Rationale; NUMERACY ACROSS THE CURRICULUM POLICY It had been identified nationally that there is a need to strengthen pupils ability to use mathematics in unfamiliar contexts. Developing the use of mathematical skills in other subjects will support progress and understanding in both mathematics and in those subjects. Ofsted emphasises the need for teaching to enable pupils to develop skills in reading, writing, communication and mathematics. This is referred to in each grade descriptor for the quality of teaching and achievement of pupils. What is numeracy Numeracy is much more than an ability to recall the times tables and perform mental arithmetic. Perhaps one of the best definitions of numeracy comes from the Cockroft Report Mathematics Counts which suggests that being numerate implies two attributes:..an at-homeness with numbers and an ability to make use of mathematical skills to cope with everyday demands and..an ability to have some appreciation and understanding of information presented in mathematical terms, e.g. graphs, charts, tables,.. Numeracy is a key skill in students learning and all students are entitled to quality experiences in this area. All teachers should consider pupils ability to cope with the numerical demands of everyday life and provide opportunities for students to: handle number and measurement competently, mentally, orally and in writing; use calculators accurately and appropriately; interpret and use numerical and statistical data represented in a variety of forms. Whilst recognising that there are some students who will not achieve all of these, all students will be encouraged to develop these skills: to read numbers and to count; to tell the time; January 2016
to pay for purchases and to give change; to weigh and measure; to understand straight forward timetables and simple graphs and charts; and to carry out any necessary calculations associated with these. The teaching of numeracy is the responsibility of all staff and the school s approaches should be as consistent as possible across the curriculum. Curriculum areas should endeavour to ensure that materials presented to students will match their capability both in subject content and in numerical demands. All students should be encouraged to present their work as neatly as possible so that their achievements and methods can be properly recognised. 1. The guidelines to aid the development of consistency across the curriculum are included in this document. 2. Subject areas should encourage students to employ appropriate methods whether these are mental, written or using a calculator. 3. Where calculators are to be used their correct use may have to be taught. 4. Not all students in a teaching group will have the same numerical skills and where unsure of an appropriate numerical level teachers should consult with the Maths staff. 5. All teachers should discourage students from writing down answers only and encourage students to show their numerical working out with the main body of their work not on scrap paper, nor at the back of their books, nor anywhere else. 6. The use of estimation particularly for checking work needs to be encouraged. 7. All staff should encourage students to write mathematically correct statements. (See guidelines attached). 8. We have to recognise that there is rarely only one correct method and students should be encouraged to develop their own correct methods where appropriate. 9. Wherever possible students should be allowed and encouraged to vocalise their maths a necessary step towards full understanding for many students. 10. All students should be helped to understand the methods they are using or being taught students gain more and are likely to remember 2
much more easily if they understand rather than are merely repeating by rote. This document should provide information and guidelines to help produce consistency across the curriculum it is not intended to be a prescription for teaching although some advice is given. Guidelines to aid consistency in numeracy across the curriculum Methods: Where a student is gaining success with a particular method it is important that he or she is not confused by being given another method. This does not disallow the possibility of introducing alternatives in order to improve understanding or as part of a lesson deliberately designed to investigate alternative methods, provided students can manage this without confusion. Presentation: In all arithmetic, the importance of place value and neat column keeping should be stressed. Increase 3.50 by 80% This is poor practice: 3.50 x 0.80 = 2.80 +3.50 = 6.30 This is good practice: 3.50 x 0.80 = 2.80 2.80 + 3.50 = 6.30 Words to watch out for: Similar The word similar in maths is used to describe objects that are exactly the same shape but not necessarily the same size. Two shapes that are similar and exactly the same size (i.e. identical) are called congruent. Decimals When referring to decimals say three point one four rather than three point fourteen. Averages It is important to use the correct mathematical term for the type of average being used, i.e. mean, median or mode. (See definitions later in guidelines) Product This means multiply Sum This means add Difference This means subtraction. 3
Circles Circumference Segment Sector Radius Chord Diameter Checking Encourage students to check divisions by multiplication and subtractions by adding. Arc Billions Remember that: in the United Kingdom a billion was one million x one million, whereas in the USA one billion = one thousand x one million. N.B. But now we use the USA version. Units Some useful conversions. Metric 10mm = 1cm 1000mg = 1g 1000ml = 1litre 100cm = 1m 1000g = 1kg 1000cm³ = 1litre 1000m = 1km 1000kg =1 metric tonne 100cl = 1litre Imperial 12 inches = 1 foot 16 ounces = 1 pound 8 pints = 1gallon 3 feet = 1 yard 14 pounds = 1 stone Rough Conversions between Metric and Imperial 1 inch 2.5cm 1 yard 1m 1kg 2lbs 5 miles 8kms 4oz 100g N.B. means approximately equal to. Rounding N.B. Decimal places not decimal points To the nearest decimal place (d.p.) means you start counting from the decimal point e.g. 5.096782111 = 5.097 to 3 d.p. 0.005589112 = 0.006 to 3 d.p. 5.096782111 = 5.10 to 2 d.p. 0.005589112 = 0.01 to 2 d.p. 5.096782111 = 5.1 to 1 d.p 0.005589112 = 0.0 to 1 d.p 4
To the nearest Significant Figure (sig. fig. or s.f.) means you start counting from the first non-zero digit (starting from the left) e.g.: 1 357 450 = 1 360 000 to 3 s.f. 0.005589112 = 0.00559 to 3 s.f. 1 357 450 = 1 400 000 to 2 s.f. 0.005589112 = 0.0056 to 2 s.f. 1 357 450 = 1 000 000 to 1 s.f. 0.005589112 = 0.006 to 1 s.f. (For both types of rounding when the next digit is a 5 or more, we round UP). Percentages To express one number as a percentage of another: Example: Express an exam result of 34 out of 60 as a percentage. 1. Rewrite as a fraction: 34 = 34 60 (This answer will be a decimal) 60 2. Multiply by 100 to change into a percentage: 34 100 or 34 100 60 or 34 60 100 60 To find a percentage of something: Example: To find 45% of 25. Either: Or: Change the percentage to a decimal (by dividing by 100) and then multiply together: 0.45 x 25 = 11.25 Change percentage to a fraction (45/100), which implies dividing by 100, and then multiply by 25: 45 100 x 25 or 25 x 45 100 or 25 100 x 45 = 11.25 or 45 x 25 100 To Increase or decrease by a percentage: Example: Increase 35 by 13% Either: Find 113% of 35: 1.13 x 35 or 35 x 113 100 = 39.55 Or: Find 13% of 35 and add this answer onto 35. 0.13 x 35 = 4.55 35 + 4.55 = 39.55 5
Equations The terms cross-multiply and swap sides swap signs can lead to misunderstandings, as part of any explanation of how to solve equations and so should be avoided. To teach solution of linear equations we use the balancing method or a flow diagram see example below. To solve: 3x 7 = 5 Balance Method: 3x 7 = 5 (add 7 to both sides) You start 3 x 7 + 7 = 5 + 7 with the last 3x = 12 (divide both sides by 3) thing that 3x 3 = 12 3 happened to x = 4 x. Flow Chart Method using function machines: START: x x3-7 5 3 +77 END: 4 5 x = 4 Guidelines for Construction/Using Graphs and Charts Students should be encouraged to: use sharp, hard, at least HB, pencils label both axes and give a title put the independent (the more important) variable on x-axis, and dependent variable on the y-axis, e.g.: if graphing temperature of a cooling liquid, time should go on the x-axis and temperature on the y-axis. (The temperature of the liquid is dependent on the time of the reading). label lines not spaces, unless a bar-chart with discrete data. use equally spaced intervals use convenient scales mark points by a small cross not a dot draw graphs on squared or graph paper draw graphs of a sensible size (they tend to make them too small) join one set of points before proceeding to next set of data. 6
In more detail: Bar-Charts 100 80 60 40 East West North 20 0 1st Qtr 2nd Qtr 3rd Qtr 4th Qtr The bars should be of equal width and equally spaced The bars do not touch for discrete data Frequency should be on the y (vertical) axis We strongly recommend the use of square paper as opposed to graph paper, which can prove difficult to use/read accurately at some levels. Pictograms Must include a key 7
Pie Charts Must include a key Must be constructed using compasses and protractor. 1st Qtr 2nd Qtr 3rd Qtr 4th Qtr Working out should be shown, but not on the pie chart. Percentages are not an integral part of constructing pie charts. Here are some examples of how to calculate the angles for a pie chart: Example 1 Example 2 Favourite Colour Method of Transport Blue 12 Car 177 Red 18 Bicycle 54 Yellow 40 Walk 105 Green 36 Bus 238 Total 106 Total 574 1. Add up data to obtain total frequency: Eg 1 12 + 18 + 40 + 36 = 106 Eg 2 177 + 54 + 105 + 238 = 574 2. To find the number of degrees for each person : Eg 1 360 106 = 3.396 Eg 2 360 574 = 0.627 (do not round here) 3. Eg 1 To find the angle for blue: 360 x 12 106 or 3.396..x 12 8
or 12 x 360 106 or 12 106 x 360 Eg 2 to find the angle for each car: 360 x 177 574 or 0.627 x 177 or 177 x 360 574 or 177 574 x 360 4. Do not bother to introduce percentages unless specifically required. 5. Sectors should be labelled (e.g. Car, Blue.) or there should be a key. 6. Do not write angles on the pie chart. 7. Do not be surprised if the total of all the angles is 360 plus or minus one or two degrees. This will almost certainly be due to the rounding that may be necessary. Scatter diagrams and correlation Scatter graph a diagram that is used to see if there is a connection between two sets of data. Correlation is a measure of how strongly connected two sets of data are. There are different types of correlation. 9
Line graphs Students need to be taught when these are appropriate. (This is very important, as students will generally produce the type of graph they last met without much thought to appropriateness). Other Terms Averages: Mean Median Mode Range Total of values in the sample sample size Middle value of sample when sample values are arranged in order of size. Sample values which occur most frequently. The difference between the highest and lowest sample values. In Maths this should be given as a number not as a sum, i.e. the range is not 36 11 (nor 36 to 11), the range is 25. Range and mode occur at Level 4, mean, median, range and mode at Level 5. Mean for grouped data occurs at Level 7 as does comparing mean, median, mode and range of appropriate frequency distributions. Discrete: Data is described as discrete if specific values only can be used. Shoe sizes, colours of cars, or favourite flavours of crisps are examples of discrete data. Shoes sizes such as 4.8 and 5.77 cannot exist. Continuous: Data is described as continuous if all values can exist, eg, height and weight are continuous data as potentially any value could be measured. Scaling: If axes do not start from zero, a break represented by a zig-zag line should be shown on the axis. 10