Rich mathematics year 3/4

Jane Haddock and Vicki Giffard
6 December 2013
National Centre for Excellence in Teaching of Mathematics. NCETM
Three main aims in new curriculum
1. Fluency
2. Reasoning
3. Problem solving
Fluency is important due to the deep level of reasoning and problem solving required in the curriculum.
Key idea is not to have dollop of maths, but to be always making connections between the different areas of maths and number.
Non routine problems are not just word problems but deep and rich investigative tasks.

Looking back at planning, how does it already meet the curriculum aims? How does it meet fluency, problem solving and reasoning?

Years 3 and 4 will be the first cohorts through the new tests in 2016. Summer 2014 will give us some examples of the test, but not until summer 2015 for a full example test. The floor standards will be set at 85% for new national standards. Probably equivalent to a very good level 4 (more likely to be a high 4b nearer to a 4a) but difficult to predict.

Advice to try to plug the gaps that will have been missed in years 1,2,3 from new curriculum that year 4 children won’t have done.

Less prominence to data and probability removed all together. A steer away from calculators, still need to use them. Levels are going, but keep things that are working well, if APP works then stick with it. Should be looking to get children in year 4 to a secure 3a at the end of the year to meet the raised expectations. Expect that there will be an arithmetic test which includes written and mental maths and a written test very much like the current version but harder with no separate level 6 type questions, but included in the main paper.

Expectation to be able to read, write and spell the vocabulary of maths. Can refer back to the old national strategies vocabulary book but will need moving down to meet the levels as changed and use that as a reference as a school.

Higher attaining children should be given deeper and richer problems before moving to the next level of content. So not necessarily move chn to year 5 content but deepen their level of confidence and understanding to apply their content.

Recommend focussing on one or two areas, like fractions or number facts to meet higher expectations in those in order to deal with what is coming.

Number facts: addition and subtraction – all sorted by end of year 2
Number facts: multiplication and division – all sorted by end of year 4

Number facts activities
Big maths beat that timed tests .
Also use handsets with software with mental maths to do 110 questions in 15 minutes in morning maths.
Children make times tables games and then play each other’s games
Target numbers on table, ask chn to do operation to different number. Differentiated easily.
Empty clock face to write timetable on them.
Turn tables, set of cards for each one. Array, answer, coaching partner to time against turning over cards.
Multisensorymatters.com
Counting sticks for times table for the key facts.
Number bones times tables and addition and subtractions.
Bingo for timestables
Flip charts – can they stick the flaps down as they know them?
Use hundred square as image to show number bonds to 100.

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Division tables – start at 10x so 60/6=10
Sum, product difference excel sheet – start with product and work through what the difference and sum and can be.

Roman numerals
Occur in lots of different places, clocks, history, Super Bowl, king and queens names, dukes, chapters and prefaces.
Will tie in with Roman history studies. Apparently hopscotch was invented in the Roman army – go put it on the playground!
Is a minimal objective.

Time
Duration and recorded time. Hard maths, 5, 60, 20, 5, 7, 12s. Need to be good at fractions. Parents are the time keepers.
Using a one handed clock can chn see and estimate the time between the hours. See objective sheet for covering time in yeah new curriculum.

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Note that time graphs are now included in year 4 statistics. Time on one access, anything else on the other.

Measures
Still work through progression, direct comparison (wind up cars which one lasts the longest), using non standard measures (sand or water time).
Time keeper of the day, wear the class watch and tell us when we need to go to assembly, break, end guided reading.
Number lines for adding time or finding differences in duration. Use a timeline to show school day (not starting at one).
Annotate clock face with 5 minutes.
Nrich activity, stop the clock. Work in pairs, move half and hour or an hour at a time, start at 6 o’clock and then take it in turns to move time on and one wins is the one to get to exactly 12 o’clock.

Planning considerations
Some people have started on new curriculum. Must to have found that year 3/4 needed a lot of number facts plugging before they could start the problem solving or reasoning.
Number facts is an easier place to start. Important to link into other subjects – for example in science, geography or history.

School based tasks –
1. Feedback to subject leader
2. Bring overview planning to next session for discussion

Guidance on curriculum on NCETM

Place value and number
Representations and model ps are very important and need to be progressed through the school.
Key problems – 47 is 4 and 7 and not 40 and 7
Seeing the size of number
Writing numbers with zero as place value
Understanding zero as a number not just a place holder (divide between positive and negative)

Ask chn to show you how they represent numbers
How would you show 369?
How would you show 5.4?

Suggest having a place value policy as part of calculation policy.
Models images and and for place value can then carry through to calculation policy:
Arrow cards, Dienes, straws, bead strings, counting sticks, hundred squares, money, moving children! Place value charts, place value base boards.
Progress – straws> dienes as chn need to know there is hundred in the large squares.
Number line and counting sticks to show continuous nature of numbers.

Game: 5 digit number into a calculator (no zero). Zap! Get rid of your number by subtracting that number 56798 teacher rolls dice and chn subtract that so, 6 would subtract 6000 from my number. First to zero.

Decimals
Need to be extension of place value and for representing fractions
Part of a unit, shape or set
Point of a numberline
Result of a division
Can show that 3/10 is 3 divided into 10 pieces. Each piece is 0.3.

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Use hundred squares for tenths and hundredths.
Bead strings can represent one, so use with decimals.

Zoom numberline is very powerful to show there are always smaller and smaller numbers.
Recommend measures over money to use as a concept for decimals.

Place value games
Nice and nasty – roll dice and put number in the grid closest to a target number.
Closest to a target number: 35
Work out differences, use two decimal places and tens and units. Works well for 2dp.
Insist on using language – model I am putting my nine in your tenths column.
Nasty version – give your number to someone else.
Friendly – as a table work together to make target number. Works nicely for a team game.

Everybody counts
Use +10 -10 +100 -100 so start counting in ones from a number and then point to one of the steps and switch to count in a different way, work from adding first then counting back.

Calculator constant function
Use ++ or — to count in steps of equal size. 10 ++ and keep pressing equal to count in those steps, ask chn what happens to numbers? Count in tenths, again what happens?

Ordering numbers
Numberline progression
Number track (no zero)
Numberline from zero
Partial numberline line, every number as a mark but only 10s marked on
Then just tens and no marks in between
Finally blank line.

Use paper clip and blank mini number line, ask chn to show you numbers using paper clip if ends of numberline are different.
Ask chn to put 3 numbers on a numberline, watch how they draw.

Ask chn to write 6 numbers that lie between 9 and 11. Order them in ascending order and then closest to 10. Could do with fractions.

20 cards ITP use for generating numbers

Yoikes as a ladder – 8 rungs and use decimals, bin. From 0 to 10.
Three in a line. Line from 0-10, use digit cards, turn over 2, choose which decimal to make (3.2 or 2.3). Try to get three numbers in a line.
Importance of the equal sign, meaning the same value, not the answer goes here.
Start with 50=50, then show 50=25*2 and then 23 + 27 = 25*2
Use equal sign in different places.
Remember to use greater than and less than signs throughout ks2

Nrich advent calendar is a great resource. Can be used for morning maths🙂

Subtraction
Not just about taking away. Need to be able to find the difference between – a way of comparing numbers.
Singapore bar to show how to work out difference: 173-85.
Different ways to look at subtraction: taking away (removal, permanent or temporary), counting back, decreasing, partitioning a set (class of 36 and 10 are girls, how many are boys?). Comparisons of 2 sets, comparison of 2 items, how many more / how many fewer?

Calculation policies need images, photos and models not just the strategy (Ofsted looking for that now).
Splitting in to parts – children need to know when to count up and when to count back.
Singapore bar works very well in Singapore but it is used very consistently by everyone, including adults in later life.

Mental methods
Still very important, new curriculum seems to skim over them, strategies are not prescribed. Need to decide as a school which mental strategies are going to be used and how they are taught.
87-13 take away 10, then 3. Show physical models first.
Compensation, take 10 then add one – money is good for that, children soon want a pound back if you take ten away!

Give two sets of numbers – choose a sum, then put it in a box – decide on mental, use jottings or use written method.
Look back to the teaching to children to calculate mentally booklet – published at the end of the national strategies time.

Written methods
Working with larger numbers for written methods – just choose one and track it back. Appendix of national curriculum shows suggested methods, these are NOT prescribed. Methods will not be scrutinised if your results are ok!
Children should always ask themselves if they can work mentally first.
Introducing a formal written method, use simpler numbers, on a numberline.
When moving to a decomposition method, use the word exchange NOT carry, borrow…
Use dienes equipment to show exchange of tens physically.
If using decomposition in year 6, make sure children can decompose numbers in different ways early on, exchange tens and hundreds.

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