Module 6  Graphs & Sequences 
Graphs  Sequences  Teacher Notes
Solvemymaths has produced pages like this for lots of topics
NRICH Tasks
Shifting Times Tables  Click here
Pose the question as to whether say 72 is in the sequence Once you have worked through this activity get the pupils to plot the term against the value and explore the gradient, intercept, equation of the line etc ...


Interactive Resource Fantastic range of interactive sequences tasks including shifting tables  Click here

Visual Sequences
Seven Squares Task –
Click here
This can be seen as a square and then 6 inverted C’s
so 4 + 6x3 These clearly all equal 22 and can lead onto the generalised forms of:
4 + 3(n1) and 1 + 3n and 2n + (n+1) then pupils can
show these are equal. 
Border Squares How many different methods can you identify for working out the number of shaded squares around the border? Or explain why 4x10  4, 4x8 + 4, 4x9, 2x10 + 2x8 and 10^{2}8^{2} give the number of shaded squares

Investigative Starting Point Simple present them with an image (or a series of images) and ask them to find the number of matches for the 100^{th} diagram of one of the sequences. Then ask them whether they can find another way, compare their method with others in the class, can they show the two ways are equivalent?

Sequences Bingo
Pupils select answers from this grid 
Click here
Preprepared Grids Finding the nth term
If you are teaching a method you might like to flag
up the method of using the So to find the nth term of the sequence 3, 7, 11, 14 we often teach it is going up in 4's therefore it begins 4n but 4x1=4 so we need to take 1 so the formula is 4n  1 but consider whether it is clearer to use 3 + 4(n1) instead as the logic for many might be a lot clearer. FlashMaths: Randomly generated linear Sequences – nth term – Click here FlashMaths: Enter your own sequence given an nth term and check or just find the nth term of a sequence = Click here Investigative Tasks Investigational Tasks in the Algebra Section  Click here Visual Patterns
An extensive Mix of Linear, Quadratic and Cubic
visual patterns which provide an excellent starting point for
discussions on sequences, finding general terms and whether a
certain number is in a sequence 
Click here
Answers 

3 Act  Dan Meyer You’ve heard of pile patterns? There are variations but generally you have three snapshots of a growing shape like this:
Questions follow regarding future piles, past piles, and a general form for any pile. What would this old classic would sound like with the addition of video. Video adds the passage of time. I added a red bounding box to the video, which was an attempt to make the question, “Where will the pattern break through the box, and when?” perplexing to students. I also added different colours, which allows students to track different things or ask themselves, “What colour will be the first colour to break through the box?” Different questions require different abstractions. If you care about total tiles, you’ll model the total. If you care about the breakout, you’ll model the width and height. Each one will require linear equations, which is nice:  Click here for online link and Click here for downloaded files


Coordinate Patterns – Click here What are the coordinates of the centre of the 10^{th} square? … What about top left?
This task can be tackled by exploring the sequence of
numbers or finding the equation of the line. There are lots of
possible avenues such as finding the equation 

Robot Stepper: Idea taken from here Imagine that you have several robots: a twostepper, a threestepper, a fourstepper, all the way to a ninestepper. Ask pupils how this task might develop and give them ownership of the task Some suggestions might be: ü Are there any patterns in the units place ü Are there any patterns in the tens place ü What about Even and odd number patterns ü If all robots start from 0 how many will land on 100 ü What if you take two robots on which numbers will they both land? ü What about 3 robots? ü What if they don't all start from 0? ü Can you find a way of predicting which numbers a threestepper will land on if it begins on the 2? ü Are there any patterns in the sums of the first and second number, the third and the fourth number, the fifth and the sixth number, and so on.If you look at a single robot you get the shift patterns like Shifting Times Tables  Click here.Jumping along a Number Line – Click here


An alternative approach might be to use this following Inquiry Task
Intersecting sequences inquiry Click here for source and further details. 

Random Sequence Generator You can generate random questions of varying difficulty and display either the nth term or the sequence and use this to discuss how to find the one you have hidden  Click here


Quadratic Sequences Representing Quadratic Sequences Visually and then illustrating related quadratic graph – Click here You can generate random sets of 10 Quadratic Sequences with the first 5 terms and the 20th term visible. I have used this to display the 5 terms and required pupils to find the formula for the nth term and then check it gives the correct value for the 20th term  Click here Don Steward http://donsteward.blogspot.co.uk/search/label/quadratic%20rules Don Steward http://donsteward.blogspot.co.uk/2014/03/squareygrowth.html Comparing Alternative generalisations derived from diagram: 

Number Spirals

Linear Graphs Interactive Resource – Click here

Number Shacks – Click here

Graphs and Transformations  Draw graphs to force the marble
to go through the stars:
Marble Slides

Practicing Coordinates Instead of providing learners with a list of random pairs of coordinates, the teacher might instead invite them to produce their own coordinates by employing a rule such as ‘the second coordinate is twice the first coordinate’ or ‘the first coordinate is three times the second coordinate, minus two’. Such a task means practice of a particular technique is taking place, but something with far richer mathematical possibilities is happening at the same time. I have observed in the classroom that learners are much more enthusiastic about plotting their points (and, consequently, perform the procedure many more times) when it is with a particular purpose in mind. They want to get the points in the right places, because their location matters for some wider purpose. Learners may find the patterns that result (e.g. straight lines) surprising and intriguing, and thus might be inclined to pose their own ‘What if?’ questions in relation to other possible rules. They will notice if a point does not fit the pattern and perhaps check whether they might have plotted the values the wrong way round or overlooked a negative sign. Such work is also more readily extendible for learners who finish early or who are particularly confident with the technique, since they can employ more complicated rules (e.g. ‘the second coordinate is the square of the first coordinate’). It is more interesting for both the learners and the teacher, much less tedious for the teacher to mark, and represents a worthwhile mathematical task in its own right. It also has the potential to illustrate to learners why coordinates might be useful in mathematics and why plotting them correctly could be important.

Mini Golf Marble Slide  DESMOS  Click here
Coordinates Inquiry:
Click here 
Fractions and Gradients I suggest you plot numerator along the xaxis and denominator on the yaxis. From this we can see that . It could also be seen that the gradients of the lines are the reciprocal of the fractions … What if two fractions are equal? 

Screen Match Pupils are presented with a single or multiple graphs and asked to come up with suggestions as to what the equations could be.

Puzzle

Marble Slides Find the equation of sets of lines that get you through a maze

Real Life Graphs Great animation showing different vessels filling and their graphs – Click here Fantastic DESMOS Filling different containers and predicting graph – Click here Graphing Stories  http://www.graphingstories.com/ Filling containers  real life graphs
http://nrich.maths.org/7419
Speed  Distance Time Graphs  Vi Hart Video


Race Commentary

Language of Functions and Graphs  Shell Centre
Section A involves sketching and interpreting graphs arising from situations which are presented verbally or pictorially. No algebraic knowledge is required. Emphasis is laid on the interpretation of global graphical features, such as maxima, minima, intervals and gradients. This Unit will occupy about two weeks and it contains a full set of worksheets and teaching notes. Section B is where the emphasis is laid on the process of searching for patterns within realistic situations, identifying functional relationships and expressing these in verbal, graphical and algebraic terms. Full teaching notes and solutions are provided. Section C is a collection of materials which supplements the material presented in Units A and B. It is divided into two sections. The first contains nine challenging problems accompanied by separate selections of hints which may be supplied to pupils in difficulty. The second section contains a number of shorter situations which provide more straightforward practice at interpreting data Lesson Plans  Click here Photocopy Masters  Click here This Module aims to develop the performance of children in tackling mathematical problems of a more varied, more open and less standardised kind than is normal on present examination papers. It emphasises a number of specific strategies which may help such problem solving. These include the following: try some simple cases, find a helpful diagram, organise systematically, make a table, spot patterns, find a general rule, explain why the rule works, check regularly Lesson Plans  Click here Photocopy Masters  Click here


Function Auction I explained that there would be an auction, and each team would be given $1,000. The team that purchased the most functions with their $1,000 would win candy – Click here 

Transformation of Graphs Great GeoGebra based set of 14 problems to identify the equations of the graphs: Quadratic, Cubic and Reciprocal – Click here Extension Idea A Novel Take on Graphing  SourceHere’s the pitch: 1. Draw a horizontal axis and label it with the numbers 1, 10, 100, ... (What numbers should go to the left of 1?) 2. Draw a vertical axis, labelled similarly. 3. Draw a downward sloping line at a 45 degree angle through the “origin” (which seems to now be the point (1 , 1) The equation y = x no longer seems appropriate. Can you write an equation that fits this graph? 4. Try going in the other direction. If you write down the equation that would normally give a circle and try to graph it on these axes, what do you get? 5. Try lines, parabolas, circles, exponential functions, logarithmic functions, rational functions, etc. 6. Try other ways of labelling your coordinate axes. Note on Notation: My students and I reserve and for the standard axes labelled in the standard way. We generally use to represent the strange values along the horizontal axis and to represent the strange values along the vertical axis.

Graphical Inequalities

Quadratics Interactive Parabola Explore the relationship between the equation and the graph of a parabola using this interactive parabola. Just type in whatever values you want for a,b,c (the coefficients in a quadratic equation) and the the parabola graph maker will automatically update! Plus you can save any of your graphs/equations to your desktop as images to use in your own worksheets

Quadratics Free Mathspad Resource – Click here A complete course on Quadratics @jamestanton – Click here Finding the Vertex of any Quadratic

Trigonometric Graphs

A Template
for Solving Quadratics
Twitter: @mathequalslove 