Transcript Topic 4 ppt - edp245MathsMateGroup20

Explain what place value is and why.... -
Sue v
Give examples of common problems and misconceptions.... - Pauline
Explain patterns and relationships.... - Su
Give examples of learning activities to develop ‘trading’ or ‘renaming’.... - Anna
Use the FSiM documents as a resource to select learning activities.... - Danielle
Outline important learning in place value for children to develop.... - Suzanne
Place Value......
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is the key to understanding how we say, read, write and calculate with whole numbers. It is the patterns in
the way we put the digits together that enables us to write an infinite sequence of whole numbers and to
easily put any two whole (or decimal) numbers in order.
makes it easy to split numbers into parts and this helps us calculate with numbers.
should not be taught in isolation.
use a variety of mental, diagrammatic and informal written strategies…
encourage students to use the patterns in the way we write and say numbers to split numbers into parts in
helpful ways.
There are many important mathematical skills that children need to make sense of and one of
these is understanding the importance of place value. Reys et al (2009) states that “place
value is critical to the understanding of the number system”.
Place value is the value of a digit, the value depends on its place or position in the number.
 There are many aspects to being able to understand place value and some are crucial for a students’
complete understanding of place value. Knowing the place value of a digit is needed to allow you to read
numbers accurately.
 EXAMPLE: the number 444
400
40
4
Hundreds
Tens
Ones
4
4
4
The famous mathematician Gauss said “without place vale, we would
get no place with numbers”.
JUST AS NAMES ARE IMPORTANT TO PEOPLE, PLACE VALUES ARE
IMPORTANT TO NUMBERS.
Common problems and misconceptions for
children
learning
place
value
We read from left to right, yet the numbers 13 to 19 must be read from right to left.
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This can cause children to reverse numbers, for example, 14 may be recorded as 41
(Booker, Bond, Sparrow, & Swan, 2010).
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The written numbers eleven and twelve give no indication of its value of tens and
ones. Twelve may be confused with twenty because of the way they sound (Booker et
al., 2010).
 Unlike the numbers 14, 16, 17, 18, 19, 40, 60, 70, 80, and 90, the way we say the
numbers 13, 15, 20, 30, and 50 sounds dissimilar to the numbers 1-9. Therefore,
Booker et al. (2010) recommend teaching students multiples of tens in the order of
90, 80, 70, 60, and 40. Thereafter, 50 and 30, which sound like the ordinal numbers 5 th
and 3rd, should be introduced, and then 20. Lastly, students should be taught the teen
numbers in the order of 19, 18, 17, 16, 14, 15, 13, and finish with 12 and 11.
 Students may not understand that the value of the tens place represents multiples of
10, and therefore may think that the ones value is larger in numbers such as 37, 28,
59 etc. (Curtin University, n.d.).
 When reading numbers with 3 or more digits, students may not realise that the
positioning of the digits indicates their value. For example, students may think that
298 is greater than 311 (Reys, Lindquist, Lambdin, & Smith, 2009).
 Understanding the concept of zero can also cause difficulties. Booker et al. (2010)
contend that using zero as a ‘placeholder’ may confuse students. This is contrary to
the notes provided by Curtin University (n.d., slide 11). Because of this difficulty,
Booker et al. (2010) recommends that students do not work with numbers containing
a zero place value until they have had some experience with multiple digit numbers.
Some activities to support children’s ‘correct’ learning.
Concrete materials and manipulatives are vitally important for teaching place value to students.
As students record the results of these learning activities, they begin to represent quantities
of objects numerically, and with the correct place values (Gluck, 1991).
Figure 3 (Gluck, 1991, p. 12) shows a board that has been divided into 3 sections. Gluck
(1991) recommends that the sections are not labelled as this can be distracting for the
students. Above each section is a flip-set of numbers valued from 0-9. Students begin by
rolling a dice and placing items to the value shown on the face of the dice onto the ‘ones’
section. They then flip over the numbers to show how many items are in that section.
Students repeat these actions, and as soon as the ones section amounts to 10 or more, the
students remove 10 units, and trade it for a single tens bar which is placed in the tens
column. The flip cards are also changed to show how many items are in each section. As
students continue to roll the dice and move around the items, they can see how the items
are represented numerically by looking at the flip cards.
Unifix trains
A group of students compete to join as many unifix cubes together as they can to
make a long train. Once all the unifix cubes are used, the students must break
down their trains into tens and ones and position them on their place value mats.
They then use paper and pen to record the number of unifix cubes they had in their
train (WGBH Educational Foundation, 1995).
 Compare, order and sequence three digit numbers using materials
to help provide meaning (Booker et al., 2010).
Place these numbers in order from smallest to largest:
 Count forwards and backwards by ones, tens and hundreds to
consolidate understanding of place value (Booker et al., 2010)
Explain patterns and relationships in the place
value system
 The key to developing numbers sense is to have an understanding of place value and ordering
of numbers (Reys, Lindquist, Lambdin & Smith, 2009). Furthermore, place value is the
organisational structure for counting and is required when working with whole numbers and
decimals (Reys et al., 2009).
 To develop place value understanding students need to learn whole numbers are in a particular
order and there are patterns in the way we say them which helps us to remember the order
(Curriculum Council of Western Australia,1998).
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Hands-on activities, including the use of manipulatives help establish and develop place value
understanding. A 100 chart, calculator and the use of ten blocks are great way to help
students recognise patterns and relationships. Students are able to see relationships
between places and how place value is represented by using 1-9 and how 0 is a place
holder that demonstrates lack of quantity. With experiences students learn the constant
multiplicative relationship between the places, with the values of the positions increasing in
powers of tens, from left to right.
 Through experiences and language students can develop place value understanding and recognise
patterns and relationships. Consequently, students are able to see there are many ways a
number can have equal representations e.g. 120 may be 1 x hundred + 2 x tens and is the
same as 12 x tens or 10 x tens + 20 ones.
 Teachers can help develop student’s understanding of patterns in number with activities such
Number Jigsaw and Jumbled Charts, whilst Number Peg Up or Washing Line and Number
Ladder help support order relation understanding.
Trading (or grouping) – grouping numbers into groups of 1s, 10s, 100s, 1000s (etc) in order to perform simple additions and subtractions
For example, you get 10 of a particular unit and then trade it for one of the subsequent unit (i.e 10 ones are traded for 1 ten. 10 tens are
traded for 1 hundred). It is important for teachers to make sure the students understand the link between the blocks, the actions and the
words – than merely move the blocks as directed by the teacher without making the connections with the concept of trading and the
addition process.
A simple trading example is:
34 + 21 = ?
= 55
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Renaming – Numbers need to be renamed in a variety of ways rather than being
understood in terms of counting or even place value (Booker et al, 2010, pg 81).
For example, 89 can be viewed as 8 tens and 9 ones, as well as 89 ones.
568 can be interpreted as 5 hundreds, 6 tens and 8 ones; or 56 tens, 8 ones; or 568
ones.
**** RENAMING AND TRADING GO HAND-IN-HAND ****
Explain why these understandings are
essential for children’s number learning.
• Renaming and trading numbers are crucial number processes in developing number sense
(Booker et al, 2010, pg 82). Renaming of numbers is used everyday.
• It is important that students understand trading and renaming in order to create meaningful
mathematical experiences.
• Renaming is a crucial way of understanding numbers. Booker et al state that understanding
how numbers can be renamed is important for comparison and rounding, counting on and back,
and, later, students will need an understanding of renaming in order to understand the
algorithms for subtracting and diving large numbers (2012, pg 119).
• These concepts lay the foundations to a students future mathematical education, for
example, once an understanding of renaming and trading has been gained, children can then
extend this thought process to counting on and back by hundreds, tens and by ones (especially
with larger numbers) Booker et al, 2010, pg 121).
• Students must understand when renaming is appropriate, for example, in Stage 1 Mathematics
(NSW), students may be shown that you only need to rename sometimes.
An example is: 65-32 = ? (show working and explain why you do/do not have to rename a
number). Answer: Renaming is the same thing as borrowing. In this particular problem, 32 can
be subtracted from 65 without renaming (65-32=33). However, if the problem was 62-35, you
would have to rename the 2 to 12, by borrowing from the 6, which would then become 5.
5 62 12 35
27
Examples
Activity 1: Using pop-sticks to rename
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Ensure that students are familiar with using pop-sticks and rubber bands to
represent three-digit whole numbers, as an extension of work with 2-digit
numbers. For example:
 43 is made with 4 bundles of ten and 3 singles,
 143 is made with 1 group of ten bundles of ten (i.e. a hundred group), 4
bundles of ten and 3 singles.
 Then ask them to make 143 using only bundles of ten and singles, (i.e. 14
bundles of ten and 3 singles). Give them practice with other three-digit
numbers. Students can make challenges for each other to complete. The
advantage of using the pop-sticks is that all the individual units are easily
seen and can be bundled and unbundled readily; a disadvantage is that many
sticks are required for larger three-digit numbers. Students can prepare
bundles of ten and of ten tens (100) to keep for use on many occasions.
Activity 2: Using MAB to rename
Ensure that students are familiar with using MAB to represent numbers. For
example: 43 is made with 4 longs and 3 minis
43 can also be made with 43 minis
143 is made with 1 flat, 4 longs and 3 minis
Note to Teachers: Emphasise that 43 separate minis is cumbersome compared
with the convenience of using 4 longs and 3 minis instead. Highlight, that
there are still the same number of blocks (really, the total volume is still the
same). Then ask them to make 143 using only:
longs and minis (14 longs and 3 minis)
flats and minis (1 flat and 43 minis)
Source: State Government of Victoria. (2009). Department of
Education and Early Childhood Development. Mathematics
Developmental Continuum. Retrieved from
http://www.education.vic.gov.au/studentlearning/teachingres
ources/maths/mathscontinuum/number/N22501P.htm#a1
Give students practice with other three-digit numbers. Students can also make challenges for each other to complete. MAB are useful
because quite large numbers can be represented. A disadvantage is that a long, for example, has to be exchanged for 10 separate minis,
rather than broken up into 10 minis.
Examples (cont’d)
Trading
Number expanders are a common tool used in the classroom
to help explain the concept of renaming.
Blank, to show hundreds, tens and ones.
236 = 2 hundreds + 3 tens + 6 ones
236 = 2 hundreds + 36 ones
Adding numbers
 Numbers of any size can be added together easily.
 When adding 2, 3 and 4 digit numbers using a
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236 = 23 tens + 6 ones
written method, write the numbers in a vertical
list.
You will need to properly line up the place value
columns so you get the correct total.
Example:
Step 1: Make sure each number is carefully listed
in neat place value columns to avoid errors.
Step 2: Working from right to left, add each
column, starting with the units column.
You may have to trade to
the next column.
236 = 236 ones
Subtracting numbers
When we subtract, we take away one of the two numbers from the
other. Make sure the same place value columns line up underneath
each other.
6 ones cannot be taken away from 5 ones. We need to add ten ones
to make 15, and change the 3 at the top of the tens column to 2. 15
ones less 6 ones, equals 9 ones.
Step 2: (Tens column)
4 tens cannot be taken away from 2 tens. Trade 1 hundred to make
12, and change the 4 at the top of the next column to a 3. 12 less 4
equals 8.
Step 3: (Hundreds column)
3 hundreds less 2 hundreds leave 1 hundred.
Answer = 189
9 ones + 5 ones = 14 ones.
Trade ten ones for one ten
6 tens + 1 ten = 7 tens
2 hundreds + 7 hundreds = 9
hundreds
Answer = 974
Use the first steps as a resource to select
learning activities suitable for students of
particular achievement levels
Place value and numeration
By the end of the matching phase (level 1)
• Can say the number names in order into the teens
• Activity - Jack in the box
By the end of the quantifying phase (Level 2)
• Can use the decades up to and over 100
• Activity – Number scrolls
By the end of the partitioning phase (Level 3)
• Students will use the names of the first several places from the right(ones, tens, hundreds,
thousands)
• Activity – Bicycle odometer
By the factoring phase (level 4)
• Students understand and use the cyclical pattern in whole numbers
• Activity – comparing numbers and place value
By the operating phase (level 5)
• Students can say and read any decimal number
(Department of Education and Training of Western Australia, 2004; First steps in Mathematics, 2004).
Important learning in place value for
children to develop
Suzanne Akrap
Professional
Judgement:
Knowledge,
experience and
evidence
Pedagogy: Decide on
learning activities and
focus questions
 Your lessons are prepared productively so your
Planning
Cycle
DETWA, 2004
Mathematics:
Decide on the
mathematics
needed to move
students on
 The planning cycle is the first key to ensure
 Students will understand place value.
Students:
Observe
students and
interpret what
they do and say
Children need to understand:
*The order of digits makes a
difference
*The position of a digit tells us
the quantity it represents
*Zero is used as a placeholder
*There is constant multiplicative
relationship between the places
with the values of the position
increasing in powers of 10 from
right to left
*The value of the digit multiplied
by the value of the place tells us
the quantity a digit represents.
HELPING CHILDREN LEARN PLACE VALUE
•The use of varieties of mental, diagrammatic and informal
written strategies help to visualise the meanings of place value.
•Encourage students to use the patterns in the way we write and
say numbers to split numbers into parts in helpful ways.
•Visual reminders can be used in the classroom such as:
Place value mats
Grouping and trading of objects
Counting objects and understanding the patterns
Reading and writing numbers
References
Booker, G., Bond, D., Sparrow, L., & Swan, P. (2010). Teaching primary mathematics (4th ed.). Frenchs Forest, NSW:
Pearson Australia
Curriculum Council of Western Australia (1998). Curriculum framework for kindergarten to year 12 education in Western
Australia. Osborne Park, W.A.: Author. Retrieved from
www.curriculum.wa.edu.au/internet/Years_K10/Curriculum_Framework
Curtin University. (n.d.). Place value and numeration [PowerPoint slides]. Retrieved from https://lms.curtin.edu.au
Department of Education and Training of Western Australia. (2004). First Steps in mathematics: Number –
Understanding whole and decimal numbers; Understanding fractions. Port Melbourne: Rigby Heinemann.
First steps in Mathematics(2004)Retrieved from
https://lms.curtin.edu.au/webapps/portal/frameset.jsp?tab_tab_group_id=_4_1&url=%2Fwebapps%2Fblackboard%
2Fexecute%2Flauncher%3Ftype%3DCourse%26id%3D_63397_1%26url%3D
Gluck, D.H. (1991). Helping students understand place value. The Arithmetic Teacher, 38 (7), 10-13. Retrieved from
http://search.proquest.com.dbgw.lis.curtin.edu.au/docview/208778114/fulltextPDF
Reys, R., Lindquist, M.M., Lambdin, D.V., & Smith, N.L. (2009). Helping children learn mathematics (9th ed.). New
York, NY: John Wiley and Sons.
State Government of Victoria. (2009). Department of Education and Early Childhood Development. Mathematics
Developmental Continuum. Retrieved from
http://www.education.vic.gov.au/studentlearning/teachingresources/maths/mathscontinuum/number/N22501P.htm#
a1
WGBH Educational Foundation. (1995). Place value centres [Video recording]. Retrieved from
http://www.learner.org/resources/series32.html
Zevenbergen, R., Dole, S. & Wright, R.J. (2004). Teaching Mathematics in Primary Schools. Retrieved from
http://www.allenandunwin.com/teachingmaths/secure/mabadd.htm#with