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? 

Discussion of Algebra 2 End of Course Exam from ADP/Achieve/Pearson - Continued...

Unfortunately, I did not receive permission from Pearson Educational Measurement to reproduce any of the released items from the Achieve website. I will respect their wishes. However, they understand that I plan to discuss some of the items indirectly without specifics. For this to make sense to my readers, you will need to download the pdf document as suggested in an earlier post and have that in front of you as I refer to individual items. Here is the link to the Achieve website that contains the released items (in the sidebar).

For this post, however, I plan to discuss the implications of this exam and related issues.

I would like to invite comments about the issues raised by a common exam that will be administered to students in 14 states (up from the original 9). I consider this to be a highly significant development in the movement toward more standardization for all students. Up to this point, the only similar kind of interstate standardized test covering Algebra 2 topics has been the SAT Subject Test - Math I. Several states now give their own end of course exams in Algebra I and Algebra 2, but the Algebra 2/ADP exam from Pearson is impacting on students from many states and this is just the beginning of this trend. For example, a similar exam for Algebra I is already under development (if not already completed) and, here in New Jersey, it will become operational shortly.

Furthermore, textbook publishers such as Pearson have already begun to publish texts (e.g., Pearson, Algebra I, 2007) that correlate with the American Diploma Project's Algebra standards (download this). Imagine that! Instead of inserts in the text that correlate to fifty different sets of state standards, we are now going to see some consistency. Glory Hallelujah!

Here are just a few of the issues that each state will have to confront as these exams proliferate:
1) How will a student's grade be determined in the course? Will the exam be worth a percentage of the final grade? What if the results do not come back in a timely fashion?
2) If a student falls below the minimum level of proficiency on the exam (great euphemism/edu-jargon for 'failing'!), how will this be recorded on their transcript?
3) Will the exam be required for graduation just as current graduation tests do in some states?
4) Can a student re-take the exam if they don't make it the first time?
5) Should the Algebra I exam have been developed and implemented before this exam?
6) Will different groups and consortia now compete to develop curricula and assessments independently as a result of the recommendations from Achieve/ADP, the National Math Panel and NCTM's Curriculum Focal Points? More splintering?
7) Will there be more than one administration of these exams each year? For the Algebra 2 Exam, the answer is found on the Achieve website:

The exam will be administered at the end of fall—December and January and at the end of spring—May and June beginning with the 2008-2009 school year.

Ironically, while this exam improves consistency of curriculum, there might not be as much consistency about how these issues will be addressed. Of course, one can hope...

Are these exams intended to impact on instructional methods, emphases, strategies, techniques just as currently occurs on the AP Calculus Exam? If you read through the pdf document and examine the Released Items document (p.4), the answer is clear for the Algebra 2 Exam (the following is an image which will appear blurry);













Your thoughts...

Nine States to Administer Standardized Algebra 2 Exam

NJ Commissioner of Education, Lucille E. Davy, announced on 4-10-07, that New Jersey is one of nine states that will administer a common exam for Algebra 2 students in May 2008. A link to the actual Algebra standards that will be used is given near the end of this post.

Here is an excerpt of her statement:

New Jersey Joins Nine-State Partnership to Administer New Algebra II Exam

New Jersey has agreed to join a partnership of nine states in the American Diploma Project Secondary Math Partnership to administer a common exam with common standards for Algebra II students beginning in May 2008. The eight states joining New Jersey in the partnership are Arkansas, Indiana, Kentucky, Maryland, Massachusetts, Ohio, Pennsylvania and Rhode Island. The project is an initiative of the ADP Network, a group of 29 states that educate more than 60 percent of all American public school students.

“This new exam will help to ensure that our children are learning the math skills that are becoming more and more essential in an increasingly competitive job and secondary education marketplace,” said Commissioner of Education Lucille E. Davy. “Our work in the American Diploma Project (ADP) Secondary Math Partnership complements our efforts in New Jersey to re-design our schools to meet the challenges of preparing our young people for the demands of the 21st century.”

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Those of you who have been reading my posts for the past four months know that I am elated at this news. I had mentioned the possibilities of this happening a couple of months ago and now it's gaining momentum. Common exam, common standards, common curriculum,...

Of course, we will need to see how this plays out. When it becomes mandatory, will all NJ high schoolers be required to pass this in order to get credit for the course? Does this mean that all NJ students will eventually be expected to take Algebra 2 in high school and does this imply that those students taking more basic level mathematics will have to reach much higher than now? Will it follow a Regents model as in NY? If a student does not pass the exam, what appears on their transcript? NYS educators can probably provide insight into how this is handled in their state. From my understanding of the American Diploma Project, developed by Achieve.org, ALL youngsters need to and will be strongly urged to take Algebra 2. For those intending to continue their education, this is not an option.

I do believe this is a big step in the right direction, however, Commissioner Davy must surely recognize that there is a demographic of youngsters who, at this time, do not yet have the skills to tackle this course. Perhaps she and the others in the consortium believe that this will force the math curricula in Grades K-8 to be significantly upgraded and compel high schools to begin to phase out low-level math courses for most students. Not that there's anything wrong with that but we're certainly not there yet!

For those interested in the Algebra benchmarks (standards) developed by Achieve.org, read this. I looked it over and I like its structure, clarity, examples and content. I need to consider more carefully whether it encompasses all of the important topics in Algebra 2, but, on first glance, it looks good. Their goal and mine is to make our children more competitive and to upgrade the quality of education for ALL of our students, particularly underrepresented groups in our society. After looking these over, let me know if you see any omissions in these benchmarks.

Update on my view of the benchmarks: After reading both the geometry and algebra 'standards' and some of the sample postsecondary problems more carefully, I detect a more traditional flavor with some newer content sprinkled throughout. Those who remember learning from or teaching from the Houghton Mifflin Dolciani series for Algebra 1/2 (Structure and Method if I recall correctly) may feel nostalgic. This is more of an instinct than a careful analysis, so feel free to correct me....