Lecture 10 – Powers and Integers

Key ideas
The concept of the convention of free choice
·  Rounding five up – why have we chosen to do this?
·  Why is 60 = 1?
·  Why is 16½ = √16?
·  Why is 2-3 = 1/23?
·  Why is the result positive when we multiply two negative numbers together?
·  Why does subtracting a negative number result in an increase in value?
Required teacher knowledge - Exponent rules for multiplication and division
Exponent of zero
Negative exponents
Fractional exponents
Adding/subtracting Integers
Multiplying/dividing Integers
Materials
·  Squared paper
·  Multilink cubes
·  Number lines
·  Red and blue multilink cubes (red / positive / hot and blue / negative / cold)
·  A see through container
·  Source games
Vocabulary
·  Indices/exponents
·  Negative/positive
·  Integers
·  Vector for integers – magnitude and direction
Points of confusion
·  Integers – the first number specifies the position on the line and the second number specifies the movement. The operation sign between the two numbers means “then do” or “followed by”.
·  Students have looked at decimal fractions where after the decimal point (going to the right) the number gets smaller but on a number line the smaller numbers are to the left.
·  A negative attached to a number does not mean subtract and a positive attached to a number does not mean add. It denotes the size of the number.
·  The concept of the difference between two numbers needs care when dealing with integers e.g. -5 - 4 = -9 whereas the difference between -5 and 4 is 9.
Discussion Questions
Exponent of zero
Why is 100=1 a sensible free choice?
23 x 2-3 = 23-3 = 20 = 1
Negative exponents
Why is 2-3 = ⅛ a sensible free choice?
2-3 = 20-3 = 20/23 = ⅛
Fractional exponents
Why is 24½ = √24 a sensible free choice?
24½ x 24½ =24½ + ½ = 241
o x o = 24
√24 x √24 = 24
Adding/subtracting Integers
Subtracting a negative is a difficult concept to teach. Discuss using the Equal Addition strategy to calculate these problems.
5 - - 3 = (5 + 3) – (-3 + 3) = 8 – 0 = 8
Multiplying/dividing Integers
Multiplying two negatives to get a positive answer is a difficult concept to teach. Consider
-5 x 0 = 0
-5 x (3 + -3) = (-5 x 3) + (-5 x -3) = -15 + 15 = 0
Ideas to try
Integers
·  Cold chilly bin or container. Adding cold and hot items to the chilly bin/container and deciding whether the overall temperature increases or decreases. E.g. if you take away a cold pack from the chilly bin, the overall temperature increases (subtracting a negative results in an increase)
·  Hills and Dales
Links
AC – EA / EA – AA
AA – AM
Integers
Dollars and Bills (Bk 5, p 50)
Hills and Dales (Bk 5, p 50)
Material Master 4-39
FIO Number 7-8 Bk 4 p 8 “It’s a Try!”
FIO Number 7-8 Bk 4 p 10 “Integer Zap”
FIO Number 7-8 Bk 4 p 11 “Integer Slam”
FIO Number 7-8 Bk 4 p 12 “Integer Links”
FIO Number 7-8 Bk 4 p 15 “Shifty Subtraction”
FIO Number L3–4 p 22 “Walking the plank”
FIO Number Bk 3 L3–4 p 22 “Chilly Heights”
FIO Number Bk 3 L3–4 p 20 “Volcanoes Erupt”
Exponents
Powerful Numbers (Bk 6, p 73)
FIO Number 7-8 Bk 4 p 13 “Family Trees”
FIO Number 7-8 Bk 4 p 16 “Calculator Power”
FIO Number 7-8 Bk 4 p 17 “Cubic Capacity”
FIO Number 7-8 Bk 4 p 18 “Growing Pains”
FIO Number 7-8 Bk 4 p 24 “Superior Side Lengths
FIO Number L3–4 p 22 “Using Exponents” / AM - AP

Lecture 1 – Starting Place Value

Key ideas
·  Teaching of Place Value Houses
·  Canon of Place Value
o  Largest number in any column is 9
o  10 for 1 (addition and multiplication)
o  1 for 10 (subtraction and division)
·  Materials
o  Bundled sticks
o  Unifix / Multilink
o  Tens Frames
o  Place Value Blocks
o  Numeracy Money
·  Vocabulary
o  ty and teen
·  Linking words, symbols and meaning “say, see, show”
o  Seventeen, 17, one ten and seven ones
·  Writing words for numbers
Discussion Questions
1.  996 + 4 = 1000 (Using numeracy money)
2.  $2000 – 4 (Using numeracy money)
3.  85 + 67 (Using algorithm with meaning)
4.  Solve 15 + 6 in Base Eight (Use Eight Frames)
Ideas to try
·  Place Value Houses
·  Using numeracy money with whole numbers
·  Modelling 9999999 + 1 in front of class
·  Place Value Diagnostic sheets
Links
AC – EA
·  Make Ten (Bk 5, p 26)
·  Subtraction in Parts (Bk 5, p 27)
·  Up and over the Tens (Bk 5, p 28)
·  Adding in Parts (Bk 5, p 29) / EA – AA
·  Jumping the number line (Bk 5, p 33)
·  Don’t subtract – add (Bk 5, p 34)
AA – AM
·  Problems like 23 + o = 71 (Bk 5, p 35)
·  Problems like 37 + o = 79 (Bk 5, p 36)
·  When one no. is near 100 (Bk 5, p 37)
·  Problems like 73 – 19 = o (Bk 5, p 38)
·  Near Doubles (Bk 5, p 41) / AM - AP
·  Introducing Decimal Fraction PV Bk 5, p 46)
·  The Decimal Fraction Point (Bk 5, p 47)
·  Adding with decimal fractions (Bk 5, p 48)
·  Subtraction with tenths (Bk 5, p 48)

Lecture 2 – More Place Value

Key ideas
Discussion Questions
Ideas to try
Links
AC – EA
·  Make Ten (Bk 5, p 26)
·  Subtraction in Parts (Bk 5, p 27)
·  Up and over the Tens (Bk 5, p 28)
·  Adding in Parts (Bk 5, p 29) / EA – AA
·  Jumping the number line (Bk 5, p 33)
·  Don’t subtract – add (Bk 5, p 34)
AA – AM
·  Problems like 23 + o = 71 (Bk 5, p 35)
·  Problems like 37 + o = 79 (Bk 5, p 36)
·  When one no. is near 100 (Bk 5, p 37)
·  Problems like 73 – 19 = o (Bk 5, p 38)
·  Near Doubles (Bk 5, p 41) / AM - AP
·  Introducing Decimal Fraction PV Bk 5, p 46)
·  The Decimal Fraction Point (Bk 5, p 47)
·  Adding with decimal fractions (Bk 5, p 48)
·  Subtraction with tenths (Bk 5, p 48)