Representations and Proof Learning Focus Participants will deepen mathematical content knowledge of algebraic reasoning develop awareness of key concepts associated with algebraic reasoning ID: 642860
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Slide1
Algebraic Reasoning as Equality,
Representations
and
ProofSlide2
Learning Focus
Participants will:
deepen mathematical content knowledge of algebraic reasoningdevelop awareness of key concepts associated with algebraic reasoning, specifically: Equality as a relationship between quantitiesRepresentationsProofdevelop pedagogical knowledge for teaching algebraic reasoning
2Slide3
Session Norms
Be engaged in the tasks and discussions as this enriches everyone’s experience.
Embrace the learning!Actively seek connections between the research and your classroom experience.By better understanding student thinking we better understand the impact we can have on their learning.Slide4
Agenda
What is Algebraic
Reasoning?Exploring Equality – a relationship between quantitiesExploring Representation – double number lines and symbols
Exploring Proof – generalizing
mathematical properties and relationships
Consolidate
Use of Symbols
ResourcesSlide5
What is Algebraic Reasoning?
Paying Attention to Algebraic Reasoning
Algebraic reasoning permeates all of mathematics and is about describing patterns of relationships among quantities – as opposed to arithmetic, which is carrying out calculations with known quantities. In its broadest sense, algebraic reasoning is about generalizing mathematical ideas and identifying mathematical structures. 5Slide6
THINK
: to yourself
PAIR: in your group
SHARE
: with the large group.
Mathematical GeneralizationSlide7
“Generalization is the heartbeat of mathematics and appears in many forms. If teachers are unaware of its presence, and are not in the habit of getting students to work at expressing their own generalizations, then mathematical thinking is not taking place.”
(John Mason, 1996)
Generalization Slide8
Equality
8Slide9
The Equal Sign
Consider the following:Slide10
How do many students respond?
The answer is
What does = mean? Slide11
S
tudents were asked to
determine if the following number sentences are true or false.Predict their responses: 3 + 4 = 7Most students stated true.Exploring Equality as a Relationship between
Quantities - 1Slide12
7
= 3 + 4
Students said false because it is backwards. 3 + 4 = 5 + 2Students initially considered this expression to be nonsense.Exploring Equality as a Relationship between Quantities - 2Slide13
5 + 1 = 7
Students
say true if considering the format or false if paying attention to the calculation.7 = 7Many young students claim this is false and correct it by substituting 7 + 0 = 7.Exploring Equality as a Relationship between Quantities -3Slide14
It has been well documented that students do not recognize that the equal sign denotes equality… Most students see the equal sign as a signal to do something – to carry out a calculation and put the answer after the equal sign.
Exploring
Equality as a Relationship between Quantities - 4Paying Attention to Algebraic Reasoning p.6Slide15
The Equal Sign - RevisitedSlide16
Instruction
in primary classrooms should encourage students to explore equality as a relationship between two quantities.
Reasoning Algebraically about ArithmeticSlide17
Q
uestions posed
by Grade 2 students 58 + ___ = 34 + 6017 + 13 = 18 + ________ + 1580 = 1582 + 400Exploring Equality as a Relationship between Quantities - 5Slide18
Paying Attention To
Algebraic Reasoning
When students work with equations, it is imperative that they understand that the equal sign represents a relation between quantities … Students who develop this understanding can compare without having to carry out the calculations. They can focus on the equivalence … (algebraic reasoning) rather than comparing … answers (arithmetic reasoning). p.7
18Slide19
Something to think
about:
How can varying the position of the equal sign, and providing students with an opportunity to define the equal sign, support them with algebraic reasoning?Slide20
Representation
20Slide21
Sunny’s
Jumps
When Sunny jumps 4 times and takes 11 steps forward, he lands in the same place as when he jumps 5 times and takes 4 steps forward.How many steps long is Sunny’s jump?All jumps are assumed to be equal in length. All steps are also assumed to be equal in length.2007 Catherine Twomey Fosnot from Contexts for Learning Mathematics
(Portsmouth, NH: Heinemann)Slide22
Sunny’s
Jumps: Double Number Line Representation1 jump = 7 stepsWhen Sunny jumps 4 times and takes 11 steps forward, he lands in the same place as when he jumps 5 times and takes 4 steps forward.Slide23
Paying Attention To Algebraic Reasoning
p.7
At the heart of algebraic reasoning are generalizations as expressed by symbols. 23Slide24
1 jump = 7 steps
4
j + 11s = 5j + 4s * 4j – 4j + 11s = 5j – 4j + 4s
11
s
=
j
+ 4
s
11
s
– 4
s
=
j
– 4
s
7
s
=
j
When Sunny jumps 4 times and takes 11 steps
forward
, he lands in the same place as when he jumps 5 times and takes 4 steps
forward
.
Sunny’s
Jumps: Algebraic RepresentationSlide25
The Pairs
Competition -1
In the competition, two frogs jump. Each team gets two jumping sequences.The length of the jump for each frog is then determined and the lengths are added together for an overall result.The winners are the pair with the longest combined jumping distance (one jump each).Referee’s Rule: Each frog’s jumps are assumed to be equal in length. All steps are assumed to be equal in length.2007 Catherine Twomey Fosnot
from
Contexts for Learning Mathematics
(Portsmouth, NH: Heinemann)Slide26
The Pairs
Competition -2
Team # 1:When Huck jumps three times and Tom jumps once, their total is 40 steps, but when Huck jumps four times and Tom jumps twice, their total is 58 steps.Slide27
Huck
and Tom’s
Results –Double Number Line RepresentationHuck + Tom = 18 stepsWhen Huck jumps three times and Tom jumps once, their total is 40 steps, but when Huck jumps 4 times and Tom jumps twice, their total is 58 steps.
58 steps
40 stepsSlide28
Huck
and Tom’s
Results – Algebraic Representation4H + 2T = 583H + 1T = 401H + 1
T
=
18
Huck’s jum
p
+
Tom’s jump
= 18 steps
When Huck jumps three times and Tom jumps once, their total is 40 steps, but when Huck jumps 4 times and Tom jumps twice, their total is 58 steps.Slide29
The Pairs
Competition -3
Team # 2:Hopper and Skipper have a different technique than the other pairs. First Hopper takes three jumps and lands in the same place as Skipper does when he takes four jumps. Then Hopper takes six jumps and nine steps to land in the same place as Skipper does when he takes nine jumps.Slide30
Hopper and Skipper: Double Number Line RepresentationSlide31
Hopper and Skipper:
Algebraic
RepresentationSlide32
What the students
did:
3h = 4s 6h + 9 = 9s
subtract
3
h
= 4
s
3
h
+
9 = 5
s
s
ubtract
3
h
=
4
s
a
gain
9 = 1
s
3
h
=
4
s
3
h
= 4
x
9
3
h
=
36
1h
=
12
1
h +
1
s
=
9
+
12
1
h +
1
s
=
21 steps
32Slide33
Something to think about
Why
is it important for students to ‘see’ solutions in a variety of representations? Slide34
ProofSlide35
Choose three
consecutive
numbers and add them together. Try another set.What do you notice?Will this always be true?
How do you know?
Something to Prove!Slide36
Video
:
The Sum of Three Consecutive NumbersListen to what this student is saying. What generalizations is she making?Are there other conjectures you can make about this sum?
Is
this a proof
?
One way of thinking about itSlide37
How do these symbols represent the problem?
What does it prove?
Another way of seeingSlide38
Instructional programs from prekindergarten through grade 12 should enable all students to
Recognize reasoning and proof as fundamental aspects of mathematics;
Make and investigate mathematical conjectures;Develop and evaluate mathematical arguments and proofsSelect and use various types of reasoning and methods of proof -NCTM
Notion of ProofSlide39
Quote:
“Convince yourself, convince a friend, convince a skeptic.”
Cathy HumphreysSlide40
Process of Proving
Convincing - justifying and proving (the WHY)
Specializing - trying specific cases Generalizing - detecting a pattern Conjecturing - articulating a pattern (the WHAT)
Mason, Burton,
Scacey
,
Thinking MathematicallySlide41
Exploring Properties and Relationships
Specific Calculations
Conjecturing Properties of numbersProving a Conjecture
2
+ 4 = 6, 10 + 20 = 30
5 + 3 = 8, 13 + 1 = 14
5 + 8 = 13, 2 + 9 = 11
Even
+ Even = Even
Odd + Odd = Even
Odd + Even = OddSlide42
Offering and Testing Conjectures
A guess or prediction based on limited evidence
Justifying and ProvingIs this always true? How do you know?Example vs. Counter-Example
Actions to Develop Algebraic ReasoningSlide43
Using Symbols, Including Letters
,
as VariablesAlgebraic reasoning is based on our ability to notice patterns and generalize from them. Algebra is the language that allows us to express these generalizations in a mathematical way.43Paying Attention to Algebraic Reasoning p.3Slide44
Variables as Changing Quantities
44
The Property of Adding Zero Slide45
Variables as
Unknowns
45Slide46
Solve for ?Slide47
An algebraic expression can be treated as an
object:
Even if students are comfortable with x + 5 = 9, it is more difficult for them to think of the expression x + 5 as an object by itselfIn our example (horse + butterfly) can be treated as an object to solve the problem horse + butterfly = 5Something More to Think AboutJacob, Fosnot; 2007Slide48
http://www.edugains.ca/resources/LearningMaterials/ContinuumConnection/SolvingEquations.pdf
Student
Solutions Across Grades p.23-26Slide49
Resources – Adobe Presenter
49Slide50
50
mathies.ca
EduGAINS MathematicsResources