Download Workshop-PlaceValue

Something to Think About

The Zeros Problem
In the number 203,500, the last two
zeroes are called terminal zeros. The
zero after the digit 2 is not a terminal
zero. How many terminal zeros does
the product of the first 30 counting
numbers (1 x 2 x 3 x 4 x . . . x 30) have?
Place Value Workshop
“We can start you off
with a weekly salary
in the four figures
…
two if you don’t
count the decimal.”
Overview
Warm-up
 Traditional Place Value
 Conceptual Place Value
 What do the students’ need to know?
 How do we teach it?

Goal: Long Term Understanding
“We remember what we understand!”
Warm-Up: Introducing tenths
Stages: AA(6) -> AM(7) (Bk 5 p. 46-47)
 Problem:
Henry has 6 bars of chocolate to share
among 5 friends (including himself).
How much does each friend get?

What is Place Value?

What is traditional place value instruction?
– Write 372 in expanded form.
– What number is in the tens place?


Goal: support the development of standard
written algorithms
Can be done without a lot of understanding:
HUNDREDS
BLIPS
3


BLAPS
TENS
7
BLEEPS
ONES
2
372
+ 7+710
+ 2  1+ 2 x BLEEPS
372==33x100
BLIPS
x BLAPS
The
place
The77isisininthe
thetens
BLAPS
place.
Conceptual Place Value
Supports the development of students’
intuitive arithmetical strategies.
 Handout:
Knowledge framework Stages 4 - 8

– Number Identification (Stages 4-6)
– Number Sequence and Order
– Grouping and Place Value
– Addition and Subtraction
Knowledge Test Questions

Number Sequence and Order
What number is one more/less than
499, 840, 30 099, 24 000, 989 999, 603 000

Place Value
– Which of these numbers is the largest / smallest?
• 4650
• 352 097
• 0.76
5046
5406
4506
90 325 79 532 297 320
0.657
0.7
– A radio costs $270. How many $10 notes do you need to
pay for it?
– You have $26,700 in $100 notes. How many notes do you
have?
– What number is 3 tenths less than 2?
– In 78.912 the 7 is in the tens column. Which number is in
the tenths column?
– Write a number that lies between 7.59 and 7.6.
– What is 137.5% as a decimal?
Place Value Teaching Progression


Place Value Knowledge builds in a cycle
Students need to work through the cycle
before moving to the next stage
Phase 1:
Number Identification/Sequence/Grouping
Phase 2:
Addition
Phase 3:
Subtraction
What Do Students Need To Know?
Number Lines – Bridging to Add/Sub
Number Word Sequences
Forwards and backwards
 By 1s, 10s, 100s, 1000s,
 By 2s, 3s, 5s
 On and off the decade & the multiple
Efficient, flexible add/sub strategies: 48+25


Jump: 48  50, 50  70, 70  73

Split:
40+20=60; 8+5=13;  60+13=73
Research shows most successful learners use a jump strategy.
Structure of the Place Value System
Names of places
 Moving to the left: each place is 10
times bigger than the previous one


Moving to the right: each place is one
tenth of the previous one
Symmetry

Pattern to the place names

Pattern continues to decimal fractions

Where is the centre of symmetry?
The Two “Laws of Place Value”

Law 1:
– You can only have up to nine in any one
place.
– Once you have ten of something, you must
replace them with one of the next size up.

Law 2:
– If you want to break something up, you
must always break it into ten of the next
size down.
Decimal Fractions and Place Value

Knowledge Framework:
Decimals do not appear at all until stage 6 (AA)
– Students know the number of tenths and hundredths in
decimals (up to 2 dp)
• tenths in 7.2 =
• hundredths in 2.84 =
– Students round decimals (up to 2 dp) to the nearest whole
number
Decimal Fractions and Place Value

Knowledge Framework:
Decimals do not appear at all until stage 6 (AA)
– Students know the number of tenths and hundredths in
decimals (up to 2 dp)
• tenths in 7.2 = 72
• hundredths in 2.84 = 284
– Students round decimals (up to 2 dp) to the nearest whole
number
Decimal Fractions and Place Value

Knowledge Framework:
Decimals do not appear at all until stage 6 (AA)
– Students know the number of tenths and hundredths in
decimals (up to 2 dp)
• tenths in 7.2 = 72
• hundredths in 2.84 = 284
– Students round decimals (up to 2 dp) to the nearest whole
number

Students do not add / subtract with decimals until
stage 7 (AM)

This means most of us will be working on whole
number place value with most of our students
Fractions before Decimal Fractions

Stage 4:
– Know symbols for basic unit fractions to fifths
– Does
represent 1 ?
3

Stage 5:
+ symbol for tenths

Stage 6:
+ symbols for hundredths & thousandths
Check Understanding
Leave the size of the piece the same
one half:

one third:
one fourth:
one fifth:
one tenth:

If students’ understand that one tenth means
one of 10 equal pieces, then they are ready
for decimal fractions.
Cake Factory - DeciMats
For Stages 5 – 7
 Mat represents 1 whole – cake ate a
party
 Cut one mat into tenths – 10 kids at the
party
 Cut one mat into hundredths (later) –
take your piece home and share with
the 10 members of your family
 Use canons of place value to trade up
or down

Numeracy Teaching Progression

Find out what they know already

Work out what they need to know next

Teach them using progression:
Materials  Imaging  Number Properties

Reinforce it through practice activities
– Starters (Bk 4) and Games

Provide extension for able students
– Enormous Numbers & Other bases
Numeracy Teaching Progression
Your Turn
In your groups you are going to be
given materials and an outline of an
activity.
 Discuss how you would use the material
in a lesson.
 What prior knowledge the students
would need.
 How you would extend it to use imaging
and a generalization.
