# MATCH-UP

The Mathematics Challenge for Undergraduate Prospects (MATCH-UP) is a mathematics competition hosted by Sterling College, located in Sterling, Kansas. The competition hopes to challenge high school scholars to creatively solve mathematics problems and find great joy in the problem-solving experience.

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**Note to Teachers and Faculty**

Many of the mathematics problems contained in the MATCH-UP competition are challenging problems. However, we encourage all teachers and faculty to give your student(s) an earnest chance to play with each of the problems and see what patterns or insight can be gained.

Mathematics is often a messy stop-and-start process with bursts of inspiration scattered between long bouts of frustration and confusion. There is great power in these “Aha!” moments, and we hope these questions can create that for your students. This competition gives students (and teachers) a look at delving into and playing with some particularly interesting and innovative questions that will stretch their minds.

While there may be long periods of frustration, and it may seem as if no one is able to answer any of the questions, we deeply express the desire for you to give it a chance and encourage your students to persevere! Whether you give it for individuals to work on solo or decide to work on it as a class, we believe that you and your students will learn to appreciate the true depths of mathematics, love its complexity, and utilize creativity that is often unexplored at the high school level.

Thanks for being willing to compete! Best of luck!

**Rules and Guidelines**

**Who Can Compete?**

All students in grades 9-12 in Kansas schools are eligible to play. During Round 1, students can work individually or in groups. If you’d like, you can even have entire classes work together! We actively encourage group collaboration during Round 1 and using any resources at your disposal. Of course, seniors are especially encouraged to take part as a taste of a challenging mathematics pathway in college!

You can submit each attempt as an individual or as a group (such as a class).

**Time and Location:**

The competition will begin on Monday, November 16, 2020 and all entries to the first round must be submitted by 11:59 PM on Friday, January 15, 2021. The second round will commence afterward.

The competition will be hosted entirely online at www.sterling.edu/academics/matchup. Sign-ups also occur on the website and require a quick registration process. The cost is free.

**Rules (Round 1):**

On the website above, on the starting date mentioned above, the competition form will open. Round 1 consists of 12 primarily fill-in-the-blank questions. There is no time limit on each question, and outside resources and collaboration is actively encouraged. (Note: Teachers please only serve in a facilitating manner; do not give answers. Additionally, only 9th-12th grade participants are allowed to collaborate on the problems.)

You may submit answers for Round 1 as many times as you’d like. After each submission, you will be told if your submission was correct, but the correct answers will not be revealed. At the conclusion of Round 1, the highest number of problems answered correctly will be your qualifying score. The highest 0.5% or top 5 qualifying scores (whichever is greater) will advance to Round 2.

**Rules (Round 2):**

After the conclusion of Round 1, in the following days, those teams advancing to Round 2 will be contacted. There will be only 4 questions in Round 2, and each one will focus on questions requiring explanatory answers. Submissions in Round 2 will be upload-based or paragraph answers. In general, the questions in Round 2 are expected to be more open-ended and push individuals and groups a bit more than in Round 1.

Round 2 will be open for 14 days, beginning Monday January 25, 2021 through Sunday, February 7, 2021 . Rubrics will be used to grade each question. A link to Round 2 will be sent to qualifying participants. You may submit answers for Round 2 as much as you’d like, but only the last submission will be graded.

**Winners:**

Of those that participated in Round 2, we will award prizes to the top 3 submissions. Prizes include the following:

- A (trophy/plaque) sent, acknowledging the achievement (for 1st place)
- Listing on the MATCH-UP website (for top 3 places)

**Hints:**

During Round 1 of the event, we will offer hints to individuals or teams that need it. Simply use the Get a Hint button on the competition page and fill out the short form. A Hint will then be emailed to you.

**Tie-Breakers:**

In the event that one or more teams are tied in score at the end of Round 1, the following are used as tie-breakers (in order):

- The number of submissions, with a lower number breaking the tie.
- The number of hints given, with a lower number breaking the tie.
- The time of highest qualifying submission, with a quicker time breaking the tie.

**Problem-Solving Strategies**

High school mathematics is often dominated by the “solve for x” type of situation. In this problem, the explanation, execution, and expectations are usually known to the student and teacher. However, real and exciting mathematics is rarely this linear; indeed, genuine problem-solving in mathematics is much more akin to creating a painting or musical composition than a step-by-step algorithm. This short guide will highlight some key strategies that may help you as you attempt to work yourself (or your class) to a resolution of a particularly tricky problem.

**Strategy 1:** Get Your Hands (or Paper) Dirty

This is one of the most important problem-solving strategies. Often, a starting point isn’t clear, and a defined path is not known. However, sitting and staring at the paper will often only end up confusing and intimidating you more! Start “playing” with the math. Write down some numbers. Look at some examples. Try some ideas on it. In some mathematical problems, the best insights are only gained by those who trudge into the deep end and begin playing with it, and only then does the answer begin to reveal itself. Don’t be afraid to get your hands (or paper) dirty!

**Strategy 2: ** Create a mathematical model or picture

Many times, the written explanation of a mathematical problem can be confusing and not very illuminating. Drawing a picture that illustrates the problem, or creating a model of the situation on your paper, can help to visualize what’s going on. The dominant human sense is sight, so any way we can use our eyes to read and solve a problem, the more we can get our neurons cooking on it.

**Strategy 3:** Construct arguments to support your reasoning.

Do you really know the math behind a phenomenon if you can’t support your reasoning? If you have an intuition or gut feeling that you think may be correct, see if you can find mathematical evidence to support your ideas. Intuition can be fooled, but it is often also a powerful tool in your toolbox. Alternatively, being able to construct an argument as a counterpoint to a possible solution is a great way to eliminate possibilities.

**Strategy 4:** Look for patterns or repeated reasoning.

Math often has an underlying reasoning or structure to it; exploit this! Many challenging problems can be made easier if a pattern or repeated reasoning can be found and used. A question that often seems to be impossible or impractical to solve may indeed have a repeating pattern that lights the way to a solution. Patterns are everywhere; you just have to look!

**Strategy 5: **Don’t be afraid to go back to Square One.

One of the biggest enemies people encounter in solving real mathematical problems (not just “solve for x” style problems) are dead-ends. Many times, your prevailing strategy and path leads to a wrong answer or even a brick wall. True mathematicians understand that frustration is actually part of the process, and this productive struggle is a key to really conquering the beast (aka, the problem). Head back to the beginning, retrace what you think worked, and discard what didn’t. Re-evaluate the information in the problem and see if you can try something new. But don’t give up! Frustration is natural and expected. That’s not a sign that you aren’t capable of solving it or that you aren’t meant to get the answer – it means that you’re becoming a real mathematician! Embrace it and enjoy the emotional rollercoaster you may take to create the solution.

**Strategy 6:** Use logical reasoning.

Working in as logical a manner as possible will pay off dividends in the end. Logical reasoning is key for all types of mathematical situations – for instance, if presented with a finite number of choices, eliminating impossible possibilities will steer you closer to victory. And don’t immediately shake off implausible choices – math often has striking and surprising results. In line with logical reasoning, make sure the steps you are taking and the results you are finding are reasonable and make sense: If you are measuring a distance and get a negative number, for instance, that doesn’t make sense, and something is wrong. Make sure every step you make is a logical one. Even if a guess-and-check strategy is called for, strategically guess; don’t just blindly dance around, as that is rarely fruitful.

**Strategy 7:** Take a break.

Sometimes, your brain’s neurons just need some time to settle and finish building neural pathways. It is very often the case that a problem that has you stumped reveals an answer after taking a break and coming back to the problem later. Your sub-conscious can be one of your greatest allies in solving math problems. Real mathematical problem-solving is usually a start-and-stop process, not a continuous exposition in one sitting. Again, get away from the idea of “solve for x” where you are expected to be done in 1 or 2 minutes. In real problem-solving situations, figuring out an answer may take many minutes or hours or even days or more. This is okay! Take your time, and you may well be more successful.

Hopefully these strategies will equip you with some of the tools necessary to become a real mathematician! Enjoy the winding road, the journey that each problem takes you on. With any luck, you may well see that math is so much more than just “solve for x”!

**Sample Lesson Plan**

Sample Lesson Plan Download Option

*This sample lesson plan is designed to be utilized for any of the questions in MATCH-UP and can be used on multiple days. For problems that (potentially) could be solved in sooner than one class period, the entire lesson format could be repeated for a second problem.*

**Standards:** __Common Core State Standards:__ CCSS.MATH.PRACTICE.MP1, CCSS.MATH.PRACTICE.MP2, CCSS.MATH.PRACTICE.MP3, CCSS.MATH.PRACTICE.MP4, CCSS.MATH.PRACTICE.MP5, CCSS.MATH.PRACTICE.MP6, CCSS.MATH.PRACTICE.MP7, CCSS.MATH.PRACTICE.MP8

__2017 Kansas Math Standards:__ N.Q.1, N.Q.2, N.Q.3, N.RN.1, N.RN.3, A.SSE.1, A.SSE.2, A.SSE.3, A.CED.1, A.CED.2, A.CED.3, A.CED.4, F.IF.7, Standards for Mathematical Practice 1-8

**Before/Icebreaker/Bellringer:** Consider any of the following riddles with your class. If you are planning to re-use this lesson plan for multiple days, you might want to use one on Day 1 and use another on Day 2, etc. Answers are given at the bottom of the lesson plan.

- What is the most number of Friday the 13ths that can occur in any given year?
- If one is writing down the digits 1 through 1000, what is the least common digit one will write down?
- For some operation #, if 13#11 = 158, 5#6 = 19, (-4)#3= 13, and (-8) # (-5) = 69, can you find 10 # 10?
- Jan tells only lies while Ben tells only truth. A jar contains 20 Red marbles. Jan and Ben each take a different number of marbles out of the jar and write that number down on a piece of paper, handing both slips of paper to Greg. One slip of paper says “I have 11 marbles.” The other says “I have 9 marbles.” Greg looks and sees 2 marbles left in the jar. How many marbles does each person have?
- Consider a quadratic equation with a, b, and c all equal to some integer n. Can you find the product of the solutions of x?

These problems are meant to get the students’ minds open and thinking mathematically. This will prepare them for the “main course” of the competition problems.

**During/Main Content: ***We do feel that some beneficial time could first be spent on going over with the students the eight Standards for Mathematical Practice that are in the Common Core State Standards and also the 2017 Kansas Math Standards. We feel that these eight concepts are very much highlighted in this competition. *

- Make sense of problems and persevere in solving them.
- Reason abstractly and quantitatively.
- Construct viable arguments and critique the reasoning of others.
- Model with mathematics.
- Use appropriate tools strategically.
- Attend to precision.
- Look for and make use of structure.

Look for and express regularity in repeated reasoning.

At this time, it is encouraged to present the students with one of the 12 questions from the MATCH-UP competition. After presenting the problem, there are several paths one could take at this point, or a combination of all:

- Give students time to think and reason solo. As each is ready to do so, you may look at their work, and if it is insightful or possibly even correct, you may lead the student to present their work to the class.
- Break out into small groups and circulate the room offering help and advice and students go over ideas and strategies. After a sufficient time has passed, re-unite the class and summarize the plots and plans developed. If possible, lead one or more students to present their work.
- Lead the class in a large overall group discussion. Act as the leader, facilitating ideas and hints, while still letting the students ponder and puzzle over the question. Allow students to interject insights and ideas they have, and pursue rabbit holes, as desired, to see if they lead anywhere. After a sufficient time has passed, summarize what has been gleaned, and if possible, lead one or more students to present their work.

*Of course, we know you do not necessarily have the correct answer yourself! While this can be an uncomfortable position as a teacher, we feel there is valuable material to be learned at this juncture. If both you and your students are totally stumped, you may also use the Get A Hint feature on the website. We encourage you to do this rather than give up!*

**After/Closing/Ticket-Out-the-Door: **To close the lesson, give the students a chance to experience formalizing their ideas as mathematicians. Have each student write a paragraph explaining their reasoning or the reasoning of the class in as clear detail as possible, in complete sentences. Make sure they pay attention to their notation and attend to precision and detail. Have the students turn in this explanation as they leave class.

*Of course, as time permits, you may be able to repeat this lesson twice in a class period if the class quickly is able to answer a question (not guaranteed, but possible). In this case, we recommend tackling a second question from the competition.*

*In all, the competition comprises 12 questions in Round 1. We highly encourage individual students who are interested or entire classes that are interested to submit their answers as often as possible – the highest of your scores is what will be taken, so there’s no harm in many submissions. Try it out, play with it, and have fun learning to be mathematicians!*

*Answers to riddles: 1) 3; 2) 0; 3) Most natural answer is 90; 4) Ben – 11, Jan – 7; 5) 1*

**Frequently Asked Questions**

**1. What is the cost to participate in the MATCH-UP Competition? **

There is absolutely no cost at all! What’s better – the entire competition is held online, untimed, over a period of weeks, so you have as much opportunity as possible to participate as individuals or classes.

**2. Am I allowed to use any resource?**

Great question! We encourage the use of other participants and using your resources to adequately answer each question. We do ask, however, that help from teachers remains limited, and that communication about competition questions only occurs between other participants.

**3. Can I submit as many times as I want?**

Yes, although you should be aware that submission count is one of the tiebreaker criteria. Therefore, a person/team submitting twice and scoring the same as someone else who submitted 18 times has the advantage in a tiebreaker. Your highest qualifying submission will be accepted in Round 1, so there’s no harm in submitting multiple times, beyond the aforementioned caveat.

**4. Do I have to register for an account to compete?**

Yes. Registering for an account will give you/your team a User Code. This User Code is necessary to Submit Answers and to Get A Hint.

**5. Can we submit one answer at a time to see if it’s correct?**

Unfortunately, this feature is not supported by the software we are using. You will need to complete an entire submission before getting the score back for any of the questions.

**6. Will I see correct answers after submitting?**

No, you will only receive information about how many questions you answered correctly. This way, you can submit again to get a better score. After Round 1 has concluded, you may send an email to receive correct answers.

**Contact us at sterlingmatchup@gmail.com with questions not answered elsewhere! Please allow up to 24 hours for a response.**

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