# Mathematics Challenge International Contest Preparation

We are working on strategies to solve challenging logic word-problems leading up to the Noetic Learning Math Contest (NLMC). The NLMC  is a semiannual problem solving contest for elementary and middle school students including tens of thousands of children worldwide. The goal of the competition is to  encourage students' interest in math, to develop their problem solving skills, and to inspire them to excel in math. During the contest, students are given 45 minutes to solve 20 problems. Many problems are designed to challenge students and to enrich their problem solving experiences.

Students take the test at their grade level or one above because the questions are extremely challenging. The tests are so challenging that students who scored 10/20 or 50% on the test can be in the top 10% of all participants. Examples of these questions are as follows:

1st and 2nd Grade: Debbie has 47 cookies and Emily has 25. How many cookies does Debbie need to give Emily so that they have the same number of cookies? Strategy: add 47 + 25 = 72 and divide that by 2 = 36; now we know that Debbie has to give Emily 11 cookies (47 - 36 = 11 or 25 + 11 = 36).

3rd and 4th Grade: Mr. Wyles told his students, “My age is a 2-digit number. If you add the 2 digits, you will get 12. My age is a multiple of 6. If you subtract 1 from my age you will get a multiple of 5.” How old is Mr. Wyles? Strategy: We know that a multiple of 6 is even and a multiple of 3,  so all possible sums of 12 are 8 + 4, 6 + 6, and 4 + 8. If you subtract 1 from 84, 66, and 48, you get 83, 65, and 82, so Mr. Wyles is 65.

5th and 6th Grade:    Six teachers can grade all final tests in 4 hours. If each teacher works at the same rate, how many hours would it have taken for 8 teachers to grade the test? Strategy: at first I thought this was a proportion 6/4 = 8/5.33, but then a student, Zach, pointed out that 6 teachers working for 4 hours each was a total of 24 “man-hours” so if 8 teachers were working at the same rate, it would take them 3 hours each for a total of 24 man-hours. Zach was correct.

One particularly difficult concept for the children was to find the area of a rectangle made up of congruent (equally sized) squares. They are only given the perimeter (outline of the rectangle). For example, if I gave them three squares side by side and told them that the big rectangle had a perimeter of 16 units, I asked them to calculate the area of the large rectangle. In order to find the dimensions of each little square, you have to look at how many equal sections make up the perimeter of the large rectangle (here there are 8 equal sections). So, we take the total perimeter of the large rectangle of 16 and divide it into 8 equal sections or 16 divided by 8; in other words for those students who maintain, “I don’t know division,” I tell them to just count by 8 until they reach 16; yes, exactly 2. So the side length of each small square is 2 so the large rectangle has dimensions of 2 by 6 so the area is 2 x 6 = 12 square units. Another way to approach this is to find the area of each small square (2 x 2 = 4) and since there are three small squares, 4 x 3 = 12 or 4 + 4 + 4 = 12.

I particularly liked the different ways children approached problems and the way they challenged my solutions if they did not agree. When they self grade their tests, they should refer to their first practice score as “x” and look at their next score as “x + 3.” For example, if they scored a 9/20 in the first practice test, their goal is 12/20 for the next. I also encouraged them not to share their scores with other students nor should they compare their scores against others (good luck on that, but …).

There is no concept of failure here. There is only failure to try. Mathletes will get the same number of Mathlete Dollars if they score a 5/20 as a 20/20 as long as they show their work and show their effort. Since I am giving them the answers to check self-grade, if they do not show their work, we have to assume that they copied down the answers, even if they did not. I hate making those kind of assumptions so they should just try to show their work. I did acknowledge that there many be a few problems where they actually will not know how to show work and that is OK.

Our goal is to go over each question on the test on which they struggled and focus on mastery throughout the next months leading up to the Spring contest.

I have attached pdfs for the Grades K-1, 2-3, 3-4, and 5 practice problems with solutions. There may not be time for them to take these tests this week and that is OK. But they should try as many as they can.

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