To Build a Repertoire of Teaching Activities Which Develop

Creativity

Aim

To build a repertoire of teaching activities which develop:

Participation in pair and group work

Quality student dialogue

Explicit learning and teachingof the process skills and functional skills

Links across mathematical concepts

Awareness of the relative mathematical difficulty and complexity of activities/questions

Meta cognition

Quality hard and ephemeral evidence for APP

To build a bank of ideas/resources for all to draw on

One way to use the resource for cpd

Step 1

Start with an 'easy' activity (simple to prepare, simple to organise, does not take long to complete)

Run the activity with the staff in pairs:

Same/Different

25 75 displayed

at speed, write down 3 ways in which these are the same, 3 ways in which they are different

– 6 ways as fast as possible (prevents maths teachers searching for complicated ways)

to one pair: now tell me your easiest way they are the same

to rest – do you have an easier way

what about most complicated way … most interesting way

prompt using: 25p can be made with 2 coins, 75 can't … 25 has symmetry on a calculator – or does it?

ask them to find more interesting/challenging ways

ask them to share their best for same and different

Step 2

Discuss

how their students might engage with the activity

how their thinking changed during the activity

what difference working as a pair made

how they could apply this to algebra … shape … data

look for ideas involving diagrams, properties, statements, definitions as well as numbers

Step 3

Use the prompt sheet below to begin a discussion on best classroom organisation, types of teacher prompts that develop student engagement and understanding,

Step 4

Staff to try the activity at least once a day for the next 2 weeks and either

post their activities on the department noticeboard or web page

keep a learning journal of the activities with evaluations

submit a list of 10 of the activities with jottings

bring to catch-up interim meetings

Step 5

Review the progress at the next meeting, and either move to a more challenging activity that takes both teacher and student thinking further, or revisit this activity in the light of the team's experience

Justifying decisions

Same/Different
In the classroom

Students in pairs to encourage justifying.
Teacher prompting/giving a single example to push the thinking a step further.
Teacher prompting for more accurate use of mathematical language (Can anyone say that a better way? More efficiently?).

/ Questions/prompts

Find at least three ways/as many ways as you can to describe how these are the same, and how they are different.
Find an obvious way to describe what is the same and what is different. What is the most surprising way you can find? Easiest? Most difficult? Most mathematical?

25 75 / Same
both have 5 units
both multiples of 5
both factors of 150 / Different
25 is a square number, 75 isn't
25 has symmetry on a calculator
25p can be made with two coins, 75p can't
/ Same
both are triangles
both have at least two equal sides
both have an acute angle on the right hand side and at the top
neither have an obtuse angle
both have at least one line of reflection / Different
one has a right angle
one looks equilateral
one could be the face of a square-based pyramid (or could they both be??)
one probably has rotation symmetry
one has more than one line of reflection
/ Same
both have red as ¼
both are about colours
neither have any white (or black etc)
both have red and green together as ½
both have 5 sectors / Different
one has more yellow than the other
one has grey instead of blue
one has a more even set of sectors, the other has a bigger difference between largest sector and smallest
Mean
Median / Same
both begin with m, e and end with a, n
both are an average
both give an idea of the typical value
both involve using all the numbers in the data set / Different
one is a calculation, the other is about ordering
one takes longer to work out and sometimes needs a calculator
one is affected by outliers, the other isn't
one could be the minimum/maximum, the other must be inbetween

Justifying decisions

Odd one out
In the classroom

This is Same/Different but with 3 items. It is more challenging for students as they look for something two with something the same and one different simultaneously (& a little harder to prepare).
Students in pairs to encourage justifying.
Teacher prompting/giving a single example to push the thinking a step further.
Teacher prompting for more accurate use of mathematical language (Can anyone say that a better way? More efficiently?).

/ Questions/prompts

List one way for each to be the odd one out.
Reverse your reason – what do the other two have in common?
Choose one item. How many ways can you find to justify it being the odd one out?
What is the most interesting way to justify an item as being the odd one out?

/ 1 appears to be … equilateral … regular … rotation symmetry … has a centre
2 appears to have a right angle (or is it 2 and 3? How could you check?) … 'sharp' … a door wedge … no symmetry
3 appears to be isosceles … reflection symmetry
16 20 25 / 1 is less than twenty, the others are twenty or more … isn't a multiple of 5, the others are … isn't a factor of 100 … isn't a coin in old or new money …
2isn't a square number, the others are square … can be made with one coin, the others need more … has a zero …
3 is odd, the others are even … isn't a multiple of 4, the others are …
y = 3x + 4
2y = 6x – 4
y +3x + 4 = 0 / 1 is in the standard equation-of-line form … doesn't have a negative y-intercept …
2has common factor 2, the others have no common factor … doesn't have x-intercept -4/3 …
3 doesn't have gradient 3 … has negative gradient … isn't parallel to the other two …
y = mx + c
y = ax² + bx + c
y = ax³ + bx² + cx / 1 is a straight line not a curve … doesn't have a point where the gradient is 0 unless m = 0 … has the same gradient along it's length … only has one x-intercept (but what about quadratic touching the x-axis at minimum?) … doesn't have tangents (as they are the same as the line itself)
2 has reflection symmetry in a vertical line … only has one point where the gradient is 0 …
3 definitely goes through the origin, the others don't unless c = 0 … is cubic, the others are not … has a point where the gradient is infinity …

Justifying decisions

Agree/Disagree
In the classroom

No more than 4 statements for a lesson starter.
Students in pairs to encourage justifying.
Teacher prompting/giving a single example to push the thinking a step further.
Teacher prompting for more accurate use of mathematical language (Can anyone say that a better way? More efficiently?).

/ Questions/prompts

Which statements do you agree with? Which do you disagree with? Give reasons.
Can you adapt each statement so you agree with them all? Disagree with all?
Can you write similar statements? Can you reverse the agree/disagree by tweaking them?
Can you write more general statements and explain why they are true/untrue?

3·74 × 10 = 3·740
18 ÷ 10 = 1·8 / 1 Disagree: putting a 0 on the end doesn't work. I know because … 3·740 = 3·74 … the 3 units must become 3 tens …
2 Agree: the 1 ten becomes 1 unit …
1 Generalising:the tens number becomes units, the units becomes tenths etc … (10a + 1b) ÷ 10 =1a + 0·1b
The median is always larger than the mean
The mean is always positive
The range is always positive / 1 Disagree: 1 1 1 1 6 has median 1 and mean 2
2 Disagree: -5, -4, -1, 0, 5 has mean -1
3 Agree: maximum subtract minimum means bigger take away smaller so bound to be positive
0·4 is the same as 40%
3 + 7(2 + 9) = 80
0·3 × 0·2 = 0·6 / Disagree: 40% means 40/100 which is 0·4, but one is a proportion and the other could be a measurement – I could say 40% has the value 0·4, but I can't say 0·4 has the value 40%
Agree: must work out brackets first so 3+7(11), then must multiply before adding (BIDMAS) so 3+77 which is 80
Disagree: 3 × 0·2 = 0·6 and this can be checked by adding three lots of 0·2, so 0·3 × 0·2 = 0·6 can't be the same
4x is always less than x²
x²  0
x ÷ 1  x / Disagree: when x = 1, the statement would say 4 < 1 which is not true
Agree: multiply 2 positive or 2 negative numbers, it will always be positive and 0 × 0 = 0
Debate: as x ÷ 1 =x, does that mean this is true and could be improved, or is it false because it contains an untruth as x ÷ 1 is never>x?
Generalising 1: when x is greater than 2 it is true, when x is 2 or less it it false

Justifying decisions

True/False/Iffy
In the classroom

Stimulus (graph, chart, working steps) plus 2 to 8 statements.
Students in pairs/4s to encourage justifying.
Teacher prompting to push the thinking a step further, or challenging: Are you sure? How do you know?
Teacher prompting for more accurate use of mathematical language (Can anyone say that a better way? More efficiently?).

/ Questions/prompts

Which statements are true and which are false? Are there any where you can't be sure? Give reasons.
Can you adapt each statement so they are all true? All untrue?
Write your own statements and challenge a friend.
What extra information do you need to help you with the 'iffy' statements?

College students travel further
than School students.
(in km) / College / School
Mean / 5•7 / 0•8
Median / 4•3 / 0•75 / None of the School students
travel more than 10km, but at
least one College student does.
Range / 12 / 5•5
1 is True – mean and median much larger for college
2 is True – College: range is 12, must be minimum 0 and max 12 or larger numbers so max is more than 10. School: median plus range is < 10, so even if median is nearly the max still < 10
Percentage bar for 30 College students / More College students travel
by bus than school students.
Percentage bar for 100 School students / A bigger proportion of College students travel by car because they are old enough to drive
1 is False – College: looks about half the chart, which means about 15 students. School: looks 20% at least, which is 20 students. In this case, definitely not true.
2 is Iffy – First part is true, a bigger proportion travel by car, but there isn't enough evidence in the chart to support the reason. Might be true, might not.
y = x³ – 3x / The graph goes through the origin.
The graph has a local maximum at (1, 3).
The graph has rotation symmetry about the centre.
1 is True – substitute x = 0 to get y = 0
2 is False – local maximum at x = 1, but the y value isn't 3 at that point it is -2
3 is Iffy – the rotation centre isn't defined, so does it mean the origin (so not true) or the point midway between the local maximum and local minimum?

Justifying decisions

Always, Sometimes, Never
In the classroom

No more than 4 statements for a lesson starter, more for a main.
Can be more difficult than True/False/Iffy (with the generalisation in the statements) or can be easier (using specific examples, Sometimes can be easier than justifying Iffy)
Students in pairs/4s to encourage justifying.
Teacher prompting to push the thinking a step further, or challenging: Are you sure? How do you know?
Sort statements into piles/Create a poster/Give two examples to support your placing.
Convince yourself/a friend/a penpal/a teacher
Play bingo with playing cards of statements, A/S/N.
Use voting pods/whiteboards for class votes.
Selection of statements to each group, justify to the class why the statements are on the A/S/N wall.

/ Questions/prompts

Which statements are always true? Sometimes true? Never true? Give examples/reasons.
What if (for example)… you can only use positive numbers? Fractions? Decimals? Negatives?
What if (for example) the shape is drawn to scale? Isn't drawn to scale? Doesn't have these measurements?
What if (for example) the axes have a different scale? Don't have a scale?
Which statements are trying to fool you?
Which statements were easy? Which did you do first/last? Why?
Can you give a better example? A more general example? An algebraic example?
Is the statement worded clearly, or can you improve it to help other people place it?

An isosceles triangle has one line of symmetry
A triangle has exactly 1, 2 or 3 lines of symmetry
A triangle has rotational symmetry of order 2
The hypotenuse is opposite the right angle
The sloping side is the hypotenuse
Double the lengths of the triangle’s sides = double the size of the angles
The area of a triangle is greater than its perimeter / To bear in mind for this set of statements:
1: does this mean exactly one, or at least one? Does isosceles include or exclude equilateral, by definition?
What mathematics is needed? Do all students need the same set of statements?
Does it matter if statements have some overlap?
For the last two: Do you need the diagram? Does it make it more difficult/easier?
What resources could be available to help students test the truth of each statement?
2 + ∆ > 2
 + 2 > 
2 -  < 2
 – 2 < 
 × 2 < 
2 ×  > 2
 ÷ 2 < 
2 ÷  < 2 / 1 +  > 1
 + 1 > 
1 –  < 1
 – 1 < 
 × 1 < 
1 ×  > 1
 ÷ 1 < 
1 ÷  < 1 / 0 +  > 0
 + 0 > 
0 –  < 0
 – 0 < 
0 ×  > 0
 × 0 < 
0 ÷  < 0
 ÷ 0 <  /  + 
 –  = 0
 × 
 ÷  = 1

Where's the maths in that

What's the story?
In the classroom

No more than 1 stimulus.
Students in pairs/4s to encourage imagination.
Teacher modelling and/or restricting the context.
Teacher prompting to keep stories linked to the stimulus.
Teacher prompting for more accurate use of mathematical language (Can anyone say that a better way? More efficiently?).

/ Questions/prompts

Your story must be about school … shopping … food … the environment … linked to geography …
Who has the shortest story? The most interesting story?
Who has the best reason for this part of the graph looking like this?
What happened at this point? How does your story link to this part of the graph?
What happened next/before? What will the new part of the graph look like?
What would the graph look like the next day … for a friend … in another country …?
What can't this graph be about?

/ Best contexts:
Travelling– distance and time or speed and time eg a cycle ride away from home and back
Rates – time against any measurable factor such as height above ground eg height of a soufflé above the time over a 4 minute period
Cannot be:
Countable eg number of bees in view
Something that changes steadily eg temperature
/ Best contexts:
Rates – emotions, sound levels, volume in a bath, something involving rapid changes eg motion on a roller coaster
Cannot be:
Countable eg number of bees in view
Something that changes steadily eg temperature

Where's the maths in that

Show me the maths
In the classroom

One picture displayed as a stimulus.
Students in pairs/4s to encourage imagination.
Teacher modelling and/or restricting the context.
Teacher prompting to keep questions linked to the stimulus.
Teacher prompting for more accurate use of mathematical language (Can anyone say that a better way? More efficiently?).

/ Questions/prompts

Who has the simplest question? The longest question? The most interesting?
Who has a question that could be answered just from this picture?
Who needs to do research to answer their question?

/ How tall are the trees?
How old are the trees?
How many trees are there in the picture?
What is the circumference of the trunk and of the canopy? Do all the trees of the same type have the same trunk-canopy ratio?
How long would it take to climb to the top of one tree? How long would it take a squirrel?
How many leaves are on one branch? The whole tree?
How much compost does each tree generate a year?

/ What proportion of the pier is still standing?
How much metal is in the picture?
How tall are the legs underwater?
How deep are the foundations?
How long did it take to build?
How much force are the waves exerting on the frame – in a normal tide – in a storm?
How much would it cost to salvage the whole structure? How long would it take? Is the scrap metal value worth it?

Where's the maths in that

What's the question?
In the classroom

Brief set of information as a stimulus.
Students in pairs/4s to encourage imagination.
Teacher modelling and/or restricting the context.
Teacher prompting to keep questions linked to the stimulus.
Teacher prompting for more accurate use of mathematical language (Can anyone say that a better way? More efficiently?).

/ Questions/prompts

Your questions must be about school … shopping … food … the environment … linked to geography …
Who has the simplest question? The longest question? The most interesting?
Who has a question that could be answered just from this information?
Who needs to do research to answer their question?

It takes 1·75 metres of denim
to make a pair of jeans.
Denim costs £3·50 per metre. / If the denim can only be bought in an exact number of metres, how much extra will you pay for the denim you do not use?
How many pairs would you need to make to ensure that there is no wastage of denim?
What is inflation at the moment? What would happen if you used that figure instead of 5%?
How much discount would you need to get the price back to where it was before the price increase?
What would the discount need to be if the increase was 10%, or 20%?
How much does the person sewing jeans in Asia get paid an hour?
How do you decide where to buy your jeans?

/ Is the mode 8?
How many people responded in this survey?
Can you find the three averages and the spread?
What is missing from the graph?
Why are there gaps between the bars?
What survey questions might have these four items as the answer? What additional labels could be on the horizontal axis?
What might the hypothesis be if you collect this information?
/ Name as many faces of this solid as you can.
How many edges, vertices and faces does the solid have?
What is the volume of the solid? What assumptions do you need to make to work it out?
Assume the centre is a regular pentagonal prism and it's volume is half the volume of the entire shape. What volumes/areas/lengths can you find from this information?
This is part of a tessellating design in a rectangle 1m by 2m. How many stars can fit in the rectangle? What it the most efficient design if you want to (a) keep all the stars at the same orientation (b) fit in the maximum number of stars? Are hexagonal stars more efficient?