Module 6 Graphs & Sequences
Module Content Lesson Resources Lesson Plans Exam Questions Probing Questions

Graphs - Sequences - Teacher Notes

Solvemymaths has produced pages like this for lots of topics

NRICH Tasks

NRICH: Grouping Goodies
NRICH: Lost
NRICH: A Cartesian Puzzle
NRICH: Route to Infinity
NRICH: 
Coordinate Patterns
NRICH: Diamond collector
NRICH: Spaces For Exploration
NRICH: Parallel Lines
NRICH:
Shifting Times Tables
NRICH: Seven Squares
NRICH: Triangle numbers
NRICH: Squares in rectangles
NRICH: 
Seven Squares
NRICH: 
Coordinate patterns
NRICH: Route to Infinity
NRICH:
Tablecloths
NRICH:
 How Far Does it Move?
NRICH: Speeding Up, Slowing Down 
NRICH:
 Up and Across
NRICH: Motion Sensor
 
NRICH: Interactive Number Patterns
NRICH: 
Fibs
NRICH: 
Sequences and Series
NRICH More Sequences and Series
NRICH 
Painted Cube
NRICH:
Creating Pictures
NRICH:
 Parallel Lines 
NRICH: 
How Steep is the Slope
NRICH: 
All About Ratios
NRICH: 
Noughts and Crosses
NRICH: Cubics
NRICH: Minus One Two Three
NRICH: 
Spaces for Exploration
NRICH:
Creating Pictures
NRICH
 Perpendicular Lines
NRICH: Trigonometric Protractor
NRICH: Sine and Cosine                     
NRICH
: Growing Surprises
Extras
Random Sequence from nth term
Online Card Matching
Number of Blocks
Drawing Graphs Stories
Random Questions - Find Gradient
Video: Catenary Curves

 

Shifting Times Tables - Click here


The numbers in the four times table are  4,8,12,16...36,40,44...100,104,108...
I could shift the four times table up by 3 and end up with
7,11,15,19...39,43,47...103,107,111...
What do you notice about the differences between consecutive terms in each sequence?

The interactivity displays five numbers from a shifted times table.
On Levels 1 and 2 it will always be the first five numbers.
On Levels 3 and 4 it could be any five numbers from the shifted times table.


Use the interactivity to generate some sets of five numbers.
Can you work out the times table and by how much it has been shifted?

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 is a superb task that requires students to visualise the development of a matchstick sequence and explain how they see the diagram being constructed and then they are asked to use this to find the number of matches for larger diagrams.  This leads onto finding the nth term: For example:

This can be seen as a square and then 6 inverted C’s so 4 + 6x3
                           or a single match and 7 inverted squares so 1 + 7x3
                           or two horizontal rows of 7 and 8 vertical lines so 2x7 + 8x1

These clearly all equal 22 and can lead onto the generalised forms of:

4 + 3(n-1) 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 102-82 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 100th 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     Pre-prepared Grids
Teacher projects these questions -
Click here       Answers

Finding the nth term

If you are teaching a method you might like to flag up the method of using the
a + (n-1)d formula that we normally teach at A level.

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(n-1) 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

 

Co-ordinate Patterns – Click here

What are the co-ordinates of the centre of the 10th 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
y = 1/3x + 1 and finding one of the ordinates, for example the 10th square would have a y ordinate of 11, so 11 = 1/3x + 1 giving x = 30.

Robot Stepper: Idea taken from here

Imagine that you have several robots:

a two-stepper, a three-stepper, a four-stepper, all the way to a nine-stepper.

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 three-stepper 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/squarey-growth.html

Comparing Alternative generalisations derived from diagram:

Number Spirals

Linear Graphs Interactive Resource – Click here

Number Shacks – Click here

 

 

Practicing Co-ordinates

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.

 

Co-ordinates Inquiry Click here

Fractions and Gradients

One way of comparing two fractions is to plot them as co-ordinates. So for example if you had the two fractions: then these could be represented on some axes.

I suggest you plot numerator along the x-axis and denominator on the y-axis.

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

 

 

 

Maze

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
http://nrich.maths.org/6424
http://nrich.maths.org/6425

 

 

Speed - Distance Time Graphs - Vi Hart Video

 

   
 

Race Commentary

       Click here

Language of Functions and Graphs - Shell Centre


This is a 6 week block of lesson ideas split into 3 sections:

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 - Source

Here’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

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