1. 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?

2. Place Value Workshop
“We can start you off with a weekly salary in the four figures … two if you don’t count the decimal.”

3. “We remember what we understand!”
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!”

4. Warm-Up: Introducing tenths
Stages: AA(6) -> AM(7) (Bk 5 p )
Problem:
Henry has 6 bars of chocolate to share among 5 friends (including himself). How much does each friend get?

5. 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 TENS ONES
372 = 3    1
The 7 is in the tens place
BLAPS
BLEEPS
BLIPS
372 = 3 x BLIPS + 7 x BLAPS + 2 x BLEEPS
The 7 is in the BLAPS place.

6. 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

7. Knowledge Test Questions
Number Sequence and Order
What number is one more/less than
499, 840, , , ,
Place Value
Which of these numbers is the largest / smallest?
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 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?

8. 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

9. What Do Students Need To Know?

10. 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: =60; 8+5=13;  60+13=73
Research shows most successful learners use a jump strategy.

11. 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

12. Symmetry
Pattern to the place names
Pattern continues to decimal fractions
Where is the centre of symmetry?

13. The Two “Laws of Place Value”
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.

14. 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

15. 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

16. 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

17. Fractions before Decimal Fractions
Stage 4:
Know symbols for basic unit fractions to fifths
Does represent ?
Stage 5:
+ symbol for tenths
Stage 6:
+ symbols for hundredths & thousandths

18. Leave the size of the piece the same one half: one third: one fourth:
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.

19. 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

20. 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

21. Numeracy Teaching Progression

22. 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.