Maths – Suggested task.
This task allows you to assess the children’s current understanding of fractions, enabling you toevaluate what pupilsneed to experience next in order to develop a broader and deeper understanding. This task also looks at a ‘whole’ which is divided into pieces which are not the same size. The more that children can physically play with different concepts of the ‘whole’ and ‘parts’ (including parts which look different but are equivalent) the deeper their conceptual understanding of fractions will be.
Throughout the activities, encourage pupils to devise their own questions by saying ‘What questions do you have about this so far? Is there anything else you are wondering about?’ This will help you assess against the reasoning section of the ‘Working Mathematically Document’
You will need:
- Sets of tangrams (at least one between two, see attached pdf- cut up in sets, plus a few extra sets)
- Sugar paper
- Marker pens
Getting Started:
You may wish to share the background story of tangrams to the children, if they haven’t seen them before.
This is a simple story of the tangram – all the characters in the video are made from the tangram pieces.
Familiarisation
Allow them to explore and play with the shapes:
- What do you notice about the shapes?
- Which shapes can you name?
- Can you sort the shapes? Can you sort them in another way?
- Can they make any of the characters/objects/ shapes from the story?
- Can you make a different picture with your tangram? Can you explain to your friend how to make the picture (using shape and positional language)?
Warning: most children will try to make the large square out of the seven shapes, some will be successful, others not so. Please limit the time for this and provide an image of the solution so that all the children know how the shapes fit together to make the square. This prevents children becoming overly frustrated with making the ‘whole’ rather thanthinking about fractional parts.
Introduction:
Ask the children to find the small triangle and the small square. What is the relationship between these two shapes? Allow time for discussion, encourage the children to rotate the shapes and place them on top of each other etc.
The small triangle is ……(half)…….. of the small square
Or
The small square is ….. (double)…… the small triangle
If the small square is worth ….. (4), what is the small triangle worth? (2)
If the small triangle is worth 6, what is the value of the small square?
Ask the children to work in pairs to order the shapes.
Assessment point:
Watch the children carefully to see how they do it – do the children put shapes on top of each other, rotating them etc. Possibly ask some children to explain their methods and reasoning focusing on the parallelogram.
Task 1: Understanding fractions
1) The children need to find out the fractional value of each shape in the tangram if the small square is worth 1, so the small triangle is worth …(half) - model this for the children if needed.
2) They are then going to order them on a number line (border roll stuck on to sugar paper, or a line drawn on sugar paper) and write the fractional values for each of the shapes and explain how they found out. Model with the small square and small triangle.
Independent/ paired/ small group learning: Depending on how confident the children are (some children may stay with the original modelled task above)
Encouraging further exploration questions:
If the parallelogram is 1/8 – can you find any combinations of shapes that are equivalent to 1/8?
Show me how you knew.
What fraction of the whole square would 2 parallelograms be?
Can you find a shape that also represents ¼? Are there any more?
Going deeper and broader ideas
If you need something more challenging start by telling the children the value of the original large square, for example, is 2 ½ . What are the values of the other shapes?
Alternatively, there are many different forms of tangram available on the internet. Allow the children to explore the fractional possibilities in the more complex ones (some have very intriguing names such as the Tormentor, the Cross Breaker) – attached.
Task 2: Finding fractional quantities
In each case, ask the children to calculate with the fractions they have been working on, for example. They can add this to sugar paper/number lines diagrams – stressing that you want to know how they have worked it out.
If you have been using the small square = 1 Task then…
Imaginethat the Medium triangle sells for £2.
What is the value of all the other shapes, including the original large square?
If you have been using the large triangle = 1 Task then…
Imagine that the large triangle sells for £8.
How much would I have to pay for the large square?
If you have been using the small square = ¼Task then…
Imagine the original large square sells for £1.60p.
What is the value of each shape?
Extension suitable for all:
Ask the children to give a value to one of the shapes. Calculate the total for the large square. Then challenge a friend to find the value of the shapes given their total for the large square.
Assessment points:
Working Mathematically
This activity provides opportunities to assess a number of strands within ‘Working Mathematically.’Can the children explain their thinking and the steps they took to solve the puzzles? Some children might be able to show you how they solved it, but may need support in writing this down.
NOTE: it doesn’t have to be written sentence explanations, drawings/images etc. are acceptable – allow them to explain it in their own way, some children may only be able to verbally explain – using the tangram and number line to support them (try to capture these conversations and support them in written/diagrammatic explanation’s)
Including:
Does the pupil make suggestions of ways to tackle the problems? Possible evidence for - Ideas, questions and lines of enquiry
Did the pupil raise further questions themselves or as a part of a group that they wanted to explore? Possible evidence for - Ideas, questions and lines of enquiry/ Reasoning
How securely was the pupil able to make connections to prior learning? E.g. did they begin to refine predictions based on knowledge of former tasks during the sequence? Possible evidence for - Ideas, questions and lines of enquiry/ Reasoning
How effectively was the pupil able to justify responses made? How well do they support their justification using the tangram shapes to convince others?How accurately are they able to use mathematical language when explaining? Possible evidence for – Reasoning / Represent and communicate
Is the pupil beginning to check for accuracy? Possible evidence for - Represent and communicate
Understanding of fractions
Write the fractions correctly? Possible evidence for F1 – recognise, find, write, name and count fractions.
Order them ?Possible evidence for F3 – comparing and ordering fractions
Do they put the shapes along the number line in approximately proportionally accurate places? If you ask them to, can they (this is 0 and this is 1, where do they go?)? Possible evidence for F3 – comparing and ordering fractions
How do the children manage shapes that have the same value but look different? Can they explain it? Do they use the language of equivalence/equal? Possible evidence for F2 – equivalent fractionsCan they think of another way of saying 1/2, do they know the equivalents? Do they know 2/4? Can they take it further? Do they know another way, and another etc….Possible evidence for F2 – equivalent fractions
How are pupils using the size of other pieces to discover the size of the one they are working on? Are the children seeing the relationship of the sizes in the pieces? Possible evidence for F3 – comparing and ordering fractions
How do they get to the value of the large square?Possible evidence for F4 – add/subtract fractions
How do they calculate the value of the shapes when they are given a monetary value? Possible evidence for F10 solve problems with fractions and decimals and/or F1 recognise, find, write, name and count fractions, particularly with reference to year 2 – write simple fractions (e.g. ½ of 6 = 3) and year 3 - find fractions of a discrete set, but do consider year 4 (non-unit fractions where the answer is a whole number).
Strand / EYFS 40-60+ months / National curriculum reference Year 1 / National curriculum reference Year 2 / National curriculum reference Year 3 / National curriculum reference Year 4 / National curriculum reference Year 5 / National curriculum reference Year 6F1
Recognise, find, write, name and count fractions / ELG 11 they solve problems, including doubling, halving and sharing / 1F1a
Recognise, find and name a half as one of two equal parts of an object, shape or quantity / 2F1a
Recognise, find, name and write fractions 1/3, 1/4, 2/4 and ¾ of a length, shape, set of objects or quantity / 3F1a
Count up and down in
tenths; recognise that
tenths arise from dividing
an object into 10 equal parts and in dividing one-digit numbers or quantities by 10 / 4F1
Count up and down in hundredths; recognise that hundredths arise when dividing an object by a hundred and dividing tenths by ten
1F1b
Recognise, find and name a quarter as one of four equal parts of an object, shape or quantity / 2F1b
Write simple fractions [e.g.: ½ of 6 = 3] / 3F1b
Recognise, find and write
fractions of a discrete set
of objects: unit fractions
and non-unit fractions
with small denominators
F2
Equivalent fractions / 3F1c
Recognise and use
fractions as numbers:
unit fractions and non-unit fractions with small
denominators
2F2
Recognise the equivalence of 2/4 and 1/2 / 3F2
Recognise and show, using diagrams,equivalent fractions withsmall denominators / 4F2
Recognise and show,using diagrams,families of common
equivalent fractions / 5F2a
Recognise mixed numbers and improper fractions and convert from one form to the other; write mathematical statements >1 as a mixed number [e.g.: 2/5 + 4/5 = 6/5= 1 1/5] / 6F2
Use common factors to simplify fractions; use common multiples to express fractions in the same denomination
F3
Comparing and ordering fractions [KS2] / 5F2b
Identify name and write equivalent fractions of a given fraction, represented visually, including tenths and hundredths
3F3
Compare and order unit fractions and fractions with the same denominators / 5F3
Compare and order fractions whose denominators are all multiples of the same number / 6F3
Compare and order
fractions, including
fractions >1
F4
Add / subtract fractions [KS2] / 3F4
Add and subtract fractions with the same denominator within one whole [e.g.: 5/7 + 1/7= 6/7] / 4F4
Add and subtract fractions with the same denominator / 5F4
Add and subtract fractions with the same denominator and denominators that are multiples of the same number / 6F4
Add and subtract fractions with different denominators and mixed numbers, using the concept of equivalent fractions
F10
Solve problems with fractions and decimals [KS2] / 3F10
Solve problems that involve 3F1–3F4 / 4F10a
Solve problems involving increasingly harder fractions to calculate quantities and fractions to divide quantities, including non-unit fractions where the answer is a whole number / 5F10
Solve problems involving
numbers up to three
decimal places / 6F10
Solve problems which require answers to be rounded to specified degrees of accuracy
4F10b
Solve simple measure and money problems involving fractions and decimals to two decimal places
Answers:
Small square = 1
Small triangle = ½
Parallelogram = 1
Medium triangle = 1
Large triangle = 2
Large triangle = 1
Small square = ½ or 2/4 or …
Parallelogram = ½ or 2/4 or …
Medium Triangle = ½ or 2/4 or …
Small triangle = ¼
Large Square = 3 ¾
Small square = ¼ or 2/8, ….
Small triangle = 1/8
Parallelogram = ¼
Medium triangle = ¼
Large triangle = ½
Large square = 2
The parallelogram is worth 1/8. What is the value of the large square?
If you need something more challenging start by telling the children the value of the original large square, for example 2 ½ .