Number and Place Value Ideas
Think about all the transition times that occur in a typical school day. Here are just some: lining up; waiting for the whole class to dispose of, or gather, coats and paraphernalia at the start or end of the day; waiting for parents to pick up; walking to or from various parts of the school; and I am sure you can think of others. So how could these times be used to do some maths?
Simple number games can be played at these times and not only will the pupils be consolidating some maths, but they will also be less distracted. Examples of games can be found in the next section.
When you are out and about with your class of younger students, and are checking that you haven't lost anybody, try using a couple of pairs of students to start at each end to count the pupils. Not only will they be practising counting in twos, but they will also see how it is much easier to count when pupils remain in pairs! Depending on their stage, pupils could be counting in tens, hundreds, fours or other multiples.
Lots of number work can be reinforced as pupils carry out their usual activities, which is why we start here with a section on transitions; what maths can we be doing during transition time?
Start at any number of choice (depending on where your pupils are) and pupils take it in turns to continue from there. A pupil could decide where to start and the intervals for counting don't have to be ones...
This is useful for pushing their learning just a little bit further so for example, starting in the nineties and going into the hundreds when this is new for pupils.
For older pupils you could count in hundreds, starting at one thousand, but alternate between thousands and hundreds so pupils would say; one thousand, eleven hundred, one thousand two hundred, thirteen hundred. This encourages flexibility with numbers since pupils often do not see the different ways in which a number can be expressed.
Start at a number of choice and pupils take it in turns to count backwards.
We often find counting backwards more difficult but then we do not practise it often in the way that we do when learning to count forwards. Counting back can help both younger and older pupils with subtraction and the jumps can be any number that you choose. For example, starting at 1234 and counting back in tens makes pupils focus on the value of the columns.
Allowing one pupil to keep a visual record of the counting on the board can help less confident pupils to see a pattern rather than relying on visualising the pattern in their heads.
Odds On or Every Even
Start at any number and only say the odd or even numbers after this, whether counting forwards or backwards. This can be useful even for larger numbers to emphasise the fact that whether a number is even or odd depends only on the final digit.
Say a list of numbers (from any starting point and with an interval of choice) but miss some numbers out. Pupils need to spot which numbers have been missed out. It is worth noting these numbers down as you are saying them so that you don't forget if there is a sudden distraction...
Choose a starting number and then instruct pupils to add an amount to a particular position such as tens, hundreds, thousands or higher. This can be quite challenging when dealing with larger numbers but older pupils need to be comfortable with numbers up to one million and therefore need to be happy with the positions greater than one thousand, which they don't often encounter. To add a further challenge, a number could be subtracted from a certain position. You could swap between addition, subtraction and position depending on which person has been reached and their confidence level.
Choose a starting number that is close to zero and count forwards and backwards with an option for you (or a pupil) to say, Switch so that pupils change the direction of counting. This allows the focus to remain on crossing zero.
Ascend or Descend
Provide numbers that are in the incorrect order (these could be written on small whiteboards for speed) and pick pupils to order the numbers, either in ascending or descending order. Do make a point of using the words ascend and descend since these words are often used infrequently so do not necessarily get embedded into pupils' understanding.
This can be useful for numbers with the same digits such as 101 and 110, or 1100 and 1010, so that pupils really need to focus on the values in the columns.
Greater Than & Less Than
Provide one pupil with a card (or small whiteboard) that all can see. Provide another child with a different card so that the second number can be compared with the first: is the second number greater than or less than the first number? If you have two cards each containing an inequality card, a pupil could choose which sign belongs between the two numbers.
Manipulatives such as base ten materials, place value sliders, place value mats, place value arrow cards and place value disks all help pupils to build better understanding by providing visual images for what is happening. Do make sure to link the use of these with the abstract calculation so that the manipulatives can be removed when pupils are confident and have a sound understanding. However, try to avoid them being removed too quickly because this can result in pupils moving forward with a shaky understanding of place value, which underpins further number and calculation work.
There seems to be a tendency when using position headings to use words (tens, hundreds, thousands) but then use fractions as headings after the decimal point. In my experience, it is more effective to use symbols for all headings: 10,000, 1000, 100, 10, 1, 1/10, 1/100. This also emphasises the value represented by the digit in that column and how many zeroes it possesses.
When warming up in PE, there are lots of opportunities for counting with younger pupils. If you are timing an activity, try counting forwards or backwards (in intervals of a particular number) for twenty seconds and pupils need to remember which ones were missed and tell you at the end.
Use songs such as Ten Green Bottles (it doesn't have to start at ten and doesn't have to go down in ones...) where pupils crouch for the fall and stretch as they sing about how many are left.
When practising throwing in pairs or groups, counts don't have to progress in ones and pupils can count backwards to see whether they can get to zero.
In pairs, pupils run towards each other as one is holding a number card. The partner writes another number on the laminated card or whiteboard held by another member of the team when they reach each other: for young pupils this could be one greater or one less than the number but for older pupils it could be the nearest prime number or one hundred less than the number. Alternatively, they could each have a number and when they reach each other, they have to insert an inequality sign in between them. Basically, you can decide what you want to reinforce at that point in time. Pupils could also be asked to round a number to the nearest ten, hundred or thousand.
During cricket warm-ups, teams of pupils could be running to various points and back, to practise their runs, and once they have all completed their running the team has to sort the cards. This could involve ordering certain types of numbers (including negative and decimal numbers) in ascending or descending order. The numbers could be in a specific sequence so that to add a twist, pupils need to trade numbers with other teams by noticing which one is not in the sequence and running to a drop point where they can pick a number from another team.
It would be worth building up a collection of cards that could be laminated so that pupils are choosing cards from several possibilities rather than having to write things down.
Zero as a place holder is a concept that takes time to develop.
When we are counting objects, we obviously do not start at zero but when we are reciting numbers from the number line, counting from, and back to, zero (or even below zero) can remind pupils that this is an even number that sits on the number line along with all the other numbers.
Make sure to include zero, plus some negative numbers, on any number lines displayed in the classroom so that pupils can see the continuous nature of the number line which can continue forever in either direction.
To emphasise the need for zero in our place value system (where each numeral's value depends on its position, or place, within the number) you could give pupils numbers (on large cards or whiteboards) and put them in front of the class with uneven spaces between them. Ask the class for ideas on what number is represented. They will usually give you the number as seen, and will be puzzled as to why their suggestions are incorrect so allow time for them to discuss this. Eventually, introduce some zeroes in the spaces to show that without zero the number displayed could be a range of numbers.
The teens clearly don't follow an overall counting pattern when written down - we say sixteen, seventeen, eighteen and nineteen but write a ten followed by a six (unlike in China, where children follow the pattern of ten one (11), ten two (12), ten three (13) followed by the twenties: two ten one (21), two ten two (22)...) Fourteen, fifteen and sixteen also sound like forty, fifty and sixty so make sure to give pupils plenty of exposure to these numbers.
Allow time for pupils to make up stories involving the teen numbers which could be combined with allocating a specific number to a pupil for a day or week so that there is a lot of focus on that number. For example, when lining up, split the line after fourteen pupils (the number fourteen pupil will be the fourteenth in line). Have a number fourteen hunt where pupils look for hidden cards (on which fourteen is written as a word and a numeral) in the playground.
Note language that isn't used every day such as less, fewer and greater than.
How often do we use the word fewer in the classroom? Less is heard more frequently (although pupils do tend to talk about someone having more than them rather than them having less...) but both words need to be understood by pupils.
As an aside, when explaining the difference between fewer and less to older pupils, perhaps the easiest way is to substitute:
smaller number of for fewer
smaller number of cars: fewer cars
smaller amount of for less
smaller amount of money: less money
where amount is the total, so five items or less in supermarkets refers to the total number of items rather than the individual items (even though they can be counted).
Looking at how other number systems worked can help pupils understand our own place value system better.
Roman Numerals (which are part of the primary maths curriculum in the UK) provide a useful comparison to our own place value system. Pupils could consider what is the same and what is different about the Roman Numeral system and our own, and which one they feel is more effective and why. They could also consider whether the position of numerals was important in the Roman system by considering IV and VI, for example.
Exploring alternative bases can help strengthen pupils' understanding of our own base 10 (decimal) system. It can be surprising how many older pupils do not think about each position being ten times greater than the previous column. Binary and hexadecimal bases are used within the computing industry: computers use binary and hexadecimal is used by programmers to shorten binary. Web page colours are expressed in hexadecimal and it is also used in error messages.
It is worth emphasising the grouping effect of any other systems - whereby each grouping is represented by a symbol, which allowed large numbers to be written - and compare it to how we use groupings of multiples of ten. Groups of pupils could be invited to invent their own numeral systems to see how easy they are for other pupils to understand.
Using negative numbers in context is important for pupils' understanding. This can start with young pupils seeing negative numbers on both vertical and horizontal number lines, although they are most likely to grasp the idea on vertical number lines
initially since they may have been in an underground car park or experienced being underwater. This could be acted out by using the desk top as the ground floor level and bending down to travel below ground level.
The physical world within geography topics provides scope for exploring negative numbers by looking at cooler climates around the world (including the Arctic and Antarctic) and comparing these with much hotter climates as well as our own. Depths of oceans is another area that can be used as a context where the range of depths throughout the world can be investigated, or the difference in height between the lowest parts of the ocean and highest mountains.
Using thermometers in cold water with ice added can show the temperature dropping and what exactly is the temperature of the freezer at home or in the staff room? Thermometers are useful for illustrating that as the temperature increases the numerals for the negative numbers decrease so -1 degrees is warmer (and higher on the number line) than -7 degrees.
Looking at the symmetry within the number line can help to demystify negative numbers for pupils. We often don't refer to the numbers above or to the right of zero as being positive numbers before introducing negative numbers, which can then make them seem very bizarre if taught out of context.
Using historical timelines can help pupils to see the idea of negative numbers getting later or higher as the numerals reduce, but take care to point out that, unlike the number line, there is no zero year in the Gregorian calendar so it is -1 BCE and +1 CE.
Pupils can have problems recognising and reading large numbers but they need to be able to eventually work with numbers to ten million by the end of primary school. Ensure that pupils see place value headings that go beyond one thousand and that they are written in numerals. Spend time exploring how the place value system works so that they understand that each successive place is multiplied by ten when numbers are getting larger. Initially, suggest that struggling pupils replace all the digits, apart from the first, with zeroes so that they can start to work out the magnitude of the number.
Use other subjects, such as science or geography where pupils could be looking at bacterial reproduction or populations of animals or people in countries around the world.
Pose questions involving large numbers such as whether anyone in the class has lived for ten million seconds (a ten year old has lived for 315 360 000 seconds plus a day for each leap year...) or estimate how many seconds, minutes or hours there are in the average person's lifespan.
See Number and Place Value for KS1 in Resources