CCSS-M: Non-negotiable Skills at the Middle School: Grades Seven and Eight CCSS-M: Non-negotiable Skills at the Middle School: Grades Seven and Eight

• thinking about variables as measurable quantities that change as the situations in which they occur change;
• understanding that variables are not usually significant by themselves but only in relation to other variables and context.
And finally, apart from considering algebra as generalized arithmetic, students should see algebra as a system with a basis in the concept of relations and functions and understand that the most useful algebraic idea for thinking about relations is the concept of function. This helps students to relate one set of representations, ideas, and properties to another and their relationships (linguistic expression, iconic, tabular, graphical/spatial, equation and systems of equations, quantitative/abstract). For example, the typical relation among two or more varying quantities may look like:
• As time passes, the depth of water in a tidal pool increases and decreases in a periodic pattern.
• As bank savings rates increase, the interest earned on a fixed monthly deposit also increases, but when the interest earned is compounded, the new amount increases exponentially.
• In a sequence of squares having sides 1, 2, 3, 4, 5, …,n, …, the areas of those squares are 1, 4, 9, 16, 25, …,n2, …and the perimeters are 4, 8, 12, 16, 20, …., 4n,….
• For any rectangle of base b and height h, the perimeter p is 2b + 2h.
Students should know the difference between a conjecture, definition, and a theorem. They should know the difference between proof, example, and counter example; between direct and indirect proof; between justifying and providing a counter example, etc. With strong numeracy skills and access to these tools, our students—the future mathematicians, can search for patterns in much the same way that scientists explore results from experiments by systematically manipulating variables. The experimental data of mathematics—calculations are made using appropriate algorithms and tools, and then data and calculations are displayed graphically to reveal patterns, regularities and variations. This data can be sorted and analyzed; and then patterns are observed and inferences are made. Further calculations are made to prove or refute these inferences. They should understand and appreciate that ultimate standard for verification remains a formal proof by reasoning from axiomatic foundations. Further, students should be familiar with and able to use, when necessary and appropriate, computational capabilities of machines—both existing and envisioned. These tools suggest some exciting curricular possibilities. Calculators and computers have a profound effect on students’ understanding of the nature of mathematics. Thus, calculators and computers can be efficient means to generate understanding and interest in algorithms, in particular, and mathematics concepts in general. In this way, the role of tools (calculators, computers, sketchpad, geogebra, software apps, etc.) could be to enhance mathematical thinking rather than detracting from understanding and mastery of arithmetical and algebraic algorithms or just mindlessly learning or applying procedures. Finally, CCCSS-M has a provision for meeting the needs of talented students in middle school. Students with high aptitude and ability in mathematics are offered pre-algebra in the seventh grade as an accelerated course that provides a transition from arithmetic to algebra and a challenging algebra course (see CCSS-M Algebra One course) in eighth grade. CCSS-M recognizes that these students are our future as they are going to invent more mathematics to provide the language for science, technology, and engineering fields. The framers understand all students should be challenged to realize their potential, but these students in particular should be provided the challenging mathematics they are capable of handling.

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Webinars

Tuesday Mathematics Education Webinars (Free)
For teachers, parents, and curriculum coordinators.

By Professor Mahesh Sharma
Assisted by: Sanjay Raghav January 18 8:00 AM US EST

Topic:Trajectory of Multiplication across the grades: Its language, conceptual models, and procedures.

Zoom ID: 5084944608
PC: mathforall