went on to look at investigating how many factors each number has and whether there is a way to figure this out. I didn’t expect anyone to get far with this.

I was genuinely surprised when two of the students working together came up with the standard algorithm for finding the number of factors using the indices within the product of prime factors.

For example, to find how many factors the number 30 has.

Step 1: Break 30 into a product of primes:  30 = 2¹ x 3¹ x 5¹

Step 2: Add one to each of the indices and then multiply these numbers together.

(1 + 1) × (1 + 1) × (1 + 1) = 2 × 2 × 2 = 8 factors

Of course, by finding the product of prime factors, you’re essentially finding the fundamental building block of each factor. Still, quite impressed that they’d found an algorithm. I personally don’t know any other way of doing it. Anyone else know a different way?

I told them that as a mathematician – our aim is to understand why an algorithm works but that might be something they’d be able to investigate in a few years once they’d learnt more mathematics. In this case that boils down to why we add one to each of the indicies and then multiply? I spent part of break with a coffee making sure I understood why the algorithm works (on a similar, but less rigorous line to Erdos; “A mathematician is a machine for changing coffee into theorems”). Not hugely difficult to make sense of/justify for numbers with only one base prime factor (e.g 8 = 2³; 2 is the base prime factor) or two base prime factors (e.g 15 = 5 x 3; 5 and 3 are the base prime factors) and then intuitively extend from there.

With one base prime factor, e.g. 8 = 2³. The factors are 2, 2², 2³ and then 1 (you could think of this as 2° if you fancied it). Hence why there are 3 + 1  = 4 factors. So for N=a^x, there will be x + 1 factors in total.

With two base prime factors, take the number 100 = 2² × 5²

By the algorithm, we know that this number has (2 + 1)(2 + 1) = 9 factors.

If you expand out the brackets (2 + 1)(2 + 1) = 2×2+1×2+2×1+1×1 = 4 + 2 + 2 + 1 (*) then we can see that these 9 factors are split up into groups. i.e. 4 factors + 2 factors + 2 factors + 1 factor.

Expanding out any bracket of the form (a + 1)(b + 1)(c + 1)… will always give you a last term which is just a load of one’s multiplied together. So the last 1 in the calculation above represents the fact that 1 is a factor of every single number.

So where do the 4 factors + 2 factors + 2 factors come from in equation (*) above. I think the geometrical representation below sheds the most light on this.

factors

It’s clear to see that the 4 factors are made up of all of the combinations of the two base prime factors. The (2 factors + 2 factors) can be seen in the picture as factors which are only made up from one of the base prime factors.

So for any number with 2 base prime factors, we have:

fac3

Of course if each exponent is equal (i.e. x = y), then we have the second binomial expansion (x + 1)². With three base prime factors all being equal (x = y = z), then you have the third binomial expansion (x + 1)³, and so on. There’s the connection to Pascal’s triangle and hence the lead into the different combinations of all of the factors.

So an interesting change in my approach to factors and primes. I’m still not sure whether I’d teach the algorithm above – wouldn’t be too hard to add in some understanding of this but then usually I’d teach factors and primes in Year 7 (Grade 6) which would be before looking at breaking numbers into a product of prime factors. It’s therefore difficult to know where this would all be placed in the curriculum. Possibly worthwhile as a closed investigation in Year 9 as a follow on from expanding brackets.

Note: Bear in mind that for the above example, we couldn’t use the calculation 4C2 (4 things choose 2 of them) for the 4 factors part because you can’t put 2 and 2² together or 5 and 5² together. You have to see the base prime factors as separate groups (2, 2²) and (5, 5²) and pick an element from one group to pair with an element from another group.

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Shape Review: Pint glasses, Supermarkets, Coding, Population Density, Islamic Art and Bolivia

It’s not easy finding the time nowadays with so much going on (not least with the new puppy – Sir Isaac Newton – to play with) but it’s great to sit down with a cup of tea and write a blog post.  I’m hoping to write another post next week but this post is mainly about some of the shape stuff I’ve been doing this term. Hopefully you’ll find at least one of the ideas interesting and I hope that a few people will be able to add to these ideas or provide new ideas that I can try out next year.

1) A great question when you’re at the pub with friends…

Which is bigger, the length of a pint glass or the circumference of the top of the glass? I bet most people will say the length. Of course, in a maths lesson, I don’t need to use the word “pint” to get my students hooked on a perplexing question. I used my water flask instead…

c4

It turns out in this case that the circumference of the top is slightly longer than the length. Once students have a full understanding of where the circumference formula comes from, they’re amazed to check the results using mathematics (it’s quicker to measure the diameter and calculate over cutting out a piece of string and directly measuring which is where the maths comes in). In future, I might open this up to an investigation into different prisms. E.g.  If the top cross section wasn’t a circle, would this still happen? If so when and why?

2) Surface Area and Structure: Why this shape at the supermarket?

sa

This is the “separating thingy” that separates people’s shopping at the supermarket. We had a great discussion on why they would use this particular shape over any other object. Why this over a cuboid? Nice intro to minimising surface area whilst maintaining the structural importance of the shape.

3) Ipads, Exterior Angles and Hopscotch – Discovery

After doing some work with interior angles, I thought I’d get the students discovering the integral fact about exterior angles of polygons through the use of the iPad app, Hopscotch. Hopscotch is essentially the new Logo. It is a beginner’s coding app in which you can use “ready-made” commands to make a code.

hop2

I made this worksheet to go along with the activity.

4) Understanding Area Conversions and Population Density

In a study conducted by the European Commission in 2011, Europeans were asked what they thought the World’s 10 biggest problems are. Obvious stuff like climate change, the economic crisis and terrorism were up there in the top 5. One of the top 10 was the exponentially increasing population of the human species. Hence, I really like doing stuff with population density because it’s relevant and connects to so many areas of mathematics (namely ratio, proportion, standardised measures, area, converting between metric units of area). To introduce this, I like to start with getting the students to internalise how big 1 m² is. How many students can fit inside 1 m²? With the square against a wall, I was quite amazed that 11 students managed to get in there!

arae

5) Understanding 3-D Shapes and Prisms before Volume

If students don’t know the difference between prisms and other shapes then they’ll simply start memorizing formulae for volumes of different shapes with little understanding. Hence, to start this topic I simply got the 3-D shape sets out and asked them in groups of three to organise the shapes in any way they thought was most logical. After 5 minutes, we had a little class discussion about what they’d done and then I encouraged them to try to categorise the shapes into 3 groups. Low and behold, all but 1 group sorted into Prisms, Pyramids and Spheres. This allowed us to move on to properly define prisms using the applet I made a while ago.

3D

5) Islamic Artwork and Transformations Project – Discovery

I decided to get Year 7/Grade 6 discovering transformations on the co-ordinate grid using the hook of Islamic Artwork this year. Since one of my “secondary objectives” is to get students familiar with Geogebra, it was a great way to get them discovering, using and effectively communicating transformations.

I started with an introductory lesson (type “Islamic Artwork” into google to find some interesting pics) which got them hooked into thinking about the artwork from a mathematical perspective. We talked about rotational symmetry and then I provided a summarised historical perspective using information from Wikipedia:

Islamic Art began around the 7th century. It was designed and made by people who lived within the Islamic territory that was inhabited by or ruled by Islamic populations. It is thus a very difficult art to define because it covers many lands and various peoples over some 1400 years.

There are often repeating elements in Islamic art which is used to symbolize the infinite nature of God. Mistakes in repetitions are sometimes on purpose to show humility by artists who believe only God can produce perfection.

With it being an introductory lesson, I then allowed for the all important “tinker phase” of introducing new tech/software – you certainly cannot expect much structured learning of mathematics during this important phase(although of course it’s massive for discovery)! They had to perform a rotation, enlargement and reflection without any initial class discussion. Obviously, some of them have almost no idea what’s going on at first but they get there eventually. If anyone found out a good way of using the software, they were encouraged to write it down on sugar paper for the whole class to share solutions. The all important part of this project is how you bring in meaningful mathematical learning without students becoming so memorized by the technology and artwork. That’s where effective classroom pedagogy takes over in planning, class discussions and quality questioning. Furthermore, making the students write ever step down as instructions (using correct mathematical terminology) focuses them more on the mathematics rather than just the artwork (60% grade for instructions, 40% artwork marked with Art teacher). The great thing about the discovery element is that students who understand quickly and finish their work can investigate translations, negative enlargements and dilations. Since I don’t have any of the students’ work available electronically, take a look at a previous post on this.

6) What do tessellations have to do with the Salt Flats in Bolivia? Previous post.

So there’s a load of shape stuff. If anyone has any ideas to improve on these or any extra ideas on anything to do with shape, please share.

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