Download Moved In - Schd.ws

Triangle High Five High School Math Summit 2016
Panther Creek high School
July 20, 2016
The development of the Algebra Strand
from 6th grade through Math III
John Pritchett, Athens Drive HS
[email protected]
The impetus
• I asked the questions:
• What can I expect them to know?
• What do I need to be sure they have before going
to the next course?
The approach
• Mathematical thinking is the focus of instruction.
• Mathematics is to take much more of a modelling approach than an
operational approach.
Models
• Equations strands
Threads•in
the Algebra and Function
&
Inequalities
• Functions as mathematical
models
Construct
a
Determine
model to
when a model
• Operations
represent
fulfills a given
the problem
condition
• Equations & Inequalities
• Systems
Determine when
two or more
models fulfill the
same condition
• Systems
Solve more
complex
equations
• Operations
Mathematical models – Planting roots
• Students in 6th grade begin to analyze relationships between
quantities.
• How much does it cost to buy 6 tomatoes if tomatoes cost $0.69 each?
• Grows to the total cost of tomatoes is $0.69 times the number of tomatoes.
• Students in 6th grade convert between units
• These unit conversions can become standardized
• How many groups of 12 inches are in a given number of feet?
inches
• Grows to feet 
12
Models growing vertically - Slope
• 6th grade – students use ratio and rate to solve problems
• 7th grade – students find unit rate
• 8th grade – students recognize unit rate as slope
• Math I – students calculate and interpret average rate of change over
an interval
Building a toolkit of mathematical models
• 7th grade – recognize proportional relationships as equations (direct
variation)
• 8th grade – interpret y = mx + b as a linear relationship
• Math I – calculate and graph models to represent linear, exponential,
& quadratic relationships
• Math II – calculate and graph models to represent quadratic, square
root, & inverse variation relationships
• Math III – calculate and graph models to represent absolute value,
polynomial, exponential, & rational relationships
Graphs and other representations
• Students should be able to describe features of
graphs
• Math I – intercepts, intervals of increase/decrease,
intervals of positive/negative range,
maximum/minimum
• Math II – AND domain/range, symmetries, rate of
change, end behavior
• Math III – discontinuities, periodicity, effect of
parameters on sine/cosine
Functions analyzed:
•
•
•
Linear, exponential,
quadratic
Quadratic, square
root, inverse variation
Absolute value,
polynomial, rational,
sine/cosine
Solving equations/inequalities in one variable
• 6th grade – understand a solution as the value(s) that make a math
sentence true, solve one-step linear equations, interpret x < c, x > c
• 7th grade – solve two-step linear equations and inequalities in one variable
• 8th grade – solve linear equations with rational coefficients,
solve x2 = p and x3 = p with roots
• Math I – understand the relationship between factors of a quadratic and
the roots of the equation
• Math II – solve quadratic, square root, inverse variation, right-triangle trig
• Math III – absolute value, polynomial, exponential, and rational
relationships.
Operations to support solving equations
• 6th grade – write expressions using variable, translate words to algebraic
expressions, write & evaluate expressions with whole number exponents
• 7th grade – simplify and expand linear expressions
• 8th grade – operate with scientific notation
• Math I – use function notation to evaluate linear, exponential, and
quadratic expressions; operate with polynomial expressions
• Math II – explain how expressions with radical exponents can be re-written
using radicals, re-express quadratics by completing the square,
add/subtract/multiply polynomials
• Math III – divide polynomials, analyze piecewise-defined functions
Interpret expressions with multiple parts as a
combination of entities
• Math I – linear, exponential, quadratic
• Math II – quadratic, square root
Systems of equations/inequalities
• 8th grade – model real-world problems with systems of linear
equations, solve systems of linear equations
• Math I – represent the solution to a linear inequality/system of linear
inequalities as a shaded region of the plane, connect the intersection
point to the solution of f(x) = g(x), approximate solutions to systems
of quadratic/exponential functions using technology/tables of values
• Math II – create systems of linear, quadratic, square root, inverse
variation to model situations in context, solve with
technology/tables/algebra
New standards of 2016
• The following changes were included in the NC Math Standards for
Math I – III
• Middle school math and the 4th math course are to be reviewed
beginning in 2016 - 2017
Math I
Moved In
Moved out
• Solve quadratic equations by
square root and factoring only
• Systems involving quadratics and
exponentials using tech/tables
• Geometric sequences
• Translating between explicit and
recursive
• Rational exponents
• “Use structure to re-write”
• Rational functions
• No more “constraints”
• Transformations
Math II
Moved In
Moved Out
• Introduction of radical
exponents
• Examine the closure of rational
numbers under +, , , 
• Introduce  as in  49
• Introduce inverse variation
• Complete the square to derive
the quadratic formula
• Advanced functions – cube root,
step function, absolute value,
piecewise-defined functions
Math III
Moved In
Moved Out
• Introduce absolute value
function and piecewise-defined
• Write equivalent forms of
exponential functions
• Review of solving radical
equations
• Closure of rational numbers (II)
• Operations on complex numbers (IV)
• Complete the square(II)
• Sequences & series (I & IV)
• Proving polynomial identities (IV)
• Pythagorean Identity (IV)
Placement
• For each question on the following slides, choose the course in which
you would use the question
• As an enrichment question
• As a regular practice or test question
• As a remedial question


8
x
y
1. Which expression is equivalent to
4
a) 128 x
b)
c)
28
3
y
35
3
11
3
56
4
x y
3
9
11 3
19
3
22
3
128 x y
d) 128 x
y
xy
7
5 3
?
1. Rational Exponents
• Enrichment – Math II
• Practice – Math II
• Review – Math III or beyond
• Rational exponents are not introduced until Math II
2. Find the x-coordinate of the solution to this
system: x + 2y = 11; 3x – 4y = 3
a)
b)
c)
d)
x=5
x=3
x = 3.5
No solution
2. System of linear equations
• Enrichment – 8th grade
• Practice – 8th grade
• Review – Math I and beyond
3. Construct a quadratic function with zeros
at x = 3 and x = 2.
a) y = x2 + 5x – 6
b) y = x2 + x – 6
c) y = x2 – x – 5
d) y = x – 1
3. Zeros of quadratic functions
• Enrichment – Math I
• Practice – Math I
• Review – Math I and beyond
4. Which of these makes 3x + 11 = 5x + 7
true?
a) x = 2
b) x = 2
c) x = 2.25
d) x = 0.5
4. Solution to a linear equation
• Enrichment – 6th grade
• Practice – 7th or 8th grade
• Review – 8th grade and beyond
5. Convert 96 inches to feet.
a) 8 feet
b) 288 feet
c) 16 feet
d) 1152 feet
5. Convert units
• Enrichment – 6th grade
• Practice – 6th grade
• Review – 6th grade and beyond
6. Find the average rate of change for
2
f(x) = 3x – 10x + 5 from x = 2 to x = 6.
a) 3
b) ½
c) 14
d) 12.5
6. Average rate of change
• Enrichment – Math I
• Practice – Math I
• Review – Math I and beyond
7. Create a mathematical model for an initial
deposit of $2000 invested at 4% compounded
annually.
a) y = 2000(1.4)x
b) y = 2000(1.04)x
c) y = 2000 + 0.04x
d) y = 2000 + 1.04x
7. Construct an exponential model
• Enrichment – Math I
• Practice – Math I
• Review – Math II and beyond
8. Which of these functions will be largest in
the long run?
Function 1:
y = x2 + 5x – 4
Function 2:
x
2
4
6
8
y = 0.1(2)x
0.4
1.6
6.4
25.6
a)
Function 1 will be larger
b)
Function 2 will be large
c)
They will be equal
d)
Not enough information
8. Compare two functions in different
representations.
• Enrichment – Math I
• Practice – Math I
• Review – Math II and beyond
Questions? Comments?
• I would greatly appreciate any feedback at
• [email protected]
• Thank you!!!!