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A Closer Look at Changes
in Math 3
Questions
If you think of questions during
today’s meeting, please write them
down and place them on the
parking lot poster.
At-A-Glance Document
Algebra, Functions & Function Families
NC Math 1
NC Math 2
NC Math 3
Functions represented as graphs, tables or verbal descriptions in context
Focus on comparing properties of
linear function to specific nonlinear functions and rate of
change.
• Linear
• Exponential
• Quadratic
Focus on properties of quadratic
functions and an introduction to
inverse functions through the
inverse relationship between
quadratic and square root
functions.
• Quadratic
• Square Root
• Inverse Variation
A focus on more complex
functions
• Exponential
• Logarithm
• Rational functions w/ linear
denominator
• Polynomial w/ degree < three
• Absolute Value and Piecewise
• Intro to Trigonometric Functions
A Progression of Learning of Functions through Algebraic Reasoning
The conceptual categories of Algebra and Functions are inter-related. Functions describe situations in which one
quantity determines the other. The difference between the Function standards and the Algebra standards is that the
Function standards focus more on the characteristics of functions, e.g., domain/range, function definition, etc.
whereas the Algebra standards provide the computational tools and understandings that students need to explore
specific instances of functions. As students progress through high school, the coursework with specific families of
functions and algebraic manipulation evolve. Rewriting algebraic expressions to create equivalent expressions
relates to how the symbolic representation can be manipulated to reveal features of the graphical representation of
a function.
Note: The Numbers conceptual category also relates to the Algebra and Functions conceptual categories. As
students become more fluent with their work within particular function families, they explore more of the number
system. For example, as students continue the study of quadratic equations and functions in Math 2, they begin to
explore the complex solutions. Additionally, algebraic manipulation within the real number system is an important
skill to creating equivalent expressions from existing functions.
Geometry
NC Math 1
NC Math 2
NC Math 3
Analytic and Euclidean
Focus on coordinate geometry
• Distance on the coordinate plane
• Midpoint of line segments
• Slopes of parallel and perpendicular lines
• Prove geometric theorems algebraically
Focus on triangles
• Congruence
• Similarity
• Right triangle trigonometry
o Special right triangles
Focus on circles and continuing the
work with triangles
• Introduce the concept of radian
• Angles and segments in circles
• Centers of triangles
• Parallelograms
A Progression of Learning
Integration of Algebra and Geometry
• Building off of what students know from
5th – 8th grade with work in the coordinate
plane, the Pythagorean theorem and
functions.
• Students will integrate the work of
algebra and functions to prove geometric
theorems algebraically.
• Algebraic reasoning as a means of proof
will help students to build a foundation to
prepare them for further work with
geometric proofs.
Geometric proof and SMP3
• An extension of transformational
geometry concepts, lines, angles, and
triangles from 7th and 8th grade
mathematics.
• Connecting proportional reasoning from
7th grade to work with right triangle
trigonometry.
• Students should use geometric reasoning
to prove theorems related to lines,
angles, and triangles.
It is important to note that proofs here are not
limited to the traditional two-column proof.
Paragraph, flow proofs and other forms of
argumentation should be encouraged.
Geometric Modeling
• Connecting analytic geometry, algebra,
functions, and geometric measurement to
modeling.
• Building from the study of triangles in
Math 2, students will verify the properties
of the centers of triangles and
parallelograms.
Statistics & Probability
A statistical process is a problem-solving process consisting of four steps:
1.
2.
3.
4.
Formulating a statistical question that anticipates variability and can be answered by
data.
Designing and implementing a plan that collects appropriate data.
Analyzing the data by graphical and/or numerical methods.
Interpreting the analysis in the context of the original question.
NC Math 1
NC Math 2
Focus on analysis of univariate and
bivariate data
• Use of technology to represent,
analyze and interpret data
• Shape, center and spread of
univariate numerical data
• Scatter plots of bivariate data
• Linear and exponential regression
• Interpreting linear models in
context.
Focus on probability
• Categorical data and two-way tables
• Understanding and application of the Addition
and Multiplication Rules of Probability
• Conditional Probabilities
• Independent Events
• Experimental vs. theoretical probability
•
•
NC Math 3
Focus on the use of sample data to
represent a population
• Random sampling
• Simulation as it relates to sampling
and randomization
• Sample statistics
• Introduction to inference
A Progression of Learning
A continuation of the work from
middle grades mathematics on
summarizing and describing
quantitative data distributions of
univariate (6th grade) and bivariate
(8th grade) data.
•
•
A continuation of the work from 7th grade where
students are introduced to the concept of
probability models, chance processes and
sample space; and
8th grade where students create and interpret
relative frequency tables.
The middle grades work is extended to develop
understanding of conditional probability and
independence and rules of probability to
determine probabilities of compound events.
•
•
•
•
Bringing it all back together
Sampling and variability
Collecting unbiased samples
Decision making based on analysis of
data
First Year Implementation – NC Math 3
Function Families Progression
Gr 8
M1
Linear
Linear
Quadratic
M2
M3
M4+
Quadratic
Polynomial
Polynomial
Square Root
Exponential
Radical
Exponential
Inverse
Variation
Logarithmic
Logarithmic
Rational
Rational
Linear Denominators
Trigonometric
KEY
Introduced
Fluency
Generalized
Trigonometric
Gr 8
M1
Linear
Linear
Quadratic
M2
M3
4th Level Math
Quadratic
Polynomial
Polynomial
Square Root
Exponential
Exponential
Inverse Variation
Introduce
Fluent
Maintain
Radical
Logarithmic
Logarithmic
Rational - Linear
Denominators
Rational
Trigonometric
Trigonometric
Function Wall and Function Finder
(for review)
Parking Fees-Piecewise Functions
Putting Essential Understanding of Functions into Practice in Grades 9–12
Function Wall
Goal
Understand the definition of a function.
Instructions to teacher: Place the letters A, B, C, and D along a wall (write each letter, large and bold,
on a separate sheet of paper, taped to the wall, or on the board). Give students instructions on where
to stand based on different characteristics. Explain that they will be extracting the definition of a
function from this experience.
Question
Where do you stand?
1. Mode of traveling to school
A. If you walked to school today, stand under A.
B. If you rode your bike to school today, stand under B.
C. If you drove or rode in a vehicle today, stand under C.
D. If you got to school any other way, stand under D.
2.
Time of travel to school
A. If it took you between 0 and 10 minutes to get to school today, stand under A.
B. If it took you between 10 and 30 minutes to get to school today, stand under B.
C. If it took you between 30 minutes and an hour to get to school today, stand under C.
D. If it took you more than an hour to get to school today, stand under D.
4.
Clothing
A. If you’re wearing blue, stand under A.
B. If you’re wearing red, stand under B.
C. If you’re wearing black, stand under C.
D. If you’re wearing white, stand under D.
5.
Birthday
A. If you were born in January, stand under A.
B. If you were born in February, stand under B.
C. If you were born in March, stand under C.
D. If you were born in April, stand under D.
6. Height
A. If you’re shorter than five feet tall, stand under A.
B. If you’re between than five feet and six feet tall, stand under B.
C. If you’re between six and seven feet tall, stand under C.
D. If you’re taller than seven feet tall, stand under D.
Copyright © 2014 the National Council of Teachers of Mathematics, Inc., www.nctm.org. All rights
reserved. This material may be used for personal and classroom use only.
Putting Essential Understanding of Functions into Practice in Grades 9–12
Parking Fees
Unlike many airports that have two different parking rates, one for short-term parking (for example, for
passenger pickup) and the other for long-term parking, the local airport adjusts its parking fees automatically
according to the time parked, to accommodate the different types of use. In the first 12 hours, parking costs
$1 for each hour, and after 12 hours, parking costs $10 per day.
Problem
a.
Write a piecewise function to describe parking costs at this airport.
b.
Graph your piecewise function.
Questions
1.
How do the short line segments in your graph differ from the longer segments?
2.
What does the graph indicate that the fee is at 4 hours?
3.
Note that the graph seems to overlap at 12 hours. What happens at the parking ticket booth then?
What happens at 10 hours?
4.
What happens from 12 to 24 hours? Why is this line segment a different length from the segments up
to 12 hours and the segments after 24 hours? What does this line describe about the parking fee?
5.
How does the fee change with the length of time that a vehicle is parked?
6.
What would be the fee after x hours?
7.
What would be the fee if you parked for 3 days? For 10 days?
Centers of Circles
Origami Geometry
Folding Medians, Angle
Bisectors &  Bisectors
C
Step 1: Draw a large acute scalene triangle on your paper. Then label each
of the angles A , B, and C.
A
Step 2: Trace your triangle on 2 other sheets of patty paper and label the
angles on all of them for a total of 3 triangles on 3 separate sheetsof patty
paper.
“fold &
crease”
Step 3a: To fold a PERPENDICULAR BISECTOR;
Fold the sides of the triangle onto each other and “crease” the
paper to construct the perpendicular bisectors of the three sides
of the triangle. Shown here Point “A” gets folded to point “B” so
that the left side of AB lies on itself (the right side of AB).
Step 3b: To fold an MEDIAN;
Fold the sides so that they overlap each other just like the
perpendicular bisector, but don’t crease the fold, simply “pinch”
the fold mark on the side (shown here on side AB).
Next, unfold that piece and make a new fold that contains the
midpoint just created with the pinch and the opposite vertex (shown
her with vertex “C” and midpoint of AB.
Step 3c: To fold ANGLE BISECTORS;
Fold one side (CB as shown) so that it overlaps and folds onto the
side that forms the other ray of the angle (side AC shown here).
Crease this fold to make the angle bisector (of ACB shown here).
Step 4: Repeat on each triangle so that you have made 3 Medians, 3
Perpendicular Bisectors, 3 Altitudes, and 3 Angle Bisectors on
their respective triangles.
BB
C
A
A
“fold” &
“pinch”
A
B
C
A B
“pinch”
C
B
A
“pinch”
C
B
A
B
Origami Geometry
Points to Ponder ! – Perpendicular Bisectors
o Look at the triangle featuring the Perpendicular Bisectors. What do you notice about all three of them
once they have all been folded? Compare your results with others near you.
Definition: The point of the intersection of the three perpendicular
bisectors of the sides of the triangle is called the CIRCUMCENTER
of the triangle.
Circumcenter
Perpendicular Bisectors
o Measure the distance from the CIRCUMCENTER to each vertex of the triangle. What do you notice
about this measurement? Compare your results with others in your triangle group.
Write a Conjecture:
Complete the sentence with a conjecture about the Perpendicular Bisectors of a triangle:
The Perpendicular Bisectors of a triangle
1
Origami Geometry
Points to Ponder ! – Medians
o Look at the triangle featuring the Medians. What do you notice about all three of them once they have
all been folded? Compare your results with others near you.
Centroid
Definition: The point of the intersection of the three medians of
the triangle is called the CENTROID of the triangle.
Medians
o Measure the distance from the CENTROID to each vertex of the triangle. Then measure the
distance from the centroid to each of the midpoints on the sides of the triangle (where the medians
intersect the sides).What do you notice about this measurement? Compare your results with others
in your triangle group.
Write a Conjecture:
Complete the sentence with a conjecture about the Medians of a triangle:
The Medians of a triangle
1
Origami Geometry
Points to Ponder ! – Angle Bisectors
o Look at the triangle featuring the the BISECTED ANGLES. What do you notice about all three of them
once they have all been folded? Compare your results with others near you.
Definition: The point of the intersection of the three angle bisectors
of the triangle is called the INCENTER of the triangle.
INCENTER
Angle Bisectors
o Measure the distance from the CENTROID to each vertex of the triangle. Then measure the
distance from the centroid to each of the midpoints on the sides of the triangle (where the medians
intersect the sides).What do you notice about this measurement? Compare your results with others
in your triangle group.
Write a Conjecture:
Complete the sentence with a conjecture about the Angle Bisectors of a triangle:
The Angle Bisectors of a triangle
Place all three of your sheets of patty paper on top of each other so that the three triangle coincide.
What do you observe?
http://www2.powayusd.com/teachers/smiddleton/Geo/Triangles/Oragami%20Geometry%20Centers
.pdf
4
Fractal Doodles and Functions
Fractal Doodles
Fractal doodles can be found in abundance on the
web. The start of a popular fractal doodle is shown
below. Begin with a segment of length x in Stage 0.
Stage 1 adds two perpendicular segments on the ends
of the segment in Stage 0 that are the length of the
original segment. This pattern continues, adding new
segments to the ends of the last additions. The new
segments are always the length of the previous
stage’s.
Use graph paper to draw
Stage 0 through Stage 6.
Create tables and find formulas for the patterns below:
a. The lengths of the successive new segments
b. The total length of segments for each stage
Summarize
Share the drawings of Stage 0 through Stage 6.
Discuss the results of the tables and formulas.
What other questions could you ask about the fractal
doodles?
This task was adapted from:
Schrock, C., Norris, K., Pugalee, D. K., Seitz, R., & Hollingshead, F.
(2013). NCSM great tasks for mathematics 6-12: Engaging activities
for effective instruction and assessment that integrate the content
and practices of the Common Core State Standards for
Mathematics. Denver, CO: National Council of Supervisors of
Mathematics (NCSM).
Zeros of Polynomials
Factors and Polynomials
Illustra tive
Ma the m a tics
A-AP R Gra p h in g f ro m Fa c t o rs III
Ta s k
Mike is tryin g to ske tch a gra p h o f th e p o lyn o m ia l
f (x) = x 3 + 4x 2 + x − 6.
He n o tice s th a t th e co e fficie n ts o f f (x) a d d u p to ze ro (1 + 4 + 1 − 6 = 0) a n d sa ys
This m eans that 1 is a root of
f (x), and I can use this to help factor f (x) and produce the graph.
a. Is Mike righ t th a t 1 is a ro o t o f f (x)? Exp la in h is re a so n in g.
b. Fin d a ll ro o ts o f f (x).
c. Fin d a ll in p u ts x fo r wh ich f (x) < 0.
d. Use th e in fo rm a tio n yo u h a ve ga th e re d to ske tch a ro u gh gra p h o f f .
Typ e se t Ma y 4, 2016 a t 23:18:24. Lice n s e d b y Illu s tra tive Ma th e m a tics u n d e r a Cre a tive Co m m o n s
Attrib u tio n -No n Co m m e rcia l-Sh a re Alike 4.0 In te r n a tio n a l Lice n s e .
1
Volumes
Illustra tive
Ma the m a tics
G-GM D Th e Gre a t Eg y p t ia n
P y ra m id s
Ta s k
Th re e o f th e gre a t Egyp tia n p yra m id s a re p ictu re d b e lo w. Ea ch is a sq u a re p yra m id .
Illustra tive
Ma the m a tics
Ca lcu la te th e m issin g in fo rm a tio n fo r e a ch th e 3 in d ivid u a l p yra m id s b a se d o n th e
give n m e a su re m e n ts:
i.Th e gre a t Pyra m id o f Me n ka u re h a s a h e igh t o f a b o u t 215 fe e t a n d a b a se sid e le n gth o f a b o u t
339 fe e t. Wh a t is its vo lu m e ?
ii.Th e gre a t Pyra m id o f Kh a fre h a s a vo lu m e o f a b o u t 74,400,000 cu b ic fe e t a n d a b a se sid e
le n gth o f 706 fe e t. Wh a t is its h e igh t?
iii.Th e gre a t Pyra m id o f Kh u fu h a s a vo lu m e o f a b o u t 86,700,000 cu b ic fe e t a n d a h e igh t o f 455
fe e t. Wh a t is th e le n gth o f its b a se ?
b. Th e Gre a t Pyra m id o f Kh u fu o n ce sto o d 26 fe e t ta lle r th a n it is to d a y. Ca lcu la te th e o rigin a l vo lu m e
o f th e Gre a t Pyra m id . Sim ila rly, th e Pyra m id o f Kh a fre h a s e ro d e d o ve r tim e a n d lo st so m e o f its
h e igh t. If th e o rigin a l vo lu m e o f th e p yra m id wa s a p p ro xim a te ly 78,300,000 cu b ic fe e t, wh a t wa s its
o rigin a l h e igh t?
G-GMD Th e Gre a t Egyp tia n Pyra m id s
Typ e se t Ma y 4, 2016 a t 23:27:08. Lice n s e d b y Illu s tra tive Ma th e m a tics u n d e r a Cre a tive Co m m o n s
Attrib u tio n -No n Co m m e rcia l-Sh a re Alike 4.0 In te r n a tio n a l Lice n s e .
3
Radians: What Are They?
Explore
Which Baby is Which?
Scenario: Suppose that one night at a certain hospital, four
mothers give birth to four baby boys. The tired hospital staff
returns the babies to their mothers completely at random,
accidentally of course. We want to examine the probabilities
of the mothers receiving their own babies.
Design a simulation for the scenario. Describe your
simulation.
Perform 10 trials of your simulation. For each trial, record the
number of babies assigned to the correct mother.
Trial #
Number of babies Correctly Assigned
What minimum number of babies might be assigned to
the correct mother?
What maximum number of babies might be assigned to
the correct mother?
Are there any numbers of mismatches that are not
possible?
What is the empirical (experimental) estimate of the
probability that one mother will receive the correct baby?
Two mothers? Three mothers? All four mothers?
Summarize the results from the trials of the entire class.
Number Correctly Assigned
Count
Relative Frequency
0
1
2
3
4
What do the relative frequencies tell you?
Let’s simulate the scenario again using an applet. Use the “Random Babies” applet
at the
http://www.rossmanchance.com/applets/randomBabies/RandomBabies.html
web site to perform 1,000 trials of the computer simulation. Indicate the results
below:
Number Correctly Assigned
0
1
2
3
4
Count
Relative Frequency
How do the results from the class’ simulations and
the applet simulation compare?
What is your empirical estimate (based on the applet
simulation compared to the class’ simulation) of the
probability of all four mothers receiving the correct
baby?
Often the first step in probabilities is to consider the
sample space of all possible outcomes. Can you list
all of the possible outcomes (sample space) for this
situation? List them. Here is a start:
1234
1243
1324
1342
How many outcomes are there?
How many of them indicate that all four mothers received
the correct baby?
What is the theoretical probability that all four mothers will
receive the correct baby?
How does this compare to your empirical probability that
all four mothers will receive the correct baby?
For each outcome you listed in the sample space,
determine how many mothers received the correct
baby. Use this to find the probability that 0, 1, 2, 3, or
all 4 mothers receive the correct babies.
Indicate the probabilities for each number of correct babies below:
Number Correctly Assigned
P(x)
0
1
2
3
4
Above, you have the probability distribution of this discrete random variable
(the list of all possible values and the probability assigned to each value).
Use the probability distribution just found to
determine the probability that at least one mother
receives the correct baby.
How does this relate to the probability that no
mothers receive the correct baby?
This task was adapted from:
Chance, B. L. & Rossman, A. J. (2006). Investigating Statistical
Concepts, Applications, and Methods. Belmont, CA: Thomson
Brooks/Cole.
https://vimeo.com/35891677
Dan Meyer 3 Act Videos
What is your guess? Share your guess with your
neighbor and say why you think it is.
What information is important here?
How would you get it?
How do you find the
Volume of a cylinder?
5.5 cm
7 cm
10 cm
3 cm
5.5 cm
7 cm
10 cm
3 cm
Using the Math of Volume and Cylinders to
Get More Soda Than My Sister
Thanks for attending!
Have a great school year!