# Combinations and The Lottery

This Nrich Lottery Simulator was a great way to introduce Combinations and Permutations to my Stats class. We started with students choosing 1 number out of a possible 4 to see who could win (and several did). I then allowed them to choose from 5 numbers. Fewer students won. Then I asked them to pick 2 numbers between 1 and 5. Even fewer won.

Then, following the British National Lottery, they had to choose 5 numbers between 1 and 49. A couple got 2 numbers correct at most. This led nicely into a conversation about how many combinations are possible, whether order matters and how we can work these things out. They were also hooked for the whole time.

This also helped to convince them that their chances of winning anything on the lottery were very slim. I think this will also lead very nicely into talking about expected value.

# Desmos is Awesome

I love Desmos. I love how user-friendly and clean the whole thing is. I want my students to love it too, so I used the first lesson back after midterms to let them play.

### The Aim

For my students to get a feel for the shape of various functions and relations through using Desmos to create a piece of art. (CC Standard F.BF.3)

### The activity

• Students take a look at http://www.desmos.com/art to see what is possible just by typing in equations (Great hook)
• I explain to students that they have this lesson and a homework to come up with a piece of art of their own using desmos.com/calculator.
• I have a sheet ready with some example functions (linear, quadratic, circular relations, radical and rational) for them to use if they are struggling. I also introduce the idea of sliders for them to use.
• They have around 50 minutes plus a homework to come up with a piece of art of their own.

### Some of the resultant artwork

#### What went well

• Any lesson where students are crying out for the Math is a good thing. It was amazing to be asked how to draw a smiley face using a parabola and domain and range and how to draw circles and ellipses. I had one student ask how to do a ‘diagonal porabola!’ I had to look that one up.
• Students were constantly engaged. Sometimes frustrations got the better of them and they needed some encouragement to keep going but generally, the lesson went really fast.
• It was great for all abilities. Students that normally struggle got the chance to play around with linear and quadratic functions, helping them to understand what changing the numbers did to the graph.

#### Even better if

• This was too early in the year to do this lesson. I would like to do this next time at the end of the year when students had more functions and tools at their finger tips. I did like how it cemented the need for domain and range, though.
• This lesson is leading into our quadratics unit. Going back I would have really liked to focus far more on parabolas so that our next lesson on Vertex Form would make sense from the start. I still think it will help, I just think I may have missed an opportunity to go deep rather than broad.
• This lesson relies heavily on technology. Being a Bring-Your-Own-Device school, some students had tablets that were very tricky to use (this also happened with Khan Academy). I may have to rethink how I do this and use it possibly for homework.

### Student Reaction

Here are results from a mini survey I did at the end of each class

### Students also said:

This activity allowed me to visualize what adding variables does to the shape of an equation.

I loved the creativity involved with it, but also the brain work involved when trying to make different shapes and move them around.

Technical difficulties were frustrating, but I realize this is something that is hard to fix.

# What my students think of Khan Academy

Recently, I used this and this Khan Academy activity to test my students’ understanding of graphing inequalities as part of our Linear Functions Unit. Having not broadly used KA before, I wanted to gauge my students’ reaction and try and work out if I should use it more regularly to give me and the student, good feedback. After the activity, I gave a quick Google Form Survey to ask my students what they thought. The results are below:

I think this really helps envision the problem, and it really helped me. I would thoroughly enjoy doing this again, although it is kind of hard to maneuver around on it.

I liked the system, but I really hated the fact that we had to get three right in a row, because if you accidentally make a mistake, you can’t go back, you’ve just permanently failed.

It was good for practice but too easy to make technical mistakes. Graphing is harder to do on Khan academy.

This feedback, coupled with talking with students during the activity told me these things:

• We are a Bring Your Own Device (BYOD) school with many varieties of device. Laptops were fine but some tablets were pretty difficult to use with KA. It was hard to move things across the screen
• The instant feedback is awesome and students really enjoyed/were frustrated at (in a good way) knowing immediately whether they got things wrong or right.
• I think in the future, I will mainly use KA for homework where students can do things at their own pace and use their home computers rather than tablets.
• The idea is great but I’m going to have to think about the most useful way to implement it as part of the wider experience, for the student.

Have you used Khan Academy in the classroom. Any good tips on how it can be integrated without the frustrations above?

# Seven Squares – The Essence of Mathematics

This post is my attempt at being part of the ExploringtheMathTwitterBlogosphere community.

This week (or last weeks) challenge is to post your favorite rich/open ended math task.

Without doubt mine is nrich’s Seven Square’s problem. To my mind, this captures the essence of what mathematics is all about: Patterns. I love starting my Algebra 2 course with this to give them a sense of why we do the Math.

In class I will give each group a set of toothpicks. Some use them, others go straight to the drawing but all seem to get into it at their level/pace straight away. It’s amazing what comes up from students of all abilities.

A great extension to this is Dan Meyer’s toothpick activity. There is no end to shapes that students can investigate. It is one of those ideas that is great for all ability levels and really does help when you are dealing with the content skills throughout the course. I find that I refer back to this lesson, often.