Red+Thinking

Summaries
We had found that when we worked on the direct investigation we collaborated a lot and we communicated more as a group. When we where doing the cubic regression and the quartic regression, we only got an approximate answer because we just rounded up to 3 decimals points. We used a few different resources. "__The Book of Numbers__" had confused us there was the information we needed but it didn't show us the formula. The teachers where one of the best resources we had because they explained all the mathematics such as the matrices, while Sonya helped us with the quartic equation and the combination notation
 * Izabella**

We found the teachers were a great help in explaining unfamiliar mathematical processes. Sonya showed us the method of finite differences and helped us work out it was a quartic equation. Jo taught us about the matrices to find the unknown constants. And Sonya also helped Jonathan to teach us combination notation.
 * Audrey**

We used our available resources well. These included the internet, The Book of Numbers and our general brainstorming. Although I did find that the book was harder to understand than the others. I feel that brainstorming worked best because it helped us to all learn at a similar pace and help those who needed help. Also getting Daniel to do a workshop helped me a lot in understanding the quartic regression and method of finite differences because I had been absent on the day we had the lesson. When I gave the workshop, I think I did pretty well in explaining combination notation and factorial notation. Sonya helped with the explanation of combination too.
 * Jonathan**

**Daniel** For this project, our fist type of resource was the directed investigation. Our investigation made us think about how to tackle the given problem. We needed to experience the problem so that we know how we can explain it in our wikispace. All of the resources that we used were very effective and it helped us gain more knowledge on how to tackle these type of problems. The most helpful resource that we found was the direct conversation with Sonya and Joe, both teachers explained the different strategies involved with solving this formula.

What __WE__ Did!
**Week 1** Because our team consists of 2 returnees and 2 newbies, Jonathan and I had to explain to our group what Surprise is all about because we had done it last year. For a start, we asked Audrey and Izabella to try do the activity sheet “Surprise” and once they are stuck, we will collaborate. It didn’t take long before they did. So we are up to the next step which was finding the formula. We figured that we could use regressions to find the formula; since the first term was 1, the second term 2, the third term 4, and the fourth term 8 and so on and so forth; we assumed that the formula was exponential so we tried an exponential regression and surprise! We finally have a formula! BUT! Surprise again because it was the wrong one. We compared the results of the exponential regression against the written example (i.e. our own investigation with the Activity Sheet “Surprise”). And it shows that the sequence does not follow an exponential pattern. Time for our secondary type of resource: Our secondary type of resource was a book called __The Book of Numbers__ by Conway/Guy. This book shows how the formula works but it doesn’t show what the actual formula is. As part of Audrey’s delegated tasks, she had to read a chapter of this book which concerns our given investigation. Funny thing is that the teachers actually got the research task from that book! Unfortunately, the book didn’t have the actual answers that we are looking for so we didn’t get much done.

It's been a week of investigating, now it's math time again. Our group decided to collaborate in finding out what the equation is. We kept on trying to use different strategies on finding the formula but nothing seemed to be working. Out of desperation we asked Sonya why neither exponential regression nor quadratic regression was working. Because we asked, she showed us a whole new strategy, the "Method of Finite Difference". Using this strategy we came to the conclusion that the formula may be a cubic formula or a quartic formula. We checked both formulas using different points to see which one fits best (this is due to the fact that regressions are an approximate of the given coordinates). Using this technique we found that the quartic regression works best. We had an approximate formula! Since we know the approximate formula, Jonathan used Microsoft Excel and found that the formula is pretty much the right one the only problem it has way too many decimal places.
 * Week 2**

Because Izabella has a different calculator that none of the students knew how to use, we resorted to asking Joe for some help. (Izabella was really keen on finding out how to make the calculator find the quartic regression.) We told her that we had the formula but it's a really long one and it's still an approximation. Because we had done a lot of mathematical learning, she then showed us a different way of finding formulas. Matrices. She showed us how to do it in our calculator and then we found it, the same formula but lesser decimal places because it was in fraction form! Since we had the formula, the group now had to do the delegated tasks which are the different quadrants.
 * Week 3**

We were happy that we had the formula, we worked on our delegated tasks and all we need to do now is polish it. But maths learning never stops, Jon found a new formula that works effectively. Combination formula. This type of formula was much easier to understand because all you need to do is substitute the numbers then your done! The only problem is that we had to do factorials which involved a lot of numbers to multiply. Sonya, who must have seen how desperate we were for help. Showed us an easier way of multiplying and dividing the factorials. Cancellation. Who knew, obviously, she did!
 * Week 4**

In the end we found four different solutions for our direct investigation which gave our team a lot of new mathematical knowledge and a better understanding in finding formulas. All we have to do is polish everything and write on our learning journal.