An inequality is a mathematical sentence that shows the relationship between quantities that are not equal. The symbols used are "not equal to" (≠), "less than" (<), and "greater than" (>).  (Learn Alberta)

Learning through problem solving should be the focus of mathematics at all grade levels. When students encounter new situations and respond to questions of the type How would you …? or How could you …?, the problem-solving approach is being modelled. Students develop their own problem-solving strategies by listening to, discussing and trying different strategies.

A problem-solving activity must ask students to determine a way to get from what is known to what is sought. If students have already been given ways to solve the problem, it is not a problem, but practice. A true problem requires students to use prior learnings in new ways and contexts. Problem solving requires and builds depth of conceptual understanding and student engagement.

Problem solving is a powerful teaching tool that fosters multiple, creative and innovative solutions. Creating an environment where students openly look for, and engage in, finding a variety of strategies for solving problems empowers students to explore alternatives and develops confident, cognitive mathematical risk takers.     (Alberta Mathematics Program of Studies)

note: this is emphasized in the Alberta Mathematics Program of Studies and talked about, but it is not explicitly defined in the Program of Studies.   

Mathematical reasoning helps students think logically and make sense of mathematics. Students need to develop confidence in their abilities to reason and justify their mathematical thinking. High-order questions challenge students to think and develop a sense of wonder about mathematics.

Mathematical experiences in and out of the classroom provide opportunities for students to develop their ability to reason. Students can explore and record results, analyze observations, make and test generalizations from patterns, and reach new conclusions by building upon what is already known or assumed to be true.

Reasoning skills allow students to use a logical process to analyze a problem, reach a conclusion and justify or defend that conclusion.

It is difficult to articulate what exactly reasoning is but, drawing on the above, here are some suggestions about what we do when we reason:

• Evaluate situations
• Select problem-solving strategies
• Draw logical conclusions
• Develop solutions
• Describe solutions
• Reflect on solutions

This list is not exhaustive   (University of Cambridge NRICH project)