Rich tasks, Problem Solving, Connecting the Curriculum and Using Data

Helen Walters

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Overview

• Problem Solving Cycle

• Rich Tasks/problem solving

• Language of Learning

• Reflection

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How are you going to solve this?

• Working in pairs

• You cannot use the number five (5) in any place

The ‘5’ button does not work???

How am I going to solve this problem?

!"#$%&$#'()!*+&'*

55 x 7

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Things could be different and maths could be a source of great pleasure and confidence

There is a huge gap between what we know works for children and what actually happens in most classrooms.

What are Rich Tasks?

In essence, rich tasks:

• encourage children to think creatively,

• work logically

• communicate ideas

• synthesise their results

• analyse different viewpoints

• look for commonalities

• evaluate findings.

However, what we really need are rich classrooms: communities of enquiry and collaboration, promoting communication and imagination.

The nature of a rich

task involves "letting go" and preparing for the range of needs of your own learners and

where they are likely to go.

What is a problem?

• In groups of four, discuss this question and come up with a definition for a problem.

One person is to record the discussion, - another, not the recorder will report back.

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A problem has...

A START

A BLOCKAGE

Solver, needs a desire to solve!

An END

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Problems or Skills?

• 6 + 5

• Find the average of 9, 10 and 11

• A rectangle has a length of 5cm and a width of 3cm, find the perimeter

Change each of the above into problems with the idea of a start, a blockage and an end.

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Problems or skills?

• Two numbers add to 11 what could the numbers be?

• The average of three numbers is 10 what could the three numbers be?

• A rectangle has a perimeter of 24 what could the lengths of the sides be?

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A four step problem solving approach

1. Understand the problem

• Read it carefully

• What information is important, un-necessary?

• Rephrase the problem in your own words?

• Have you seen a problem like this before?

• Research

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We do not spend enough time analysing problems but rush in. That is why we often get children saying...

“I don’t know what to do!”

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Temple of Gloom

DefineLee Crockett

DiscoverLee Crockett

Solution Fluency

2. Decide on Strategy (way)

• If it is similar to a problem I know I will use the same strategy

• What problem solving strategies might I use? - • Envisage range of creative solutions to real life situations

• What is it that I need to do? + - x ÷

• Draw a plan

• What materials do I need - counters?

We must spend time with this step!!!

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DreamLee Crockett

DesignLee Crockett

3. Do it!

• Use your strategy(ies) and your understanding of the problem to try and find the solution(s)

We spend most of the time on this area but if the previous steps are not used properly it is like ‘pushing garbage up hill!’

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DeliverLee Crockett

4. Reflect back

• Have I a solution that I can justify?

• Have I answered the problem?

• Have I found a number of solutions?

• Is there now another problem I would like to investigate?

• What if?- questions

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“Holding high standards” is not about making work ‘more difficult’ but about motivating through relevance and personalisation” Capacity Building Series, Ministry of Education, Ontario

DebriefLee Crockett

How to work on the rich tasks?

• In order to improve BOTH teachers' and students' abilities to work on the rich tasks, it is necessary to take account of, and find ways of supporting:

The challenges involved in learning mathematics through rich tasks

• valuing the joint voyage of discovery - the journey is more important than the final answer

• accepting that students may have to get used to floundering sometimes - part of the experience is likely to involve getting stuck

• highlighting the value of strategic thinking

• accepting that ideas will be jotted as they are discovered; it does not matter if work is not always recorded in a neat and orderly fashion

The use of a range of lesson structures

• allowing for one problem to be the focus of a whole lesson

• using a number of mini-plenaries where students feed back

• using group work

• students working on fresh problems with little initial guidance

•The use of effective questioning to promote thinking

• Tasking questions that require students to think

• Challenging students to justify their findings

• Asking questions that probe the depth of students' understanding

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Cats

This is not a trick question. This is a real maths problem so don't say that a bus has no legs.

• There are 7 girls in a bus.

• Each girl has 7 backpacks

• In each backpack, there are 7 big cats.

• For every big cat there are 7 little cats.

• Question: How many legs are there in the bus ?

You will be asked to explain how you worked this out.Justification Clarification Proof of thinking!!!!

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Start with weights that differ by 2kg. What whole number

averages can you make?

Then try weights that differ by 3kg, then by 4kg...

Dotty 6

3

4

2

1

535

1

2

What are the rules and how do you

win?

Three ways to

make it more challenging to suit

your students

!FIRING UPImag ine you are liv ing 200 years ago and you are a trader at the produce market. Using the gu ide lines be low, you barter for fru it and vegetab les for your fam ily.

Two hundred years ago peop le swapped and traded th ings w ithout using money. Th is type of trad ing was ca lled ‘bartering’. (Today, many of us use the internet to trade th ings in the same way.) W hat e lse do you know about life 200 years ago?

How can you use mu ltilink cubes, co loured b locks or draw ings to represent the fru it and vegetab les to he lp you so lve the prob lem?

1 On Monday you take some p ineapp les to the market to trade . Work out how many p ineapp les you need to trade for the fo llow ing:

12 bananas 12 app les 36 carrots 15 oranges2 On Tuesday you take 120 carrots to the market to trade for other fru its. Work

out how many of the fo llow ing you wou ld get for your carrots: oranges bananas app les p ineapp les3 How many app les w ill you need to trade on Wednesday to take home: 5 p ineapp les 36 carrots 12 bananas 15 oranges

FINDING OUT FURTHER

Th ink of three d ifferent ways you cou ld get 60 carrots by bartering other fru its.

FOCUSING FORWARD Deve lop an a lgebra ic expression to create new prob lems for

yourse lf or others to so lve , e .g:

12 carrots + apples = 2 pineapples

Th ink of, or research , other ways of using a lgebra ic equations.

What d id you find most d ifficu lt? The most easy? What d id you learn?

At the barter market 200 years ago

Super Tip!An algebraic

expression is a maths sentence

that uses numbers and

letters.

5Maths Activ ities: Plants and Trees, prepared by He len Wa lters and Len Cooper © 2009 Interact Curricu lum

=

=

=

=

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Platonic solids and Archmedian polyhedra

• Tetrahedron

• Cube

• Octahedron

• Dodecahedron

• Icosahedron

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Rhombicuboctahedron

Investigate how many:✴faces✴edges✴equilateral triangles✴squares

•If each tooth pick is 6cm what is the total length of all the edges added together?

Discover

Define

Dream

Design

Debrief

Deliver

• 26 total faces: 8 equilateral triangles and 18 squares

• 24 vertices: 1 triangle and 3 squares

• 48 edges

How Big is a Foot? Rolf Myller

• Set the scene

• Why was the bed too small?

• http://playpen.meraka.csir.co.za/~acdc/education/Dr_Anvind_Gupa/Learners_Library_7_March_2007/Resources/books/combination/kingsfoot.pdf

Things to think about?

• Where did the standard “foot” measurement come from?

• Where did “inch” measurement originate from?

• Where did the measuring tool “ruler” originate?

•

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!  Trace around your foot/shoe on paper. Cut it out.

!  Investigate and make comparisons with other body parts.

!  What other body parts are the same or are proportioned to your foot?

!  Think multiplicatively…..

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How Big is a Foot?

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!  Cut out 6 feet in total and sellotape together (heel to toe)

!  Investigate and make comparisons with other body measurements

!  Investigate relationships between other body measurements

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Proportions

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!  What is your palm in relation to the width of your fingers?

!  What is the relation of the width of your palm to the length of your foot

!  What is a cubit?

!  a cubit is the width of six palmsHow many palms is a cub it?

!  What is your heigh in relation to your cubit?

!  What is your height in relation to other measurements of your body?

!  What is the distance from the hairline to the bottom of the chin in comparison to your height?

!  What is the distance from the top of the head to the bottom of the chin in comparison to a man's height?

!  What is the maximum width of the shoulders to your height?

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Proportions of Vitruvian Man

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!  a palm is the width of four fingers

!  a foot is the width of four palms

!  a cubit is the width of six palms

!  a man's height is four cubits (and thus 24 palms)

!  a pace is four cubits

!  the length of a man's outspread arms is equal to his height

!  the distance from the hairline to the bottom of the chin is one-tenth of a man's height

!  the distance from the top of the head to the bottom of the chin is one-eighth of a man's height

!  the maximum width of the shoulders is one-fourth of a man's height

!  the distance from the elbow to the tip of the hand is one-fifth of a man's height

!  the distance from the elbow to the armpit is one-eight of a man's height

!  the length of the hand is one-tenth of a man's height

!  the distance from the bottom of the chin to the nose is one-third of the length of the face

!  the distance from the hairline to the eyebrows is one-third of the length of the face

!  the length of the ear is one-third of the length of the face

Paper Planes - group competition

You will need:• A4 and/or A3 paper

• stop watch/phone

• measuring tape

In pairs or groups of 4 construct paper planes/darts

• What type of dart performs the best and why?

• Investigate which one flies the:• fastest?• furtherest?• straightest?• longest in the air

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Discover

Define

Dream

Design

Debrief

Remember the ‘what if...?’

Deliver

Where could you go with this?

Discover

Define

Dream

Design

Debrief

Deliver

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Connecting the Curriculum

In groups identify:

• What could the dart theme have been?

• What Curriculum areas could this relate to?

• What other activities can be used to support a theme?

• Share back

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• You will need:

• Pen

• paperclip

• leaf

• rubber band

How hard can you blow?

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How hard can you blow?

• Mark out a start line and a finish line 1 metre apart

• Place object on start line.(repeat with each object)

• Estimate how many blows it will take to blow the object from start to finish lines.

• Blow each object across to the finish line. Count the number of blows

• Who has the strongest blow in your group?

• Two things we often measure are Force and Motion. What is the force in this activity and what is the motion?

• If you had a straw to blow the objects would it take more or less blows? Why?

paperclip,

pen,

rubber band

leaf

You could create and make a ‘Wind Yacht’ to move by your puff or the wind. “Think” It would need to have wheels and a sail!How large would it be? How would you steer it?

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What is learning? What is a successful learner?

• Students often do not know what learning actually is let alone what you have to do to be successful at it!

• Responses:“Successful learners sit up

straight.” (Year 2 student)

“Successful learners do as the

teacher tells them.” (Year 5)

Successful learners pick up the rubbish.

(Year 1 student)?

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What do life-long learners do?

• ask questions

• take risks

• set goals/plans

• make connections

• self-motivate

• perservere

• reflect

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Key Competency?

Managing Self

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Three Monkeys and some bananas!

!  Three monkeys worked together all day to gather a pile of bananas and then went off to bed.

!  In the night one of the monkeys decided to take one third of the bananas and hide them in a safe place.

!  The monkey had to eat one of the bananas before she could take exactly one third of them. She then went back to bed.

!  As the night progressed the other two monkeys did the same as the first, eating one banana and hiding away exactly one third. In the morning they were able to share the remaining bananas equally between each other.

!  How many bananas were in the original pile?

Discuss:What are the key areas of

assessment?What are you going to assess?

Where have you got your AO’s/LI’s from?How are you going to manage the assessing

and recording ?How are you going to identify

‘next learning?

Discuss:What are the key areas of

assessment?What are you going to assess?

Where have you got your AO’s/LI’s from?How are you going to manage the assessing

and recording ?How are you going to identify

‘next learning?

13 Away

• Take turns to remove 1,2 or 3 counters

• The last person to take the last counter is the loser.

• What if?

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Reflection

• What have you found useful?

• What are one or two things that have really struck you?

• How might this session impact on what you may do in your classroom?

• What might you do differently when you plan next?

• What short -term and long-term action in regards to your PLD and your classroom programme do you need to make a defensive OTJ and integrate maths more effectively.

What?

So What?

Now What?

Think about how you might assess multiple areas with

one rich task!

Questions?

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Helen Walters

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