# Enhance Your Mathematical Skills Through Our Exercises

Complementary Mathematics / Problem Solving

## Problem Solving – Applying and Reasoning

The realm of problem-solving spans a gamut that includes multiple-choice items (and other types of student-selected answers) as well as short traditional/routine questions, extended response items, open-ended questions, surveys, portfolio entries, projects, presentations, and various other student-generated response scenarios.

The National Research Council’s 5-strand proficiency model focuses on two aspects of the problem-solving process: Strand (3) “Applying: Being able to formulate problems mathematically and to devise strategies for solving them using concepts and procedures appropriately” and Strand (4) “Reasoning: Using logic to explain and justify a solution to a problem or to extend from something known to something not yet known.”

We have referenced some of our favorite problem-solving resources and websites using the grade level groupings of primary (grades K-2), elementary (grades 3-5), and middle (grades 6-8). One of our favorite sources of challenging and unique problems comes from the world of math competitions/challenges. Please note that we are sharing these sources, not for the competitive aspect (although you may want to offer that option to interested students), but solely for the quality, variety, and diversity of the problems presented.

### Each Problem, From Any of the Resources Cited

Can be individually isolated and used as an opportunity to analyze the mathematics involved, explain the approach taken to obtain the solution, as well as present a justification of the solution itself. Another beneficial exercise for both teachers and students is to scan a page of these contest problems and identify those items which address the topic of mathematics currently being studied and/or to classify the topic(s) of mathematics being addressed across the set of contest questions.

As a quick example of this type of resource, we have provided a link to the free sample section of The Math League—a long-time favorite of ours for use in grades 4-8. The questions get progressively more difficult as one proceeds through the contest, thereby allowing for some differentiation within the 3-page format when items are used as a single question assignment option.