
Photo credit  Rebecca Gammill
Bowl by Baltic by Design
It has been a very busy but
exciting school year! There have been so many times when I said to
myself, "Well, that will have to wait until summer." As I look at my
ToDo list, I think that I need for my summer break to last as long
as a school year just to get started on all of these tasks. (I may
have a procrastination problem.) I have so many contacts to make,
books to read, topics to research, new ideas to ponder, and
activities to create. I thought that as I accumulated more years of
teaching experience that eventually my need to learn new things
would slow down. The opposite appears to be true. The more
experience I gain, the more I want to learn and think deeply about
ideas. I am also realizing that many of my previous understandings
are full of misconceived notions. After attending the 2018 NCTM
Annual Conference this year, I added one more book to my summer
reading list  Catalyzing Change in High School Mathematics:
Initiating Critical Conversations. With his
President's Address, Dr. Matt Larson launched the release of
this researchbased resource that identifies and addresses critical
challenges in high school mathematics to ensure that each and every
student has the mathematical experiences necessary for his or her
future personal and professional success.
I was also impressed to discover
the 2018 NCTM Annual Conference included a heavy representation of
Georgia educators in various capacities. We had a GCTM member on the
2018 NCTM Program Committee, more than 25 presenters, 3
representatives at the Delegate Assembly, and countless participants
in Washington, D.C., for this tremendous event. It was great to see
so many familiar faces and hear those Southern accents in the
sessions that I attended. Their participation reflected the
dedication of Georgia educators to help our schools and students
continue to grow and improve.
GCTM is excited to showcase
several of the nationallyrecognized speakers from the NCTM Annual
Conference at the 2018 Georgia Mathematics Conference. Dr.
Thomasenia Lott Adams, 2018 NCTM Program Committee Chair from the
University of Florida, will be kicking off the Georgia Math
Conference on Wednesday night. Dr. Matt Larson, NCTM Immediate Past
President, will be the keynote for Thursday night. Dr. James Tanton,
an ambassador for the Mathematical Association of America and a
purveyor of joy in mathematics, will be the Closing Keynote on
Friday afternoon. While attending the NCTM annual conference, I also
had the privilege of meeting and attending a session led by one of
our featured speakers, Dr. Kristopher Childs, from Texas Tech
University. Needless to say, with this many bright minds converging
again at GMC, I highly recommend that you make your plans to attend.
Try to arrive early as we have special plans for Wednesday
afternoon, and stick around Friday afternoon to hear Dr. Tanton. For
more details about GMC, visit
www.gctm.org.
GCTM is also providing
opportunities for professional growth over the summer. Don't forget
to register for the 2018 GCTM Summer Academies that take place in
June and July. Now is a great time to watch the gradelevel videos "What
Do Mathematics Standards Look Like in the Classroom?" released
by the Georgia Department of Education. Look for more of these
videos to be released this summer.
With all these activities going
on this year, it will be important to find some time to relax,
refresh, and spend meaningful time with family and friends! I hope
everyone has a great summer, and I hope to see you all at some of
these events. If you have any questions about what GCTM has to
offer, please contact your GCTM Regional Representative or any
member of the
Executive Committee.
We wish to better serve you and your
interests as a mathematics teacher.
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2018 Legislative Session
Report
In the early hours of the morning
on March 30th, 2018 the Georgia General Assembly completed its
40day legislative session and adjourned "Sine Die." After
adjournment, the Governor has 40 days to sign or veto bills. If the
Governor does not sign or veto a bill, it will automatically become
law. The Governor has the power of lineitem veto over the budget
bills. Throughout the legislative session, the advocacy team
identified and analyzed legislation of interest to GCTM and worked
with the appropriate decisionmakers to determine what action (if
any) should be taken.
Earlier in the session on February 13th, 2018 the advocacy team
facilitated an opportunity for a group from the Georgia Council of
Teachers of Mathematics to come to Atlanta for Math Day at the
Capitol. The day began in the Senate Chamber where Senate Finance
Committee Chairman Chuck Hufstetler (RRome) recognized GCTM with a
resolution and invited Bonnie Angel to address the members of the
Senate. This provided an opportunity for Bonnie to thank the members
of the Senate for their support of mathematics teachers across the
state and remind them of the critical work that GCTM does to support
quality mathematics instruction in Georgia. The group then hosted a
lunch in the capitol where numerous key influencers (listed below)
came by to learn more about GCTM's role in Georgia and discuss key
education issues that are facing the state. This provided a critical
opportunity to update these influential leaders on the work of
GCTM's members, thank them for their leadership, and discuss
relevant legislation that is pending in the House/Senate. Below is a
list of the key influencers who attended the GCTM lunch at the
capitol.
Senator Brian Strickland (RMcDonough); Senator Josh McKoon
(RColumbus); Senate Higher Education Chairman Fran Millar
(RAtlanta); Senator Larry Walker III (RPerry); Senator Ed Harbison
(DColumbus); Senator Kay Kirkpatrick (RMarietta); Senate Minority
Leader Steve Henson (DTucker); Senator Blake Tillery (RVidailia);
Senator Horecena Tate (DAtlanta); Senator Matt Brass (RNewnan);
Senator John Albers (RAlpharetta); Senator Chuck Payne (RDalton);
Rep. Wes Cantrell (RWoodstock); Rep. Debbie Buckner (DJunction
City); Rep. Tim Barr (RLawrenceville); Rep. Buzz Brockway
(RLawrenceville); Tim Cairl, Education Policy Director Metro
Atlanta Chamber of Commerce; and Joshua Roye, Governor's Legislative
Liaison.
Below is a detailed summary of the path that each of the most
relevant pieces of legislation took during the 2018 session of the
legislature followed by a comprehensive grid of all legislation that
the advocacy team tracked and analyzed throughout the session.
K12:
HB 787: Rep. Scott Hilton (RPeachtree Corners) filed this piece
of legislation that seeks to bridge the gap between existing funding
sources for local charter schools/public schools and state charter
schools. Currently, state charter schools receive money from the
state along with a small supplement equivalent to the average
perstudent local funding of the five poorest school systems in the
state. HB 787 would increase the supplement: 1) to the state average
perstudent funding level for schools with statewide attendance
zones and 2) for those with smaller attendance zones to the greater
of local district revenue or the average of the state's five poorest
systems. After this bill cleared the House a number of significant
changes were made in the Senate Education Committee at the request
of public education advocates although they were later reversed on
the Senate floor at the urging of Lt. Governor Cagle who is a strong
charter school advocate. After some additional changes were made on
the House floor, the Senate agreed to the changes and the bill was
transmitted to the Governor for his signature or veto.
HR 1162: This resolution will create the House Study Committee
on the Establishment of a State Accreditation Processthis study
committee was originally part of the major education reform
legislation that was approved in 2017 but the committee was never
appointed. The committee, if appointed, will study the costs,
benefits, and potential issues associated with creating a statewide
accreditation process for all K12 schools (primary/secondary public
schools and local school systems) in Georgia.
SB 330:Senator John Wilkinson (RToccoa) filed this bill that
seeks to create a state pilot program for agriculture education in
elementary schools. The legislation authorizes the Department of
Education to establish, using the nationally recognized
threecomponent model of schoolbased agricultural education, a
pilot in a minimum of six public elementary schools across six
regions beginning in the school year 20192020. The legislation
further outlines the parameters through which the program would be
developed. The measure made its way through the legislative process
and was recently signed by Governor Nathan Deallead sponsor
Senator Wilkinson is a former agriculture teacher and was
instrumental in securing the final passage of this measure.
SB 362: This measure, filed by Senate Education Chairman Lindsey
Tippins (RMarietta), seeks to provide for the establishment of an
innovative assessment pilot program for local school districts.
Specifically, the bill states that beginning with the 20182019
school year, the State Board of Education shall establish an
innovative assessment pilot program to examine one or more alternate
assessments and accountability systems aligned with state academic
content standards. The pilot program would span from three to five
years in duration, as determined by the state board and may include
up to ten local school system participants. Of note, during a
hearing on the measure Chairman Tippins told the committee that his
motivation behind the bill is to test out an assessment program that
would allow realtime feedback that can be used to guide instruction
to be available several times throughout the year. He acknowledged
that a number of districts are already doing this above and beyond
the required Milestones assessments (such as Gwinnett County Schools
who testified in support of the measure) but noted that he hopes to
prove out through the tendistrict pilot program that it can be
expanded statewide. Additionally, State Supt. Richard Woods
testified briefly in favor of the bill and remarked that the
Department of Education convened an innovative assessment task force
that will begin meeting internally to discuss how to implement the
bill should Gov. Nathan Deal sign it into law.
Higher Education:
SB 405: Since early 2017, Senate Higher Education Committee
Chairman Sen. Fran Millar (RAtlanta) has been working with the Bill
and Melinda Gates Foundation and the Southern Regional Education
Board to create a program to allow Georgia students who do not
qualify for the HOPE or Zell Miller scholarships  and cannot afford
the difference between a Pell Grant and the rest of the costs
associated with attending collegeto receive a grant to cover some
of those costs. The bill's parameters limit the grant to students
whose family income does not exceed $48,000. The bill creates an
application process whereby students can apply for up to a $1,500
per semester grant so long as certain conditions are met. In order
for the program to go into effect, the General Assembly would have
to appropriate the necessary fundsno such funds were appropriated
in the AFY 2018 and FY 2019 budgets. Although this measure easily
cleared the Senate, it was held up in the House Higher Education
Committee when Chairman Jasperse declined to call for a vote.
Senator Millar subsequently attached the measure to HB 787 in the
Senate Education Committee where it remained through final passage.
Fiscal Year 2019 Budget:
Governor Nathan Deal signed the conference committee report on the
FY 2019 budget although lineitem vetoes are not likely to be
released until May 8th.

After language related to
dual enrollment programs was added in the House and subsequently
removed in the Senate, the final conference committee report on
the FY 2019 budget added new language. This language directs the
Georgia Student Finance Commission to implement a 15credit hour
per student per semester cap; require ongoing professional
development for adjunct faculty teaching dual enrollment courses
to the same degree that is required for fulltime faculty; and
implement admission standards for dual enrollment students at
private postsecondary institutions to be in parity with that of
the University System of Georgia for degree level transferable
courses and with the Technical College System of Georgia for
courses leading to a diploma or certificate effective July 1,
2018. Additionally, language was added directing the Georgia
Student Finance Commission to develop a list of approved dual
enrollment courses that prioritizes courses leading to a degree
or indemand certificate or diploma and report findings to the
House and Senate Appropriations Committees by December 1, 2018,
to be implemented in FY 2020.

Two notable changes were made
by the House to the Governor's Office of Student Achievement FY
2019 budget that was included in the final conference committee
report on the budget. $750,000 in state funds was added to
increase funds for one nonSTEM AP exam fee for lowincome
students. Additionally, budget instructions were added that
direct GOSA to increase existing grant funds for birthtofive
literacy/numeracy in rural centers located in the lowest
performing K12 school districtsthis recommendation came out
of issues that were brought to light as the House Rural
Development Council traveled the state over the past year.

Governor Nathan Deal
increased the FY 2019 revenue estimate by more than $194 million
over initial projections that brought the budget total to over
$26.2 billion. As a result, Deal amended his budget
recommendation to include an additional $167 million for K12
education. These funds will ensure the state is fully funding
the Quality Basic Education (QBE) funding formula and providing
local school systems with 100% of the state's share in financing
for local schools.
Bill 
Sponsor 
Committee 
Status 
Description 
HB 702

Rep. Heath Clark (RWarner Robbins) 
HC: Higher Education 
Did not pass 
Service cancellable loan program for STEM 
HB 728 
Rep. Brooks Coleman (RDuluth) 
HC: Ways & Means
SC: Finance 
Did not pass 
Repeals the sunset on the tax credit for the Public
Education Innovation Fund 
HB 740 
Rep. Randy Nix (RLaGrange) 
HC: Education
SC: Education and Youth 
On Governor's desk 
Revises expulsion/suspension protocol 
HB 763 
Rep. Randy
Nix (RLaGrange) 
HC: Education
SC: Education & Youth 
On Governor's
desk 
Expands
student protocol committees to include school climate 
HB 778 
Rep. Terry
England (RAuburn) 
HC: Higher
Education 
Did not pass 
Transfer CTAE
from DOE to TSCG 
HB 781 
Rep. Kevin
Tanner (RDawsonville) 
HC: Education 
Did not pass 
Adds
maintenance and operations to ESPLOST 
HB 787 
Rep. Scott
Hilton (RPeacthree Corners) 
HC: Education
SC: Education & Youth 
On Governor's
desk 
Charter
Schools substantive revisions in Title 20 
HR 1039 
Rep. Dave
Belton (RBuckhead) 
HC: Special
Rules 
Did not pass 
Study
Committee on Ennobling the Teaching Profession 
HR 1162 
Rep. Brooks
Coleman (RDuluth) 
HC: Education 
House
Passed/Adopted 
Study
Committee on State Accreditation Process 
SB 3 
Sen, Lindsey
Tippins (RMarietta) 
SC: Education
and Youth
HC: Education 
On Governor's
desk 
CONNECT Actto enhance industry credentialing for some programs in high
school 
SB 330 
Sen. John
Wilkinson (RToccoa) 
SC: Ag and
Consumer Affairs
HC: Education 
Signed by
Governor Nathan Deal 
Agricultural
education pilot 
SB 362 
Sen. Lindsey
Tippins (RMarietta) 
SC: Education
HC: Education 
On Governor's
desk 
Local
district innovative assessment pilots 
SB 377 
Sen. Brian
Strickland (RMcDonough) 
SC: Higher
Education
HC: Industry and Labor 
Signed by
Governor Nathan Deal 
Transfers
State Workforce Development Board to TCSG 
SB 401 
Sen. Lindsey
Tippins (RMarietta) 
SC: Education
and Youth
HC: Education 
Did not pass 
Flexibility
for guidance counselors to develop career assessment plans 
SB 405 
Sen. Fran
Millar (RAtlanta) 
SC: Higher
Education
HC: Higher Education 
On Governor's
deskadded to HB 787 
Creates
lowincome grants for students attending USG institutions 
SR 1068 
Sen. Steve
Gooch (RDahlonega) 
Senate
Passed/Adopted 
Senate Adopted 
Senate Study
Committee on the school start date 
T.J. Kaplan is a legislative consultant who provides real time reporting and guidance while facilitating relationship development on behalf of GCTM.
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The GCTM Middle School Math Tournament was held
at Tattnall Square Academy on April 21, 2018. Middle schools across
the state were invited to register up to eight students to compete.
The tournament consisted of a 30 question multiplechoice test with
a 45minute time limit; 10 individual ciphering problems, each
problem with a twominute time limit; 3 rounds of four pair
ciphering problems (in which students from a school formed teams of
two), each round with a fourminute time limit; and a fourperson
team "power question," in which the team solves a complex problem
with a 10minute time limit.
The tournament is designed to challenge middle school students and
to reinforce classroom skills. However, we also make sure the
students have fun! At the conclusion of the tournament, students
participate in a fun "Frightnin' Lightnin'" Round, where students
must be quick on the draw to answer math problems posed orally. The
winners of this round get candy!
Trophies went to the top five teams and the top ten individuals. The
top teams are below.
TOP TEAMS:

South Forsyth Middle School, Cumming

Fulton Science Academy, Alpharetta

Tattnall Square Academy, Macon

Southeast Bulloch Middle School, Brooklet

Stratford Academy, Macon
Sixtyone students from twelve schools
participated. Sponsors that are members of GCTM only had to pay a
$10 registration fee or submit five multiplechoice questions for
possible inclusion in a future tournament. The next GCTM middle
school tournament is scheduled for April 20, 2019.
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The 42nd annual GCTM State Math Tournament was
held at Middle Georgia State University in Macon, Georgia on April
28, 2018. Schools are invited to the state tournament based on their
performance on previous Georgia tournaments throughout the 20172018
school year. Thirtysix invited schools attended this year's state
tournament. Four students are selected by their school sponsor to
represent each school (one school brought a team of two). Twenty
individuals were also invited to tryout for the statewide Georgia ARML team, making a total of 162 participants.
The tournament consisted of a very challenging
written test of 45 multiplechoice questions and 5 freeresponse
questions with a 90minute time limit; 10 individual ciphering
problems, each problem with a twominute time limit; and a team
round. The team round consisted of 12 problems for each team to
solve while working together within eighteen minutes.
The student with the best improvement at the state tournament over
the previous year was given the Steve Sigur Award for Most Improved
Performance. This award, named in honor of the great mathematician,
teacher, and mentor Steve Sigur, went to Ty Bondurant of Columbus
High School. Each participant and their school sponsor was given a
2018 State Tournament Tshirt.
The top five teams and the top fifteen individuals are listed below.
TOP 5 TEAMS:
Fulton Science Academy
Walton High School
Kennesaw Mountain High School
Westminster
Northview High School
The classification winners are the schools which were not in the top
5, but, except for the top 5, placed above all other schools in
their classification. We call these "classification champions."
Unfortunately, there was no Class AA Champion this year, as all AA
schools that qualified for the State Math Tournament declined to
participate.
CLASSIFICATION CHAMPIONS:
Class A: Wesleyan School
Class AAA: Greater Atlanta Christian School
Class 4A: North Oconee High School
Class 5A: Chamblee High School
Class 6A: Dunwoody High School
Class 7A: Peachtree Ridge High School
NonGHSA Class: Eureka Scholastic Academy
TOP 15 INDIVIDUALS:

Daniel Chu, Kennesaw Mountain High School
(second consecutive year)

Holden Watson, Fulton Science Academy (second
consecutive year)

Shawn Im, Peachtree Ridge High School

Anup Bottu, Westminster

Vishaal Ram, Milton High School

Ramanan Abeyakaran, Chamblee High School

Russel Emerline, Walton High School

Daniel Shu, Walton High School

Alex Eldridge, Dunwoody High School

Cade Lautenbacher, Dunwoody High School

Charlie Furniss, Fulton Science Academy

Aaron Yu, Westminster

Ty Bondurant, Columbus High School

Jayson Wu, Walton High School

Johnny Fang, Westminster
An item analysis of the competition problems was
completed at the state tournament. The responses analyzed included
the 45 multiplechoice problems on the written test and the 10
problems from the individual ciphering round. Before we discuss what
the item analysis revealed, some background information would be
useful. The problems on the written test are designed to increase in
difficulty. Thus, theoretically, problem 1 is the easiest
multiplechoice problem and problem 45 the most difficult
multiplechoice problem on the test. Below are those problems.
Test Problem #1: What is 15% of 0.75 of 48/30?
a) 9/100 b) 3/25 c) 9/50 d) 9/25 e) 3/5
According to the analysis, problem 1 was actually the easiest, as
160 students out of 162 answered it correctly! Problem 1 was a
straightforward problem that required a simple product to be
computed. Since graphing calculators are allowed on the written
test, the product was easily computed.
In contrast, the analysis revealed that it was Problem 44, not
Problem 45, that was the most difficult. Exactly 21 students
answered Problem 45 correctly, while only 4 students answered
Problem 44 correctly! Below are both problems.
Test Problem #44: Let ABCD be a square and let F be
the midpoint of segment BC. Points G and H are
chosen randomly and uniformly on the sides of segments AB and
DC, respectively. The probability that angle GFH is
acute can be written as the reduced fraction (bln(c))/a,
where a, b, and c are integers. Compute a
+ bc.
a) 3 b) 4 c) 5 d) 8 e) 10
Test Problem #45: Consider the following array.
For integers i^{3} 1 and j^{3} 1, define D_{i,i}
= D_{I1,j+1}D_{i1,j}.
If D_{0,j} is the jth Fibonacci number, find S^{5}_{i=0}
[(5Ci) D_{i,1}].
a) 5 b) 5 c) 8 d) 34 e) 89
Problem 44 seems like a geometry problem, but it
is, in fact, a trigonometry problem, which turns into a calculus
problem! For angle GFH to be acute, the sum of the other two
angles along the side of the square (angles BFG and CFH)
must be greater than 90 degrees. Since these other two angles are in
right triangles, we can express these angles as inverse tangents of
the portions of the side of the square opposite the angles; call
these portions x and y. From the inequality arctan(x)
+ arctan(y) > p/2 then leads to the inequality y >
cot(arctan(x)). But since the cotangent of the inverse
tangent of x is 1/x, we see that y > 1/x.
Hence, we want the area of the square which lies above the hyperbola
y = 1/x, and this is calculated using a definite
integral. The probability is the area above the hyperbola which lies
in the square over the area of the square. This is (3ln(4))/4.
Problem 45 can be solved by noticing that the
entries in the array are the finite differences of the Fibonacci
sequence; that is, an entry in the array is the difference of the
two terms above it. Adding all the combinations of all the leading
differences yields D_{0,6} which is the sixth Fibonacci
number: 8.
As for the ciphering, there is no particular order of difficulty for
the questions, so it is always interesting to see which problems are
answered correctly and quickly. The easiest ciphering problem,
judged by the fact that 148 participants gave the correct answer, is
the following. (Recall that each of the problems below should be
answered in less than two minutes, without a calculator.)
Ciphering Problem #9: Let x, y, and z be real
numbers with y < 0. Define the function deb_{x}(y)
to be equal to z when y^{z} = x.
Compute deb_{64}(4).
This function "deb" was a logarithm in disguise, which was easily
recognized by the participants. Therefore, since (4)^{z}
= 64, it is easy to see that z = 3.
For the most difficult ciphering problem, there
was a tie! Only 12 participants gave the correct answer to the
following two problems.
Ciphering Problem #3: The distance between the foci of the conic 16x^{2}
+ 4y^{2}32x + 64y + 208 = 0 can be
written as
k, where k is a positive integer. Compute
k.
Once again, it is a conic section problem that is
considered a difficult question. This is normal over the last few
years; apparently the teaching and learning of conic sections is no
longer a priority. The answer to this problem is 12.
Ciphering Problem #5: A function from a set X
to a set Y is said to be onto, or surjective, if for all y
in Y, there exists an x in X such that y =
f(x). Compute the number of onto functions from the set {1, 2,
3, 4, 5} to the set {A, B, C, D}.
Two elements in X must be mapped to the
same element in Y; there two elements in X can be
chosen in 10 ways. These two elements can be mapped to any of the 4
elements in Y, and this leaves 3! = 6 ways to map the
remaining elements. The answer is 10 ' 4 ' 6 = 240.
State Tournament registration is free, but schools must be invited.
The next State Mathematics Tournament is scheduled for April 27,
2019.
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Have you ever wondered whether what you ate
caused you to have food poisoning? The CDC uses twoway tables
to determine a contaminant of foodborne outbreaks. At NCTM in
Washington, D.C., Michelle Mikes and I lead an engaging STEM
simulation for teachers to help students understand the purpose
and use of twoway tables. We adapted our presentation from a
STEM conference session that we attended led Dr. Ralph Cordell.
This lesson was designed for 8thgrade students to learn how
ratios, twoway tables, and probability can be used to determine
details about food poisoning contaminants. But, you can take
this lesson and adapt it for use in elementary or high school
classrooms.
Participants first learn some background
information on food poisoning: what defines an outbreak,
symptoms and causes of food poisoning, the highest causes of
foodborne illnesses and deaths, the early thoughts about
diseases, types of foodborne germs, and information from
articles and radio posts. For example, did you know that food
should not sit out at room temperature for more than 2 hours or
that you shouldn't eat any leftovers that are more than 4 days
old? This information was provided by Dr. Sanjay Gupta, Chief
Medical Correspondent for CNN, in a radio post on June 9, 2014.
Participants are then given an example of an outbreak
investigation from Saudi Arabia in 1979 and asked to discuss how
twoway tables could be used to organize data and determine the
attack rate for each food item.
From the attack rate, students determined a
relative risk for each food and also calculated which food had
the highest relative risk. In this case, the meat was the cause
of the foodborne outbreak.

Attack Rate Exposed 
Attack Rate Unexposed 
Relative Risk (RR) 
Rice 
66.7 
100 
0.66 
Meat 
71.6 
14.3 
5.01 
T.S. 
65.8 
73.7 
0.89 
Using this background knowledge, participants
then chose from a list of six food items: chocolate cake, angel
food cake, apple pie, chocolate ice cream, vanilla ice cream,
and strawberry ice cream. Using clear plastic cups, participants
simulated "eating" these foods taking one spoonful from each
"food" and leaving the spoons within the cups as to not
contaminate the other foods. Once all participants collected
their "foods", they gathered data to determine who was sick.
Participants filled all six "food" cups with water, even though
only one cup was contaminated with baking soda. Phenolphthalein
is added to each participant cup, and those cups that turn a
purple/fuchsia color were sick.
Participants then recorded
their results the following handout to tabulate the data for
each food item based on who ate or didn't eat the food and who
was sick or not sick afterward.
After all data had been collected per table,
participants collected the whole group data using larger chart
paper.
After the group collected all
data, they calculated the attack rate for those exposed and not
exposed to each food item by taking the total number of those
sick and dividing it by the total within the exposed row and the
not exposed row. Then, participants used these two rates to
determine the relative risk by dividing the attack rate of those
exposed by the attack rate of those not exposed. Finally,
participants analyzed the data to determine which food was the
contaminated food.
Towards the end of the task,
participants discussed how the simulation could be used across
all grade levels. For example, elementary students may count to
collect the data and use fractions and decimals when creating
their attack rates. Middle school students can focus on rates,
fractions, decimals, percents, proportions, and twoway tables.
Further, high school students could analyze twoway tables with
joint and marginal frequencies, statistical significance,
confidence intervals for estimates, pvalues, and causation
versus correlation.
This task may also be tied in
other related events. In one example, power outages caused by
inclimate weather are related to food poisoning prevention.
Students learn that placing a frozen mug of water with a quarter
on top is an excellent tool to determine when refrigerated food
is no longer safe for consumption. During a power outage, the
water melts and the quarter slowly sinks to the bottom of the
mug. As the quarter's movement to the bottom of the mug provides
an interesting measurement of time. When the quarter reaches the
bottom of the bottom of the mug, it becomes clear that the
refrigerated or frozen food may no longer be suitable to eat.
Disease outbreaks and the CDC are another an interesting context
for this activity. For example, there is concern about certain
romaine lettuce and eggs, so we shared the information from the
CDC in relation to these outbreaks. These warnings could be used
to ignite a conversation about food poisoning in class, and thus
more deeply contextualizing this topic of mathematics for
students. (Some examples are provided below.)

Ashley Clody is an assistant supervisor for
the Division of Instruction and Innovation Practice in Cobb
County. Previously, Mrs. Clody taught middle school math for ten
years. She has been a member of GCTM for the past 13 years as
either a student member or teacher member. She was the recipient
of the Dwight Love award in 2017. 

Michelle Mikes has over 20 years of teaching in middle and high
school mathematics. She was recipient of the John Neff Award in
2012. She is currently the Mathematics Supervisor for the
Division of Instruction and Innovation Practice for Cobb County
Schools. 
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Using fractions as operators means applying
fractions to numbers, objects, or sets as if they are themselves
functions (Behr, Harel, Post, & Lesh, 1993). Studies suggest
that deep understanding of fractions as operators supports
students' flexible reasoning with fractions in later mathematics
(Behr et al.,1993; Hackenberg & Lee, 2012). The operator
construct, however, seems easily forgotten in school curricula
(Post et al., 1993; Usiskin, 2007). We implemented tasks
intended for 7thgrade students in a class of K8 preservice
teachers to elicit their thinking on fractions. We used tasks
across a unit to intentionally provide preservice teachers with
opportunities to encounter situations in which fractions can be
used as operators and to use fractions as operators in their
problemsolving. Selected student strategies for each task show
a variety of approaches utilized by students in their
problemsolving.
Task 1 shown in Figure 1 was purposefully given to students
prior to percent lessons to stimulate student thinking and
reasoning about the use of fractions.
Task 1: An Incredible Discount
An electronics store is having an incredible Black
Friday deal. The price on a model of a smartwatch had
been reduced by 30%, and the store is now taking 30% off
their alreadyreduced prices. At what percentage of the
original price is this smartwatch selling? 
Figure 1. Task 1: An
Incredible Discount
We designed Task 1 to provoke
student thinking through the use of fractions as operators; that
is, solving the task involves applying a fraction (e.g., the
final cost is 30% off the sales cost) to a fractional amount
(e.g., the sales cost was 30% off the original cost). We
subsequently chose student samples of incorrect solutions (see
Figure 2) to illustrate different levels of problemsolving
strategies and misconceptions.
Student Strategies to Task 1: An Incredible Discount
Student 1

Student 3

Student 2

Student 4

Figure 2. Student Strategies
to Task 1: An Incredible Discount
Student 1's strategy, adding
percentages, reveals a common misconception in which the
fractional amounts (of different wholes) are simply added and
subtracted from 100%. Student 3's and Student 2's strategies
show a slightly higher level of reasoning in that they
understand that multiplication is necessary, but incorrectly
connect the product to the context of the task. That is, both
fail to recognize the product as a "percent off" or difference.
Although both are on the right track, Student 3 should subtract
21% from 70%. Student 2 adds 21% to 30% (the percentage off the
original price) but should subtract 51% from 100%. Hence Student
2 finds the "percent off" the original price rather than the
"percent of" the original price. Among these four students,
Student 4 shows the highest and most valid fractional reasoning.
He reaches a correct number but does not utilize formal
notations and correct units.
Task 1 (Figure 1) entails a
composition of operations and is an example of using fractions
as operators. So what does "fractions as operators" really mean
and why should we emphasize this concept with upper elementary
and middle grades students?
Fractions As Operators: What Does It
Mean?
Five fraction constructs are
recognized in mathematics education: partwhole, measurement,
division, operator, and ratio (Behr et al., 1983). While the
partwhole construct is an effective way to develop initial
fraction understanding (Cramer & Whitney, 2010), previous
research emphasizes that various fraction meanings introduced in
later instruction improve fraction understanding (Clarke, Roche,
& Mitchell, 2008; Lamon, 2012; Siebert & Gaskin, 2006). Behr and
colleagues (1993) suggested that understanding fractions as
operators can strengthen students' grasp of fraction
multiplication, and also lead students' to translating word
problems into number sentences more easily. Connecting algebraic
thinking to understanding fractions as operators supports
students' writing equations to represent multiplicative
relationships between two unknown quantities (Hackenberg & Lee,
2012). Nonetheless, the operator construct often lacks emphasis
in school curricula (Post et al., 1993; Usiskin, 2007).
Evidence of students'
understanding the operator interpretation of fractions includes
students ability to: (a) interpret a fractional multiplier in at
least two ways, (b) use one fraction for a composite operation,
(c) relate outputs and inputs, and (d) identify a single
composition of compositions (Lamon, 1999 and 2012; Marshall,
1993). We describe the four skills briefly. Lamon described (a)
as thinking of a fraction (e.g., 3/4) in two ways: as multiple
copies of one fractional part of the unit (e.g., 3 of [1/4 of
the unit]) or as a fraction of multiple copies of the unit
(e.g., 1/4 of [3 units]). In using (b), Lamon explained that
students could use a fraction to describe a composite operation;
that is, a fraction used to multiply or divide can be thought of
as two operations performed as one. For example, multiplying a
unit by 3/4 is the same as dividing the unit by 4 and then
multiplying by 3. The third skill that students need, (c), is to
understand the relationship between input and output: For
example, a 3/4 operator defines a 3for4 exchange, transforming
an input quantity of 4 into 3. Finally, Lamon described (d) as,
for example, recognizing that 3/4 of (4/5 of a unit) is
precisely equal to 3/5 of a unit.
So how do we help improve
students' understanding of fractions as operators?
Enhancing Understanding Fractions as
Operators
For the brevity of the paper,
we will focus on the first (a) and last (d) skills described in
the previous paragraph. To help students interpret a fractional
multiplier in more than one way, we first gave them the
following task (Figure 3):
Task 2:
Jill is planning to run 3 miles. When she has run
twothirds of this distance, how far has Jill run? 
Figure 3. Task 2
To obtain the answer for Task
2, students typically partitioned this distance (3 miles) into
three equal groups, resulting in a quantity of 1 mile in each
group. Their answers came from taking two groups of 1 mile.
Hence, 2/3 of 3 miles is 2 miles. As shown in Figure 4, while
each of the four students represented three individual groups,
they thought about it differently: Students 5 and 7 used
diagrammatic representations, while Students 6 and 8 used
graphical representations.
Student 5

Student 7

Student 6

Student 8

Figure 4: Student Strategies
for Task 2
While the strategies in
Figure 4 are common for this task because the context supports
breaking three miles into three equal parts, there is another
way of finding 2/3 of 3 miles to i
Alternative Approach to Task 2

Figure 5. Alternative
Approach to Task 2
Let us compare the strategies in Figures 4
and 5. In the student strategies (Figure 4), twothirds of three
miles is interpreted as two copies of onethird of three miles,
or equivalently 2 x [1/3 x 3 miles]. In the second approach
(Figure 5), the operation is interpreted as onethird of two
copies of three miles, or equivalently 1/3 x [2 x 3 miles].
Students could also see that 2 x [1/3 x 3 miles] equals 1/3 x [2
x 3 miles] because of the commutative and associative properties
of multiplication. Helping students see the connections between
the approaches in Figures 4 and 5 might enhance their
understanding of fractions as operators.
In Tasks 2 (Figure 3), "2/3
of" is an example of an operator in which both multiplying and
dividing may occur during the process of applying the operator.
The result of a 2/3 of the operator, Jill's running distance,
shortens the length because the operator does more shortening
than lengthening. In particular, it lengthens Jill's distance by
a factor of two and shortens it by a factor of 3. Whereas, in an
example such as 5/4 of eight marbles, the 5/4 of operator
increases the number of objects; that is, it does more
increasing than decreasing because it increases by a factor of
five and decreases by a factor of four. Thus, understanding
fractions as operators means knowing a result of applying an
operator such as 2/3 of three and 4/5 of eight can be decreasing
or increasing, which will help address the misconception that
multiplication "always makes bigger" and division "always makes
smaller" (Clarke, Roche, & Mitchell, 2008; Karp, Bush, &
Doughtry, 2014).
A Composition of Compositions
After our students completed
Task 2 and we discussed connections between alternative
strategies for the task, we asked them to complete Task 3 so
that we could explore their understanding of fractional
operators with regard to the commutative property. Task 3 was
adapted from Beckmann (2014) and is shown in Figure 6.
Task 3 (Adapted from Beckmann,
2014):
Option 1: The price of a portable speaker
is marked up by 10% and then marked down by 30% from the
increased price.
Option 2: The price of a portable speaker
is marked down by 30% and then marked up by 10% from the
discounted price.
1. At what percentage of the original price is this
portable speaker selling in option 1?
2. At what percentage of the original price is this
portable speaker selling in option 2?
3. Which, if either, of the two options above will
result in the lower price for the portable speaker?
Please explain.

Figure 6. Task 3
Two student strategies were
selected and are shown in Figure 7. Both Students 1 and 3
(Figure 2) complete Task 1 incorrectly but solve Task 3
correctly. Student 3 multiplies the two percentages as she does
in Task 1, but this time she correctly multiplies the first time
and second time markedup/markeddown price percentages and
provides correct answers with units included. Student 1
incorporates all the operations correctly and in the correct
order. Additionally, Student 1 displays a more detailed written
description of her reasoning, which indicates some substantial
improvement in her/his conceptual understanding of percentages.
This is in contrast to the understanding we observed in Task 1,
in which she considers the result of a sequence of percentage
changes as a result of an addition operation.
Figure 7. Student strategies
to Task 3
Conclusion
Understanding fractions as
operators should not be forgotten in school curricula. Through
the implementation of these tasks, students were exposed to
situations in which fractions were used as operators and used
fractions as operators in their problemsolving. When working on
Task 3, students were also able to write equations representing
a multiplicative relationship among quantities, which supports
findings from Hackenberg & Lee (2012)'s study that understanding
fractions as operators can support students in writing equations
to represent multiplicative relationships between quantities.
With a goal of exploring variations of student problemsolving
strategies and enhancing student understanding of fractions, we
encourage teachers to implement these tasks and share student
solutions and discussions. Fostering students' reflection about
making sense of their problemsolving strategies through
discourse supports the type of deeper understanding that we want
our students to have.
References
Beckmann, S. (2014).
Mathematics for Elementary Teachers with Activity Manual.
4th ed. New York: Pearson.
Behr, M., Lesh, R., Post, T.
& Silver, E. (1983). "Rational number concepts." In R. Lesh and
M. Landau (eds.), Acquisition of Mathematics Concepts and
Processes, pp. 91125. Academic Press, New York.
Behr, M. J., Harel, G., Post,
T. R. & Lesh, R. (1993). "Rational numbers: Toward a semantic
analysisemphasis on the operator construct." In T. P.
Carpenter, E. Fennema & T. A. Romberg (Eds.), Rational
numbers: An integration of research, pp. 1347. Hillsdale:
Lawrence Erlbaum Associates
Cramer, K. A., & Whitney, S.
(2010). "Learning rational number concepts and skills in
elementary classrooms: Translating research to the elementary
classroom." In D. V. Lambdin, & F. K. Lester (Eds.), Teaching
and learning mathematics: Translating research to the elementary
classroom, pp. 1522. Reston, VA: NCTM
Clarke, D., Roche, A., &
Mitchell, A. (2008). 10 practical tips for making fractions come
alive and make sense. Mathematics Teaching in the Middle
School, 13(7), 373380.
Hackenberg, A. & Lee, M.
(2012). Prefractional middle school students' algebraic
reasoning. In L.R. Van Zoest, J.J. Lo, & J.L. Kratky (Eds.),
Proceedings of the 34th annual meeting of the north American
chapter of the international group for the psychology of
mathematics education, pp. 943950. Kalamazoo, MI: Western
Michigan University
Karp, K. S., Bush, S. B., &
Dougherty, B. J. (2014). 13 rules that expire. Teaching
Children Mathematics, 21(1), 1825.
Lamon, Susan J. (2012).
Teaching Fractions and Ratios for Understanding. 3rd ed. New
York: Routledge.
Lamon, Susan J. (1999).
Teaching Fractions and Ratios for Understanding. Mahwah, NJ:
Lawrence Erlbaum Associates.
Marshall, S.P. (1993).
"Assessment of rational number understanding: A schemabased
approach." In T.P. Carpenter, E. Fennema and T.A. Romberg
(eds.), Rational Numbers: An Integration of Research, pp.
261288. Lawrence Erlbaum Associates, New Jersey.
Post, Thomas, Kathleen
Cramer, Merlyn Behr, Richard Lesh, and Guershon Harel. (1993).
"Curriculum Implications of Research on the Learning, Teaching,
and Assessing of Rational Number Concepts." In Rational
Numbers: An Integration of Research, edited by Thomas
Carpenter, Elizabeth Fennema, and Thomas Romberg, pp. 32761.
Hillsdale, NJ: Lawrence Erlbaum Associates.
Siebert, D., & Gaskin, N.
(2006). Creating, naming, and justifying fractions. Teaching
Children Mathematics, 12(8): 394400.
Usiskin, Z. (2007). Some
thoughts about fractions. Mathematics Teaching in the Middle
School, 12(7): 370373.

Dr. Ha Nguyen is an Assistant Professor of
Mathematics Education at Georgia Southern University
and a Blue'10 Fellow in the Mathematical Association
of America's (MAA's) Project NExT (New Experiences
in Teaching) for mathematics faculty. She is
interested in students' understanding and thinking
of mathematics and how to make mathematics relevant
to students. She earned her Ph.D. in Mathematics
from Emory University. 

Dr.
Heidi Eisenreich is an Assistant Professor of
Mathematics Education at Georgia Southern University
and a 2017 STaR (Service, Teaching, and Research)
fellow through AMTE (Association of Mathematics
Teacher Educators). Her interests lie in finding
meaningful tasks that push beliefs about mathematics
teaching and learning through discourse and
reflecting on those tasks with preservice teachers,
inservice teachers, and parents. She earned her
Ph.D. in Mathematics Education from the University
of Central Florida in Orlando. 

Dr. Eryn M. Stehr is an Assistant Professor of
Mathematics Education at Georgia Southern
University, and a 2018 fellow in the Association of
Mathematics Teacher Educators' (AMTE's) Service,
Teaching, and Research (STaR) program for
mathematics education faculty. Her research
interests focus on developing teacher autonomy and
decisionmaking in mathematics teaching and
learning, with a special focus on integrating use of
technology with rich tasks and mathematical
discussion. She earned her M.A. in Mathematics from
Minnesota State University and her Ph.D. in
Mathematics Education from Michigan State
University. 

Dr. Tuyin
An is an Assistant Professor of Mathematics
Education at Georgia Southern University. Her main
research interest is preservice secondary
mathematics teachers' conceptions of geometry
theorems. She is a Service, Teaching, and Research (STaR)
Fellow of the Association of Mathematics Teacher
Educators (AMTE) and a Scholarship of Teaching and
Learning (SoTL) Fellow at GSU for 20182019. 
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When asked to name the most hated subject in
school...mathematics often wins. Why, I ask myself, is something so
basic and necessary in life, so despised? Could it be that math is
perceived as boring? Tedious? An unnecessary evil? Possibly. I
postulate that math is not only, necessary, but it is anything but
evil. So, let's look at the adventure that is mathematics.
One plus one is two you say.
Will it always be this way ?
Yes, yes, yes....
Math has rules, no hidden tricks,
Unlike spelling that can make you sick.
Do you hear or here you do.
Are you there or is their with you.
See what I mean, or is it sea.
Math has rules that do not flee, or flea?
Addition, subtraction, multiplication, division!
Master them once. They maintain their position. 



Ah, but there, or their, is so
much more, you say.
Agreed, but the rules are the rules and will always
stay.
Even advancing beyond the basics to the harder stuff..
Squares, cubes, taking roots...easy enough.
Three squared is always nine,
Never any other value that would blow your mind.
The same is true for other values as well,
No reason to panic or crawl in a shell. 



Without math, what would we do
when we watch our weight?
How would we know if we were late?
There would be no rich or poor,
No one would know who has more.
How would the stores know what to charge,.
Should the price be small or should it be large? 



I understand that math is not
loved by all,
But all of us need it, both large and small.
Without math, our growth could not be measured,
And the thrill of growing taller, we could never
treasure.
Miles per gallon would be a complete mystery
As would the dollar amount in our treasury.
You might feel all this is way too simple to justify
Any math beyond the basics on which we all rely. 



That may be true but then again,
maybe not.
Do you enjoy all the luxuries that you have got?
Geometry is to be thanked for the shapes we see,
Circular tables, rectangular beds, square desks, and so
much more.
All solids that we use every day to hold the things we
need to store.
Do you need large or do you want small?
Know the volume and you will make the right call. 



And think of all the trips you
take. The highways and bridges you are on each day.
Without math calculations, they could not have been
built that way.
Even the mechanics who keep your car in shape and
running smooth
Could not work their wonders without measuring tools.
And the yummy desserts we all enjoy after a hardy meal,
Must be baked to precision or they would hardly appeal. 



So the next time you are opening
your math book
And grimacing and frowning with that "I don't want to"
look,
Stop for a minute and think about a world without
mathematics.
Pretty awful, don't you think? But no need to panic...
Math is here to stay, and all the rules stay the same,
So, apply yourself, do all you are asked, and soon math
will be as easy as saying your name. 

Mrs.
Coley is a retired math teacher who taught high school and college
mathematics in both private and public institutions including
Woodward Academy. While at Woodward, Mrs. Coley received the
Presidential Award For Excellence in Mathematics and Science
Teaching. Although she has taught topics from Algebra through
precalculus, she adores teaching Geometry. Currently residing in
Woodstock, Georgia, Mrs. Coley enjoys spending time with her five
grandchildren, traveling, and writing poetry.
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Editor's Foreword:
Sun
Hong, a Cobb County School District Elementary Art teacher,
submitted the following lesson plan. She has been a sweet friend
of mine since middle school, and somehow we both found ourselves
enjoying our professions in this wonderful world of education.
While our students and content areas are quite different, our
passion for education remains the same.
As she shared her success of the following
lesson through social media, the creative and mathematical
nature of her lesson struck a chord. (Pun intended, Geometry
folks.) Through the promotion of early mathematical vocabulary,
creativity, and navigation, this STEAM lesson illustrates that
the field of mathematics is not mutually exclusive with other
disciplines. Mathematics can be used to make other subjects
interesting and engaging.
Although Mrs. Hong may differentiate this
activity with most elementary grade levels, I challenge our
readers to think about how they could use the lesson in their
classroom. Could this activity be used to discuss relationships
between inscribed angles, cyclic quadrilaterals, or even
trigonometry? Could instructors vary the circle size or have a
different number of notches along the perimeter for students to
explore? Think of how you and your students could use cardboard
circles and string to explore mathematics! And, if you do branch
off and adapt this lesson for your classroom, don't forget to
let us know how it goes. We would love to hear from you. Have
fun!
Fourth/Fifth Grade Art: String Art and
Geometry  3 (45 minute) sessions required
Power Standard(s):

Makes interdisciplinary connections,
applying art skills and knowledge to improve understanding
in other disciplines

Demonstrates how shape/form can have
radial balance or symmetrical balance

Uses terminology with emphasis on the
elements of art: space, line, shape, form, color, value,
texture

Creates compositions using traditional
and/or contemporary craft methods

Describes how repeated colors, lines,
shapes, forms, or textures can create pattern and show
movement in an artwork
Essential Question(s):

How do I create geometric shapes and
patterns with string?

How can I use my knowledge of a compass
and clock to create a beautiful geometric pattern?
Vocabulary:

geometric vs. organic

circumference

radial

symmetry

triangle, square, pentagon, hexagon, etc.
Instruction:
Day 1:

Review geometric shapes together as a
class.

Show examples of geometric string art.

Discuss the various geometric shapes you
see in the examples. Point out that some of the examples
have 8 notches (like a compass) around the circumference and
some have 12 (like a clock). Count the notches together and
discuss vocabulary terms such as circumference and radial
symmetry.

Give each child a piece of paper with 46
circles on it, some should be marked like a compass and some
like a clock.

Demonstrate how to generate ideas by draw
squares and triangles within the circles (utilizing the dots
marked on the circumference). Students may use colored
pencils to draw the shapes.

Pass out the materials: colored pencils,
rulers

Students will spend 1520
drawing/sketching their design ideas.
Closing of Day 1: Conduct a brief critique of
the work done today. Collect all the work and use the document
camera to show the various patterns that were created today.
Discuss what worked well and what could use improvement. Explain
that we will be recreating our best patterns with tag board and
yarn during the next art session.
Day
2:

Review concepts and terminology from
previous session.

Demonstrate how to create the string art:
Trace the circle stencils on the tag board and mark the
circumference like a compass or clock (student choice). Cut
out notches carefully along the circumference. Tape the
beginning of the yarn ball to the back of the circle and
begin wrapping the yarn in pattern you have chosen. Once the
shape or pattern has been achieved, cut the yarn and tape
the end piece to the back. Shapes and patterns can be
layered and multiple colors may be used.

Pass out the materials: scissors, sharpie
markers, tag board, circle stencils, (yarn and tape, if time
allows)

Students will cut out 2 circles, mark
them, and cut out notches. (This may take up to 20 minutes.)
Closing Day 2: Discuss our goals for the next
session and any problems we may have encountered during today's
session.
Day 3:

Review the steps and procedure to create
the string art.

Pass out materials: yarn, tape, scissors,
(sharpie markers, tag board, and circle stencils if
necessary)

Students will continue to create their
compass/clock based designs with tag board and string. By
the end of this class period, students should have at least
2 string art designs to display or take home.

*The 12 pointed star shape is simple to
do, but may require a video tutorial (possibly a separate
lesson on a separate day). The teacher may choose to
demonstrate the steps under the document camera and the
students can follow along step by step.
Closing of Day 3: Conduct a brief critique of
student artwork. Which designs catch your eye and why? Does
craftsmanship play a role in the success of the work? Do you
feel like you have a better understanding of geometric shapes
and patterns? Do you feel like you have a better understanding
of the markings on a compass or a clock?
Differentiation:
Struggling scholars will start with simple
shapes and practice until they feel comfortable. Advanced
scholars will explore more complex patterns, possibly using more
than one color string to enhance the design.
Technology/Resources:
smartboard, laptop, and document camera will
be used for instructional purposes
Formative Assessment:
Verbal assessment in the form of questions
and discussion.
Visual assessment of sketches and artwork.
Summative Assessment:
Scholars will be assessed based on a standard
rubric, which takes into consideration the goals met (Did the
student create at least 4 sketches? Did the student create at
least 2 string art designs?) and craftsmanship.

Sun Hong is a fulltime art teacher at Bryant
Elementary School in Mableton, GA. She has finished
her fifth year teaching art K5. She received her
bachelor's degree from the University of Georgia in
Art Education, with a focus on drawing and painting.
Her goal as elementary art teacher is to expose
students to a wide variety of art media, techniques,
styles, and subjects. Choice and critical thinking
are important elements she strives to include in
every art project she teaches. You can
follow her on Twitter at @mshongsclass. 
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GCTM would like to
congratulate 2017's award and grant recipients. As a GCTM
tradition, these recipients were recognized at the 2017 GMC.
Click here to download a Powerpoint of the GCTM 2017 awards
presentation.
Do you know a teacher of mathematics that
goes above and beyond their job description to assure their
students are successful? Now is the perfect time to stop and
recommend this person for a welldeserved GCTM honor/award.
The rules for making a nomination make it easier than ever to
submit the name of a special educator that truly makes a
difference in the lives of their students for a GCTM
honor/award. No longer does the person making the nomination
need to be a member of GCTM, except in the case of the Gladys M.
Thomason Award. This means any teacher, coach, administrator,
parent, or student is now eligible to submit a fabulous
candidate for any of the other appropriate honors/awards.
The deadline for
nominations for the following awards is Labor Day of the current
year.
Gladys M. Thomason Award for Distinguished Service
Each year, GCTM selects one outstanding individual as the
Gladys M. Thomason Award winner. Selection is based on
distinguished service in the field of mathematics education
at the local, regional, and state levels. To be eligible for
the award, the nominee must be a member of GCTM and NCTM; be
fully certified in mathematics, elementary or middle grades
education at the fourth year level or beyond  or if the
nominee is a college professor, be at least an assistant
professor; and have had at least five years teaching or
supervisory experience in mathematics or mathematical
education in Georgia.
Dwight Love Award
This award is presented to a teacher in Georgia who models
excellence in the profession and in life and gives much to
others beyond the classroom as mentor, teacher and leader.
The awardee is a master teacher, professionally active, and
promotes GCTM and its mission.
John Neff Award
This award is presented to a member of GCTM who demonstrates
excellence as a full time post secondary educator and/or
district supervisor. The recipient is someone who is an
inspirer, a mentor, and an advocate of mathematics and
mathematics education.
Awards for Excellence in the Teaching of Mathematics
Three awards, one each for elementary, middle, and secondary
levels, are given to excellent teachers who have strong
content foundations in mathematics appropriate for their
teaching level, show evidence of growth in the teaching of
mathematics, and show evidence of professional involvement
in GCTM and NCTM.
Teacher of Promise Award
GCTM recognizes one outstanding new teacher/ member in the
state each year who has no more than 3 years experience at
the time of the nomination and who demonstrates qualities of
excellence in the teaching of mathematics.
GRANTS
Do you have marvelous
ideas for activities and lessons for your students, but just
do not have the materials to implement them in your
classroom because there is no money available through your
school, system, or PTA? GCTM can help!
Be sure to make your
rationale simple for those voting on your grant to
understand the purpose of your lesson, why you need the
items you are requesting, and why you need help with
funding. GCTM wants to help YOU!
Click here to find out more!
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Before we know it, mathematical
educators across the state will be making their trek to Rock Eagle
for the Georgia Mathematics Conference. GMC 2018 will host inspiring
speakers and offer engaging sessions to help refresh and renew your
professional perspective this Fall. Click on the image below to
learn more about this year's keynote speakers. There will be so much
to explore, that you will likely want to come for all three days.
Check out the
Conference Overview.
Here are just a few images from our GMC
2018 Scrapbook.
Why don't you join us? Better yet, consider being
a speaker. Complete the
speaker form present your own lesson. We can't want to see you!
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Here,
you will find a PDF of
Matt Larson's presentation from the most recent NCTM conference. Within this presentation, he introduces new
board members, presents the NCTM's Strategic Framework, and
discusses NCTM's major initiatives.
Among these initiatives, NCTM is responding to the suggestions
of its members to simplify the membership process while making
it more affordable. As such, NCTM will offer the following
membership plans.
Essential
$89  Membership with one gradeband journal (print or digital)
and its archive, and a 20% discount for the online bookstore and
meeting registrations. MyNCTM, Illuminations, and Problems of
the Week are now member benefits.
Introductory
Firsttime members get the Essential for $59.
Premium
$139  full access to Essential level benefits plus all journals
and JRME, print or digital ( a $250 value) as well as all
journal archives, and a 30% discount for online bookstore and
meeting registrations. One free ebook annually available upon
renewal.
Students, Emeritus, and Life members
Get the Premium level for $49
NCTM continues to recommend that
"High School mathematics should discontinue the practice of tracking
teachers as well as the practice of tracking students into
qualitatively different or deadend course pathways." Therefore,
encourage this change, NCTM also offers a plethora or resources to
address making mathematics more accessible, promoting equity in the
classroom, and empowering students through mathematics.
Your NCTM membership includes
access to resources, new networking opportunities via email and
discussion boards, and chances to professionally grow through
various national conferences. Continue renewing your membership
today!
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Summer Means Renewal
As
we move into a slower pace and some personal time, being a teacher
also means a time of renewal. Teacher summers are always busy, even
without the daily planning for classes. You might find yourself
taking graduate courses, planning for the upcoming year, or revising
lessons used in the past. It is still a busy time, but one you can
control better than during the academic year.
As you RENEW, please put GCTM
RENEWAL on your list! $20 will continue to bring you the benefits of
GCTM and enrich you and your students. Think of how often you go
through a drivethrough and spend $10 or more. GCTM uses your
membership fees VERY wisely, and we appreciate your loyalty. We need
you! Drive through to GCTM today!
Our membership count stands at 1245!  There
are 457 lapsed members in 2017.
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President 
Bonnie Angel
President Elect 
Denise Huddleston
VP for Advocacy 
Brian Lack
VP for Competitions

Chuck Garner
VP for Constitution
and Policy  Joy Darley
VP Honors and Awards
 LaTonya Mitchell
VP for Regional
Services  Kristi Caissie
Membership Director

Susan Craig
Executive Director 
Tom Ottinger
Conference Board
Chair  Nikita Patterson
Treasurer
 Nickey Ice
NCTM Representative

Michelle Mikes
Secretary 
Kim Conley
REFLECTIONS Editor 
Becky Gammill
IT Director 
Paul Oser
IT Intern 
Bill Shillito

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Table of Contents
President's Message  by Bonnie
Angel, GCTM President
Advocacy Update
 by T.J. Kaplan, Advocacy
GCTM Middle
School Math Tournament News  by
Chuck Garner, VP for Competitions
GCTM State Math
Tournament News  by
Chuck Garner, VP for Competitions
GCTM Summer Mathematics
Academies
Have You Ever Had Food
Poisoning?  by Ashley Cody and Michelle Mikes
Fraction as Operator:
Not Forgotten by
Ha Nguyen, Heidi Eisenreich, Eryn Stehr, and Tuyin An
Why Hate Math? by
Sybil E. Coley, Retired Math Teacher
Fourth/Fifth Grade Art: String Art and Geometry by
Sun Hong
Awards and Grants
GMC Update
NCTM
Report  by Michelle Mikes
GCTM Membership Report  by
Susan Craig, Membership Director
GCTM Executive Board
