Written By: Brian Devine
If no cuts on a pizza yields one slice, and one cut on a pizza yields two slices, and two cuts on a pizza yields four slices, what is the maximum number of discretely sized slices of pizza that are possible for ten cuts?
Does that paragraph above scare you? In over 35 years of teaching mainstream secondary and undergraduate mathematics, and especially over the last 10 years of working with Mr. Hamilton and within his mission of holistic enrichment, I have found that the one thing most all math students fear deeply, above all else, is the dreaded word problem.
A problem such as the one stated above can be a student’s worst nightmare: All one sentence, but a paragraph long; all words, even the numbers are words; looks easy but…slightly ambiguous wording, and in a strange cadence… does slice mean to cut or a piece, is it a verb or noun?!
I know well the reactions to this brief initial sentence/paragraph because I have used this same problem, or ones terribly similar in design, as an educational diagnostic tool for over 45 years. It scares people. And yes, this problem was designed. I cannot claim authorship. It has been in use in math education for over 2500 years. As the resident “Math Doctor” at Hamilton Education, I am familiar with the issues our students have with ambiguously worded problems via their experiences in SAT/ACT and AP Calculus prep. I got a cure for that! I always present them with a recipe for success, just as my dear father told it to me, when I, too, was afraid of the dreaded word problem. He said that:
“Solving any problem is just like milking a cow!”
And to milk a cow, you must properly manipulate the udder to receive a life-giving solution. Thus, I give you The Udder Method!
The UDDER Method is a recipe, an algorithm, or the key steps when attempting any word problem … be it in mathematics, the sciences, or the humanities. UDDER is a mnemonic acronym for these key steps:
- Understand the question (?)
- Determine what is given/not given.
- Devise a plan.
- Engage in the plan.
- Reflect and Check.
Now…That is pretty much all of it, right there. How could it be any easier than that? You are now ordained as a full-fledged problem solver. But maybe I should elucidate a bit…
1. Understand the question… by reading the question first!
“Understanding the question” is often neglected as being obvious, escaping initial focus. When one must read through a word pile of information, it is easy to get lost. Some “paragraph problems” leave students reticent to attempt solution due to strange syntax and being buried underneath the words. To remedy this mental speedbump, I emphasize the importance of reading the question first. Avoid the perceived mud-puddle of confusion. By reading the question first, you get a big hint at what to do and what to find without having to stress at the start. I often proffer a quick method of self-prompting to ensure understanding:
- Do you understand all the words in the question?
- What are you asked to find or show?
- Can you restate the problem in your own words?
- Can you think of an experience, or draw a diagram that relates to the question?
- Is there enough information to enable you to find a solution?
- Do you need to ask or answer another question to get what you need?
2. Determine what is given … and what is not given
The question gives you an important piece of information: the thing you need to find. Understanding the question first allows you to pick up the information within the wording – the given data – more quickly and to define a variable as dictated by the question – the not given.This is the skill of mathematics: building of an algebraic model via common sense, write an algebraic form, and then manipulate it using long-practiced techniques. Guaranteed success comes from practice and experience.
3.Devise a plan! (… Experience, Fact, Formula, Theorem)
There are many reasonable ways to solve any problem.This is the art of mathematics. Choosing an appropriate and efficient strategy is best learned by solving many different types of problems. There is no one “silver-bullet” strategy; there are many strategies that generate a correct solution. Again, practice is the key. You will find choosing a strategy increasingly easy the more you use the UDDER recipe.
A partial list of strategies include:
- Make an orderly list of given information; Look for a pattern.
- Draw a picture.
- Use direct reasoning.
- Remember an applicable formula; Use a formula.
- Solve a simpler problem. Use a familiar model.
- Guess and check.
- Eliminate least likely possibilities.
- Work backwards/reverse engineer.
- Be creative: apply any or all these paths
4. Engage in the plan
This step is usually easier than devising a plan. Persist with the plan as often you will be on the right path via experience. If your plan reaches impasse, adjust it, or discard it and choose another. This feedback loop is important in developing confidence through persistence. This is how mathematics is done. It is a game of discovery each time; and with practice and an attitude of play, you can win far more times than you will lose.
5. Reflect and review/extend the idea
Much can be gained by taking the time to reflect and look back at what you have done, what worked and what did not, and extending that thinking to other problems where this could be useful. The few seconds you take to reflect and check the solution works will save you years of tears later down the line.
Dad was right when he said that “solving any problem is just like milking a cow!”And like milking a cow it is not easy, at first. Through some dedicated practice is becomes second nature to draw forth a life-sustaining solution What I have learned about problem solving from father, and from every one of the incredible teachers I have experienced – and then try to instill in every student — is that The UDDER method is the true basis of all mathematics.
How to use the problem-solving techniques of pattern recognition, analogy, generalization, induction, variations and similarities, symmetry, decomposition and recombination, reverse engineering, and other heuristic tools… These are valuable life-lessons I teach through the vehicle of mathematics.
You were probably wondering about that pizza problem. Maybe you decidedon 20 slices after 10 cuts through the center. That would be incorrect. The question asked for the maximum slices possible for 10 cuts. The words “discretely-sized” means different size. Cutting so every successive cut traverses all previous cuts, the maximum possible pieces of discrete size can be found. This can be modeled algebraically as a quadratic sequence:
For n cuts, the maximum number of slices possible is found as s(n) = [n(n + 1)]/2 + 1. So, for 10 cuts, you get a maximum of 56 slices of discrete size.
My name is Brian Devine. I am the Senior Mathematics Instructor at Hamilton Education. I am fortunate to be a member of the amazing Hamilton team of hyper-educated experts who go all-out every day to propel students to higher levels of excellence.
