Something to Think About

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. 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. 46-47) • 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 3 7 2 • 372 = 3  100 + 7  10 + 2  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 BLAPSplace.

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, 30 099, 24 000, 989 999, 603 000 • Place Value • Which of these numbers is the largest / smallest? • 4650 5046 5406 4506 • 352 097 90 325 79 532 297 320 • 0.76 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?

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:40+20=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” • 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.

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. 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.