The ABCs of XYZ

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The ABCs of XYZ
Integers A Need for Negative Ideas
Patterning and Algebra
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How many s to balance?
Problem of the day Lets Doodle In silence.
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The Focus is on

• Patterns
• Functions
• Ability to analyze using symbols

The Result is

• Algebraic Reasoning
• HOTS in Mathematics

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When Mathematicians need to solve a problem, they
often use a letter to represent the number they
are seeking. This technique, called
Algebra, enables many different problems in the
real world to be solved. Today, Algebra is
necessary not only in Mathematics, but also in
Science, Economics and Technology as it promotes
a way of thinking.
Presentation by our moderators
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Ministry Guidelines
Grade 4 Recognize math relationships ? (before /
after a translation) Demonstrate equivalence ? 13
7 19 1 Identify / create / extend pattern by
changing an attribute ? Use a pattern strategy
to solve a problem ?
Grade 5 Grade 4 use patterns in a table.
Grade 6 Grade 4, 5 determine a value of a
missing term / factor
Grade 7 Grades 4,5,6 describe pattern in a
variety of sequences / predict
Grade 8 Grades 4,5,6,7 write algebraic
equations with nth term of sequence Use
inequalities / solve 1st degree equations /
translate complex statements
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Traditional vs. Patterningm m m m 4m

• Memorize a series of symbol manipulation
• Little connection to the real world / abstract
• Preconceived ideas of disaster
• Is the biggest issue the teacher?
• BFI

• a vigorous active science of patterns (NCTM)
• Makes / extends connections
• Promotes mathematical thinking
• Game of manipulation

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Issue 1 The Role of Language

• Technical Reading
• Compact
• Conceptually dense
• Several ideas in a small space
• Slower reading rate / attention to detail
• Persistence, not speed, is most critical
• Aggressive attitude

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Language Strategies

• Teacher read / student re-reads /student
  verbalizes
• Count with fingers / point with fingers / move
  lips when reading
• Show me / Tell me
• Varied eye movement including regression
• Slower reading rate
• Be careful with some vocabulary.
• Build meaningful patterns when possible

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O C O H A N L T P L I A O S C
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Issue 2 Sources of AnxietySheila Tobias Stress
In The Math Class
Until teachers recognize that they are
handicapping students by forcing them to think
while racing the clock, mathematics learning will
continue to be a fearsome and negative
experience.

• Time to Assimilate Ideas
• Humiliation
• Emphasis of Right Answer
• Working in Isolation

Key Ability is to Analyse
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Issue 3 A Need for Strategies

• Strategies need to be taught!
• Strategies need to be practiced.
• Multiple solution paths need to be sought.

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Tools of Algebra

• Whole Numbers
• Decimal ( Fractional) Concepts
• Integer Operations

Algebraic ideas develop along a K 12 scope. Why
does formal teaching of Algebra not appear in
curriculum until Grade 7?
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Teaching Integers

• Development starts in Grade 6 / addition and
  subtraction in Grade 7 / Multiplication and
  Division in Grade 8
• Extends from knowledge of positive rational
  numbers
• Needs reference to real-world
• negative not minus

N 6-19
Ways to demonstrate ...
Thermometers / elevators / planes and submarines
/ money sports like golf and football
T1
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Modeling Integers
Number Line
'Electric Charge'

• Use 2 colours to represent positive and negative
  values
• Zero Property One of each colour ( and - )
  cancels

• Use position on a number line that has 0 in the
  middle.
• Show the effect of positive and negative
  positions of the same magnitude.
• Order on Line (rope)

T1-opposite T2-markers
Now on to operations ...
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Assembly / Disassembly of IntegersPart 1
Subtracting Integers Only One Model
applies Electric Charges

• Adding Integers
• Both Models apply
• Electric Charges
• Number Line
• with arrows
• c. The Rope

4 - 5 -1
How could we arrive at the same place using
addition? (pattern search)
-2 5 3
N 7-13
T2-NumLine1 T2-Markers2
T3-Markers E1E5
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Assembly / Disassembly of IntegersPart 2
Division Develop with inverse examples and then
Pattern Search
Multiplication Pattern Search in charts
N 8-12
T4-Neg x Neg
T5-Inverse Ex1
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Getting ready to start Algebra
Core ideas and language that need time to
assimilate

• Check
• order of operations ?
• commutative, associative,
• distributive properties ?
• same arithmetical operations apply ?
• Where are my students in the development stages
• Covering up hidden numbers

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Look for Patterns .
T1-Alg-Trick1-Ex
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Variables

• Why x ?
• Misconceptions that can develop
• Variable stands for an object (a ? apple)
• Changing the variable changes the problem
• Combining whole numbers and variables 3 4b 7b
• Make variable names meaningful (real words)
• 8c 6t ? 8 cabbage 6 turnip
• Show mathematical relationships
• Encountering new notation

PA 7-4
3 x a 3a
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Pattern Searching
PA 6-9
How many toothpicks are used to make any given
number of squares?
How will you lead students to resolve this
pattern? Try a table of values.
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n
PA 7-6
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Using Pattern Blocks
4.
1.
2.
3.
1.
2.
3.
4.
PA 5-8
2.
1.
3.
4.
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(x 3) (x2)
x2 5x 6
PA 9-8
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Formulas

• First encounters with algebraic expressions
• A l x w
• Learn the mechanics of substitutions and
  formatting
• A l x w
• 5 m x 4m
• 20 m2

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The Art of Manipulation

• Using undoing operations to isolate a term of
  an equation 3m 5 13
• Using a variety of approaches
• n 3 8 ? n 3 3 8 3 ? n 3 -3 8
  -3
• Stress CLARITY

Play
Clean-up
y 3 2y 20 3y 3 20
PA 8-8
PA 7-10
T2-Eq-Tiles T3-Eq-Tiles
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Observe Equality
2x 4 10 2x 6 x 3
10 - 3m -5 -3m -15 m 5
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Try It Out
8 4 b 32 8 8 4 b 32 8 (subtract 8
B.S.) 4b 24 4b 24 (divide B.S. by 4) 4
4 b 6
8 4 b 32 8 -8 4 b 32 -8 (add -8
B.S.) 4b 24 4b ? 4 24 ? 4 (divide B.S. by
4) b 6
A
A
3 k 15
12 4 2 x
B
Two unique solutions for each / Be very linear
and describe the process.
C
Technical Writing (Journal) 1. Describe a
solution in words. Test this on others. 2.
Reverse the process. Make a problem . x 2 6
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Functions
not a formal component . try making a
function machine begin developing from
Grade 4
Functions lead to Truth Tables which lead to
Graphing concepts
Value of one thing depends on another
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Connecting Algebraic Ideas To Other Data
If water is poured in at a consistent rate
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Literacy ? NumeracyThe Technical Reading

• Provide lots of modeling and practice
• Make exercises that require only the translations
• Get the problem into workable form
• Dont use 2 variables if one will do
• Give students the equation / they write the
  problem(s)

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• 5 Home teams (A, B, C, D, E)
• Select one person each for 1, 2, 3, 4, 5
• c. Disassemble into expert groups / share
  solutions
• d. Reassemble home teams share problems and
  results.

Jigsaw
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Sources
Puzzle and Riddle Books
The Internet
Text and Reference Books
Teachers that share
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30 Second Warm-up

• Set the beat with your right foot - every second
• Tap left index finger to alternate with your
  right foot
• Trace small circles with your right arm.

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Recording Relations .

• Tables
• Charts
• Graphs

T5-Graph Relations-Ex1
Students need many opportunities to design,
build and interpret information displayed with
these tools. Teachers must demonstrate
construction techniques.
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Pythagoras (540 B.C.)
C2 B2 A2
C2
B2
A2
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b
b
a
a
c
a2
a
a
a
c
b
c
c2
c
b2
b
c
b
b
c
a
a
b
b
a
Area a2 b2 4 triangles
Area c2 4 triangles
C2 B2 A2