Vol X

No. 2


Photo credit - Rebecca Gammill
Bowl by Baltic by Design

President's Message
by Bonnie Angel, GCTM President

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 To-Do 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 research-based 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 nationally-recognized 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 grade-level 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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Advocacy Update
T.J. Kaplan, Advocacy

2018 Legislative Session Report

In the early hours of the morning on March 30th, 2018 the Georgia General Assembly completed its 40-day 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 line-item 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 decision-makers 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 (R-Rome) 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 (R-McDonough); Senator Josh McKoon (R-Columbus); Senate Higher Education Chairman Fran Millar (R-Atlanta); Senator Larry Walker III (R-Perry); Senator Ed Harbison (D-Columbus); Senator Kay Kirkpatrick (R-Marietta); Senate Minority Leader Steve Henson (D-Tucker); Senator Blake Tillery (R-Vidailia); Senator Horecena Tate (D-Atlanta); Senator Matt Brass (R-Newnan); Senator John Albers (R-Alpharetta); Senator Chuck Payne (R-Dalton); Rep. Wes Cantrell (R-Woodstock); Rep. Debbie Buckner (D-Junction City); Rep. Tim Barr (R-Lawrenceville); Rep. Buzz Brockway (R-Lawrenceville); 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.


HB 787: Rep. Scott Hilton (R-Peachtree 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 per-student local funding of the five poorest school systems in the state. HB 787 would increase the supplement: 1) to the state average per-student 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 Process--this 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 K-12 schools (primary/secondary public schools and local school systems) in Georgia.

SB 330:Senator John Wilkinson (R-Toccoa) 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 three-component model of school-based agricultural education, a pilot in a minimum of six public elementary schools across six regions beginning in the school year 2019-2020. 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 Deal--lead 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 (R-Marietta), seeks to provide for the establishment of an innovative assessment pilot program for local school districts. Specifically, the bill states that beginning with the 2018-2019 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 real-time 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 ten-district 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 (R-Atlanta) 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 college--to 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 funds--no 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 line-item 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 15-credit 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 full-time 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 in-demand 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 non-STEM AP exam fee for low-income students. Additionally, budget instructions were added that direct GOSA to increase existing grant funds for birth-to-five literacy/numeracy in rural centers located in the lowest performing K-12 school districts--this 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 K-12 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 (R-Warner Robbins) HC: Higher Education Did not pass Service cancellable loan program for STEM
HB 728 Rep. Brooks Coleman (R-Duluth) 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 (R-LaGrange) HC: Education

SC: Education and Youth

On Governor's desk Revises expulsion/suspension protocol
HB 763 Rep. Randy Nix (R-LaGrange) HC: Education

SC: Education & Youth
On Governor's desk Expands student protocol committees to include school climate
HB 778 Rep. Terry England (R-Auburn) HC: Higher Education Did not pass Transfer CTAE from DOE to TSCG
HB 781 Rep. Kevin Tanner (R-Dawsonville) HC: Education Did not pass Adds maintenance and operations to E-SPLOST
HB 787 Rep. Scott Hilton (R-Peacthree Corners) HC: Education

SC: Education & Youth
On Governor's desk Charter Schools substantive revisions in Title 20
HR 1039 Rep. Dave Belton (R-Buckhead) HC: Special Rules Did not pass Study Committee on Ennobling the Teaching Profession
HR 1162 Rep. Brooks Coleman (R-Duluth) HC: Education House Passed/Adopted Study Committee on State Accreditation Process
SB 3 Sen, Lindsey Tippins (R-Marietta) SC: Education and Youth

HC: Education
On Governor's desk CONNECT Act--to enhance industry credentialing for some programs in high school
SB 330 Sen. John Wilkinson (R-Toccoa) SC: Ag and Consumer Affairs

HC: Education
Signed by Governor Nathan Deal Agricultural education pilot
SB 362 Sen. Lindsey Tippins (R-Marietta) SC: Education

HC: Education
On Governor's desk Local district innovative assessment pilots
SB 377 Sen. Brian Strickland (R-McDonough) SC: Higher Education

HC: Industry and Labor
Signed by Governor Nathan Deal Transfers State Workforce Development Board to TCSG
SB 401 Sen. Lindsey Tippins (R-Marietta) SC: Education and Youth

HC: Education
Did not pass Flexibility for guidance counselors to develop career assessment plans
SB 405 Sen. Fran Millar (R-Atlanta) SC: Higher Education

HC: Higher Education
On Governor's desk--added to HB 787 Creates low-income grants for students attending USG institutions
SR 1068 Sen. Steve Gooch (R-Dahlonega) 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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GCTM Middle School Math Tournament News
by Chuck Garner, VP for Competitions

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 multiple-choice test with a 45-minute time limit; 10 individual ciphering problems, each problem with a two-minute time limit; 3 rounds of four pair ciphering problems (in which students from a school formed teams of two), each round with a four-minute time limit; and a four-person team "power question," in which the team solves a complex problem with a 10-minute 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.


  1. South Forsyth Middle School, Cumming

  2. Fulton Science Academy, Alpharetta

  3. Tattnall Square Academy, Macon

  4. Southeast Bulloch Middle School, Brooklet

  5. Stratford Academy, Macon

Sixty-one students from twelve schools participated. Sponsors that are members of GCTM only had to pay a $10 registration fee or submit five multiple-choice questions for possible inclusion in a future tournament. The next GCTM middle school tournament is scheduled for April 20, 2019.


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GCTM State Math Tournament News
by Chuck Garner, VP for Competitions

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 2017-2018 school year. Thirty-six 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 try-out for the state-wide Georgia ARML team, making a total of 162 participants.


The tournament consisted of a very challenging written test of 45 multiple-choice questions and 5 free-response questions with a 90-minute time limit; 10 individual ciphering problems, each problem with a two-minute 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 T-shirt.
The top five teams and the top fifteen individuals are listed below.


Fulton Science Academy
Walton High School
Kennesaw Mountain High School
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.

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
Non-GHSA Class: Eureka Scholastic Academy


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

  2. Holden Watson, Fulton Science Academy (second consecutive year)

  3. Shawn Im, Peachtree Ridge High School

  4. Anup Bottu, Westminster

  5. Vishaal Ram, Milton High School

  6. Ramanan Abeyakaran, Chamblee High School

  7. Russel Emerline, Walton High School

  8. Daniel Shu, Walton High School

  9. Alex Eldridge, Dunwoody High School

  10. Cade Lautenbacher, Dunwoody High School

  11. Charlie Furniss, Fulton Science Academy

  12. Aaron Yu, Westminster

  13. Ty Bondurant, Columbus High School

  14. Jayson Wu, Walton High School

  15. Johnny Fang, Westminster

An item analysis of the competition problems was completed at the state tournament. The responses analyzed included the 45 multiple-choice 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 multiple-choice problem and problem 45 the most difficult multiple-choice 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 (b--ln(c))/a, where a, b, and c are integers. Compute a + b--c.
a) 3 b) 4 c) 5 d) 8 e) 10

Test Problem #45: Consider the following array.

For integers i3 1 and j
3 1, define Di,i = DI-1,j+1--Di-1,j.
If D0,j is the jth Fibonacci number, find S5i=0 [(5Ci) Di,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 (3--ln(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 D0,6 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 debx(y) to be equal to z when yz = 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 16x2 + 4y2--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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GCTM Summer Mathematics Academies


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Have You Ever Had Food Poisoning?
by Ashley Clody and Michelle Mikes


Have you ever wondered whether what you ate caused you to have food poisoning? The CDC uses two-way 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 two-way tables. We adapted our presentation from a STEM conference session that we attended led Dr. Ralph Cordell. This lesson was designed for 8th-grade students to learn how ratios, two-way 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 two-way 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)













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 two-way tables. Further, high school students could analyze two-way tables with joint and marginal frequencies, statistical significance, confidence intervals for estimates, p-values, 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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Fraction as Operator: Not Forgotten
Ha Nguyen, Heidi Eisenreich, Eryn Stehr, and Tuyin An

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 7th-grade students in a class of K-8 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 problem-solving. Selected student strategies for each task show a variety of approaches utilized by students in their problem-solving.

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 already-reduced 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 problem-solving 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: part-whole, measurement, division, operator, and ratio (Behr et al., 1983). While the part-whole 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 3-for-4 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 two-thirds 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), two-thirds of three miles is interpreted as two copies of one-third of three miles, or equivalently 2 x [1/3 x 3 miles]. In the second approach (Figure 5), the operation is interpreted as one-third 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 marked-up/marked-down 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.

Student 3 Responses Student 1 Responses

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 7. Student strategies to Task 3


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 problem-solving. 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 problem-solving 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 problem-solving strategies through discourse supports the type of deeper understanding that we want our students to have.


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. 91-125. Academic Press, New York.

Behr, M. J., Harel, G., Post, T. R. & Lesh, R. (1993). "Rational numbers: Toward a semantic analysis--emphasis on the operator construct." In T. P. Carpenter, E. Fennema & T. A. Romberg (Eds.), Rational numbers: An integration of research, pp. 13-47. 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. 15-22. 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), 373-380.

Hackenberg, A. & Lee, M. (2012). Pre-fractional 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), 18-25.

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 schema-based approach." In T.P. Carpenter, E. Fennema and T.A. Romberg (eds.), Rational Numbers: An Integration of Research, pp. 261-288. 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. 327-61. Hillsdale, NJ: Lawrence Erlbaum Associates.

Siebert, D., & Gaskin, N. (2006). Creating, naming, and justifying fractions. Teaching Children Mathematics, 12(8): 394-400.

Usiskin, Z. (2007). Some thoughts about fractions. Mathematics Teaching in the Middle School, 12(7): 370-373.

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 decision-making 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 pre-service 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 2018-2019.

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Why Hate Math?
by Sybil E. Coley,
Retired Math Teacher

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 un-necessary 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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Fourth/Fifth Grade Art: String Art and Geometry - 3 (45 minute) sessions required
Sun Hong

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):

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

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

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

  4. Creates compositions using traditional and/or contemporary craft methods

  5. Describes how repeated colors, lines, shapes, forms, or textures can create pattern and show movement in an artwork

Essential Question(s):

  1. How do I create geometric shapes and patterns with string?

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


  • geometric vs. organic

  • circumference

  • radial

  • symmetry

  • triangle, square, pentagon, hexagon, etc.


Day 1:

  1. Review geometric shapes together as a class.

  2. Show examples of geometric string art.

  3. 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.

  4. Give each child a piece of paper with 4-6 circles on it, some should be marked like a compass and some like a clock.

  5. 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.

  6. Pass out the materials: colored pencils, rulers

  7. Students will spend 15-20 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:

  1. Review concepts and terminology from previous session.

  2. 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.

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

  4. 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:

  1. Review the steps and procedure to create the string art.

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

  3. 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.

  4. *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?


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.


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 full-time art teacher at Bryant Elementary School in Mableton, GA. She has finished her fifth year teaching art K-5. 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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Awards and Grants

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 well-deserved 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.


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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GMC Update

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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NCTM Report
by Michelle Mikes,
NCTM Representative

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.

$89 - Membership with one grade-band 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.

First-time members get the Essential for $59.

$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 e-book 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 dead-end 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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GCTM Membership Report
by Susan Craig, Membership Director

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 drive-through 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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GCTM Executive Board

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


Georgia Council of Teachers of Mathematics | PO Box 683905, Marietta, GA 30068