1. List2. The

3. Possibilities

Being thorough and complete without wasting time being redundant is also an important skill in mathematics. Besides promoting organization, learning how to list possibilities efficiently helps us to organize our thinking for other types of math problems, especially those that require us to choose the correct fit (such as in factoring polynomials). The key to doing these problems well is to be systematic in listing the solution.

Problems like these can be created by teachers and also by students.

Sample problems, solutions, and links to other resources for these problems are on the following slides. Note that the students will probably not begin by listing systematically, but they will learn this as they practice and you model the process.

Sample problem and solution: How many ways can we program a candy machine to recognize 25 cents if it can take pennies, nickels, dimes, or quarters?

Solution:Use p for pennies, n for nickels, d for dimes, and q for quarters.1. 1 q 2. 2 d, 1 n3. 2 d, 5 p 4. 1 d, 3 n5. 1 d, 2 n, 5 p 6. 1 d, 1 n, 10 p7. 1 d, 15 p 8. 5 n9. 4 n, 5 p 10. 3 n, 10 p11. 2 n, 15 p 12. 1 n, 20 p13. 25 pSo, there are thirteen ways to do this.Note that the way I solved this was to start with the biggest number of quarters, then the biggest number of dimes, then the biggest number of nickels. I moved them down systematically.

LIST THE POSSIBILITIES

1) LIST ALL THE WAYS YOU CAN MAKE A MEAL WITH ONE MEAT, ONE STARCH, AND ONE VEGETABLE IF YOU HAVE BEEF OR CHICKEN, ROLLS OR POTATOES, AND CARROTS, OR GREEN BEANS.

2) LIST ALL THE THREE DIGIT EVEN NUMBERS YOU CAN HAVE IF THE SECOND DIGIT IS A 3 AND THE NUMBER IS LESS THAN 500.

3) LIST ALL THE POSITIVE INTEGERS THAT DIVIDE 72.

4) FIVE DOTS ARE PLACED ON A PAPER. LIST ALL THE WAYS YOU CAN CONNECT ANY TWO OF THE DOTS

5) BY USING THE LETTERS IN THE WORD ‘MATHEMATICS’, LIST ALL THE WAYS WE CAN MAKE A TWO LETTER COMBINATION WHERE ONE OF THE LETTERS IS A VOWEL AND THE OTHER IS A CONSONANT. ORDER MATTERS, SO ‘MA’ IS NOT THE SAME AS ‘AM’.

6) LIST ALL THE WAYS WE CAN ADD TWO POSITIVE INTEGERS TO GET 15 WHERE THE FIRST NUMBER IS SMALLER THAN THE SECOND ONE.

LIST THE POSSIBILITIES SOLUTIONS

LIST ALL THE WAYS YOU CAN MAKE A MEAL WITH ONE MEAT, ONE STARCH, AND ONE VEGETABLE IF YOU HAVE BEEF OR CHICKEN, ROLLS OR POTATOES, CARROTS, OR GREEN BEANS.

BEEF-ROLLS-CARROTS BEEF-ROLLS-GREEN BEANS BEEF-POTATOES-CARROTS

BEEF-POTATOES-GREEN BEANS CHICKEN-ROLLS-CARROTS CHICKEN-ROLLS-GREEN BEANS

CHICKEN-POTATOES-CARROTS CHICKEN-POTATOES-GREEN BEANS

LIST THE POSSIBILITIES SOLUTIONS

LIST ALL THE THREE DIGIT EVEN NUMBERS YOU CAN HAVE IF THE SECOND DIGIT IS A 3 AND THE NUMBER IS LESS THAN 500.

THE FIRST DIGIT CAN BE 1, 2, 3, OR 4

THE SECOND DIGIT CAN BE 3

THE THIRD DIGIT CAN BE 0, 2, 4, 6, OR 8

130, 132, 134, 136, 138, 230, 232, 234, 236, 238, 330, 332, 334, 336, 338, 430, 432, 434, 436, 438

LIST THE POSSIBILITIES SOLUTIONS

LIST ALL THE POSITIVE INTEGERS THAT DIVIDE 72.

1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

FIVE DOTS ARE PLACED ON A PAPER. LIST ALL THE WAYS YOU CAN CONNECT ANY TWO OF THE DOTS

CALL THE DOTS A, B, C, D, AND E. THE COMBINATIONS WOULD BE: AB, AC, AD, AE, BC, BD, BE, CD, CE, AND DE

LIST THE POSSIBILITIES SOLUTIONS

BY USING THE LETTERS IN THE WORD ‘MATHEMATICS’, LIST ALL THE WAYS WE CAN MAKE A TWO LETTER COMBINATION WHERE ONE OF THE LETTERS IS A VOWEL AND THE OTHER IS A CONSONANT. ORDER MATTERS, SO ‘MA’ IS NOT THE SAME AS ‘AM’.

THE VOWELS ARE A, E, AND I THE CONSONANTS ARE M, T, H, C, AND S

AM, AT, AH, AC, AS, EM, ET, EH, EC, ES, IM, IT, IH, IC, IS, MA, ME, MI, TA, TE, TI, HA, HE, HI, CA, CE, CI, SA, SE, SI

LIST THE POSSIBILITIES SOLUTIONS

LIST ALL THE WAYS WE CAN ADD TWO POSITIVE INTEGERS TO GET 15, WHERE THE FIRST NUMBER IS SMALLER THAN THE SECOND ONE.

1 + 14 2 + 13 3 + 12 4 + 11 5 + 10

6 + 9 7 + 8

MORE LISTING PROBLEMS AND RESOURCES ONLINE

• HTTP://WWW.BASIC-MATHEMATICS.COM/MATH-PROBLEM-SOLVING-STRATEGIES.HTML

• HTTP://WWW.MATHSTORIES.COM/STRATEGIES_MAKEALIST.HTM

• HTTP://WWW.THESINGAPOREMATHS.COM/STRATF.HTML

• HTTP://PRED.BOUN.EDU.TR/PS/PS2.HTML