#### Mathematical Deductive Reasoning Problems

###### These deductive reasoning problems connect with Arizona Math Standards in the areas of Algebraic Thinking, Number Operations in Base Ten, and Mathematical Practices. What is the connection with the Golden Rule? In life, we are usually presented with many choices. Circumstances often lead us to narrow those choices to a few, often even to one. In relating to others, choosing to use the Golden Rule as a guide is usually the best choice.

**Many to One problems** allow students to work through a list of clues and perform math operations correctly to select a number which answers the given clue. Initially, a great many numbers are correct answers. Each clue reduces the group of correct options. Eventually, mathematical reasoning eliminates all but one possible correct solution. The student’s first goal is to seek a range of correct answers. Secondly, they recognize which possible number is the only correct answer. The third objective is to identify the ‘Golden Clue’ which eliminated all other possible correct responses.

**Procedure: **

- Scratch paper, a pencil, and critical thinking skills are the only materials needed.

- Students can use quarter sheets of scrap paper or notebook paper, numbering down from 1 to 10.

- They are to write a number which gives a correct answer for the clue given.

- When the next clue is given, students whose answer remains correct make no change in their selected number. If the answer becomes incorrect, students simply change their answer to a correct number. In numbers with 2 or more digits, students may change one digit to make their answer correct. Very seldom is it necessary to change more than one digit, and keeping changes to a minimum utilizes the best mental math reasoning.

- The teacher polls the students for correct answers during the early clues to establish a range of correct possibilities. As they progress through the clues,possibilities are eliminated until only one choice remains.

**The second part of the activity** is the identification of the clue that ruled out all possibilities but one. **Students circle this ‘Golden Clue.’ ** Explaining the rationale is a good practice for the first few times to build and check deductive reasoning skills.

**Grading is easy. ** The teacher just walks around the room and uses a checklist to mark those with the correct solution. One point is given for the correct answer (which all students should find by clue 10) and a second point is given for determining the Golden Clue (a more difficult task).

**Modifications:** Give a visual clue for young students by making a list of the options and having them cross off the numerals as they are eliminated. This strengthens the

concept of number structure. Having students work with a partner may help and underscores the concept of AGREEment. The rationale for the solutions is included and may be used to help students understand the process.