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## Introduction:

For this CAT we had to answer questions and find patterns in questions using factorising, sketching, and expanding. We had to watch videos to explain the starting techniques and eventually learnt how to do it ourselves, starting with using shapes to map it out, finding patterns, and answering the questions mentally. Once we had mastered the techniques we used substituting to create a graph on desmos. Below is a picture of an example of how I used the shapes and a link to two videos explaining how I used it for the distributive law and factorising worksheets which will also answer a couple of the questions from the worksheets. There is also an image of all the different shapes and the multiplication chart:

This is the photo which shows all the different shapes we used to make a model of our learning. They are on top of or around the multiplication chart we used to create the model.

This is an example of how we used the shapes to answer questions on the worksheet. This question is 2 times 3x plus 4= 4x plus 8.

## The Questions Posed:

The questions we answered were to do with applying the distributive law, expanding, factorising and sketching. Click the links below to see the worksheets we had to fill in to widen our understanding. If you don't want to click on these, here are some example questions typed out and written:

Applying the Distributive Law:
x⃣=algebraic x
4x2=
3x3x⃣=
4x2x⃣=
3x⃣x2x⃣=
y(2x⃣+y)=
2a(3b+y)=
-yx3x⃣=
-x⃣(2x⃣-3)=
2m(1-p)=
(2x⃣-1)(x-2)
(x⃣+3)(x⃣+2)

Factorising:
6x⃣+3
8x⃣+10
14x⃣+2

## PREDICTIONS:

I predicted that this would involve algebra because I saw the pronumerals x and y and because our unit was on algebra.
I predicted that this task was going to be challenging, because I have never done this before.
I predicted that it would start easy and then get more challenging.
I predicted there would be pattens to make working the answers out easier and be done mentally.
I predicted that the graphs we would make would not be linear because of what I already know and because the activity was called quadratics which are the graphs we made.
I predicted that there would be a pattern for each of the algebraic pages, because for most maths things there is a pattern that you can find which will help you to work out the answers.
I predicted that one of the patterns would be to multiply the numbers together and then add the pronumerals, as this is one of the algebraic rules I have focused on before.

## Page 1Page one dealt with multiplying whole numbers and pronumerals together. The pattern I found was that you multiply the whole numbers and write the pronumerals on the end.

Page 2 dealt with multiplying whole numbers with pronumerals together. It also involved multiplying whole numbers to equations in the brackets. The pattern I found was if there are brackets you multiply the whole number to the components in the brackets to get your answer. If it is like the top half of the page you multiply the whole numbers together, then multiply the pronumerals together and put them together to answer the question.
Page 3 encountered the same sort of questions but brought in negative numbers. I found that the same pattern applied for these questions as well, but if you had a negative answer the - mark would go out the front of the answer rather then where it came from the multiplication.
Page 4 was a challenge page that we used our knowledge of the distributive law to answer.

## Written Questions answered

The Factorising Worksheet:
Page 1 dealt with working out the multiplication sum for addition with whole numbers and pronumerals. The pattern I noticed was that you find the highest common factor for the numbers, place that number at the start of the answer and put in the brackets what the numbers are divided by that factor and add an x to the first number in the brackets.
Page 2 brought in negatives and subtracting equations. The pattern I found was the same pattern if you were subtracting, and you make it a subtract in the brackets, and if it was a negative, it was the same pattern but you added a negative on the number before the brackets and made the inside equation addition.
Page 3 brought in the same sort of questions with the same sort of pattern.

Pages 4 and 5 were challenge questions which we answered with our knowledge of factorising.
This photo are the questions handwritten above with the answers.

## The Mathematical Skills I Used:

I USED:
Addition, subtraction, multiplication and division, when part of the patterns or questions involved using this methods.
Logic, as I had to write down answers that are logical and are not random and out of this world.
Problem solving, because I had to answer lots of questions on the sheets.
Classify relationships as either linear/non-linear, as the graphs I made had to be classified.
Identify and plot points on a Cartesian plane, as I had to plot and identify points on a Cartesian Plane.
Use substitution to evaluate algebraic equations, as we had to answer substitution questions using algebra to create our graphs.
Identifying terms, expressions, equations, coefficients, variables and constants, as I had to identify the parts of the equations including expressions and variables.
Using a rule to determine a function output from a given input, as I had to find patterns and use them to work out answers to questions mentally.
Algebra conventions, because when I were substituting we had to use algebra to find out the coordinates.

## The Mathematical Strategies I Used:

I USED:
Using a similar problem, when I used patterns to do answers mentally on the booklet sheets.
Draw a picture or graph, when I created the graphs from substituting the algebraic rules.
Guess, check and improve, when I tried to answer the questions and got it wrong, then improved our answers, and learnt from it.
Make a model, when I used the paper to create a model of the problems we had to answer.
Try a simpler problem first, when I started with simple problems and worked up to challenging ones in the booklet.
Look for a pattern, when I looked for patterns in the booklets so we could work out the answers mentally.
Make a list or table, when I created a table to show the substitution of the algebraic rules, and show the quadratics.
Act it out, when I used the paper to make a model of the equations we had to answer.
Describe, continue and create number patterns, when I had find the patterns for the distributive law and use them to answer questions mentally.
Algebra conventions, when I had to do the substitution and the graph making of the questions.
Use substitution to evaluate algebraic expressions, when I had to answer all the questions using algebra.
Identify and plot coordinates on a Cartesian plane, when I had to use the algebra we substituted to make a graph.
Classify relationships as non linear, as the graphs I made were non linear and were instead quadratics.

## Results, Part 1:

Here are some photos and videos of some of the completed work sheets and my working out for some of the questions. There are also photos of the graph and the tables I made the graphs with. I used patterns to work out the challenge questions mentally and the graphs were made on Desmos.

GRAPHING:
We had to use our understanding of the distributive law and factorising to create a graph through substitution. This showed us that the graphs were non linear. The things below are a picture of the substituting table we had to fill in before making the graphs, and three pictures of the graphs I made with the substituting. The first is a graph showing the substitution rule X2 (squared) +7x +6. The second one is X2 (squared) +5x+4. The third is X2 (squared) +5x +6.

## In future I could further expand my knowledge of the distributive law and work out harder patterns.I could also create bigger graphs to further show the way the graphs turn out.If I was given a similar problem, I would use similar techniques such as make a model to help me work it out.I found this fairly challenging and will probably not investigate this further. If I did though, I would use the same techniques and strategies to help myself work the answers out, find the patterns, or create the graphs.If I were to do something like this again, I would probably want to be able to work out all the questions mentally, without having to use the model.I probably would like to create a bigger graph by substituting more numbers so I could see the quadratics better

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Emma SMITH,
Sep 1, 2016, 10:35 PM
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Emma SMITH,
Sep 1, 2016, 10:33 PM