Showing posts with label Common Core. Show all posts
Showing posts with label Common Core. Show all posts

Friday, August 15, 2014

Never ASS-U-ME in Geometry: A Triangle Problem to Get Them Thinking!



Not quite back to school for most but the problem above might prove interesting to review some geometric/deductive reasoning.

For new geometry students, replace 'a' by a value, say 40, and ask them to fill in all the missing angles. Most should deduce that angle 5 = 50, but my educated guess is that many will assume b = 40, so
angle 5= angle 6 = 50 and angle 3 = angle 4 = 40. From there to angle 1 = angle 2 = 50, so
angle 2 + angle 3 = 90. QED! Not quite...

Well, the '90' is correct but the reasoning is another story! So this is all about justifying, checking validity of mathematical arguments, sorta' like some of the Eight Mathematical Practices of the Common Core!

In fact, you might ask them to redraw the diagram, keeping the given conditions but making it clear that b does not have to be 40 and that Angles 3&4 also do not have to be 40!

Monday, May 12, 2014

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

If link doesn't work try this twitter link..

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ドル.95. Secured pdf will be emailed when purchase is verified.

Monday, May 5, 2014

Desmos Common Core Activity Linking Circles, Tangents and Linear-Quadr Systems



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ドル.95. Secured pdf will be emailed when purchase is verified.

Thursday, April 24, 2014

Parametric/Projectile Motion Simulated in Desmos - A Common Core Activity for Algebra/Precalculus



[Updated using folders to reduce amount of visible text. Click on the arrow next to the Folder icon to see the frames below. Thanks to Desmos team for this helpful hint!]

CLICK ON GRAPH TO ACTIVATE DESMOS...

The Desmos activity above is both an investigation of parametric representation and a tutorial for more advanced use of this remarkable WebApp. The The text in the side frames begins with a detailed background of the activity for the instructor and how Desmos can be used to demonstrate projectile motion using both parametric and rectangular coordinates. Some of the uses of slider 'variables' are demonstrated including animation, a powerful feature of Desmos.

In addition to showing how to use parameters in Desmos, the activity itself asks students to compare two different trajectories, representing an object dropped from some initial height, then a 2nd object two seconds later. The horizontal translation of the first graph is juxtaposed against the algebraic representations of these graphs using both system of coordinates.

The student activity starts about halfway down. There is a series of questions and actions the student needs to take in Desmos.

I'm hoping this will prove useful for both the instructor and the student. Desmos is powerful but, in my opinion, some of the illustrative examples provided by Desmos do not flesh out the ideas behind the various uses of slider 'variables'. I'm hoping this will fill in some of those gaps. I'm still a novice here so I'm sure more advanced users will be able to improve upon this...

Your comments and reactions are very helpful to me...







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ドル.95. Secured pdf will be emailed when purchase is verified.

Saturday, February 15, 2014

New video tutorials uploaded to MathNotationsVids YouTube channel

Haven't posted in awhile and haven't uploaded videos to YouTube recently. I now have the technology to make short screencasts. Here's a link to the latest video I posted, a typical medium- level standardized test question relating to a non- convex quadrilateral. As aways these tutorials are intended for both students and educators.
Your feedback is important to me. You can support my efforts simply by subscribing to my channel. Thanks...

Saturday, January 4, 2014

Three congruent isosceles right triangles walked into a bar...

OVERVIEW
Silly title but you might want to try the following problem with your high school geometry students or with middle schoolers doing a unit on right triangles. Furthermore, elementary school children need many hands-on experiences with pattern blocks, tangrams, pentominos and the like to develop their innate spatial sense. They should also be allowed to experiment with two such triangular pieces to make a square, a parallelogram, a larger isosceles triangle, etc. Then have them work with the 3 triangles to make different polygons including the trapezoid. They don't need to consider the area or the 2nd part of the question.

THE PROBLEM
Three congruent isosceles right triangles are joined to form an isosceles trapezoid having an area of 3 sq units.

(a) Draw a possible diagram.
(b) Determine the perimeter of the trapezoid.

Answer: (b) 6+2√2

REFLECTIONS
•How much time would you allow for a discussion of this problem! 10 min? 15? 20? Guess it depends on whether you see this as just an exercise or as an activity.
• How much difficulty do you think most middle and secondary students would have with drawing an appropriate diagram?
•Do you think most will need to draw several figures before arriving at the isosceles trapezoid? Do you think some will come up with a trapezoid which is not isosceles and think they're finished? Can you anticipate that some will miss one of the key words like isosceles (which occurs TWICE!).
• Do you think the spatial "puzzle pieces" part of the problem is more significant than the numerical part or about equal?
• Do you expect some students to hit a wall and express something like "I forgot the formula for the area of a trapezoid!" We should make this a teachable moment -- "WE DON'T NEED TO RECALL THAT FORMULA! WHY!"
•Do you see benefits from students working in pairs here? Would you have them work independently then come together after a few minutes? My view is the stronger spatial student will "see" the correct figure more rapidly and influence the other who may give up and wait for his/her partner to draw it. So I might ask them to draw a few figures on their own for a couple of minutes.
•Do you think any of the older students need manipulatives?
• What is our role here? Catchphrases like"guide on the side" do not tell us what interventions we should actually use? Part of knowing what to do/say comes from our experience and part from instinct but my rule of thumb was "less is more". Allowing them to struggle for awhile is critical or, to put it another way, "without irritation there would never be a pearl!"
• How would you solve this problem? When planning do you feel it's important to think of alternate solutions or let this flow from the students?
•Finally, I think it's important to identify which of the  Mathematical Practice Standards are brought to play in this investigation. All of them? A couple? Guess that depends on you...

I typically get few if any comments from these detailed investigations. That's ok. Just planting seeds I guess...

Wednesday, December 25, 2013

Reciprocals, Square Roots and Iteration -- The gift that keeps on giving!

OVERVIEW
SEASONS GREETINGS!
While gifting and regifting this holiday season, here's my gift to all my faithful readers without whom I'd have no reason to put finger to touch screen...
The following series of problems does not on its surface involve anything more than basic algebra, but it is intended to provoke students to reflect on the interconnectedness of number and algebra.
The extension at the bottom goes beyond what might be expected from the beginning of this exploration.
Math educators can adapt this for Algebra 1 through AP Calculus students...
THE PROBLEMS
What are the number(s) described in the following?
1. A number equals its reciprocal.
2. A number equals 25% of its reciprocal.
3. A number equals twice its reciprocal.
4. A number equals the opposite of its reciprocal.
5. A number equals k times it's reciprocal. Restrictions on k? Cases?
Answers:
1. 1,-1
2. 1/2,-1/2
3, √2,-√2
4. i,-i
5. k>0: √k,-√k; k<0: i√k,-i√k; k=0:undefined
OVERVIEW and much more...
• So why don't we just solve the equation x^2=k? See extension below for one reason.
• Why not ask the students what the graphs of, say, y=x and y=2/x have to do with #3. They might find it interesting how the intersection of a line and a rectangular hyperbola can be used to find the square root of a number!
• Extension to Iteration
Ask students to explore the following iterative formula for square roots:
(*) New = (Old + k/Old)/2
Have them try a few iterations for k=2:
x1=1 (choose any pos # for initial or start value; I chose 1 as it's an approximation for √2 but any other value is OK!)
x2=(1+2/1)/2=3/2=1.5
x3=(1.5+2/1.5)/2=17/12≈1.417 Note how rapidly we are approaching √2)
x4= etc
[Note: Plug in √2 into the iteration formula (*) to give you a feel for how this works!]
Students may want to explore further and they might be curious about where this formula came from, how it's related to Euler, Newton, Calculus and Computer Science. For example, they could implement this on their graphing calculator or program the algorithm themselves!

Wednesday, December 18, 2013

Two overlapping circles of radius r... - A Common Core Geometry Problem

OVERVIEW
Intersecting circle problems are always interesting and often challenging whether you find them in the text, on SATs or on math contests. The general case involves trig and formulas can be found online.

The objectives of the problem below include:

• Drawing a diagram from verbal description
• Dissecting or subdividing an unknown region into more common parts
• Applying circle theorems and area formulas
• Solving a multistep problem (developing organizational skills, attention to detail)

THE PROBLEM
Two circles of radius r intersect in two points in such a way that the overlap is bounded by two 90° arcs. If the area of the common region is kr^2, determine the value of k.

Answer: (Pi-2)/2
Note: Please verify!

REFLECTIONS FOR MATH TEACHERS
[Note: These are discussion points --- not short answer questions with simple answers!]

• Should the diagram have been given to eliminate confusion?
• Does this problem appear to have any practical application?
• Have you seen a similar problem in your geometry texts? On standardized tests like SATs?
• In similar problems, were the arcs 60° or 90°?
• How would you introduce this problem? Is it worth the time to have students cut out congruent paper or cardboard circular disks, keep one fixed and move the other until it approximates 90° arcs?
Better to use geometry software?
• Assign this for homework? As a group activity in or out of class? As a demo problem with a detailed explanation provided by you?
• How much time would be needed for classroom discussion of this problem?
• Would you plan on providing extensions/generalizations?
• Too ambitious for "regular" classes? Appropriate only for Honors?
• So what makes this a Common Core activity? Are you guided by the Mathematical Practice Standards?

Friday, December 6, 2013

The square root of x+1 equals x+1... A Common Core Investigation

OVERVIEW
Fairly straightforward radical equation in the title but there is so much hidden potential here for students in Alg 2/Precalculus.
REFLECTIONS
• The solutions to the equation above are -1 and 0. No big deal, right? The usual algorithm --- just square both sides and solve the resulting quadratic by any one of several methods. Done. Cheerio. But wait...
• We can encourage students to "make it simpler" by substituting 'a' for x+1 obtaining a^(1/2)=a, square both sides yielding a=a^2 which gives 2 easy solutions 0,1 and then x+1=0,1 producing the final result. Not that big a deal though except...
• A graphical interpretation of these equations is illuminating and illustrates multiple representations/The Rule of 4. You could demo this with the graphing calculator displayed on your smart board or have the students graph by hand or on their device. The graphs of y=x^(1/2) and y=x intersect at x=0 and x=1 then, by translation, the graphs of y=(x+1)^(1/2) and y=x+1 will intersect at x=-1 and x=0. Students should be asked for this conclusion BEFORE checking the graphs to verify!
• Is that all there is? Hardly! The current trend on assessments and hopefully in texts is to have students analyze a family of equations using a parameter. But first we can generalize numerically:
Solve
(i) (x+4)^(1/2)=x+2
(ii) (x+9)^(1/2)=x+3
Are there still 2 solutions for each of these? Solving just a couple of these and recognizing extraneous or apparent solutions would traditionally have been the WHOLE lesson! Not any more...
By the way - why the "4" and "9"? Did I change the pattern from the original equation?
• Now for the parametric form:
(x+k^2)^0.5=x+k
What questions should STUDENTS be asking themselves BEFORE WE ASK THEM?
• Students can certainly be asked to solve the latter equation for x in terms of k. Some will struggle with the procedure/algorithm. Hopefully someone in each group (or the whole class!) will obtain x=0 and x=1-2k. BUT WILL THEY CHECK THE 2nd SOLUTION! The use of a parameter goes beyond making a better standardized test question. Now the student has to recognize that, in order for there to be 2 solutions, k must be less than or equal to 1 which was suggested by the numerical examples above.
• Of course I'm anticipating most teachers' reactions to an exploration like this. I've provided much more than can reasonably fit in a 40 min lesson. Use it as you see fit or just ignore it. It will go away or will it?

Subscribe to: Comments (Atom)

AltStyle によって変換されたページ (->オリジナル) /