PLACE VALUE “We can start you off with a weekly salary in the four figures … two if you don’t count the decimal.” 1

OVERVIEW 1.Finding out what they know already 2.What do they need to know? 3.How do we teach it to them? 4.Reinforcement 5.Extension 2

1. FINDING OUT WHAT THEY KNOW 3

TRADITIONAL PLACE VALUE QUESTIONS Write 372 in expanded form. What number is in the tens place? If a student can do these questions, what does that tell us? HUNDREDSTENS ONES = 3    1 The 7 is in the tens place BLIPSBLAPSBLEEPS 372 = 3 blips + 7 blaps + 2 bleeps The 7 is in the blaps place To find out how much they really know, we need to ask better / more diverse questions which challenge students’ understanding of place value. 4

KNOWLEDGE TEST QUESTIONS (Q20 – 34) A CD player costs $80. How many $10 notes do you need to pay for it? 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? 5

ALTERNATIVE PLACE VALUE QUESTIONS Write down / hold up these numbers: Eight hundreds and nine ones Five thousands and three tens Three hundreds, sixteen tens and four ones Four ten thousands, twenty five tens and eleven ones Three and four tenths Three and forty tenths Three and forty hundredths What number is 5 more / 10 less / 100 more / 1 tenth less than …? What number is ten times as much as …? Show me a number between 1 and 2 / between 0.5 and 0.6 … Given 3-4 digits to work with : What is the largest / smallest number you can make? Put the digits in order from smallest to largest How many different numbers can you make? Can you make two numbers that are close together? How close? 6

2. WHAT DO THEY NEED TO KNOW? 7

WHAT DO STUDENTS NEED TO KNOW? Structure of place value system – names of places; each place is 10 times the size of the previous one; NOT symmetric about decimal point! Relative size of numbers / parts of numbers The two “Canons of Place Value”: Canon 1 (Needed for adding / multiplying): 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 place size up. Canon 2 (Needed for subtracting / dividing): If you want to break something up, you must always break it into ten of the next size down. 8

DECIMALS AND PLACE VALUE Place Value in Knowledge Framework (Book 1 p 18-22): Decimals do not appear at all until stage 6 (Advanced Additive) At stage 6, students know the number of tenths and hundredths in decimals (up to 2 dp) e.g. tenths in 7.2 = 72, hundredths in 2.84 = 284 Students round decimals (up to 2 dp) to the nearest whole number Addition / Subtraction in Strategy Framework (Book 1 p 15 – 17): Students do not add / subtract with decimals until stage 7 (Advanced Multiplicative) This means most of us will be working on whole number place value with most of our students 9

3. HOW DO WE TEACH IT TO THEM? 10

NUMERACY TEACHING PROGRESSION (Book 3 p 5-7) 11

NUMERACY TEACHING PROGRESSION Using Materials (Manipulating): Teacher models on materials. Students manipulate materials themselves. Using Imaging (Visualising): Teacher covers materials and describes what they are doing (e.g. adding 2 more). Students are asked to describe the result. Teacher asks students to predict what will happen, then use materials to check answer. Using Number Properties (Generalising): Once students can image problems, increase the complexity or size of the numbers involved so the use of materials (even images of materials) becomes inefficient or difficult. 12

WHOLE NUMBERS – THE CAKE FACTORY A cake factory sells single cakes, packets of 10 cakes, and boxes which contain 10 packets of 10 cakes. 1. Sarah needs 13 cakes for a party. What should she buy? 13 single cakes OR 1 packet of 10 and 3 single cakes Show both arrangements with materials – 13 single multicubes; 1 block of 10 plus 3 single cakes Record: 13  1 is the same as 1   1 What would be the easiest way to buy the cakes? 2. Tim needs 32 cakes for a party. What should he buy? 32 single cakes OR 3 packets of 10 and 2 single cakes Other possibilities – 2 packets of 10 and 12 singles; 1 packet of 10 and 22 singles Show or draw arrangements (use maths book squares) Record: 32  1 = 3   1 etc What would be the easiest way to buy the cakes? 13

WHOLE NUMBERS – THE CAKE FACTORY 3. Tim needs 100 cakes for a party. What should he buy? 4. When Tim gets home, his mother says the cakes are to be shared equally between two different parties. She sends him back to exchange the big box of cakes. Have a “banker” at the cake factory who changes the box of 100 for 10 packets of 10 Record: 1  100 = 10  Sarah has two packets of 10 cakes to share out among 5 friends. Banker – exchange 2 packets of 10 for 20 single cakes Share out equally 6. Tim has 6 packets of 10 cakes to share out among 5 friends. Does he need to change all the packets for single cakes? Banker – exchange 1 packet of 10 for 10 single cakes Share out equally 14

NUMERACY TEACHING PROGRESSION 15

DECIMALS – DECIMATS Mat represents 1 whole. Cut one or two mats into tenths (one colour). Cut one or two mats into hundredths (another colour). Keep one or two whole mats (third colour). Students use their own set of pieces to: make four tenths make four tenths and five hundredths make one whole, six tenths and four hundredths show one tenth is the same size as ten hundredths show twelve hundredths is the same as one tenth and two hundredths show ten tenths is the same as one whole decide which is bigger – 0.4 or 0.14? add 0.4 and 0.3 add 0.2 and 0.05 subtract 3 hundredths from one whole … 16

4. REINFORCEMENT 17

REINFORCEMENT - PLACE VALUE GAMES Place Value Memory (matching words with symbols) Highest Number Wins (comparing size of numbers) Place Value Challenge (making numbers given instructions) These games can all be easily adapted to include tenths and hundredths for stage 7-8 students … … Or adapted to smaller whole numbers (e.g.. up to thousands only) for lower students. 18

REINFORCEMENT - STARTERS Close to 100 Number Hangman Dice Game:Students rule up grid: Student running the game rolls dice; players place this number in the top cell. Next roll – students have a choice of either cell in the second row. Next roll must be used to complete the second row. Next roll – can choose where to position in third row … Continue filling grid one row at a time. Students must place each number before the next one is rolled! 19

REINFORCEMENT - STARTERS For example: First roll = 1 Second roll = 5 Third roll = 6 Fourth roll = 2 … and so on … When grid is full, add up: , , ,311. Winner is the person with the highest score. Variation: Students to aim for the lowest possible total

5. EXTENSION 21

EXTENSION – ENORMOUS NUMBERS How would you write this number in words? 658 vigintillion 757 novemdecillion 348 octadecillion 655 septencillion 231 sexdecillion 585 quidecillion 454 quattuordecillion 587 tredecillion 687 duodecillion 858 undecillion 758 decillion 657 nontillion 656 octillion 747 septillion 324 sextillion 468 quintillion 526 quadrillion 457 trillion 854 billion 243 million 356 thousand 8 hundred and 46! 22

EXTENSION – ENORMOUS NUMBERS Now can you: Add 1 novemdecillion, 10 quidecillion, 28 trillion, 16 thousand? Subtract 28 vigintillion, 14 octadecillion, 29 tredecillion, 45 billion? Is this number divisible by 2? Explain. Is this number divisible by 3? Explain. What other numbers can this number be easily divided by? Inspiration: “Amazing Maths Activities” – Macmillan Education 23

EXTENSION – PLACE VALUE PUZZLER Start: Write down the number six million, thirty four thousand, seven hundred and fifty two. Now: (keeping a running total) a)Increase this by three thousand b)Add on 400 c)Reduce by d) Increase by two hundred thousand e)Add on fifty more f)Decrease by g)Increase by 2000 h)Subtract i)Subtract two tenths j)Add on three hundredths k)Increase by nine thousandths l)Increase by 8 hundredths m)Add on 5 tenths n)Write the final number in words. 24

EXTENSION – BASE EIGHT In base 10: 12 = 1   1 20 = 2  = 2    = 3     1 In base 8: 12 8 = 1   1 = = 2  8 = = 2    1 = = = 3     1 = =

EXTENSION – BASE EIGHT Adding and Subtracting: The Canons of Place Value change: You can only have up to 7 in one column. Once you have 8, you must change them for one of the next size up = “9”  1   Ones column: = “9” = 1  … Put down 1, carry 1 Tens column: = 8 … Put down 0, carry 1 Hundreds column: = 2 … so =

SUMMARY 1.Find out what they know already 2.Work out what they need to know next 3.Teach them using progression: Materials  Imaging  Number Properties 4.Reinforce it through practice activities 5.Provide extension for able students 27