DEVELOPING THE BIG IDEAS IN NUMBER

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DEVELOPING THE BIG IDEAS IN NUMBER
A professional development day for Numeracy
Coordinators and Teachers of Mathematics Presente
d by Professor Di Siemon, RMIT University SA
DECS, Sunnybrae Farm 29 May 2008
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THE BIG IDEAS IN NUMBER

• Trusting the count
• Place-value
• Additive strategies
• Multiplicative thinking
• Partitioning
• Proportional Reasoning

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Additive Strategies - Overview

• Pre-requisites - trusting the count and
  place-value
• Developing addition and subtraction - Early ideas
  and strategies
• Initial recording of addition and subtraction
  facts
• Formal recording for addition and subtraction
• Applying what is known

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TRUSTING THE COUNT is evident when children

• know that counting is an appropriate response to
  questions which ask how many
• believe that counting the same collection again
  will always produce the same result irrespective
  of how the objects in the collection are changed
  or manipulated
• are able to invoke a range of mental objects and
  work flexibly with numbers 0 to ten using
  part-part-whole knowledge and/or visual imagery
  without having to make or count the numbers and
• are able to use small collections as units when
  counting larger collections (e.g., twos, fives,
  tens).

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PLACE-VALUE
Children can be formally introduced to
place-value as a system of recording numbers when
they

• can comfortably count to 20 and beyond
• are well-acquainted with the numbers 0 to 10 in
  terms of their parts (part-part-whole relations)
• can work flexibly with numerals to 10 without
  having to model the count (trust the count)
• can interpret/visualise numbers beyond ten in
  terms of 1 ten and 4 more, fourteen
• recognise numbers to 10 as countable units for
  the purposes of counting, eg, 2, 4, 6, 8 ...)

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INTRODUCING PLACE-VALUE

• Establish the new unit 10 ones is 1 ten
• Introduce the names for the multiples of ten.
• Make, name and record regular examples of the
  2-digit place-value pattern
• Make, name and record the teen numbers.
• Consolidate through comparing, ordering, counting
  forwards and backwards in place-value parts and
  renaming.

Place-value is all about pattern recognition and
use it is essentially multiplicative
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DEVELOPING ADDITION SUBTRACTION
Concepts join, combine, take-away, missing
addend, and difference
Counting strategies make-all/count-all, cover
and count on, count on from larger displayed
Mental strategies count on from larger (1, 2 and
3 only), doubles and near doubles, make-to-ten,
think of addition
Initial recording vertical to support
place-value, avoid premature use of sign
Mental computation open number lines
Extended recording 2 digits and beyond, decimals
and fractions
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CONCEPTS FOR ADDITION
Join Two birds on the grass. Three more join
them. How many birds are there now?
Combine There are 4 bananas and 3 apples in the
bowl, how many pieces of fruit are there
altogether?
Explore through action stories no recording at
this stage
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CONCEPTS FOR SUBTRACTION
Take-away 8 birds on the grass. 3 fly away. How
many birds are left on the grass?
Missing Addend Jamie-Lee has 5 Pokemon cards.
She would like to have 9. How many more cards
does she need?
? more
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CONCEPTS FOR SUBTRACTION (cont.)
Difference Kate has 3 smarties. Belinda has 7.
How many more smarties does Belinda have than
Kate?
? Difference
NB Students find this idea harder much harder
than take-away and missing addend.
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Counting Strategies

• Make all count all (physical models, one-to-one
  correspondence)
• Count on from covered (physical models and
  numerals, count on from hidden part)
• Count on/back from larger (physical models and
  numerals, count on/back from larger number)
• Skip counting (physically count by twos, then
  fives and tens)

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MENTAL STRATEGIES
Pre-requisites

• Children know their part-part-whole number
  relations (eg, 7 is 3 and 4, 5 and 2, 6 and 1
  more, 3 less than 10 etc) and doubles
• Children trust the count (that is, they have
  access to mental objects for numbers 0 to 9) and
  can count on from hidden or given
• Children have a sense of numbers to 20 and beyond
  (eg, 10 and 6 more, 16) including doubles to 20

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1. Count on from larger for combinations
involving 1, 2 or 3 (using commutativity)
For example, for 6 and 2, THINK 6 7, 8 for 3
and 8, THINK 8 9, 10, 11 for 1 and 6, THINK
6 7 for 4 and 2, THINK 4 5, 6
Scaffolded initially by ten-frames, dice and oral
counting
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For example
Cover 5, count on
paper plates
Cover 4, count on
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2. Doubles and near doubles
For example, for 4 and 4, THINK double 4,
8 for 6 and 7, THINK 6 and 6 is 12, and 1 more,
13 for 9 and 8, THINK double 9 is 18, 1 less,
17 for 7 and 8, THINK double 7 is 14, 1 more, 15
Ten-frames and bead frames (to 20) can be used to
build doubles facts
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For example
Ten-frames
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For example
Count 6 and 6 is 12, and 1 more, 13
Bead Frame (to 20) Double-decker bus scenario
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3. Make to ten and count on
For example, for 8 and 3, THINK 8 10, 11 for
6 and 8, THINK 8 10, 14 for 9 and 6, THINK
9 10, 15 for 7 and 8, THINK double 7 is 14, 1
more, 15
Ten-frames and bead frames (to 20) can be used to
bridge ten, build place-value facts
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For example
For 8 and 6
20
For example
Think 10 and 4 more, fourteen
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Extend Mental Strategies to Subtraction
For example, for 9 take 2, THINK 9 8, 7
(count back) for 6 take 3, THINK 3 and 3 is 6
(think of addition) for 15 take 8, THINK 15,
10, 7 (make back to 10)
For 16 take 9, THINK 16 take 8 is 8, take 1
more, 7 (halving) 16, 10, 7 (make back to
10) 9, 10, 16 7 needed (think of
addition) 16, 6, add 1 more, 7 (place-value)
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Can be used to consolidate mental strategies
Beat the Teacher
Game 1
Game 3
Game 2
Me Teacher ____ ____ ____
____ ____ ____ ____ ____ ____
____ ____ ____ ____ ____ ____
____ Totals
Me Teacher ____ ____ ____
____ ____ ____ ____ ____ ____
____ ____ ____ ____ ____ ____
____ Totals
Me Teacher ____ ____ ____
____ ____ ____ ____ ____ ____
____ ____ ____ ____ ____ ____
____ Totals
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Some more examples, The difference between two
numbers is 5. What might the two numbers
be? Here are two pieces of string. What is the
difference in their length? Measure your own
and your best friends height. Are you taller or
shorter than your friend? By how much?
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INITIAL RECORDING
Record once mental strategies established Record
vertically to support place-value and
commutativity (eg,count on from larger), and to
eliminate difficulties with the equal sign (which
does not mean makes). For example,
6 7 13
19 - 8 11
2 and 3 5
2 3 5
6 8 14
16 - 9 7
Strategies?
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INTRODUCE THE EQUAL SIGN
via well-known equivalences such as
Why are there two numbers on the right?
5 4 6 3
Note This requires a confident understanding of
numbers 0 to 10 in terms of part-part-whole. In
this case, that 9 is 6 and 3, 5 and 4, 7 and 2,
.
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DEVELOP WRITTEN COMPUTATION ()
Introduce when basic addition fact knowledge is
established and extended For example, for 28 and
6, THINK 30 34
Use bundling or stacking materials to model and
record 2-digit addition
Without regrouping With regrouping
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For example
tens ones
1 4 8 3 6 8 4
tens ones
Model with materials (bundling sticks and
MAB) Explore strategies
4 3 5 2 9 5
3 ones and 2 ones is 5 ones 4 tens and 5 tens is
9 tens
8 ones and 6 ones is 14 ones, regroup or rename
as 1 ten and 4 ones and record appropriately. 1
ten and 4 tens and 3 tens is 8 tens
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DEVELOP WRITTEN COMPUTATION (-)
Introduce when basic subtraction fact knowledge
is established and extended For example, for 74
take 8, THINK 74, 70, 66
Use bundling or stacking materials to model and
record 2-digit subtraction
Without trading With trading
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This method is called decomposition
For example
tens ones
4 13
tens ones
Model with materials Explore strategies
8 6 - 5 2 3 4
5 3 - 3 7 1 6
3 ones, cannot take 7 ones, trade or rename 1
ten as 10 ones, record as 13 ones and 4 tens 13
ones take 7 ones, 6 ones 4 tens take 3 tens is 1
ten
6 ones take 2 ones is 4 ones 8 tens take 5 tens
is 3 tens
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SUPPORT MENTAL COMPUTATION
Open Number Lines
Eg, for 43 and 28
add 1
43 53 63 70 71
START
make to next ten
add 2 tens
Eg, for 82 take 47
take 5 more
take 4 tens
35 40 42 82
START
make back to ten
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EXPLORE WRITTEN COMPUTATION
For example,
34 28 26 47 52 93 45
Look for make to ten facts 3 tens and 5 more,
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THINK 9 and 9, 18 2 more, 20 ... 5 more, 25
and 7 more 30 32 tens
Try doing this in other ways
3

325
NB This is not expected in Early Years!
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EXTEND ADDITION SUBTRACTION
Think about the materials and/or representations,
language and recording that you might use to
support written solutions for the following
4004 2677
45,782 56,839
-

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For example,
Why is MAB inappropriate here?
4004 2677
-
thousands
4
0
hundreds
0
4
tens
ones
4
0
0
4
tens
ones
Number Expanders are more useful for larger
numbers and decimals
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Solutions to computation problems involving
decimal fractions is a logical and natural
extension of the language and recording used for
whole numbers
9.6 4.7
6 tenths Can I take 7 tenths away? No Need to
trade 1 one for 10 tenths. 16 tenths take 7
tenths is 9 tenths record with the tenths 8
ones take 4 ones is 4 ones record with the ones
-
8 16 9.6 4.7
8 16 9.6 4.7 4.9
-
-
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WORKING WITH FRACTIONS
Think about the materials and/or representations,
language and recording that you might use to
support written solutions for the following
3? 1?
4 fifths 3 fifths
3 quarters 1 half

-

Note Students are expected to be able to solve
these problems by the end of Year 4
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Ideally, solutions to these should be arrived at
mentally via appropriate thinking strategies
THINK 7 fifths 1 whole and 2 fifths
4 fifths 3 fifths

3 quarters 1 half
THINK 1 half is 2 quarters 5 quarters
altogether so, 1 and 1 quarter

Although Fraction Bars and paper-folding are
useful aids if needed.
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Add and subtract whole numbers
23417 45809
Language Materials??? Models??
7 0 4 2 - 4 5 8 9
For subtraction 2 ones take 9 ones? No ...
Trade 1 ten for ones ... 3 tens take 8 tens? No
... Trade 1 hundred for tens ...
3 12

• 7 0 4 2
• 4 5 8 9
• 3

Use a Number Expander to rename
hundreds
7 0
13 12
6 9

• 7 0 4 2
• 4 5 8 9
• 2 4 5 3

13 tens take 8 tens? ... 9 hundreds take 5
hundreds? ... 6 thousands take 4 thousands? ...
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Add and subtract decimal fractions
32.5 17.9
Language? Materials??? Models??
7 0.4 2 - 4 5.8 9
For subtraction 2 hundredths take 9 hundredths?
No ... Trade 1 tenth for hundredths ... 3 tenths
take 8 tenths? No ... Trade 1 one for tenths ...
3 12

• 7 0. 4 2
• 4 5. 8 9
• 3

Use a Number Expander to rename
ones
7 0
13 12
6 9

• 7 0. 4 2
• 4 5 .8 9
• 2 4. 5 3

13 tenths take 8 tenths? ... 9 ones take 5 ones?
... 6 tens take 4 tens? ...
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Add LIKE and RELATED fractions
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Thinking? Language?
3 fifths and 4 fifths 7 fifths
1 ¼ 3 ¾
1
Think 1 half is 2 quarters 2 quarters and 3
quarters is 5 quarters
2 ½ 5 ¾
3
3/9
1 1/3 5 2/9
4
Thinking? Language?
6 5/9
Rename and regroup simple common fractions
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Add UNLIKE fractions

Rename and regroup simple common fractions
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THINK thirds by fifths ... fifteenths
thirds
fifths
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Subtract LIKE and RELATED fractions
3 fifths take 1 fifth 2 fifths
6 ¾ - 2 ¼
1
2
4 1/3 - 1 2/9
Think 3/9 take 2/9 is 1/9
3
2 quarters take away 3 quarters? No, need to
trade 1 one for 4 quarters 6 quarters take 3
quarters is 3 quarters 6 ones take 5 ones is 1
6 6/4
4
7 ½ - 5 ¾
1 3/4
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Subtract UNLIKE fractions
Rename and trade simple common fractions
5
THINK thirds by fourths ... twelfths
thirds
quarters
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APPLYING WHAT IS KNOWN
Tallies eg, dice sums
Highest sum? Lowest sum? Keep a record? What do
you notice?
Graphs and Charts eg, birthdays,
eye-colour Money eg, I have 2.50 in my pocket.
What coins might I have? Measurement eg, Find
something that is longer than 53 cm but shorter
than 94 cm
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Problem solving and open-ended questions also
provide a valuable means of developing and
assessing students number ideas and strategies.
Prices Large 85c Small 65c
Sally bought an apple. How much change did she
have from 1?
I have 23.65 in my pocket, what coins and/or
notes might I have in my pocket? How can you be
sure you have found them all?