The Place-Value Domain
The Origins of the Place-Value System
It is generally accepted that sometime around the fifth century CE the place-value system was
discovered, or created, in India. Three other civilisations went close to the discoverywithout quite getting
there – the ancient Chinese, the Babylonians, and the Aztecs. Given that it was discovered just yesterday in
human history, and in only one place, we can confidently assert that it is very difficult to understand.
knowing that it has literally replaced all other number systems in the world we can suggestthatit is simply
the best system. This means that teachers and parents need to understand that while students will find
learning the place-value system difficult, and time consuming it is of central importance that to become
numerate adults they need understand system is deeply and flexibly.
The Core of Place-Value
Thereare two central ideas that make place-value so effective as a number system.Firstly,there is
theplace-holder zero. Secondly, and much more importantly,there is the need to understand that if, as a
result of an addition or multiplication, the numeral in any column in a place-value table exceeds 9, ten of
these units must be replaced by a unit that is worth ten times as much, which isthen written in the next
column. And conversely, for subtraction or division problems, when a unit need to be broken down, it
must be into ten equal sub-units.
For example, adding seventy and sixty produces thirteen tens. But the maximum numeral that may be
placed in the tens column is 9, so ten tens must be swapped for one hundred. Now thereis one hundred,
and three tens. In order to indicate this, 1 is placed in the hundreds column, 3 is placed in the tens column
and 0 is placed in the ones column. So this calculation has led to the number130. This simple example
required the two great ideas of place-value, namely trading ten units for one larger unit, and the use of the
place-holder zero.
A Small Diversion About Symbols and Words
In the problem above technically this is not the number 130;rather it a numeral130 because a numeral is
symbol used to represent a number. But this distinction is somewhat pedantic; in practice number is usually
used when what is meant is numeral.
A more serious problem is whether to say 130 as one, three, zero, which is undoubtedly correct, and not
allow it to be read aloud as one, three, (letter) o. Many adults, knowing 0 is zero will say o without any
confusion. However, teachers and parents would be advised to teach 0 as zero – then at least they cannot
never be accused of teaching something thatis wrong!
Place-Value and Part-Whole Thinking
The idea that ten of an object can be equivalent to one of another object is not straightforward for
young students; teachers know that small children,when asked to choose between ten one dollar coins and
a ten-dollar note with which to buy sweets, will choose the coins because they can see more of them. The
idea that ten ones and one ten are equivalentis a profoundlyimportant idea; children need to understand
that a combination of parts can be equivalent to a whole even if they do not look the same on materials.
Without this conceptchildren will forever find numbers a mystery.
A Domain for Place-Value Materials
Within the Place-Value Domain that follows there is a subliminal domainrelating to the kind of materials
used at different stages. It begins with the most concrete representation of ones and tens in which ten
sticks, or similar material, are bundledby students into tens. In suchrepresentations students areencouraged
to view a bundled set of ten in two ways; eventually they will be ableto view the bundle as a unit while
simultaneously looking within the bundle to see there are ten smaller units namely ten ones. (In a number
of books “units” mean “ones”; this is avoided here because there are actually an infinite number of different
“units” needed for place-value.) This part-whole connection is crucial and difficult , so teachers are advised
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not to rush this work. Other representations of two-digit numbers, for example students creating bags of
ten objects, are very similar to bundling.
Later place-value blocks can be introduced.These are similar to bundled material in that ten ones is
physically the same as one ten. However, a slight complication is that, unlike a bundle that can be
unbundled to show ten ones, with place-value blocks one ten rod must be exchanged for ten small cubes.
A representation that is particularlypowerfulfor numbers between 1 and
100 is the Slavonic abacus. It consists of ten rows of ten beads. Each of the
first five rows has two groups of five beads that havetwo different colours.
Similarly, each of the nextfive rows has a different pair of colours. So rows
have a quinary (five) base, and within a row there is also a quinary base. Since
most children can identify five objectsthis is a powerful way to identify
numbers from one to a hundred. For example, in the abacus shown (Figure
1),there are six complete rows to the right – this is easy to see because the
colourings show five rows plus one more row – and eight to the right on the
Figure 1
seventh line – again this is easy because the colourings show five ones plus
three ones. So altogether there are 68beads on the right. In the same way there are 32 on the left.
Students who can reliably identify numbers from 1 to 100 on the Slavonic Abacus havea really solid
knowledge of the part-whole relationships involved inunderstanding place-value.
The last kind of place-value materials is non-representational; open abacuses and play money are
examples of this. They differ from representational materials in that ten ones does not look like one ten,
ten tens does no look like one hundred, and so on. For these materials to be used effectively, students
need to understand that ten ones is identical to one ten, ten tens is identical to one hundred, and so
on,evenwhenthey do not look identical.
The huge advantage that the non-representational materials have over representational materials is that
large numbers can bemodelled easily. For example, trying to represent ten thousand with bundled sticks
would require ten-thousand individual sticks that are bundled suitably. Creating a bundle of ten-thousand is
physically impractical, whereas with play money ten-thousand is represented by a single note.
Zero, the Number Line, and Place-Value
Students will encounter the symbol for zero, namely 0, early in their school careers. Unfortunately this
zero has two distinct meanings, which can cause considerable confusion.
Firstly,the idea that zero is a number is problematic. Even the ancient Greeks, who were rather good at
mathematics, did not believe zero is a number. And a problem like “I have six apples and grandma gives me
zero apples – how many apples do I now have” is clearly absurd. Consequently zero as a number should be
avoided as long as possible in students’ development of number concepts.
Unfortunately,many teachers of young students use number lines at a time when it is impossible for
these students to understand the zero at one end of a number line. Just reflect on what a mess they make
at this age of reading rulers, which are,in effect,number lines. Understanding of number lines also needs
understanding of fractions, so it strongly suggested that they are used only with much older students.
On the other hand, using 0 as a place-holder for place-value is unavoidablegiven its central importance in
writing numbers in digits. Its presence signals that “this column is empty”, and need not be read out loud.
For example, 306 is “three hundred and six” not “three hundred, no tens and six”.
Confusingly thezero as place-holder is not the same as zero as a number. So zero as a number should
be delayed for many years at school, but zero as a placeholder has to be introduced early.
Stories and Questions
A major problem for many studentsthat is not necessarily spotted by their teachers is that they do not
connect a story, a question arising from the story, and the related equation that represents the story and
question.For example, a diagnostic question might be:
•
Write a story for
+ 23 345 = 41 234.
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An incorrect story would be: I bought a car costing $41 234. I made a deposit of $23 345. How much
do I still owe?
The reason this is wrong is that it is a story for 41 234 – 23 345 =not + 23 345 = 41 234.
Reading the original equation from left to right a story that is correct could be:
•
I had some money in the bank and I won $23 345 on Lotto. I put the money in the bank. Now my
bank balance is $41 234.
At this point it is not necessarily obvious to students that there is another important thing to do, namely
ask a question. Here it would be:
•
How much money did I have in the bank to start with?
The creation of stories and questions is centrally important because it rapidly reveals which students
have significant holes in their understanding of place – value among many other things.
Symbols from 1 to 12
Understanding place-value, and therefore to have the number sense necessary to operate in the world
of numbers, is difficult for a number of reasons. The first problem year-one students run into, often
without their teachers being aware of it, is the problematic nature symbols for the counting numbers that
follow nine.
Students learn to count the numbers from one to nine aloud, and write them as words “one, two,
three” up to “nine” and symbols 1, 2, 3 up to 9. This is no particular problem with children learning two
“spellings”- for example, for five countersis spelt “5” and “five”. However, now comes a massiveproblem.
When saying ten, eleven, twelve and writing “ten, eleven, twelve” there is no difference in counting from
one to nine – there are just a few new words. But writing ten, eleven, twelve in symbols as 10, 11, 12 is
nonsensical for year one students. They end up thinking of these symbols as just another way to write the
words ten, eleven, and twelve. Yet 10 really means one group of ten and no ones, 11 really means one
group of ten and one singleton, and 12 really means one group of ten and two ones;this is much too
difficult for young children so it is suggested the use of the symbols 10,11, 12, …., 99 be treated with care.
Starting to Teach Place-Value
The core place-value idea is to understand that numbers from 10 to 99, whether spoken or written,
have two entirely different mental representations that the students must learn to swap between fluently.
One way is to view a number as something that students count out with objects, and the other way is that
the number represents groups of tens and ones.And teachers must not underestimate how difficult and
time-consuming this will be.For example, suppose a teacher writes 17 on the board and asks the children
to count out this number of blocks. The teacher notices those children who successfully attempt the task
and therefore understand one-to-one counting. Then the teacher provides plastic bags and asks this
question:
• If you put ten blocks in a plastic bag how many will be left over?
This now tests whether the students understand that, in addition to 17 meaning seventeen singletons, it
also means one ten and seven ones.
Read, Say, Do, Times Two
A powerful teaching model is based on recognising that there are six things that students need to
understand before they can be said to understand thoroughly the meaning of two digit numbers. (Larger
numbers come later.) For example, for seventeen or 17, students need to be able to read, interpret or
show the following:
Read:
•
they read 17 as “seventeen”
•
they read seventeen as “seventeen”
Say:
•
they say seventeen or 17 is “seventeen”
•
they say seventeen or 17 is “one ten and seven ones”
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Do:
•
they show seventeen or 17 by counting objects from one to seventeen
•
they show seventeen or 17 as one ten plus seven ones
Students who understand two-digit numbers thoroughly, and can therefore can move from any of these
six ways of reading, saying, or modelling the numbers on material to any of the other five, will have a vital
head start in developing number sense.
It is hoped that students arriving atthe stage four, i.e. counting-on, have understood the relationship
between reading, saying, and modelling numbers. Now is crunch time. Students absolutely need to
understand place-value, so should not proceed until formative assessment show that they thoroughly
understand two-digit numbers in all its representations.
Bigger Numbers
Students may now move on to representing three-digit numbers. Notice that because of the physical
size of representational materials multi-digit problems are more difficult to manage in a classroom situation.
So play money, which is not representational, may well be introduced. While students thoroughly enjoy
using play money teachers need to observe carefully whether students understand swapping ten units for
one unit that has ten times the value even though, say, ten ten-dollar notes in no way looks like a hundreddollar note.
A Critical Time: Early–Part Whole Thinking Develops
Early part-whole thinking is crucial for students to develop a deep understanding of numbers. It is
imperative that students learn to move fluently between different representations of numbers so that they
can perform operations mentally. Consider, for example, how students might attempt a problem like
72 – 7; at this stage they might do this:
• take away two from seventy-two leaving seventy
• there is a further five that need to be subtracted because two plus five equals seven
• in order to subtract five from seventya ten has to be swapped for ten ones
• seventy has been decomposed into a non-canonical form, namely six tens and ten ones
• ten minus five equals five because it is a basic fact
• there are six tens and five ones left i.e. the answer is sixty-five in words or 65 in digit-symbols
When the solution to this apparently simple problem is broken down this way it is obvious that it is
complex; this is why so many students have trouble with early part-whole thinking.
Both early and advanced additive and subtractive part-whole thinking are characterised by counting
being replaced by breaking numbers into parts and reassembling new wholes in clever ways. For example,
48 + 8 could be solved by adding ten to forty-eight to give fifty-eight, then subtract two to give forty-six.
This is clever because each of the steps is fast. Whereas solving 99 + 56 by adding 44 to 56 to give one
hundred and subtracting 44 from 99 to give 55 and adding to give 155 works but is not clever; other
methods are smarter.
After Early–Part Whole Thinking
Advanced part-whole thinking is an extension of early part-whole thinking that requires students to have
placed their basic addition facts into long-term memory and mastered “Read, Say, Do, Times Two” i.e. two
digit place-value.What makes advanced part-whole thinking much more challenging than early part-whole
thinking is the number of mental steps needed. For example 199 + 56 is an early part-whole problem
because adding 1 to 199 and subtracting 1 from 56 leads quickly to the answer, 255. Whereas 88 + 75 is an
advanced additive problem because it requires multiple steps to solve it mentally:here 88 + 75 might be
rearranged to 88 + 12 + 75 – 12, which is 100 + 63 i.e. 163.
The Place of Algorithmic Thinking
An algorithm is list of well-defined instructions for consistently correctly completing a task. Mathematics
abounds with such algorithms - computer programmers constantly produce algorithms that computers
follow - and they are extraordinarily useful. However, a crucial feature of algorithms is the user, either a
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person or a computer, does not need necessarily have to understand why the algorithm works.Historically
rows of clerks would sit all day in offices using pencil and paper algorithms to compute mainly money
problems involving adding, subtracting, multiplying, and dividing. Correct procedure, accuracy and speed
were vital, so schools needed to produce clerks with these skills. Understanding the algorithms did not
really matter so schools seldom bothered to teach for meaning. However, in New Zealand changes started
around 1970. This was in the choice of a standard algorithm for subtraction. Previous generations of school
children had been taught the method of equal additions. It was very efficient, but it was relatively difficult
for students to understand because it relied on equal addition of tensin which differences were preserved;
this idea was too difficult for students to understand at the age when the algorithm was introduced to
them. Perhaps because teaching was moving more towards understanding, the method of decomposition
started to be taught; it was obvious how decomposition worked when using materials like place-value
blocks. Unfortunately its algorithmic form is much messier than the old equal additions method. So there
was a clash between efficiency of a written algorithmic method and the need to understand why it worked.
In 1970 a truly iconoclastic algorithmic device arrived on the scene - the calculator. It is algorithmic
because, when a user appliesa set of rules about the correct sequence of pressing buttons, problems are
solved reliably. It was shattering because it now cast doubt on the usefulness of traditional algorithms and
what schools should now teach them. Many schools de-emphasised traditional methods, including the
teaching of the times tables, thereby providing commercial after-school education centres a perfect
opportunity to flourish by teaching in the past. This hashad huge appeal to many parents, and it has led to
frequent conflicts between home and schoolabout the place of old standard written methods.
So, does learning algorithms have a place in a world where virtually all calculations are made by
calculator or computer? The answer is unequivocally yes; provided they understand how the algorithms
work they are an invaluable tool in learning what is really important. And that is place-value.which is the
absolute core of number sense.
Does a student leaving school need pencil-and-paper methods? The honest answer is no. Who seriously
would work out the interest on a mortgage involving, say, computing 5·07% of $239 987·56 using a penciland-paper method? However, what is crucial is the ability to recognise wrong calculator answers by fast
estimation. This is far from easy: take for instance the question above. A numerate person might estimate
the answer mentally and quickly using reasoning like this:
•
5·07% ≈ 5% -and 5% is selected because it is half of 10%, which will be easy to work out soon
•
10% of $239 987·56 is about 10% of $240 000
•
Finding 10% of a number is equivalent to finding 1 tenth of that number
•
Finding 1 tenth of a number is equivalent to dividing the number by ten, so 10% of $240 000 is
1 tenth of $240 000, which is equal to $240 000 ÷ 10, which is $24 000
•
Half of $24 000 is $12 000, so 5·07% of $239 987·56 ≈ $12 000
If any of the steps above is not thoroughly overlearned students cannot make this estimate i.e.they are
innumerate.
Numeracy
A powerful definition of numeracy that works for most situations in the adult world to be numerate is
to be able to estimate well, and it has nothing to do with learning standard written forms.
Cents
Counting ten-cent coins is a very useful in reinforcing the fact that “x-ty” means x tens. Following this,
valuable connections can be made with other coins and their values particularly by doing addition problems.
For example, mentally a twenty cent coin plus a fifty cent coin makes seventy cents because two plus five is
seven;this is a very powerful reinforcer of the meaning of English’s tens words.
Dollars and Cents
Addition problems involving problems where the total cents exceeds a hundred are very powerful
practice in embedding place-value concepts and overlearning basic facts. For example, writing
$3·35 + $4·90 + $5·85 in the vertical form then involves adding ones and tens in cents to get 210 cents,
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which is now converted into 2 dollars and 10 cents using a base hundred notion - one dollar is a hundred
cents.
Dollars, Cents, and Decimal Numbers
Many teachers have fallen to the understandable temptation to use dollars and cents to introduce
decimal numbers. Unfortunately this leads to serious conceptual misunderstandings. The main problem is
that it encourages students to see the decimal point as separating whole number dollars from whole
number cents instead of seeing the part following the decimal point as very special fractional parts. This
issue is addressed in full in the chapter on decimal numbers. It is suffice for the moment to recommend
that teachers don’t use money to teach decimal numbers.
Canonical Forms
A canon - not cannon – that is a gun – is a rule or especially body of rules or principles generally
considered correct and fundamental in a field or art or philosophy. In many fields of mathematics there are
canonical formsthat may be useful conventions, or deep ideas. In the case of number systems it is hard to
exaggeratethe usefulness of the canonical form of numerals:
•
no column in a numeral may contain a symbol for representing a number greater than 9
ª
during calculations the canon of place-value sometimescan be broken, but only for purposes of
calculation. For example 76 – 9 may be calculated by first seeing seven tens and six ones, the
canonical form, as six tens and sixteen ones
•
the final answer to a calculation must be expressed the canonical form
It is just as well to remind teachers that the purpose of learning to make calculations using addition,
subtraction, multiplication and division is not aboutlearning how to calculate, although this is a useful side
effect, but to improve student and adult number sense or estimation ability indealing with real situations in
the world.And this involves a significantly more challenging understanding place-value at higher stages.
Multiple Alternatives to the Canonical Form of Numbers
To become fluent in multiplication and division students need to understand that three-or more digit
numbers have multiple equivalent meanings. For example, consider this problem:
•
Peter works in the cake factory packing ten cakes into a packet – Mallow Puffs will do. Today it is his
job to pack 265 cakes. How many packets does he have at the end of the day?How many cakes are
still loose?
Here it is vital that students can see to understand that this must involve creating 26 packets with 5 loose
without having to actually do the packing - they knowthis key multiplicative place-value relationship.
Continuing the problem:
•
Peter is now told to pack ten packets into a box for shipment. How many boxes, packets and loose
ones are there now?
What is being asked here is whether students understand that 265 can be viewed also as 2 boxes, 6 packets
and 5 loose cakes i.e. 2 hundreds plus 6 tens and 5 ones.
In summary 265 means these things:
• 265 objects that are counted one-by-one from one to two hundred and sixty-five
• 265 objects grouped as twenty-six tens plus five ones
• 265 objects grouped as two hundreds, six tens, and five ones
An example shows why it is significant.Work out:
• 7 x sixty
Firstly 7 x 6 = 42 – tables are vital here – and42 tens equals four hundreds and two tens, and, written in
the canonical form, the answer is 420.
For division the reverse process to multiplication is required. For example, suppose 375 apples are to
be shared out among 5 families. The problem is how many to pack for each family. The physical reality of
the apples is that there 375 single apples that someone counted one-by-one. There are not three groups of
a hundred, seven groups of ten, and five single apples; so the place-value meaning of 375 has no
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correspondence to the real situation. Now to work out 375 ÷ 5 there is one meaning of 375 that is
needed, namely 37 tens plus 5 ones - again this has no physical reality. However, this form allows the use of
tables to work out 37 tens divided by five leading to seven tens with two tens remainder. In passing it is
significant to note that in the 37 ÷ 5 calculation the 37 is not three tens and seven ones – it is thirty-seven
countable tens. Significantly thiskind of problem re-enforces the absolute necessity of students recalling
their times-tables instantly.
This idea of multiple multiplicatively-based forms for numbers needs to be extended to more digits. For
example, suppose the number of cakes in the problem above were 549 734. We could extend the meaning
of columns this way: pack ten boxes into one carton, pack ten cartons onto one pallet, and pack ten pallets
into one container. So 549734 cakes would appear as five containers, four pallets, nine cartons, seven
boxes, three packets, and four loose cakes.
The important multiplicative idea about place-value is that every multi-digit number has many different
representations. For example, here are fiveof many ways of viewing 549734:
• 54 ten-thousands and 9734 ones
• 549 thousands and 734 ones
• 5497 hundreds and 34 ones
• 54 973 tens and 4 ones
• 549 734ones
A feature of advanced multiplicative thinking around place-value is that much larger numbers need to be
introduced, like millions, billions, or even trillions.
Finally, place-value needs to be extended to decimal numbers that have whole number and decimal
fraction parts. Because understanding decimal numbers requires a deep understanding of fractions as well
as whole number place-value this is left to the chapter on decimal numbers that in turns follows after the
chapter on fractions.
Measurement and Number Sense
Numerate adults areultimately those who have threeintertwined skills.
Firstly the numerate adult would recognise incorrect measurements – a new born baby weighing 8800
grams would be ridiculous; this skill has to be based on practical experience.
Secondly numerate adults can recognise errors in computation regardless of whether the calculation
was made with penciland paper, a calculator, or an abacus.For example, 23· 3 x 19· 8 is not 46· 134 even
though the digits are correct.
And lastly numerate adults recognise calculated measurements are inappropriate even if the calculation
seems OK. For example a calculation that, in planning a party,we need to buy 48 litres of soft drink for 23
eight year-olds is absurd.
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Place Value: What to Learn
Stage 2: Counting from One on Materials
Numbers from 10 to 19:
The key idea is that students practise bundling and connecting the bundles to symbols e.g. 17 is read as
seventeen, and one ten and seven ones, and modelled as one ten and seven, and also seventeen singletons.
Students need to group materials so they eventually understand that one ten and ten ones are the same.
Materials could be:
•
sticks with rubber bands
•
beans and plastic bags
•
pipe cleaners
•
Unifix, ideally with a wrapper around ten cubes - but not Multilink as small children find them hard to
join together
•
no place-value blocks yet
Base problems on going from any of six ideas to any of the other five in the “Read, Say, Do Times Two”
model. The ideas are:
• read e.g. 13 is “thirteen”
• read e.g. thirteen is “thirteen”
• say in one way e.g. thirteen is “thirteen”
• say in the other way e.g. thirteen is “one ten and three ones”
• do in one way e.g. show 13 by counting out thirteen lollies
• do in the other way e.g. show 13 lollies as a bag of ten lollies and three loose lollies
Stage 3: Counting from One by Imaging
Numbers from 10 to 99
The key idea is that students practise bundling and connecting the bundles to symbols e.g. 37 is read as
thirty-seven, and three tens and seven ones, and modelled as three tens and seven ones, and also thirtyseven singletons. Students need to group materials so they eventually understand that one ten and ten ones
are the same.
Materials could be:
• Pipe cleaners
•
Unifix, ideally with a wrapper around ten cubes. But not Multilink as small children find them hard to
join together
• No place-value blocks yet
Base problems on going from any of six ideas to any of the other five in the “Read, Say, Do Times Two”
model. The ideas are:
• read e.g. 23 is “twenty-three”
• read e.g. twenty-three is“twenty-three”
• say in one way e.g. twenty-three is “twenty-three”
• say in the other way e.g. twenty-three is “two tens and three ones”
• do in one way e.g. show 23 by counting out twenty-three lollies
• do in the other way e.g. show 23 as two bags of ten lollies and three loose lollies
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Stage 4: Advanced Counting
Numbers from 100 to 999
Repeat writing, reading, saying, and modelling of two-digit numbers, but increase the largest numbers
involved three-digit numbers.
Materials now could be:
• sticks with rubber bands
• play money
Base problems on “Read, Say, Do Times Two” model. The ideas are:
•
read e.g. 463 or four hundred and sixty-three
•
read e.g. 463 or four hundred and six tens and three
•
say in one way e.g. four hundred and sixty-three
•
say in the other way e.g. four hundreds, six tens and three ones
•
do by imagining a count, as the number is too large.For examplefor 463 the students sayone, two,
three,….to ninety-nine, a hundred, a hundred and one,….to two hundred and ninety-nine, three
hundred …and eventually …four hundred and sixty-one,four hundred and sixty-two, four hundred
and sixty-three.
• do inanother way e.g. show 463 as four flats, six rods, and three ones on place-value blocks
Select Correct Operations for Addition and Subtraction
Example:
Write Geraldine collects 629 apples. She sells 467 apples at her farm shop. The number of apples unsold is:
467 - 629
629 + 467
629 – 467
367 + 629
Grouping With Money
Find the total amount of money using a mixture of ones, tens and hundreds with the answer less than a
thousand.For example, students are given uncounted piles of play money where there is more than twenty
ones, more than twenty tens, some hundreds so that after swapping ten ones for one ten tens for one
hundred the total amount of money is less than one thousand dollars.
This emphasises the canonical from of numbers i.e. no more than 9 can appear in any column.
Optional: Numbers from 100 to 9999
Estimating
• Estimate number of objects using ten as a benchmark
Stage 5: Early Part-Whole Addition and Subtraction
The Slavonic Abacus
The Slavonic abacus is a very powerful aid to help students understand place-value:
•
Hide the Slavonic abacus from the students, and push across, say, 67beads – see Figure 1, page 1.
Briefly show this to prevent students counting by tens. Students need to recognisethere are five
rows of ten and another row of ten i.e. sixty in all, and also see a row with five ones and two ones in
it i.e. seven ones. So altogether there are 67 beads. Repeat with more two-digit numbers until
students can recognise any two-digit number quickly.
•
Proceed to imaging e.g. the teacherhides, say, 82 and asks the students what it looks like.
•
Hide the Slavonic abacus from the students and push,say, 56 beads across to the right. Tell the
students that you can see 56 and ask them to imagine what is on the left hand side- here it is 44.
Connect 56 and 44 to 56 + 44 = 100
•
Repeat frequentlyfor pairs that add up to 100
•
Given two numbers under 100 write in words stories involving pairs like bigger and smaller,largest
and smallest
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Ordering
• Order sets of two and three digit numbers
Mixing Words and Symbols
A mix of words and numbers is a powerful indicator as to whether students understand place-value or
are merely using an algorithm that has no meaning for them. Use materials to work out the answers.
Problem do not require renaming
• 50 + 30
•
twenty-eight + forty
•
24 + sixty
• Thirty + forty + 20
•
Sixty – 20
•
70 + 20
• 90 – 10
•
24 – 10
•
Thirty + 83
• 60 + twenty-seven
•
200 plus 4 hundred
•
3 hundred + 4 hundred
• 420 + 300
•
30 + 35
•
twenty-three + fifteen
• 230 + 400
•
420 + sixty
•
430 + three hundred
Mental
Add columns of up to six single digit numbers.
With the meaning of two-digit numbers thoroughly established students can now move on to simple
mental calculations.
Materials:
• ten frames
• play money: ones, tens, hundreds
• arguablynumber lines - but note this should be introduced only whenstudents have fluency with the
use of other materials or better when students are at advanced additive thinking
Problems
Solve:
•
28 + 2 = ?
•
4 + 36 = ?
•
27 + ? = 30
•
twenty- ? = 17 ……
Continue doing these kinds of problems until students are fluent in doing them mentally.
Solve:
•
28 + 8 = ?
•
8 + 36 = ?
•
27 + ? = 33
•
23 - ? = 17
……
Continue doing these kinds of problems until students are fluent in doing them mentally.
Solve:
•
45 + 43 = ?
•
28 + 61 = ?
•
27 + ? = 77
•
94 - ? = 31
……
Continue doing these kinds of problems until students are fluent in doing them mentally.
Notice that while the problems below appear more complicated than the previous problems they all
involve the “tidy”hundreds; this considerably simplifies the calculations.
Solve:
• 98 + 8 = ?
•
8 + 96 = ?
•
97 + ? = 103 •
102 - ? = 97 ……
Continue doing these kinds of problems until students are fluent in doing them mentally.
Again these problems appear more complicated than the previous problems but they all involve the
“tidy” numbers, like 600, that considerably simplify the calculations.
Solve:
• 198 + 7 = ?
•
8 + 399 = ?
•
597 + ? = 602
•
802 - ? = 797 ……
Standard Forms
Addition
The standard algorithm for multi-digit addition in the vertical form, in which students engage in
theinternal talk of place-value,is a powerful aid that helps students understand place-value.For example, for
56 + 78 the self-talk would be something like this:
• Six ones and eight ones equals fourteen ones
• Swap this for one ten and four ones
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• One ten plus five tens plus seven tens makes thirteen tens
• Swap this for one hundred and three tens
• The answer is one hundred and thirty-four
Do up to three digits plus three digit problems
Subtraction
The standard decomposition algorithm for multi-digit subtraction in the vertical form, in which students
engage in the internal talk of place-value, is a powerful aid that helps students understand place-value. For
example, for 562 - 78 the self-talk would be something like this:
• Rename 562 as 5 hundreds 5 tens and 12 ones
• Five tens minus seven tens does not work so rename 5 hundreds 5 tens and 12 ones as 4 hundreds,
15 tens and 12 ones
• Proceed to the answer
Do up to three digits plus three digit problems
Simple Division and Multiplication Problems
Examples:
4 x 5 = 5 + 5 + 5 = 10 + 10 = 20
On materials 20 apples are shared among 4 people. Solution is by guess and check. E.g. try 4 each then
there are 4 apples to be given i.e. 1 more each. Or 20 ÷ 4 = 5 is a known fact.
Select Correct Operations
Examples:
A farmer has 110 cows. He sells some. He now has 98 cows. The number of cows he sold is
110 + 98
98 + 110
98 x 110
98 - 110
110 - 98
10 ÷ 5
Steve owns five cars. Each car has four wheels. The total number of wheels is:
4+5
5+4
4-5
5-5
4÷5
5x4
Word Problems
Students create word problems for the following equation types, then solve them. These are sharply
more challenging than previous problems because the unknown number is on the left of the equals sign; up
to now “equals” has meant “work out the number that is on the right hand side of the equal sign.
• 40 +
= 70
•
+ sixty = 90
• 90 = forty
•
- twenty = 60
• 45 +
= 51
•
+ 19 = 26
• 91 = 89
•
- 4 = 29
Division as Repeated Subtraction
Examples:
Freddie has a bag of 15 sweets. He eats three sweets every hour starting at 3 o’clock. How many
sweets does he have left at 3 o’clock, 4 o’clock and so on? At what time does he run out of sweets? How
many different times does he eat sets of three sweets?
Inbetweeness
Examples:
• Find two numbers between 35 thousand and 36 thousand
• Find two numbers between 5006 and 4989
Counting Back and Forth in Tens and Hundreds Orally
Examples:
• Count forward by hundreds: three hundred and seventy-six, four hundred and seventy-six etc
• Count backwards by tens: three hundred and seventy-six, three hundred and sixty-six etc
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Stage 6: Advanced Part-Whole Addition and Subtraction
Ordering
• Order sets of two to seven digit numbers
Word Problems
Students create word problems for the following equation types, then solve them. These are sharply
more challenging than previous problems because the unknown number is on the left of the equals sign; up
to now “equals” has meant “work out the number that is on the right hand side of the equal sign.
• 40 +
= 70
•
+ sixty = 90
• 90 = forty
•
- twenty = 60
• 45 +
= 51
•
+ 19 = 26
• 91 = 89
•
- 4 = 29
Mental Strategies
Mental multi-digit addition and subtraction problems that have multiple methods of solution are now
practiced
Material:
• number lines - notice this is the first time number lines need to be used
Problems
Solve:
•
28 + ? = 92
•
72 + ? = 91
•
27 + ? = 74
•
45- ? = 17
……
Continue doing these kinds of problems until students are fluent in doing them mentally.
Solve:
•
? + 28 = 92
•
? + 72 = 91 •
? + 27 = 74
•
? - 23 =
Continue doing these kinds of problems until students are fluent in doing them mentally.
Solve:
•
78 + 28 = ?
•
34 + 68 = ? •
88 - 39 = ?
•
123 - 36
Continue doing these kinds of problems until students are fluent in doing them mentally.
=
38
……
?
……
Stories and Calculators
Selecting the correct operation from a story then proceed to use a calculator to solve the problem is
very important.
For example: 9625 apples are send to 25 supermarkets. They each get the same number of apples.
Which calculation is appropriate for this problem?
9625 x 25
9625 ÷ 25
25 ÷ 9625
9625 – 25
Solve the problem with a calculator.
Addition and Subtraction Estimation
Estimate sums of numbers as a check to calculator answers.
Examples:
• 98 + 297 + 302 + 99 is nearest to:
600
700
800
900
Jerry uses his calculator and claims 987 + 986 + 2012 + 219 = 5205. Why is he wrong?
Harriet uses her calculator and claims 7903 – 1003 – 998 – 2001 = 4901. She must be wrong. Why?
Standard Forms
Material:
•
Play money up to billion dollar notes
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Repeat Addition and Subtraction standard forms as in the previous stage with numbers extended to the
millions.
Extend this to the vertical form with asingle digit multiplied by up to six-digit numbers. For example,
3546 x 7.
Four-digit numbers divided by a single digit:
3345 ÷ 5 is done by realising 3345 also means 33 hundreds plus 45 then proceeding to complete the
problem
Stories
Examples with stories. These are very important!Notice that students should create a word problem for
the equations, and write down their question before solving problems like these:
• 45 +
= 81
•
+ 299 = 401
• 201 = 169
•
- 84 = 186
Stage 7: Advanced Part-Whole Multiplication and Division
Large numbers
Problems
• Read and use numbers in the millions, billions and trillions.
• 1 001 – 8 = ?
•
999 989 + 14 = ?
•
2 000 001 – 1 999 987 = ?
• 3 000 023 - ? = 2 999 978
•
? + 999 999999 978 = 1 000 000000 023
Material:
• Play money
Inbetweeness
Example:
• Find two numbers between 388 999 899 and 389 999 945
Multiplicative Place-Value Relationships
Problems
•
A bank has only ten-dollar notes, and one dollar coins. How is $587 withdrawn i.e. connect the
canonical form 587 with the non-canonical form 58 tens and 7 ones.
•
Model 1947 in the canonical form on play money and show why it is also read as the non-canonical
form nineteen hundred and forty-seven – it is non-canonical as there is nineteen in the hundreds
column – and only a maximum of 9 is permitted
•
Model the non-canonical form twenty-three hundred and thirty-seven on play money and do the
exchanges that show its canonical form is one thousand, three hundred and thirty-seven (i.e. 1337)
•
Up to five-digit numbers divided by a single digit:
345 ÷ 5 is done by realising 345 also means 34 tens plus 5 ones
Long Multiplication
Material:
• Hundreds grid paper
Problems
• 34 x 29 on squared paper
Material:
• Counters arranged in grids
Quick Ways to Multiply
Problems
• Doubling and halving for multiplication. E.g. 50 x 448 = 100 x 224 = 22 400
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•
•
•
•
Doubling and doubling for division. E.g.24 000 ÷ 5 = 48 000 ÷ 10 = 4800
Multiplying by n and dividing by n for multiplication problems where n is equal to 2, 3, 4, or 5
Multiplying by n and multiplying by n for division problems where n is equal to 2, 3, 4, or 5
Multiplying and dividing by 25 e.g. 36 ÷ 25 = (36 x 4) ÷ (4 x 25) = 144 ÷ 100 = 1·44
Multiplicative Algebriac Rules
Problems
• Adding, subtracting, and multiplying odds and evens e.g. odd minus odd is even and why
• Add any three consecutive numbers and the answer is a multiple of 3.
Factors
• Factor rules e.g. an even number has a factor 2
• Factorise a number into primes
• Square root test to determine whether a number is prime or not
• Lowest Common Multiples and Highest Common Factors
• LCM (a, b) = ab ÷ HCF(a, b)
• Which numbers have exactly three factors (Answer: square numbers)
• Which years are leap years
The Pigeon Hole Principle
Problem
•
A postman has 63 letters to deliver to 10 addresses. Because he cannot read he delivers them anyold-how. Explain why there is at least one person who gets seven letters or more.
Geometric Patterns
Problems
• What is the, say, number of counters needed to make the say 100thpattern in this sequence using
multiplicative reasoning Answer: 4 x 99
99
99
99
99
•
What is the, say, 100th term in the sequence 3, 8, 13, 18, ….. Multiplicative thinkers should readily
understand that the 5 has been added 99 times, therefore the answer is 99 x5 + the first term i.e.
99x 5 + 3 = 498. Other similar multiplicative methods should also be encouraged.
Estimation to Check Calculations
Examples:
67 x 88 ≠ 5016 because 70 x 90 = 6300
56 x 74 is less than 60 x 80 = 4800 and more than 50 x 70 = 3500. Average of 4800 and 3500 is about
4150. So 56 x 74 ≈ 4150.
4789 ÷ 69 ≈ 4900 ÷ 70 = 700
Invalid Approximations
Example: Why is this wrong?
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The tour guide at the museum said "that dinosaur is 70 million and 12 years old."Asked how he could be
so exact the guide said "I have worked here for 12 years and, when I started, I was told the dinosaur was
70 million years old".
Ratios and Proportions
Most ratio and proportion problems in science and technology do not have tidy numbers that can be
solved by spotting factors - refer to the Decimal Numbers Domain. However, whole number problems are
worth doing as a preliminary to the decimal fractions types.
Problems
•
6 oranges are mixed with 8 apples in a fruit salad. How many apples will be needed for a fruit salad
containing 9 oranges?
Here 6 : 8 = 3 : 4 by dividing both parts of the ratio by 2, then 3 : 4 = 9 : 12 by multiplying both parts of
the ratio by 3. So the answer is 12 apples.
Rates
The key idea here is to understand why rates cannot be averaged.
Problems:
•
On the way to Rotorua Charlotte averages 80 km/h. On the return trip she averages 100km/h. Show
that here average speed there and back is 88·9 km/h. (Imagine the one way is 400 km. This is
selected because then the times are easy – 400 ÷ 80 = 5 hours there, and 400 ÷ 100 = 4 hours back.
So she takes 9 hours to travel 800 km. The average speed is 800 ÷ 9 ≈ 88·9 km/h.
•
A trip takes 6 hours travelling at an average speed of 60 kph. How long will the trip take at an
average speed of 90m/h?
•
John plans to run marathon at 15 km/h. Show this amounts to taking 4 minutes to run every
kilometre.
•
A marathon is42·195 km long. Show John plans to take about 2 hours 49 minutes.
How does this compare with the world record for the marathon?
Checking Calculations
A variety of strategies can help check answers; typically they cannot detect everyerrors.
Problems:
• 23 x 29 = 588 is wrong because 3 x 9 = 27 means there must be a 7 in the ones column.
• 21 x 29 = 509 is wrong even though 1 x 9 = 9 means the ones digit in the answer is correct. Here 21
x 29 ≈ 20 x 30 = 600. So 509 must be wrong.Here the estimation found by applying tables and place-value
• Notice a tables and place-value method will not detect minor errors: For example 62 x 58
equals 3586 is wrong - the answer is actually 3596 - but the estimate 62 x 58 ≈ 60 x 60 = 3600 will not rule
out 3586.
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Stage 8: Advanced Equivalent Fractions and Ratios
Ratios and Proportions
Problems
•
•
•
•
•
10 workers take 18 days to paint 12 houses. An urgent job requires 15 houses to be painted in 12
days. How many workers are needed? What assumptions did you have to make?
Murray drives half way from Auckland to Hamilton at an average speed of 90 kph, then he drives the
second half of the trip at an average speed of 60 kph. Explain why the average speed for the whole
trip is 72 kph.
3 467 309 737 is not a square number. Prove this by looking at the ones digit only. (When squaring
digits no answer has 7 in the ones column.) Explain why a ones column digit test will not work for
cube numbers.
John and Jackie decide to have a two-lap car race. They have to work out before the race what
speeds they will do for each lap so that the numbers add up to 200. So, for example, lap one at 120
kph and lap two at 80 kph is OK, but lap one at 130 kph and lap two at 60 kph is not OK.
John decides to travel the first lap at 198 kpmand the second lap at 2 kpm. Jackie decides to travel
the first lap at 100 kpm, and also travel the second lap at 100 kpm. Who will win the race. It is not a
tie!
An ace goal-shooter averages 24 goals in the first twelve games of the season and 18 goals in the last
six games. Show her season average is 22 goals per game.
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Place-Value Lesson Plans
Lesson 2.1
Key Idea
Students demonstrate that they understandthat x-teen is equivalent oneten and x oneson materials, for
fourteen and sixteen to nineteen.
Equipment
Blocks/beans and plastics bags/film canisters.
Assessment of Readiness
•
•
Write nineteen on the board and say:
Count out nineteen beans.
• Unsuccessful students revert toLesson 1.x.
Assessment Question
• Point at nineteen on the board and say:
•
How many bags of ten could you make from nineteen blocks? You can use the plastic bags if you want to.
Successful
•
Students who immediately say one i.e. they have understood that that x-teen means one ten and x
ones.
• These children do practice and extension
Unsuccessful
• Students who say one but needed to make up a bag of ten
• Students who cannot say one even with using the bundles
• These children stay for teaching.
Teaching
Key Concept
The big issue here is language. English speaking students need to know x-teen means one ten and x
ones. This problem does not occur in languages like Maori and Mandarin. In thislesson fourteen, sixteen,
seventeen, eighteen, and nineteen are the only x-teen words used because their coding is entirely reliable
whereas the numbers twelve, thirteen, and fifteen are all irregular in some way.
Teaching Ideas
•
•
•
•
Point out the “teen” code meaning ten
Write on the board and discuss the missing wordsusing the materials:
sixteen is one ten and ….ones
one ten and seven ones is .………………
eighteen is one ten and ….ones
one ten and ten ones is .………………
fourteenis .… ten and …. ones
seven ones and one ten is .………………
nineteen is .... ten and ….. ones
eight ones and one ten is .………………
Get students to show these numbers using one group of ten and loose ones:
fourteen
four and ten
ten and seven
seventeen
eighteen
eight and ten
nineteen
Get students to show one bag of ten and four, six, seven, eight or nine ones and write down how
many sticks there are in two ways. For example, if there are one bag of ten and six ones students
write one bag of ten and six ones, and sixteen.
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Lesson 2.2
Key Idea
Students demonstrate that they understandthat the exceptional words, eleven, twelve, thirteen and
fifteen are equivalent to 1ten and some ones.
Equipment
Blocks/beans and plastics bags/film canisters.
Assess Readiness
Write twelve on the board and say:
Count out twelve beans
Successful students go on to the Assessment Question below.
Unsuccessful students revert toLesson 1.x.
Assessment Question
Write: “twelve” on the board.
Then say:
Count out twelve blocks.
Then say:
How many bags of ten could you make from twelve blocks? You can make up packets of ten in plastic bags if you
like if you want to.
Teaching
TBA
Lesson 2.3
Key Idea
Students demonstrate that they understand that two-digit numbers up to 19 written as two digits means
one ten plus some ones on bundleable material.
Equipment
None
Assess Readiness
Write “16” on the board - don’t read it aloud - and say:
Count out this number of beans.
Successful students go on to the Assessment Question below.
Unsuccessful students revert to pageLesson 1.x.
Assessment Question
Write “16” on the board - don’t read it aloud - and say:
How many bundles of ten could you make with this number of sticks?
What to Look For
Successful
Observe students who immediately say one i.e. they have understood that that x-teen means 1 ten and x
ones. These children are sent to either:
• do practice and extension
• consider advancing them to the next stage
Unsuccessful
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Observe who can say one but need to make up a bag of ten. Observe who cannot say one even with
using the bundles. These children stay with the teacher.
Key Ideas
The key issue here is again language. English speaking students to know x-teen means one ten and x
ones. This problem does not occur in languages like Maori and Mandarin. In the previous Lesson fourteen,
sixteen, seventeen, eighteen, and nineteen were the x-teenwords whose coding is entirely reliable. In this
Lesson the numbers twelve, thirteen, and fifteen, which are all irregular in some way, are taught.
Teaching
Write on the board and discuss the answers using the materials:
eleven is one ten and ……. ones
one ten and two is ……….
twelve is one ten and ……. ones
one ten and five is……. ones
fifteen is .… ten and …… ones
three ones and one ten is ……….
Thirteen is ....ten and ….. ones
one and ten is ……..
Get students to show these numbers on tens-bundled sticks:
eleven
twelve
eighteen
nineteen
six and ten
ten and eight
seventeen
Get students to show one bag of ten and 1, 2, 3 or 5 ones and write down how many sticks there are in
two ways. For example, if there is one bag of ten and two ones students write one bag of ten and two ones,
and twelve.
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Lesson 3.1
Key Idea
Students demonstrate that they understandthat x-ty is equivalent x tenson bundleable material, starting
at sixty and over.
Equipment
Sticks pre-bundled with rubber bands into tens
Assess Readiness
None
Assessment Question
Write: sixty on the board.
Then say:
How many bundles of ten do you need to have sixty? You can use bundles of ten if you want to.
What to Look For
Successful
Observe students who immediately say six without counting ten, twenty, thirty, forty, fifty, sixty, i.e. they
have understood that that x-ty means x tens.
These children are sent to either:
• do practice and extension
• consider advancing them to the next stage
Unsuccessful
Observe who can say six but count ten, twenty, thirty, forty, fifty, sixty.
Observe who cannot say six even with using the bundles.
All these children stay with the teacher.
Key Ideas
The important issue in this lesson is for English speaking students to know that the –ty at the end of
number words means lots of ten. This problem does not occur in languages like Maori and Mandarin. It is a
significant reason why in many countries children do better than New Zealand children from the earliest
time at school. In this lesson sixty, seventy, eighty, and ninety are the only -tywords whose coding is entirely
reliable whereas the numbers twenty, thirty, forty and fifty are all irregular in some way.
Teaching
Point out the -ty code means tens.
Write on the board:
sixty is …. tens
seven tens is ……….
seventy is .… tens
six tens is ……….
eighty is .... tens
ninety is .... tens
Get students to show these numbers on tens-bundled sticks:
sixty
seven tens eighty
nine tens
seventy
six tens
nine tens is ……..
eight tens is ……..
ninety
eight tens
Get students to count six or more bundles of ten and write down how many sticks there are in two
ways. For example, if there are seven bundles the students are expected to count seven bundles one-byone then write seventy and seven tens.
Lesson 3.2
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Key Idea
Students demonstrate that they understandthat x-ty is equivalent x tenson bundleable material, for
twenty, thirty, forty and fifty.
Equipment
Sticks,rubber bands, with some bundled into tens.
Assess Readiness
Write seventy on the board and say:
How many bundles of ten do I need to get this number of sticks?
Successful students go on to the Assessment Question below.
Unsuccessful students revert to Lesson 3.1.
Assessment Question
Write thirty on the board.
Then say:
How many bundles of ten do you need to have thirty sticks? You can make bundles of ten if you want to.
What to Look For
Successful
Observe students who immediately say three without counting ten, twenty, thirty, i.e. they have
understood that that thirty means three tens. These children are sent to either:
• do practice and extension
• consider advancing them to the next stage
Unsuccessful
Observe who can say three but count ten, twenty, thirty. Observe who cannot say three even with
using the bundles. These children stay with the teacher.
Key Idea
The key issue continues to be, for English speaking students, to know that the –ty at the end of number
words means lots of ten. But it is irregularly applied – twenty not twoty, thirty not threety, forty not fourty,
and fifty not fivety.
Teaching Ideas
Point out twenmeans two. Point out thir means three as in thirteen, third, and thirty. Point out fif means
five, as in fifteen, fifth, and fifty. Point out the misspelling of forty as fourty.
Write on the board and discuss the answers using the materials:
twenty is …. tens
four tens is ……….
thirty is .... tens
tens is ……..
forty is … tens
two tens is ……….
fifty is …. tens
……..
Get students to show these numbers on tens-bundled sticks:
twenty
four tens
thirty
forty
two tens
fifty
tens
five
three tens is
five tens
three
Get students to count five or less bundles of ten and write down how many sticks there are in two
ways. (Do twenty, thirty, forty and fifty out of order.) For example, if there are four bundles the students
are expected to count them on-by-one then write forty and four tens.
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Get students to work out 1 plus 1 (2), 1 hundred plus 1 hundred (2 hundred), 1 dog plus 1 dog (2
dogs),1 ten plus 1 ten i.e. connect to 1 ten plus 1 ten is 2 tens i.e. twenty.
Repeat for some of 1 + 2, 1 + 3, 1 + 4, 1 + 5, etc....2 + 2, 2 + 3 etc, 3 + 3, 3 + 4 etc, and 4 + 4.
Lesson 3.3
Key Idea
Students demonstrate that they understandthat numbers up to 19 written as two-digitsmeans one ten
plus some ones on bundleable material.
Equipment
Sticks and rubber bands.
Assess Readiness
None
Assessment Question
Write: 18 - don’t read it aloud.
Then say:
How many bundles of ten could you make with this number of sticks? You can use the sticks and rubber bands if
you want to.
What to Look For
Successful
Observe students who immediately say one, i.e. they have understood that there is a one in the tens
column. These children are sent to either:
• do practice and extension
• consider advancing them to the next stage
Unsuccessful
Observe who cannot immediately says one but can count out ten ones to create one bundle of ten.
Observe who cannot say one ten even with the material. These children stay with the teacher.
Key Idea
So far the connection between English number words and their meaning on material have been
introduced. The use of the number symbols has been deliberately delayed until now. Now students begin
the arduous task of connecting number words, number symbols and meaning. Of particular importance is
the introduction the twin meaning for two-digit numbers. For example, 18 means, in effect, ”count out
eighteen objects one-by-one”. But its place-value meaning is “one ten and eight ones”. The ability to move
fluently between the words, symbols and two meanings is of the upmost importance.
Teaching Ideas
Students bundle up to 19 sticks into 1 ten and some ones. Help students to record
this as two-digits in columns e.g. 16 is shown on the right.
Repeat bundling for other numbers between 11 to 19.
Repeat an assessment question as often as is appropriate. For example, change the
number to 12.
Cycle through this process an appropriate number of times.
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Tens
1
Ones
6
Lesson 3.4
Key Idea
Students understand that a counted set up to nineteen, where the numbers are written in words, can be
grouped as one ten plus some ones on bundleable material and written as a two-digit number.
Equipment
Sticks and rubber bands.
Assessment Question
Write sixteen on the board and then say:
Count out sixteen sticks.
Then point at the sixteen on the board and say:
How many bundles of ten could you make with this number of sticks? You can use the rubber bands if you want to.
How would you write sixteen with digits?
What to Look For
Successful
Observe students who can count out sixteen sticks and immediately say one ten without needing to
group them, i.e. they have understood that there is a one in the tens column. These children are sent to do
practice and extension, or some other learning that does not need the help of the teacher.
Unsuccessful
Observe who cannot immediately say one but does so after bundling ten sticks with a rubber band.
Observe who cannot solve the problem at all. These children stay with the teacher.
Key Idea
This is the same as the previous Lesson except that students need to demonstrate that they can count
and create bundles of ten, when given a number in word form, then connect this to two-digit numbers.
Teaching Ideas
Students count out up to nineteen sticks. Then they create bundles of ten. The this
number is written in words on the board e.g. thirteen. Help students to record this as twodigits in columns e.g. thirteen is shown on the right.
Repeat for some of eleven to nineteen.
Repeat the assessment question with a different number of sticks to count e.g. nineteen
Cycle through this process a number of times.
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Tens
1
Ones
3
Lesson 3.5
Key Idea
Students demonstrate that they understandthat two-digit numbers up to 99 written as two-digitsmeans
some tens plus some ones on bundleable material.
Equipment
Sticks and rubber bands with somepre-bundled tens.
Assessment Question
Point at 28 on the board - don’t read it aloud - and say:
How many bundles of ten could you make with this number of sticks? You can use the sticks if you want to.
What to Look For
Successful
Observe students who immediately say two, i.e. they have understood that there is a one in the tens
column.
These children are sent to do practice and extension, or some other learning that does not need the
help of the teacher.
Unsuccessful
Observe who cannot immediately says one but can count out ten ones to create one bundle of ten.
Observe who cannot say one ten even with the material.
These children stay with the teacher.
Key Idea
Up until now understanding two-digit numbers has been restricted to less than twenty. It may be
reasonably argued that the “teen” numbers are significantly more difficult to understand that the “x-ty”
numbers because of all the exceptional names for the numbers. However, because students inevitably run
into the ten numbers in counting before they get to the “x-ty” numbers, two-digit numbers above twenty
are introduced after the teen numbers.
Teaching Ideas
Students get out 3 bundles of tens and 9 ones - numbers bigger than 39 need too much material. Help
students to record this as two-digits in columns e.g. 39 is shown on the right. And say
Tens
Ones
thirty-nine.
Repeat bundling for other numbers between 21 to 39.
3
9
Repeat an assessment question as often as is appropriate. For example, change the
number to 85.
Cycle through this process an appropriate number of times.
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Lesson 3.6
Key Idea
Students understand that a counted set up to 99 written as in words can be grouped as tens plus some
ones on bundleable material and written as a two-digit number.
Equipment
Sticks and rubber bands. No pre-bundled material.
Assessment Question
Write: twenty-seven.
Then say:
How would you write twenty-seven using only a 7 and 2?
What to Look For
Successful
Observe students who immediately say 27, i.e. they have understood that there is a one in the tens
column.
These children are sent to do practice and extension, or some other learning that does not need the
help of the teacher.
Unsuccessful
Observe who cannot immediately write 27. These children stay with the teacher.
Key Idea
This checks the connection between two number words and symbols. More opportunities to learn is
provided.
Teaching Ideas
Write thirty-nine on the board. Ask students to count out loose ones, and make
bundles of ten. Help students to record this as two-digits in columns e.g. 39 is shown on
the right.
Repeat for some of twenty one to ninety-nine.
Repeat the assessment question with a different number of sticks to count e.g. thirty-six
Cycle through this process a number of times.
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Tens
Ones
3
9
Lesson 3.7
Key Idea
Students understand that two-digit numbers up to 99 written as two-digitsmean some tens plus some
ones on blocks/beans.
Equipment
Blocks/beans and plastics bags/film canisters.Loose material only.
Assessment Question
Write: 47 on the board - don’t read it aloud.
Then say:
How many bags of ten could you make with this number of blocks? You can use the plastic bags if you want to.
What to Look For
Successful
Observe students who immediately say four, i.e. they have understood that there is a four in the tens
column. These children are sent to do practice and extension, or some other learning that does not need
the help of the teacher.
Unsuccessful
Observe who cannot immediately says one but can count out ten ones to create one bag of ten.
Observe who cannot say one ten even with the material. These children stay with the teacher.
Key Idea
Another opportunity to check the relationship with words and symbols is provided because the issue is
absolutely central to students eventually having good number sense.
Teaching Ideas
Students bag up to 39 blocks into tens and some ones before writing this as a two-digit number. Help
students to record this as two-digits in columns.
Repeat bagging for other numbers between 11 to 99.
Repeat an assessment question as often as is appropriate.
Cycle through this process an appropriate number of times.
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Lesson 3.8
Key Idea
Students understand the meaning of 0 in two-digit numbers like x0.
Equipment
Any structured material in which ten ones is visibly the same as one ten.
Assessment Question
Write: 80.
Then say:
You have this number of apples. There are ten apples in each bag. How many bags do you have?
What to Look For
Successful
Observe students who immediately say eight, i.e. they have understood that what the nought means in
80. These children are sent to do practice and extension, or some other learning that does not need the
help of the teacher.
Unsuccessful
Observe who cannot immediately say eight but does so after counting out eight tens typically saying
“ten, twenty, thirty,… eighty”… Observe who cannot solve the problem at all. These children stay with the
teacher.
Key Idea
Checking understanding of nought has been delayed until now. It represents the first great idea in placevalue namely that it must be present in order that a person reading a number knows which columns he
needs to read out – the “nought “ columns being ignored.
Teaching Ideas
Students count outs bundled tens materials. Then the number represented is written in
words on the board e.g. forty because we have four bundles of ten notes. Help students to
record this as two-digits in columns explaining what nought is for. E.g. forty is shown on the
right.
Repeat for some of twenty to ninety in tens.
Repeat assessment question with a different number of objects to count e
Cycle through this process a number of times.
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Tens
Ones
4
0
Lesson 3.9
Key Idea
Students link two-digit numbers written in symbols to word stories.
Equipment
Any structured material in which ten ones is visibly the same as one ten.
Assessment Question
Write: Joelene has 78 chocolatesthat she puts into bags of ten chocolates. How many are left over?
Then say:
Joelene has this number of chocolates - point at the 78 but don’t say it - that she puts them into bags of ten
chocolates. How many are left over?
What to Look For
Successful
Observe students who immediately say eight i.e. they have understood that what the 8 means in 78.
These children are sent to do practice and extension, or some other learning that does not need the
help of the teacher.
Unsuccessful
Observe who cannot immediately say eight. These children stay with the teacher.
Key Idea
Two--digit numbers have two distinct meanings. Firstly they represent a counted collection of ones.
Secondly they represent the groups of ten with ones left over.
Teaching Ideas
A story is written on the board with a two-digit number. Then students count out the two-digit number
of objects that is not too large – for example 36. Students predict how many bundles/bags of ten and ones
left over there will be, then check by packing ten into bundles/bags.
Repeat assessment question. This time the question might emphasise the tens rather than the ones.
E.g. Write: Mary has 46apples that she puts into bags of ten. How many bags of ten does she pack? Then
read it aloud.
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Lesson 3.10
Key Idea
Students link two-digit numbers written in words to word stories.
Equipment
Any structured material in which ten ones is visibly the same as one ten.
Assessment Question
Write:Repeka counts out ninety-three kiwifruit, and then packs them into bags containing ten kiwifruit.
How many bags does she pack?
Then say:
Repeka counts out ninety-three kiwifruit, and then packs them into bags containing ten kiwifruit. How many bags
does she pack? How many loose Kiwifruit are there?
What to Look For
Successful
Observe students who immediately say nine i.e. they have understood that what the nine means in
ninety-three. These children are sent to do practice and extension, or some other learning that does not
need the help of the teacher.
Unsuccessful
Observe who cannot immediately say eight. These children stay with the teacher.
Key Idea
This repeats the key idea from the previous Lesson except the two-digit numbers are written in words
not symbols. Again there are two meanings. Firstly they represent a counted collection of ones. Secondly
they represent the groups of ten with ones left over.
Teaching Ideas
A story is written on the board with a two-digit number in words not symbols. Then students count out
the two-digit number of objects that is not too large – for example 29. Students predict how many
bundles/bags of ten and ones left over there will be, then check by packing ten into bundles/bags.
Repeat assessment question. This time the question might emphasise the tens rather than the ones. E.g.
Write: Mary has thirty-eightapples that she puts into bags of ten. How many bags of ten does she pack?
Then read it aloud.
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Lesson 3.11
Key Idea
Students create stories linking two-digit numbers written as digits as ones to tens and ones.
Equipment
Any structured material in which ten ones is visibly the same as one ten.
Assessment Question
Write:Make up a story for this: 76 ones equals 7 tens and 6 ones.
Thensay:
Make up a story for this:
76 ones equals 7 tens and 6 ones.
What to Look For
Successful
Observe students who immediately say “Murray packs bags of ten marbles. He gets 7 bags of ten with 6
ones left over”.
These children are sent to do practice and extension, or some other learning that does not need the
help of the teacher.
Unsuccessful
Observe who cannot immediately say eight. These children stay with the teacher.
Key Idea
This reverses the usual problems. Here the equation is given and students need to show that they can
turn the equation into a story. This is challenging but it is a key idea to check whether they can link the
maths to the world.
Teaching Ideas
A story is written on the board with a two-digit number in words not symbols. Then students count out
the two-digit number of objects that is not too large – for example 29. Students predict how many
bundles/bags of ten and ones left over there will be, then check by packing ten into bundles/bags.
Repeat assessment question. This time the question might emphasise the tens rather than the ones. E.g.
Write: Mary has thirty-eightapples that she puts into bags of ten. How many bags of ten does she pack?
Then read it aloud.
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Lesson 3.12
Key Idea
Students create stories linking two-digit numbers written in words as ones to tens and ones.
Equipment
Any structured material in which ten ones is visibly the same as one ten.
Assessment Question
Write:Joachim counts out eighty-five cakes.
Then say
Make up a story for this:
Eighty-five ones equals 8 tens and 5 ones.
What to Look For
Successful
Observe students who immediately say “Joachim packs bags of ten cakes. He gets 8 bags of ten with 5
ones left over”.
These children are sent to do practice and extension, or some other learning that does not need the
help of the teacher.
Unsuccessful
Observe who cannot immediately say eight. These children stay with the teacher.
Key Idea
This reverses the usual problems. Here the equation is given and students need to show that they can
turn the equation into a story. This is challenging but it is a key idea to check whether they can link the
maths to the world.
Teaching Ideas
A story is written on the board with a two-digit number in words not symbols. Then students count out
the two digit number of objects that is not too large – for example 29. Students predict how many
bundles/bags of ten and ones left over there will be, then check by packing ten into bundles/bags.
Repeat assessment question. This time the question might emphasise the tens rather than the ones. E.g.
Write: Mary has thirty-eightapples that she puts into bags of ten. How many bags of ten does she pack?
Then read it aloud.
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Lesson 3.13
Key Idea
Students create stories linking two digit numbers written in words as ones then and tens.
Equipment
Any structured material in which ten ones is visibly the same as one ten.
Assessment Question
Write:Jenna buys sweets at the shop. She now has six loose sweets and seven packets of sweets. There
are ten sweets in each packet.
Then say:
Make up a story for:
Six ones and seven packets of ten equals 76
What to Look For
Successful
Observe students who immediately say “Joachim packs bags of ten cakes. He gets 8 bags of ten with 5
ones left over”.
These children are sent to do practice and extension, or some other learning that does not need the
help of the teacher.
Unsuccessful
Observe who cannot immediately say eight. These children stay with the teacher.
Key Idea
This reverses the usual problems. Here the equation is given and students need to show that they can
turn the equation into a story. This is challenging but it is a key idea to check whether they can link the
maths to the world.
Teaching Ideas
A story is written on the board with a two-digit number in words not symbols. Then students count out
the two digit number of objects that is not too large – for example 29. Students predict how many
bundles/bags of ten and ones left over there will be, then check by packing ten into bundles/bags.
Repeat assessment question. This time the question might emphasise the tens rather than the ones. E.g.
Write: Mary has thirty-eightapples that she puts into bags of ten. How many bags of ten does she pack?
Then read it aloud.
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Lesson 3.14
Key Idea
Studentscounts one dollar notes/coins given as a two digit number, and swap them ten-dollars notes.
Answer not to exceed 99.
Equipment
One and ten-dollar play money.
Assessment Question
Write: Thomas has saved 56 one dollar coins. He goes to the corner shop where the shopkeeper swaps
ten dollar coins for ten-dollar notes. How many ten-dollar notes does Thomas have now? How many one
dollar coins does he have?
Then say:
Thomas has saved 56 one dollar coins. He goes to the corner shop where the shopkeeper swaps ten dollar coins for
ten-dollar notes. How many ten-dollar notes does Thomas have now? How many one dollar coins does he have?
What to Look For
Successful
Observe students who immediately say “Joachim packs bags of ten cakes. He gets 8 bags of ten with 5
ones left over”.
These children are sent to do practice and extension, or some other learning that does not need the
help of the teacher.
Unsuccessful
Observe who cannot immediately say eight. These children stay with the teacher.
Key Idea
This reverses the usual problems. Here the equation is given and students need to show that they can
turn the equation into a story. This is challenging but it is a key idea to check whether they can link the
maths to the world.
Teaching
A story is written on the board with a two-digit number in words not symbols. Then students count out
the two digit number of objects that is not too large – for example 29. Students predict how many
bundles/bags of ten and ones left over there will be, then check by packing ten into bundles/bags.
Repeat assessment question. This time the question might emphasise the tens rather than the ones. E.g.
Write: Mary has thirty-eightapples that she puts into bags of ten. How many bags of ten does she pack?
Then read it aloud.
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Lesson 3.15
Key Idea
Studentscounts one dollar notes/coins given as a two digit number in words, and swap them ten-dollars
notes. Answer not to exceed 99.
Equipment
One and ten-dollar play money.
Assessment Question
Write: Thomas has saved seventy three one dollar coins. He goes to the corner shop where the
shopkeeper swaps ten one dollar coins for a ten-dollar note. How many ten-dollar notes does Thomas
have now? How many one dollar coins does he have?
Then say:
Thomas has saved seventy three one dollar coins. He goes to the corner shop where the shopkeeper swaps ten
dollar coins for ten-dollar notes. How many ten-dollar notes does Thomas have now? How many one dollar coins
does he have?
What to Look For
Successful
Observe students who immediately say “Joachim packs bags of ten cakes. He gets 8 bags of ten with 5
ones left over”.
These children are sent to do practice and extension, or some other learning that does not need the
help of the teacher.
Unsuccessful
Observe who cannot immediately say eight. These children stay with the teacher.
Key Idea
This reverses the usual problems. Here the equation is given and students need to show that they can
turn the equation into a story. This is challenging but it is a key idea to check whether they can link the
maths to the world.
Teaching Ideas
A story is written on the board with a two digit number in words not symbols. Then students count out
the two digit number of objects that is not too large – for example 29. Students predict how many
bundles/bags of ten and ones left over there will be, then check by packing ten into bundles/bags.
Repeat assessment question. This time the question might emphasise the tens rather than the ones. E.g.
Write: Mary has thirty-eightapples that she puts into bags of ten. How many bags of ten does she pack?
Then read it aloud.
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Lesson 3.16
Key Idea
Students are told how many one dollar coins they have and how many ten-dollar notes they have, then
they convert this to a number of one dollar coins. Answer not to exceed 99.
Equipment
One and ten-dollar play money.
Assessment Question
Write: Thomas has saved 5 ten-dollar notes and 6 one dollar coins. He goes to the corner shop where
the shopkeeper swaps each ten-dollar note for ten one dollar coins. How many one dollar coins does he
have? Write this down in digits.
Then say:
Thomas has saved 5 ten-dollar notes and 6 one dollar coins. He goes to the corner shop where the shopkeeper
swaps each ten-dollar note for ten one dollar coins. How many one dollar coins does he have? Write this down in
digits.
What to Look For
Successful
Observe students who immediately write “Joachim has 56 coins”.
These children are sent to do practice and extension, or some other learning that does not need the
help of the teacher.
Unsuccessful
Observe who cannot immediately write 56. These children stay with the teacher.
Key Idea
This reverses the usual problems. Here the equation is given and students need to show that they can
turn the equation into a story. This is challenging but it is a key idea to check whether they can link the
maths to the world.
Teaching
A story is written on the board with a two digit number in words not symbols. Then students count out
the two digit number of objects that is not too large – for example 29. Students predict how many
bundles/bags of ten and ones left over there will be, then check by packing ten into bundles/bags.
Repeat assessment question. This time the question might emphasise the tens rather than the ones. E.g.
Write: Mary has thirty-eightapples that she puts into bags of ten. How many bags of ten does she pack?
Then read it aloud.
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Lesson 3.17
Key Idea
Recognise two-digit numbers from ten-frames.
Equipment
Tens frames.
Assessment Question
Draw this diagram on the board:
Write these on the board:
thirty-six
seventy-three
Point at the answers and say:
Which of these is the total number of dots?
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37
10
Stage 4
Lesson 4.1
Key Idea
Demonstrate bundling over a hundred sticks or similar material into tens and hundreds, and writing it as
a three-digit number.
Equipment
Sticks and rubber bands.
Assessment Question
Give students over one hundred loose sticks (different amounts), and rubber bands Say:
Make up bundles of tens and hundreds that show your sticks as a number. For example your answer might look like
this: 156 (write 156 without saying it).
Lesson 4.2
Key Idea
Demonstrate that three-digit numbers up to 999 can written as three digitsmeans some hundreds, some
tens plus some ones on bundleable material.
Equipment
Sticks and rubber bands.Some Pre-bundled tens and loose ones.
Assessment Question
Point at 128 on the board - don’t read it aloud - and say:
Make up bundles that show this number (point at 128 without saying it).
Lesson 4.3
Key Idea
Demonstrate that three-digit numbers up to 999 written as three digitsmeans some hundreds, some
tens plus some ones on play money.
Equipment
Play money – one dollar, ten-dollar, and hundred-dollar notes.
Assessment Question
Point at $724 on the board - don’t read it aloud - and say:
Count out $724 (point at $728 without saying it).
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Lesson 4.4
Key Idea
Demonstrate that a mixture of some hundreds, some tens plus some ones presented out of order can
be written as a three-digit number.
Equipment
Play money – one dollar, ten-dollar, and hundred-dollar notes.
Assessment Question
Write 5 tens + 7 hundreds on the board.
Write underneath
57
75
and say:
Which of these equals 5 tens + 7 hundreds?
705
570
750
507
Lesson 4.5
Key Idea
Demonstrate that a mixture of some hundreds written as x00, some tens written as y0 plus some ones
written out of order can be written as a three-digit number.
Equipment
Play money – one dollar, ten-dollar, and hundred-dollar notes.
Assessment Question
Write 80 + 700 on the board.
Write underneath
80700
and say:
Which of these equals 80 + 700?
70080
870
780
78
87
Lesson 4.6
Key Idea
Demonstrate that a mixture of some hundreds written as x00, some tens written as y0 plus some ones
can be written as a three-digit number on play money.
Equipment
Play money – one dollar, ten-dollar, and hundred-dollar notes.
Assessment Question
Write 80 + 700 on the board.
Write underneath
80700
70080
870
780
78
87
and say:
Show 80 in play money. Show 700 in play money. Which of these numbers on the board equals 80 + 700?
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Lesson 4.7
Key Idea
Connect stories to calculations for addition and subtraction problems involving three-digit numbers.
Then allow the use calculators to find the answers.
Equipment
None
Assessment Question
Write Geraldine collects 629 apples. She sells 467 apples at her farm shop. The number of apples unsold
is:
467 - 629
629 + 467
629 – 467
367 + 629
and say:
Geraldine collects 629 apples. She sells 467 apples at her farm shop. The number of apples unsold is which of
therse? (Point at four answers).
Lesson 4.8
Key Idea
Collect piles of one dollars, ten-dollars, and hundred-dollars so that the total does not exceed a
thousand dollars.
Equipment
Play money – one dollar, ten-dollar, and hundred-dollar notes.
Assessment Question
Give out piles of more than ten ones, more than ten tens, and a few hundreds – total not to exceed one
thousand and say:
Find the total money you have and write down this total in digits.
Lesson4.9 Optional
Key Idea
Collect piles of one dollars, ten-dollars, hundred-dollars, so that the total does exceed a thousand
dollars.
Equipment
Play money – one dollar, ten-dollar, hundred-dollar notes, and thousand dollar notes.
Assessment Question
Give out piles of more than ten ones, more than ten tens, and lots hundreds – the total must exceed
one thousand and say:
Find the total money you have and write down this total in digits.
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Lesson 5.1
Key Idea
Quickly state any number from 1 to 100 when it is shown on a Slavonic abacus.
Equipment
Slavonic abacus – teacher only
Assessment Question
Arrange a number on the Slavonic abacus without students seeing what you
are doing – the diagram shows 68 and 32. Say:
I am going to show you the other side. Look for two numbers of beads.
Then students the abacus for about eight seconds then say:
What two numbers could you see on the abacus?
Lesson 5.2
Key Idea
Add and subtract up to three-digit numbers without any renaming. Numbers are presented in a mixture
of words and symbols.
Assessment of Prior Learning
Equipment
Play money.
Assessment Question
Write twenty-four + 60 on the board. Point at the numbers – do not read them aloud - and say:
Add these numbers.
Lesson 5.3
Key Idea
Add a two-digit number to a one-digit number whose answer is x0 by a part-whole method i.e. no
counting.
Equipment
Ten frames.
Assessment Question
Write twenty-seven + 3 on the board. Point at the numbers - do not read them aloud - and say:
Add these numbers.
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Lesson 5.4
Key Idea
Add a two-digit number plus an unknown one-digit number whose answer is given x0 by a part-whole
method i.e. no counting.
Equipment
Ten frames.
Assessment Question
Write forty-six + ? = 50 on the board. Point at the numbers - do not read them aloud - and say:
Find the missing number.
Lesson 5.5
Key Idea
Find x0 minus a single-digit number by a part-whole method i.e. no counting.
Equipment
Ten frames.
Assessment Question
Write forty - 4 on the board. Point at the numbers - do not read them aloud - and say:
Work out the answer.
Lesson 5.6
Key Idea
Find x0 minus an unknown single-digit number equals a given two-digit number by a part-whole method
i.e. no counting.
Equipment
Ten frames.
Assessment Question
Write forty - ? = 35 on the board. Point at the numbers - do not read them aloud - and say:
Find the missing number.
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Lesson 5.7
Key Idea
Add a two-digit number plus a one-digit number by a part-whole method i.e. no counting.
Equipment
Ten frames.
Assessment Question
Write 8 + forty-six on the board. Point at the numbers - do not read them aloud - and say:
Work out the answer.
Lesson 5.8
Key Idea
Find a two-digit number minus a single-digit number by a part-whole method i.e. no counting.
Equipment
Ten frames.
Assessment Question
Write forty-five - 7 on the board. Point at the numbers - do not read them aloud - and say:
Work out the answer.
Lesson 5.9
Key Idea
Find a two-digit number minus an unknown single-digit number equals a given two-digit number by a
part-whole method i.e. no counting.
Equipment
Ten frames.
Assessment Question
Write forty two - ? = thirty six on the board. Point at the numbers - do not read them aloud - and say:
Find the missing number.
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Lesson 5.10
Key Idea
Add a two-digit number to a one-digit number by a part-whole method i.e. no counting.
Equipment
Ten frames.
Assessment Question
Write seven + 48 on the board. Point at the numbers - do not read them aloud - and say:
Work out the answer.
Lesson 5.11
Key Idea
Add a two-digit number near 100 plus a one-digit number by a part-whole method i.e. no counting.
Equipment
Ten frames.
Assessment Question
Write 98 plus six on the board. Point at the numbers - do not read them aloud - and say:
Work out the answer.
Lesson 5.12
Key Idea
Find a two-digit number near 100 minus a single-digit number by a part-whole method i.e. no counting.
Equipment
Ten frames.
Assessment Question
Write one hundred and two minus 6 on the board. Point at the numbers - do not read them aloud - and
say:
Work out the answer.
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Lesson 5.13
Key Idea
Find a two-digit number near 100 minus an unknown single-digit number equals a given two-digit
number by a part-whole method i.e. no counting.
Equipment
Ten frames.
Assessment Question
Write 102 - ?equals ninety-nine on the board. Point at the numbers - do not read them aloud - and say:
Find the missing number.
Lesson 5.14
Key Idea
Add a two-digit number near to a hundred to one-digit number by a part-whole method i.e. no
counting.
Equipment
Ten frames.
Assessment Question
Write seven plus 96 on the board. Point at the numbers - do not read them aloud - and say:
Work out the answer.
Lesson 5.15
Key Idea
Find a three-digit number near x00 minus a single-digit number by a part-whole method i.e. no counting.
Equipment
Ten frames.
Assessment Question
Write 298 minus six on the board. Point at the numbers - do not read them aloud - and say:
Work out the answer.
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Lesson 5.16
Key Idea
Find a three-digit number near x00 minus an unknown single-digit number equals a given three-digit
number by a part-whole method i.e. no counting.
Equipment
Ten frames.
Assessment Question
Write 802 – what number equals seven hundred and ninety-eight on the board. Point at the numbers do not read them aloud - and say:
Find the missing number.
Lesson 5.17
Key Idea
Add a two or three-digit number plus a two or three-digit number in the vertical form with renaming
required, where students use the self-talk of place-value not the talk of “digits and carry”.
Equipment
Play money.
Assessment Question
Write 348 + 86 on the board in the vertical form. Point at the numbers - do not read them aloud - and
say:
Work out the answer. I will want you to explain your reasoning aloud when you have finished
Lesson 5.18
Key Idea
Find a two or three-digit number minus a two or three-digit number in the vertical form with renaming
required, where students use the self-talk of place-value not the talk of “digits and borrows”.
Equipment
Play money.
Assessment Question
Write 704 - 686 on the board in the vertical form. Point at the numbers - do not read them aloud - and
say:
Work out the answer. I will want you to explain your reasoning aloud when you have finished
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Lesson 5.19
Key Idea
Connect stories to calculations for addition, subtraction, multiplication, and division problems involving
any sized numbers. Then allow the use of calculators to find the answers.
Equipment
None
Assessment Question
Write Geraldine collects 625 apples. She decides to share the apples out among five needy families. Each
family gets how many apples:
625 - 5
5 - 625
six hundred and twenty-five ÷ 5
5 ÷ 625
5 x 625
and say:
Geraldine collects 625 apples. She decides to share the apples out among five needy families. Each family gets how
many apples? (Point at the answers).
Lesson 5.20
Key Idea
Use part-whole additions to solve multiplication problems.
Equipment
None
Assessment Question
Write:
Henry works out 4 x 5 this way:
5 + 5 + 5 + 5 = 10 + 10 = 20
Use Henry’s method to work out 5 x 3.
and say:
Henry works out 4 x 5 this way
5 + 5 + 5 + 5 = 10 + 10 = 20
Use Henry’s method to work out 5 x 3.
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Lesson 5.21
Key Idea
Use part-whole reasoning to solve division problems.
Equipment
Counters
Assessment Question
Write:
Janice works out 20 ÷ 5 this way:
She guesses the answer is 3 so she gives out 15 counters. Now there are five left. She gives one more
to each person. So the answer to 20 ÷ 5 is 4.
Use counters to show what Janice did, then be ready to explain what you did.
Now say:
Janice works out 20 ÷ 5 this way:
She guesses the answer is 3 so she gives out 15 counters. Now there are five left. She gives one more to each
person. So the answer to 20 ÷ 5 is 4.
Use counters to show what Janice did, then be ready to explain what you did.
Lesson 5.22
Key Idea
Students demonstrate that they can add a column of single digit figures rapidly. This activity can be
repeated to help overlearn basic facts and early part-whole reasoning.
Equipment
None
Assessment Question
Point at the column of single –digit numbers on the board and say:
Add these up.
What to Look For
Successful
9
4
8
6
8
+9
Students use part –whole reasoning to work out the answer.
Unsuccessful
Students use fingers or make mistakes in basic facts especially the pairs that add up to ten i.e. 1 + 9, 2 +
8,
3 + 7,… 2 + 8, 1 + 9.
Key Idea
TBA
Teaching Ideas
TBA
8/2/2011
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