Transcript Workshop 3 - Place Value

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?
BLIPS
HUNDREDS
3
BLAPS
TENS
7
BLEEPS
ONES
2
372 = 3 blips
 100++77blaps
 10 ++ 22 bleeps
1
The 7 is in the blaps
place
tens 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?
• 488
620
602
• 4650
5046
5406
• 352 097
90 325
79 532
• 0.76
0.657
0.7
• 0.478
0.8
0.39
448
4506
297 320
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?
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  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?
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  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
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
4
6 6
3 5 4
6 5 3 3
5
1
3
2
1
1
6
2
2
2
1
… 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.
20
5. EXTENSION
21
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!
22
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
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 1 000 000
d)
Increase by two hundred thousand
e)
Add on fifty more
f)
Decrease by 41 000
g)
Increase by 2000
h)
Subtract 300 000
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:
In base 8:
12 = 1  10 + 2  1
128 = 1  8 + 2  1 = 1010
20 = 2  10
208 = 2  8 = 1610
215 = 2  102 + 1  10 + 5  1
2158 = 2  82 + 1  8 + 5  1
= 128 + 8 + 5
= 14110
3461 = 3  103 + 4  102 + 6  10
+11
34618 = 3  83 + 4  82 + 6  8
+11
= 1536 + 256 + 48 + 1
= 184110
25
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.
5 + 4 = “9”  1  8 + 1  11
11
12
+ 167
201
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
26
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