Download "Big" Idea?

Dennis C. Ebersole
Northampton Community College
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Fits latest findings in cognitive science
Focuses students on core concepts
Helps students see interconnectedness of
mathematics
Promotes understanding versus memorization
Increases the likelihood of transferring the
learning to different contexts
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Prior Knowledge: All students seem to
remember P/E/MD/AS although they may
not remember that * & / and + & - are at the
same level
Most equations can be put in a form where
they can be solved using the same approach
This approach relies on the student’s ability to
indicate the order in which an expression
would be simplified and the concept of inverse
functions
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Common Equation Solving Steps
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Put the equation in a form where it is amenable to
solution using inverse functions
Write down the functions that would be used to
evaluate the expression (in order using PEMDAS)
Write down the inverse of each function in reverse
order
Compose both sides of the equation using these
inverse functions (in the reverse order)
Note: If done correctly, something will “cancel” at
each step.
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See the Linear Example on the Handout
Students should note how the x in the inverse
function is replaced by the entire expression on
each side – this is composition of functions
Your turn: Solve the quadratic, radical, and
exponential equations using this approach
Note: All have already been written in a form
where inverse functions can be used
What other equations could be solved this
way?
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There is a reason why this is the definition used
by mathematicians – it generalizes to all
branches of mathematics
Also, we do not have the revise the definition
to include functions of multiple variables
Can you think of other benefits?
See page 2 of the handout
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My Definition: Converting from one
representation to another representation
Example: Converting words to a table
representation
For common function models, translation of
axes is a powerful tool in the conversion
process
If I am correct, we need to include practice in
converting between representations in our
courses
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As a student, why should I care?
Linear Functions
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Quadratic Functions
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In general: y = ax + b (2 parameters to find)
If goes thru (0, 0): y’ = ax’ (1 parameter to find)
In general: y = a𝑥 2 + bx + c (3 parameters)
If vertex is (0, 0): 𝑦 ′ = 𝑎(𝑥 ′ )2 (1 parameter)
Absolute Value Functions
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In general: y = a|x – h| + k (3 parameters)
If vertex is (0, 0): y’ = a |x’| (1 parameter)
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Look at the examples on pages 3 through 7 of
the handout with a partner
Discuss the patterns you see
Where should students look in the translated
table to find the value of the parameter a in the
desired equation?
Are there other functions where translation of
axes will make it easier to find the
corresponding equation?
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Stretches, Compressions, Reflections
Problem Solving Heuristic
Absolute Value as Distance From the Origin
Choose a course you teach regularly
Find another participant who teaches that
course
Decide on some BIG ideas for the course
Report out to the whole group
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Comments and suggestions are welcome; I am
still learning to teach mathematics
[email protected]