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.”

“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!”

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 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 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 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?

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” 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 ? Stage 5: + symbol for tenths Stage 6: + symbols for hundredths & thousandths

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.

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.