Numeracy and mathematics: Experiences and outcomesdornochmaths.weebly.com/uploads/3/0/4/5/30459404/kn… · Web viewExperiences and outcomes. Numeracy and Mathematics Third Level

Numeracy and mathematicsExperiences and outcomes

Numeracy and Mathematics Third Level The questions and tasks listed with each of the Experiences and Outcomes at Third Level illustrate skills and knowledge which should be able to be demonstrated reliably by pupils who are secure at the particular outcome. Questions are designed to verify that basic facts and processes are known. Contexts for application of the knowledge and skills will vary. The notes whilst made with reference to Third level work, are also mindful of the demands which will be made at later levels and at National 4 and 5 and beyond and which build upon the foundations laid in Second and earlier levels.

As a working guide we are assuming that whilst a topic is being worked on, security is considered to be an accurate response at least 90% of the time, after retention for some months security is considered to be an accurate response at least 75% of the time.An accuracy of reponse between 50% and 90%, at the time the topic is being worked on, and between 50% and 75%, after some months, would indicate that the knowledge and skills required further consolidation and a rate of accurate response of less than 50% in either circumstance would indicate the particular skill or knowledge is still at the development stage.

In Numeracy and Mathematics sound progress at Third and Fourth levels and beyond depends on fluency of early skills, good retention of all earlier facts and processes and clarity of both calculations and statements showing interpretation of solutions. Clarity in written evidence, as calculations, diagrams, statements or some combinations of these, is important to demonstrate understanding and to provide a firm foundation for further learning and for application of that learning in unfamiliar contexts.

My learning in mathematics enables me to:

develop a secure understanding of the concepts, principles and processes of mathematics and apply these in different contexts, including the world of work engage with more abstract mathematical concepts and develop important new kinds of thinking understand the application of mathematics, its impact on our society past and present, and its potential for the future develop essential numeracy skills which will allow me to participate fully in society establish firm foundations for further specialist learning understand that successful independent living requires financial awareness, effective money management, using schedules and other related skills interpret numerical information appropriately and use it to draw conclusions, assess risk, and make reasoned evaluations and informed decisions apply skills and understanding creatively and logically to solve problems, within a variety of contexts appreciate how the imaginative and effective use of technologies can enhance the development of skills and concepts.

Number, money and measure

Third Questions / Activities to demonstrate Key Skills / Knowledge Notes

Estimation and rounding

I can round a number using an appropriate degree of accuracy, having taken into account the context of the problem.

MNU 3-01a

Round 3.185 to 2 decimal places

(expect to see 3.185 = 3.19 (2dp))

If 3 friends share £20, how much does each get?

(expect to see 20 3 = 6.666… so each gets £6.66 and 2p is left over)

If there are 46298 spectators at a football match, how would you expect this number to be reported in the newspaper headlines?

(expect to see 46000 as a sensible statement )

degree of accuracy used should be stated 3.185 = 3.19 (2dp) is correct, but 3.185 = 3.19 is not a true statement since 3.185 3.19

rounding appopriate to the context should be used eg here rounding to £6.67 would require £20.01 to be shared

Number and number processes including addition, subtraction, multiplication, division and negative numbers

I can use a variety of methods to solve number problems in familiar contexts, clearly communicating my processes and solutions.

MNU 3-03a

£60 is shared among 4 people. Jane has twice as much as Sarah who has £5 more than Fred. Paul has the same as Fred? How much does each have?

(expect to see a series of logical steps / constructed model, followed by a clear answer to the question posed)

eg Paul has £x so Fred has £x, Sarah has £(x + 5) and Jane has £2(x + 5) x + x + (x + 5) + 2(x + 5) = 60 5x + 15 = 60 5x = 45 x = 9

Jane has £28. Sarah has £14, Fred has £9 and Paul has £9

x = 9 is not the final answer since it does not address the question

result should be set in context

Number, money and measure (continued)


Number and number processes including addition, subtraction, multiplication, division and negative numbers (continued)

I can continue to recall number facts quickly and use them accurately when making calculations.

MNU 3-03b

Kelly is writing a 3000 word essay. so far she has written 1850 words. How many more words does she have to write? She writes 250 words per page, how many more pages does she have to write?

(expect to see number of words left to write stated and number of pages calculated with some comment about the part page still to be written)

Find the number represented by each * in these calculations and write out each calculation in full.

* . 4 * 4 9 . 2 * *+ 2 . * 8 * * . * 8 6 6 . 4 8 3 3 . 4 5 7

(expect to see correct calculations with oral explanation of process used to find them)

I can use my understanding of numbers less than zero to solve simple problems in context.

MNU 3-04a

Which is the higher temperature, – 50C or – 20C ?

(expect to see answer, with units, clearly stated)

A diver is 75 metres below sea level. A helicopter is hovering directly overhead 20 metres above sea level. How far is the diver from the helicopter?

(expect to see evidence of an addition calculation)

understand that below sea level may be stated as a negative number eg – 75 m



Multiples, factors and primes

I have investigated strategies for identifying common multiples and common factors, explaining my ideas to others, and can apply my understanding to solve related problems.

MTH 3-05a

I can apply my understanding of factors to investigate and identify when a number is prime.

MTH 3-05b

Find the lowest number which can be divided by 2 and 3 and 5.(expect to see eg multiples of 2, 3 and 5 listed until the LCM is found or multiplesof the largest number listed with clear indications of commonn multiples)

What is the highest common factor of 36 and 54?(expect to see eg factors of 36 listed until largest possible factor is identified as a factor of 54 also)

A lighthouse flashes every 5 seconds and another flashes every 6. If they flash at the same time, how long will it be before they flash at the same time again?(expect to see multiples of 5 and 6 used to find LCM)

Why is 12 not a prime number?(expect to see explanation involving factors of 12 other than 1 or 12)

Why is 11 a prime number?(expect to see stated that is is only divisible by itself and 1)

Find the prime factors of 48(expect to see a factor tree or repeated division by increasing primes 2 60 2 30 2 15 3 5 5 5 1 60 = 22 x 3 x 5

a clear understanding of the terms factor and multiple is needed before HCFs and LCMs can be used

primes, up to 100, would be found using Sieve of Eratosthenes

prime factors can be used to find HCF and LCM of larger numbers

Powers and roots Having explored the notation and vocabulary associated with whole number powers and the advantages of writing numbers in this form, I can evaluate powers of whole numbers mentally or using technology.

MTH 3-06a

What is the meaning of 42 ?

(expect to see 4 x 4)

Write 3 x 3 x 3 x 3 as a power of 3.

(expect to see 34)

What is the volume of a cube with sides 2.5 cm ?

( expect to see evidence of calculator used to find 2.53 and solution stated, including units)

60 2 30 2 15 3 560 = 22 x 3 x 5



Fractions, decimal fractions and percentages including ratio and proportion

I can solve problems by carrying out calculations with a wide range of fractions, decimal fractions and percentages, using my answers to make comparisons and informed choices for real-life situations.

MNU 3-07a

By applying my knowledge of equivalent fractions and common multiples, I can add and subtract commonly used fractions.

MTH 3-07b

Having used practical, pictorial and written methods to develop my understanding, I can convert between whole or mixed numbers and fractions.

MTH 3-07c

What is 35% of £200?

(expect to see eg use of 10% leading to 30% and 10% leading to 5%)

If 3/4 of the school take part in Sports Day, what percentage do not take part?

(expect to see eg 1/4 = 25%)

Evaluate: 1/2 + 3/5 Evaluate: 15/16 – 1/2

(expect to see equivalent fractions found prior to completion of calculation 5/10 + 6/10 = 11/10 = 11/10 )

Arrange in order from smallest to largest 2/5, 0.25, 0.2, 30%, 1/2

( expect to see conversion to common form, more usually decimal or percentage, before final list is made)

Arrange in order from largest to smallest 150%, 13/4, 1.85, 9/5

( expect to see conversion to common form, more usually decimal or percentage, before final list is made)

fluency of conversion from fraction to decimal to percentage and vice versa is to be encouraged, mental for widely used simple fractions and with a calculator for more demanding examples

may also see “kiss and smile” used followed by simplification of answer

correct order in the requested form is important to encourage careful reading and interpreting of questions



Fractions, decimal fractions and percentages including ratio and proportion (continued)

I can show how quantities that are related can be increased or decreased proportionally and apply this to solve problems in everyday contexts.

MNU 3-08a

3:2 = 24: (expect to see simple ratios calculated mentally)

If 3 apples cost 75 pence, how much would 7 cost?(expect to see cost of 1… calculated and stated as an interim step)

Colin makes lime and lemon juice by mixing 1 part lime to 3 parts lemon.How much lime juice is needed to mix with 12 litres of lemon?

(expect to see ratio stated and used lime : lemon 1 : 3 : 12 and appropriate statement made for final answer)

unitary step is important for both direct and inverse proportion

answer needs to be given in context

Money When considering how to spend my money, I can source, compare and contrast different contracts and services, discuss their advantages and disadvantages, and explain which offer best value to me.

MNU 3-09a

I can budget effectively, making use of technology and other methods, to manage money and plan for future expenses.

MNU 3-09b

Voicephone charges £24 for 600 minutes and Alltalk charges £20 for 400 minutes. Which company has the cheaper rate per minute?

(expect to see cost per minute calculated for each and followed by a clearly stated conclusion)

Several of Belinda’s relatives live abroad and send her money for her birthday.Altogether she receives $30, ¥10000, and € 40. How much did she receive in total, to the nearest £1?

(expect to see foreign currencies converted and total found)

Discuss then make a list of 5 usual monthly expenses and 2 unexpected expenses for a) an individual and b) a family of 2 adults and 3 children. How can these expenses be anticipated and budgeted for?

(expect to hear discussed regular expenses and reasonable estimates for these)

similar calculations would be needed dependant upon the context used

current exchange rates can be found from a variety of sources

expected expenses eg food, heating, clothes …..

unexpected eg car repair, new kettle ….



Time Using simple time periods, I can work out how long a journey will take, the speed travelled at or distance covered, using my knowledge of the link between time, speed and distance.

MNU 3-10a

A train leaves at 09 30 and travels 200 miles, arriving in Newcastle at 12 00.Calculate the average speed of the train.

(expect to see S = D/T used and answer expressed in appropriate units)

where appropriate time should be stated in hours and decimal fractions of an hour, in this case as 2.5



Measurement I can solve practical problems by applying my knowledge of measure, choosing the appropriate units and degree of accuracy for the task and using a formula to calculate area or volume when required.

MNU 3-11a

Having investigated different routes to a solution, I can find the area of compound 2D shapes and the volume of compound 3D objects, applying my knowledge to solve practical problems.

MTH 3-11b

If an item is 2000 mm long, is it likely to be a lorry, a table, a pencil or a football pitch?

(expect to see an answer indicating an understanding of the different units of measurement)

A tennis court is 10.8 m by 23.6 m. Calculate the area to 2 significant figures.

(expect to see use of appropriate formula, clear substitution and answer of form and accuracy requested)

Be able to find a compound area bya) splitting the shape into 2 or more smaller shapes and adding areasor b) completing the larger rectangle and then subtracting the “extra” area

Be able to find compound volume by find cross-sectional areaand multiplying by length of solid



Mathematics – its impact on the world, past, present and future

I have worked with others to research a famous mathematician and the work they are known for, or investigated a mathematical topic, and have prepared and delivered a short presentation.

MTH 3-12a

Work in a group to research the life and work of a famous mathematician and present clear findings as a talk, presentation or poster.

(expect to see contributions from all members of the group and all members able to discuss the findings. Finished presentation should indicate a knowledge and understanding of the mathematician’s work or impact)

mathematician could be chosen from those whose works are familiar by this stage



Patterns and relationships

Having explored number sequences, I can establish the set of numbers generated by a given rule and determine a rule for a given sequence, expressing it using appropriate notation.

MTH 3-13a

State the rule for continuing this sequence of numbers34, 26, 18, 10, ……..

(expect to see subtract 8 from the previous term)

A sequence is generated using the rule 3n + 2 . Starting with n = 1, list the first 4 terms of the sequence.

(expect to see terms listed)

x 1 2 3 4 5 6 7y 1 5 9 13

Find the next 3 numbers in the table and state the algebraic rule connecting y and x.

(expect to see table completed and rule y = 4x – 3 stated)

the rule must produce the lower numer from the upper number each time it is applied in the table



Expressions and equations

I can collect like algebraic terms, simplify expressions and evaluate using substitution.

MTH 3-14a

Simplify: 3x + 5y + 4x – 2y

(expect to see like terms collected)

If a = 2 and b = – 3, find the value of 4a + b2

(expect to see clear substitution made followed by calculation)

Having discussed ways to express problems or statements using mathematical language, I can construct, and use appropriate methods to solve, a range of simple equations.

MTH 3-15a

I can create and evaluate a simple formula representing information contained in a diagram, problem or statement.

MTH 3-15b

For 50p I can buy 4 pencils and and get 2p change. If a pencil costs x pence, construct and equations to show this informations and use it to calculate the cost of a pencil.

(expect to see equation stated, appropriate working steps and answer clearly stated in context asked.

A plumber charges a £50 callout charge and £25 per hour for his labour. Construct an equation to satisfy this information and use it to calculate the total cost of 4 hours of work.

(expect to see eg c = 25h + 50, followed by clear substitution and answer stated in appropriate units)

emphasis here is on developing ability to describe a situation in concise mathematical language rather than carrying out a simple calculation

Shape, position and movement


Properties of 2D shapes and 3D objects

Having investigated a range of methods, I can accurately draw 2D shapes using appropriate mathematical instruments and methods.

MTH 3-16a

Using a ruler and protractor, construct Δ ABC in which BAC = 500, AC = 7 cm and AB = 6 cm.

Using a ruler and compasses, construct Δ QPR in which PQ = 6 cm , PR = 7 cm and RQ = 5 cm.

(expect to see lengths accurate to 2 mm and angles to 2 degrees)

Construct a rhombus with diagonals of 3cm and 5 cm then measure each of the angles of the rhombus.

Construct a kite with diagonals of 3 cm and 5 cm.

(expect to see all produce the same rhombus but a variety of kites)

this could be extended to include bisection of lines and angles and construction of equilateral triangles using ruler and compasses only, with further extension to accurate drawing of 300

,, 450, 600 and multiples of these without use of protractors

properties of quadrilaterals need to be familiar before these constructions can be undertaken

Shape, position and movement (continued)


Angle, symmetry and transformation

I can name angles and find their sizes using my knowledge of the properties of a range of 2D shapes and the angle properties associated with intersecting and parallel lines.

MTH 3-17a

Having investigated navigation in the world, I can apply my understanding of bearings and scale to interpret maps and plans and create accurate plans, and scale drawings of routes and journeys.

MTH 3-17b

I can apply my understanding of scale when enlarging or reducing pictures and shapes, using different methods, including technology.

MTH 3-17c

Name and calculate the size of the angle a0.(expect to see QRK = 560, supported by clear working) Calculate the size of all the remaining angles on the diagram

(expect to see all 15 angles calculated)

Calculate bearings and distances from a simple map.

Work with others to produce a scale plan of a large known area eg school playing field and devise a trail / hunt for clues to be followed using bearings and distances.

(expect to see clear directions given as a table of bearings and distances)

The living room on an architect’s plan measures 72 mm by 66 mm.The enlargement factor used for the real room is 60.What are the dimensions of the real room?What is the area of the real room in square metres?

(expect to measurements x 60 then converted to metres for further calculation)

explanations of calculations are expected with use of correct language, eg alternate and corresponding angles, angle sum of triangle to be encouraged

bearings should be stated with 3 figures eg 0470

familiarity with mm is to be encouraged since many items of household furniture and equipment are described in mm not cm

a0

1180

Q K

R

B

850

1230

Shape, position and movement (continued)

Second and Third Questions / Activities to demonstrate Key Skills / Knowledge Notes

Angle, symmetry and transformation(continued)

I can use my knowledge of the coordinate system to plot and describe the location of a point on a grid.

MTH 2-18a / MTH 3-18a

Draw a coordinate grid with 0 < x < 10 and 0 < y < 10.Plot the points A(2, 5) B(3, 8) C(8, 8) D(7, 5) and join them up in order.What shape is ABCD?

(expect to see a parallelogram with vertices labelled)

axes should be labelled x and y and clear scale demonstrated with numbers small and to left of x-axis and small and underneath the y-axis. the origin should also be marked

I can illustrate the lines of symmetry for a range of 2D shapes and apply my understanding to create and complete symmetrical pictures and patterns. MTH 2-19a / MTH 3-19a

Show all lines of symmetry on the shape shown

(expect to see lines drawn eg 6 on shape shown)

Complete this symmetry drawing by reflecting the shape shown in both of the axes of symmetry

(expect to see shape reflected in both horizontal and vertical lines shown)

lines of symmetry are shown as broken lines and extend beyond the boundaries of the shape

Information handling


Data and analysis

I can work collaboratively, making appropriate use of technology, to source information presented in a range of ways, interpret what it conveys and discuss whether I believe the information to be robust, vague or misleading. MNU 3-20a / MNU 4-20a

When analysing information or collecting data of my own, I can use my understanding of how bias may arise and how sample size can affect precision, to ensure that the data allows for fair conclusions to be drawn.

MTH 3-20b

Gather statistical evidence from a variety of sources, eg paper media, internet sites and present findings for group discussion.

(expect to see a variety of sources – pictorial, graphical, raw data and discussion to highlight useful and less helpful aspects of each source dependant upon context)

Gather information about lunch habits of pupils in class by asking where they usually have lunch. Analyse results for a small group of friends, a whole class, several classes from different yeargroups and discuss findings.

(expect to see different patterns from larger and smaller groups and be able to explain why these differences have occurred)

.

need to be able to identify robust and misleading information presented in a variety of forms and to communicate the meaning of the information clearly

need to understand how bias can occur eg if data on filmgoing is collected from a cinema queueunderstand that a small sample size may give biased results

I can display data in a clear way using a suitable scale, by choosing appropriately from an extended range of tables, charts, diagrams and graphs, making effective use of technology.

MTH 2-21a / MTH 3-21a

Conduct a class survey and display and communicate results on an appropriate display.

(expect to see clear, unambiguous comments and displays)

graphs and charts should include a title

Information handling (continued)


Ideas of chance and uncertainty

I can find the probability of a simple event happening and explain why the consequences of the event, as well as its probability, should be considered when making choices.

MNU 3-22a

Fred has to pick a card to find out which day he is on duty as a Prefect.Calculate a) the probability that he picks Friday b) the probability that he picks another day

(expect to see P(Friday) = 1/5, P(not Friday) = 4/5 )

The probability of rain on Thursday, when Sports Day is planned, is 70%.Should Sports Day be postponed?What do you need to consider when you make this decision?

(expect to see a list of consequences from postponement and from not postponing the event)

formal notation should be used

there needs to be an understanding that all decisions have consequences which have to be given consideration

Numeracy and mathematics: experiences and outcomes 17

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Numeracy and mathematics: Experiences and outcomesdornochmaths.weebly.com/uploads/3/0/4/5/30459404/kn… · Web viewExperiences and outcomes. Numeracy and Mathematics Third Level