place value
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PLACE VALUE. “We can start you off with a weekly salary in the four figures … two if you don’t count the decimal.”. OVERVIEW. Finding out what they know already What do they need to know? How do we teach it to them? Reinforcement Extension. 1. FINDING OUT WHAT THEY KNOW. - PowerPoint PPT PresentationTRANSCRIPT
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
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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?
HUNDREDS TENS ONES 3 7 2
372 = 3 100 + 7 10 + 2 1
The 7 is in the tens place
BLIPS BLAPS BLEEPS
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.
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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?• 488 620 602 448• 4650 5046 5406 4506• 352 097 90 325 79 532 297 320• 0.76 0.657 0.7• 0.478 0.8 0.39
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?
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ALTERNATIVE PLACE VALUE QUESTIONSWrite 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?
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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 VALUEPlace 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
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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.
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WHOLE NUMBERS – THE CAKE FACTORYA 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 10 + 3 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 10 + 2 1 etc• What would be the easiest way to buy the cakes?
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WHOLE NUMBERS – THE CAKE FACTORY3. 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 10
5. 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
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DECIMALS – DECIMATSMat 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 …
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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.
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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 ofeither cell in the second row.
Next roll must be used to completethe 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: 1 + 56 + 412 + 6,632 + 35,422 + 653,311.
Winner is the person with the highest score.
Variation: Students to aim for the lowest possible total.
15 6
2146 6 3 2
3 5 4 2 26 5 3 3 1 1
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EXTENSION – ENORMOUS NUMBERS
658 757 348 655 231 585 454 587 687 858 758 657 656 747 324 468 526 457 854 243 356 846
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!
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EXTENSION – ENORMOUS NUMBERS
658 757 348 655 231 585 454 587 687 858 758 657 656 747 324 468 526 457 854 243 356 846
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
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EXTENSION – PLACE VALUE PUZZLERStart: Write down the number six million, thirty four
thousand, seven hundred and fifty two.
Now: (keeping a running total)a) Increase this by three thousandb) Add on 400c) Reduce by 1 000 000d) Increase by two hundred thousande) Add on fifty moref) Decrease by 41 000g) Increase by 2000h) Subtract 300 000i) Subtract two tenthsj) Add on three hundredthsk) Increase by nine thousandthsl) Increase by 8 hundredthsm) Add on 5 tenthsn) Write the final number in words. 24
EXTENSION – BASE EIGHTIn base 10:
12 = 1 10 + 2 1
20 = 2 10
215 = 2 102 + 1 10 + 5 1
3461 = 3 103 + 4 102 + 6 10 + 1 1
In base 8:
128 = 1 8 + 2 1 = 1010
208 = 2 8 = 1610
2158 = 2 82 + 1 8 + 5 1 = 128 + 8 + 5 = 14110
34618 = 3 83 + 4 82 + 6 8 + 1 1 = 1536 + 256 + 48 + 1 = 184110
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EXTENSION – BASE EIGHTAdding 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.
5 + 4 = “9” 1 8 + 1 11
12+ 167
Ones column: 2 + 7 = “9” = 1 8 + 1 … Put down 1, carry 1
Tens column: 1 + 1 + 6 = 8 … Put down 0, carry 1
Hundreds column: 1 + 1 = 2
… so 128 + 1678 = 2018
1
1
02
1
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