Transcript Level 5 slides (2010)

Level 5
mathematics
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Expectations and activities
Aims:
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• To become familiar with Level 5
expectations in mathematics
• To develop understanding of
Level 5 mathematical knowledge
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Are children at Key Stage 2 getting
the learning experiences they need
to meet the Level 5 expectations?
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How can activities be developed to
create these opportunities for
Level 5 children?
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Task:
• Using the Level 4 – 5 APP assessment
guidelines, identify the additional skills,
knowledge and concepts needed to
achieve a Level 5 from a Level 4
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• Highlight which of the identified aspects
the children find most tricky to learn
Here is an equilateral triangle inside a rectangle.
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Not to scale
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x
12°
Calculate the value of angle x.
Do not use a protractor (angle measurer).
Show
your method.
You may get
a mark.
º
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2 marks
It is not enough for children to know
the facts, they need to be able to
apply their knowledge to the
problem, in what may be an
unfamiliar context for them, in order
to succeed at Level 5
(argue and reason)
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Algebra
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Level 5 algebra expectations
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• Use letter symbols to represent unknown numbers or
variables
• Know and use the order of operations and
understand that algebraic operations follow the same
conventions and order as arithmetic operations
• Simplify or transform linear expressions by collecting
like terms; multiplying a single term over a bracket
• Substitute integers into simple formulae
• Use and interpret coordinates in all four quadrants
• Plot the graphs of simple linear functions
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Algebraic Conventions
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• Recognise
and0100
explain
3x + 5 = 11
p + q = 20
2l + 2b = p
y = x/2 – 7
use of symbols
Represent an unknown value in equations with a unique solution
Represent unknown values in equations with a set of solutions
Represent variables in formulae
Represent variables in functions
• Identify equivalent terms and expressions
2x + x + 5
ax + 5
7(x + 2)
(x + 2)(x + 5)
x³ × x
simple chains of operations
some with unknown coefficients
brackets (linear)
brackets (quadratic)
positive indices
• Identify types and forms of formulae
a/b = l, a = l × b
a = l × b, 2l + 2b = p
equivalence of formula
dimensions of a formula
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Mints
A teacher has 5 full packets of mints and 6 single mints.
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The 1101
number
of 0100
mints1011
inside each packet is the same.
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The teacher tells the class:
“Write an expression to show how many mints there are altogether.
Call the number of mints inside each packet y”
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Here are some of the expressions that the pupils write:
5+6+y
5y6
5y + 6
5 + 6y
(5 + 6) x y
6 + 5y
Write down two expressions that are correct.
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Jill took 2 marbles out of one of the
Jill has 3 bags of marbles. bags, and none out of the other bags.
Each bag has p marbles
inside.
Jill takes some marbles out.
Now the total number of
marbles in Jill’s 3 bags is
3p – 6
Some of the statements
on the right could be true.
Put a tick by each
statement which could be
true.
Jill took 2 marbles out of each of the
bags.
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Jill took 3 marbles out of one of the
bags, and none out of the other bags.
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Jill took 3 marbles out of each of two of
the bags, and none out of the other bag.
Jill took 6 marbles out of one of the
bags, and none out of the other bags.
Jill took 6 marbles out of each of two of
the bags, and none out of the other bag.
Solving Linear equations
• One-step linear equations
One-step
linear
equations
the unknown
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1011 in a ‘standard’ position:
x+4=7
positive integer solutions
x/4 = 6
x – 9 = 34
8x = 56
3x = 5
non-integer solutions
x + 14 = 9
negative integer solutions
One-step linear equations with the unknown perceived to be in a ‘harder’ position:
13 = 8 + x
positive integer solutions
20/x = 10
20/x = 3
non-integer solutions
13 = 8 – x
negative integer solutions
• Equations involving brackets
3(x + 4) = 27
(x – 5)/3 = 7
positive integer solutions
• Inequalities
5x < 10
–4 < 2x < 10
5x + 3 < 10
one boundary to solution set, one-step solution
two boundaries to solution set
two-step solution
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Algebra Pairs
Join pairs of algebraic expressions that have the same value when
a = 3, b = 2 and c = 6
0011 One
0010 pair
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is 1101
joined
for you.
ab
3c – 2b
3c
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2c + b
Which expressions
have the same value
when a = b = c
a2
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a+c
2a
Sort these equations into “always true”,
“sometimes true” or “never true”:
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a + 5 = 12
b + 12 = b + 16
2c + 3 = 3 + 2c
2d – 5 = 5 – 2d
f + 12 = g + 12
k + 5 < 20
4 + 2e = 6e
p2 = 10p
3(m + 3) = 3m + 3
4(3 + n) = 12 + 3n
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Sequences
• Linear sequences:
find and describe in words and symbols
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the 1010
rule for
the0001
next 0100
term and
Use the context of a sequence to generate the related numerical terms
Notice and describe how the sequence is growing term by term and relate this to the context
Notice and describe a general term in the sequence and relate this to the context
Appreciate different forms for the general term and relate each to the context
Generate different forms for the general term and relate each to the context
• Quadratic sequences:
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find and describe in symbols the rule
for the next term and the nth term of a sequence
Repeat the progression described above
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• Recognise and describe types of sequences:
for example, arithmetical sequences and multiples, triangular numbers,
square numbers…
Use knowledge of related geometrical patterns
Use differences to test for types of sequence
Sequences
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1. Use the context of a sequence to generate the related numerical terms
2. Notice and describe how the sequence is growing term by term and relate
this to the context
3. Notice and describe a general term in the sequence and relate this to the
context
4. Appreciate different forms for the general term and relate each to the context
5. Generate different forms for the general term and relate each to the context
Number Chains
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Each number chain has a similar property to the
Fibonacci sequence – that is, each term is the
sum of the previous two.
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Find the missing terms:
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4
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… … … 18
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…
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4
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Functions and graphs
• Interpret graphs of functions
• Generate graphs of functions
• Interpret graphs arising from real-life
problems
• Generate graphs arising from real-life
problems
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Level 5 expectations
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• Generate and plot pairs of coordinates
for:
y = x + 1, y = 2x
• Plot graphs such as:
y = x, y = 2x
• Plot and interpret graphs such as:
y = x, y = 2x and y = x + 1, y = x – 1
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Are children at Key Stage 2 getting
the learning experiences they need
to meet the Level 5 expectations?
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How can activities be developed to
create these opportunities for
Level 5 children?
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