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Visual Stimuli in

Mathematics

 

Dan Meyer teaches high school math outside of Santa Cruz, CA, and explores the intersection of math instruction, multimedia, and inquiry-based learning.  He first came to my attention through this video clip -  Click here

He promotes the use of real life visual stimuli and adopts the mantra that less is more.  Stripping back problems and both engaging and challenging pupils. A key question he encourages is "What Can You Do With That?"  [WCYDWT] At the same time, we shouldn’t underestimate the hookiness of “How do we find solutions to this equation?” and appreciate that visual stimuli is not the only way forward.

His videos can be found on Vimeo (free registration for downloads) and can be found by Clicking here

This clip shows how Dan engages a class - Click here

What do you notice and What are you wondering? -  Video Clip

Three-act structure works for a play, a romantic seduction, the 1st Marine Division marching up to Baghdad. It’s the architecture for a WWE wrestling match, a Frank Gehry concert hall or an infomercial.

Storytelling gives us a framework for certain mathematical tasks that is both prescriptive enough to be useful and flexible enough to be usable.

Act One

Introduce the central conflict of your story/task clearly, visually, viscerally, using as few words as possible.

The visual is clear. The camera is in focus. It isn't bobbing around so much that you can't get your bearings on the scene. There aren't any words. And it's visceral. It strikes you right in the terror bone.

With math, your first act looks something like this:

Act One is the hook. “A priest, a rabbi and a gerbil walk into a bar … ” The purpose of the first act is to engage the audience. The greatest Act One ever is a roller coaster. Up, up, up and then … over the falls! You’re hooked. Two other aspects of a great beginning: it must be unique and it must make a promise. A great fishing lure is a shiny, eye-catching object that makes the prey think, “Ah, a delicious meal!”

Act Two

The protagonist/student overcomes obstacles, looks for resources, and develops new tools.
 

What resources will your students need before they can resolve their conflict? The height of the basketball hoop? The distance to the three-point line? The diameter of a basketball?
 
What tools do they have already? What tools can you help them develop? They'll need quadratics, for instance. Help them with that.

Act Two is deepening complications. This is the meat of the project. Billy Wilder said, “In Act One, get your hero up a tree; Act Two, set the tree on fire; Act Three, get the hero down from the tree.”

The first movement of a symphony establishes the musical theme. The middle movements exhaust variations on the theme. Our middle passage—whether it’s a novel, a startup or a philanthropic venture—plays out the promise of the beginning to the point of excruciation. Think of making love. Think of a great meal. Think of middle age.

Act Three

If we've successfully motivated our students in the first act, the payoff in the third act needs to meet their expectations. Something like this:
 

Make sure you have extension problems (sequels, right?) ready for students as they finish.

Article taken from here

Act Three is the payoff. The release of tension. The climax. The resolution of the dilemma.  In the third act we learn if the defendant will be hanged or go free. Will Janie and Joey get married? Do the good guys win or lose?

For some reason, the human mind loves items that come in threes. That’s the key to laying out our structure - Extracts taken from here

 

Dan Meyer exemplifies the three stages to the 3 Act lesson with suggestions of key questions and the thought processes involved - Click here

In my opinion the following clips best exemplify his philosophy:

Mr Kraft's website has some excellent 3 Acts resources - Click here

How Many Meatballs can I put in the sauce? - Click here

Cake ribbon - Click here

Which Discount should you use? - Click here

Stacking Cups - http://blog.mrmeyer.com/?p=692    
Detailed Lesson Plan - Click here

Extension Diagrams - Click here

The Bone Collector - http://blog.mrmeyer.com/?p=1510

http://mr-stadel.blogspot.co.uk/2013/01/styrofoam-cups.html

The Worlds Largest Coffee Cup - http://threeacts.mrmeyer.com/hotcoffee/

Toilet Rolls and the Empire State Building - Click here

Pole to Pole Run - Click here

Road Lines - Speed

Domino Spiral - Rates

Fly me to the moon - Speed of Light

Mega Coin - Rates and Ratios

 

Megalodon - Ratios and Proportions

Viva Las Colas - Ratios and Proportions

TV Space -  Pythagorean Theorem

Domino Spiral - Area

Pop Box Design -  Surface area

Coca-Cola Slim - Surface Area and Volume

Really Big Coke - Volume

Mega Coin - Volume

Mmm Juice - Volume

Viva Las Colas - Volume

Mmm Juice - Rates and Ratios

Starting Points for Mathematical Discussions and Explorations are exemplified by the following set of problems under the heading "What Can You Do With This?"
 

WCYDWT

 

Dave Gale likes to explore ways of posing interesting mathematical questions - Click here

This zipped file contains a PowerPoint presentation by E Styler pulling together a variety of examples into a structured talk - Click here

The website http://www.101qs.com/ features lots of photos/videos (many from Dan the Man!) which will provoke lots of Maths questions from students....?

Graphing Stories - http://blog.mrmeyer.com/?p=10535

Shower v Bath - http://vimeo.com/30166690

Drink Ratios - http://vimeo.com/20628942

Consider a similar problem using discrete objects (e.g., playing cards. Take 10 red cards and 10 black cards face down in separate piles. Take four at random from red pile; mix into black pile. Shuffle. Return four random cards face down to red pile. Ask: more black in the red pile or red in the black pile. Try this several times. If you’re not convinced, do it with the faces showing. Apply principle to soda problem.

How long is the roll - http://vimeo.com/39044428

How big is the circle - http://vimeo.com/39047347

Will they hit - http://vimeo.com/38873347

Water Tank Filling - http://vimeo.com/9632608

Geometry Problem - http://vimeo.com/27451133

Loads of Money - http://vimeo.com/26163161

Diagonals Investigation - http://vimeo.com/32301200

MaxBox is one of many investigations which could be presented in this way

Probability Decision - http://vimeo.com/32756849

Speed of Light - http://mrmeyer.com/threeacts/speedoflight/

 

These problems are adapted from the Bowland task 'You Reckon' - Click here

Suffolk Dan Meyer Project  
Project Aims:

To increase student motivation/engagement

To give students more ownership of their own learning

To equip our students with better problem solving skills which they can use to solve a variety of problems

Underlying principles for the lessons should be:

Give minimal information about the problem

Have a visual stimulus

Include the use of multi-media

Have a key question

Situate the problem in a meaningful context

Have closure to the problem.

Those involved the project are:

Gill Larkin
gill.larkin@suffolk.gov.uk

Mark Greenaway
mgreenaway@st-albans.suffolk.sch.uk

Mohammed Ibrahim, Chantry
mib@chantryhighschool.org.uk

Gareth Jones, Sudbury Upper
gjones@sus.suffolk.sch.uk

Darren Page, Stowupland
d.page@stowupland.suffolk.sch.uk

Lydia Unwin, Leiston High
floss15@btinternet.com

Lesson Idea 1

Real Life Graphs - Filling Beakers

The key steps in this lesson are:

  1. Vocabulary

  2. Open Discussion using an unlabelled graph

  3. Pause Video Clip of 3 beakers being filled at a constant rate, get pupils to predict which will fill first with a reason

  4. Play clip and get pupils to  reflect on their thinking or on each others.

  5. Pupils to decide which beaker match graph A with reason.

  6. Decisions to be made on numbers and labelling of axes Calculate/estimate volume of beaker A.

  7. Sketch graph for the other two beakers.

  8. Draw a beaker that would generate line B.

  9. Bath Filling Activity

  10. Tackle the original exam question the graph is taken from.

  11. Produce their own exam question with a mark scheme

Parts of this PowerPoint are to be projected and other sections are to support the teacher - Click here

The PowerPoint refers to this Movie Clip - Click here

Related nrich activities

http://nrich.maths.org/7419

http://nrich.maths.org/6424

http://nrich.maths.org/6425

Lesson Idea 2

This lesson idea initially explores the optimum way of packaging 12 coke cans and is adapted from Mr Piccini's Idea - Click here

Question 1: As a customer which box would you choose to buy?

Question 2: If you were Coca Cola which box would you choose to make?

Possible initial questions that need to be considered
How many cans?
Estimate Dimensions?
Packing in the fridge
Size of pallet: 1m by 1.2m
How many stacked boxes would be the same as your height?
Volume wasted in the box
Value for money (can versus bottle)
Surface Area
Packing on lorries?
Base not printed so cheaper to manufacture
Do we consider flaps?

Possible Progression Route

1.            Estimate Can size and hence sizes of box (real size 11.5 by 6.3cm)
2.            Round can size to 12 by 6 for lower group
3.            Size of boxes?
4.            Sketch net and write measurements on and then discuss Scale
5.            Lower ability draw rectangles for each face on A4 squared paper (or photocopy squares onto    
                card) and then sellotape them together or lay them out on large sugar paper.
6.            Draw nets to scale (1:2) will fit on A3 easily
7.            Surface area?
8.            Fit onto pallet?
9.            What about slim cans
10.          Cost of packaging (Note base cheaper)
11.          Extra 100% or %0% free recalculate
 

Extension Ideas

Boxes that are not cuboids: PowerPoint  - Click here
Design a Container which holds 300ml - Click here
Using Similarity with Drinks Cans - Click here

Related nrich activities

http://nrich.maths.org/5888
http://nrich.maths.org/2664
 

Lesson Idea 3

Packaging Salad

This activity looks at the redesign of some packaging for Salad

Act 1 - What is the question? - Click here
Act 1b - Some additional information - Click here

Surface Areas

Act 2 - Measurements for calculating Surface Area - Click here
Act 3 - Calculation of the two Surface Areas - Click here

Volumes

Act 2 - Measurements for calculating Volume - Click here
Act 3 - Practically comparing the two volumes - are they equal? - Click here
Act 3 - Calculation of the two Volumes - Click here

Lesson Idea 4

Projectile Motion - Modelling with Quadratics

Using this applet keep the speed at 18ms-1 and set the angle to 800 and note the maximum height and the range, repeat for 700 and 200 and pose the question - Note that the initial height is not 0 as the cannon is above the ground.

What must the angle be if I want the range to be 18m

One approach would be to plot angle against range and find the quadratic that fits the 3 given points by transforming the basic quadratic y=a(x-b)(x-c).  This can then be used to read of the angle needed to give a specific range.
A second approach would be to solve the quadratic equation rather than reading off the graph
A third approach would be to use the three points to find the equation of the curve using 3 simultaneous equations

Use the applet to test their results

Alternatively fix the angle and ask what the speed must be to achieve a given range.

Simplified Task

An alternative applet which can be adjusted to start at ground height can be found here

You could choose to fire at 900 and pose the question what must the initial speed be in order to reach 20m?
 

Lesson Idea 5

Overtaking Lorries - Clip
Watch first 1:10 and then pose the question as to how long the white truck will take to pass the oil tanker

Related nrich activities

http://nrich.maths.org/5990

 

Lesson Idea 6

Opening up exam questions

These questions consist of some Foundation Tier exam questions with the numbers covered up. pupils should decide on some sensible values and solve their problem, the slide afterwards has the numbers uncovered - Click here

These problems are taken from Foundation Tier GCSE papers where all that is left is the diagram and the challenge includes devising suitable questions to solve and then solving them before going to the next slide and looking at the exam question - Click here

This PowerPoint consists of data or diagrams taken from Foundation Tier GCSE Papers with no questions simpler contexts than the example above - Click here

These problems are taken from Higher Tier GCSE papers where all that is left is the diagram and the challenge includes devising suitable questions to solve and then solving them before going to the next slide and looking at the exam question - Click here

 

Report regarding the aims, process and outcomes of the Project - Click here