## Wednesday, May 21, 2014

### NEW Desmos Right Riemann Sum Calculus Investigation of Areas

UPDATED VERSION WITH BORDERS DRAWN BETWEEN RECTANGLES AND DETAILED EXPLANATIONS! The above is an extensive exploration of rectangular approximations to ares under a parabola utilizing the outstanding online graphing calculator, Desmos. CLICK on it to activate. In addition to a detailed overview and suggestions for the instructor, I've included a fleshed-out step-by-step activity for students which reflects my usual balance among the visual, symbolic, verbal and numerical- the Rule of Four... Please comment or use the new Contact Form in the sidebar to let me know if it's working properly and if you find it helpful. Your feedback is crucial to me. Use new contact form at top of right sidebar to contact me directly! If interested in purchasing my NEW 2012 Math Challenge Problem/Quiz book, click on BUY NOW at top of right sidebar. 175 problems divided into 35 quizzes with answers at back and DETAILED SOLUTIONS/STRATEGIES for the 1st 8 quizzes. Suitable for SAT I, Math I/II Subject Tests, Common Core Assessments, Math Contest practice and Daily/Weekly Problems of the Day. Includes multiple choice, case I/II/III type and constructed response items. Price is \$9.95. Secured pdf will be emailed when purchase is verified.

## Monday, May 12, 2014

### Desmos Advanced Algebra Exploration -- Power Functions, Inverses and Graphs

NOTE: CLICKING ON THE GRAPH ABOVE SHOULD NOW LOAD THE DESMOS activity...

Hope you enjoy this new Common Core Investigation...

Students will examine the relationships among the graphs of y = kx^n, the inverse x = ky^n and their graphs. Beginning with particular values k=2, n=2 students will observe how the graphs are reflection images of each other over the line y=x. They will be asked to observe how the number of points of intersection vary over positive integer values of n, according to whether n is even or odd.

They will then determine the coordinates of these points first by estimating from the graphs, then by obtaining exact values using a system of equations.

Finally, they will use more advanced algebra tools to solve in terms of k and n. Some students will recognize the BIG IDEA that the points of intersection must lie on the graph of y=x, therefore the algebra is simplified by using y=x to replace x=ky^n when solving. This is crucial.

Thus, there is a blend of discovery and application of exponent skills. This is to me is the best use of technology - to enhance not replace instruction.

Desmos empowers the student to probe deep beneath the surface but the teacher must carefully plan and guide this process, otherwise many students will make pretty graphs and not get beyond moving sliders left and right. My opinion of course...

Desmos is a powerful teaching/learning tool because it enables students to discover important mathematical relationships and formulate key concepts for themselves. However, it is the expertise of the instructor which will determine WHAT they are learning. This is GUIDED self-discovery!

In this activity I included a detailed overview and guide for the instructor but I left it to the professional to tailor the investigation to the students and the curriculum. In other words, I did not provide a student worksheet. I encourage the professional to modify as he/she sees fit.

Your feedback is very important to me as I continue to develop these. Feel free to comment below or contact me directly using the new Blogger contact form. Also follow me on Twitter @dmarain.

Use new contact form at top of right sidebar to contact me directly!

If interested in purchasing my NEW 2012 Math Challenge Problem/Quiz book, click on BUY NOW at top of right sidebar. 175 problems divided into 35 quizzes with answers at back and DETAILED SOLUTIONS/STRATEGIES for the 1st 8 quizzes. Suitable for SAT I, Math I/II Subject Tests, Common Core Assessments, Math Contest practice and Daily/Weekly Problems of the Day. Includes multiple choice, case I/II/III type and constructed response items. Price is \$9.95. Secured pdf will be emailed when purchase is verified.

## Monday, May 5, 2014

Detailed investigation with extensive background notes for instructor and step by step outline for students to follow. Students will be asked to use a slider to approximate the position of a tangent line of slope -1 to a circle centered at (0,0). The tangent line, x+y=k, requires use of a parameter.

Students will begin with a particular radius, 3, then solve a linear-quadratic system to determine the exact equation of one of the tangent lines. They will also be asked to enter an expression for the other tangent line of slope -1 using the same parameter k. After approximating the locations of similar tangent lines for other radii, they will be asked to solve a general system using radius r.

There are different systems offered to the instructor, depending on the sophistication of the student. Finally, a geometric solution is suggested using 45-45-90 triangles.

Use new contact form at top of right sidebar to contact me directly!

If interested in purchasing my NEW 2012 Math Challenge Problem/Quiz book, click on BUY NOW at top of right sidebar. 175 problems divided into 35 quizzes with answers at back and DETAILED SOLUTIONS/STRATEGIES for the 1st 8 quizzes. Suitable for SAT I, Math I/II Subject Tests, Common Core Assessments, Math Contest practice and Daily/Weekly Problems of the Day. Includes multiple choice, case I/II/III type and constructed response items. Price is \$9.95. Secured pdf will be emailed when purchase is verified.