Topic 4: Place value and numeration- Group 20

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......

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

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.

400 40 4Hundreds Tens Ones

4 4 4

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. This can cause children to reverse numbers, for example, 14 may be recorded as 41 (Booker, Bond, Sparrow, & Swan, 2010).

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 5th 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 trainsA 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).

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.

Explaining the concepts of ‘trading’ or ‘renaming’

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

• 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

Activity 1: Using pop-sticks to rename 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 renameEnsure 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)

Examples

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/teachingresources/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)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

236 = 23 tens + 6 ones

236 = 236 ones

TradingAdding numbers Numbers of any size can be added together easily. When adding 2, 3 and 4 digit numbers using a

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.

9 ones + 5 ones = 14 ones.Trade ten ones for one ten6 tens + 1 ten = 7 tens2 hundreds + 7 hundreds = 9 hundredsAnswer = 974

Subtracting numbersWhen 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

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).

Use the first steps as a resource to select learning activities suitable for students of

particular achievement levels

Important learning in place value for children to develop

Suzanne Akrap

The planning cycle is the first key to ensure Your lessons are prepared productively so your

Students will understand place value. Planning

Cycle DETWA,

2004

Professional Judgement: Knowledge, experience

and evidence

Pedagogy: Decide on learning

activities and focus questions

Mathematics: Decide on the mathematics

needed to move

students on

Students: Observe

students and interpret

what they do and say

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 objectsCounting objects and understanding the patterns

Reading and writing numbers

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.

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