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I am still not happy with my “factor by grouping progression” because it ends in a trick.
Our curriculum currently requires “factor cubics by grouping,” so my progression has started off with a cubic like this: x^3+ x^2 -3x -3 which we factor by grouping :
(x3+ x2 )+(-9x -9)
x2 (x + 1) -9(x+1)
(x2 -9)(x+1)
(x-3)(x+3)(x+1)The next step in my factoring progression has been to break up the middle term of a quadratic (with ANY integer coefficients)…and then factor by grouping:
4x^2 -11x – 3
4x^2 +x – 12x -3
x(4x + 1) -3(4x+1)
(x-3)(4x+1)This method eliminates student frustration from guessing and checking, but in order to break up the middle term, the students use the AC trick, “What two numbers multiplied together give you AC and add to get B?” This Khan Academy video shows more examples. At the end of the video, he explains why the trick works, but of course 99.9% of the students will zone out during the explanation. https://www.khanacademy.org/math/algebra/multiplying-factoring-expression/factoring-by-grouping/v/factor-by-grouping-and-factoring-completely
After further study of the standards, I see no reason why we would have to teach factoring a cubic by grouping because we could use graphing calculators to find integer zeros for polynomials of degree greater than 2 and divide out factors from there. If that is true, I will encourage my district to remove the requirement for “factoring by grouping cubics” and focus on trial and error for simple quadratics and completing the square for more complicated cases. Am I correct that there is no need to teach factoring a cubic by grouping or did I overlook something?
Another nice thing about the Standards is the way students are encouraged to use area models in lower grades. If the students have been using them in prior grades…examples at: https://dl.dropboxusercontent.com/u/7405693/MEGSL/Progressions%20in%20Area%20Models%20for%20Tools.doc
then surface area is easily connected to those.
I am going to chicken out at this point because I am not sure exactly how the CCSS writers want us to word this with the best (consistent) precision, and Dr. McCallum may want to address it here. In high school, we sometimes say, “larger negative” meaning the number is farther from zero on a number line. Vector magnitude and absolute value fall into this discussion as far as what does it mean to be “larger.” I’m thinking Phil Daro’s video about misconceptions might be helpful because he addresses how a concept understood at one level can best be tweaked at the upper level:
I would not use that exact worksheet at 5th grade (right now I use it in Algebra I based on our old curriculum). Here’s a quick, partial modification I envisioned for what you were asking about: https://dl.dropboxusercontent.com/u/7405693/MEGSL/multip%20and%20divide%20by%20ten%20comparisons.docx
In my opinion, it is unwise to imply decimals move as though they have wheels. I would talk about place value and relative size of the number.I see what you mean. What an interesting thought! It is important how we word things. We do not want them to think dividing 0.05 by 10 will result in fewer digits on the right! I have not observed any students making this wrong conclusion, probably because as we talk about it, we are focusing on relative value. Here’s the quick exercise they complete: https://dl.dropboxusercontent.com/u/7405693/Webpg/worksheets%20Alg%201%20SPRING/Scientific%20Notation%20comparisons.doc
I see what you mean and it does matter how we word things. The exercise I give my students involves focusing on relative value. Certainly it is incorrect to state that there are fewer digits because, for example, there may be more digits on the right side of the decimal when dividing by 10. Here is the exercise and maybe this is why I have not been aware of any of my students equating size with number of digits: https://dl.dropboxusercontent.com/u/7405693/Webpg/worksheets%20Alg%201%20SPRING/Scientific%20Notation%20comparisons.doc
In my observations, describing decimals “moving left and moving right” easily becomes something students memorize and confuse, especially with scientific notation. When converting from scientific to decimal notation, the decimal moves one way and when converting back it moves the other way. The students get it right 50% of the time. That’s not to say we should never say “the decimal moves to the left,” but I have gotten much better results by having students talk about numbers being “one decimal place larger or one decimal place smaller.” Dividing by 10 makes a number one decimal place smaller while multiplying by 10 makes a number one decimal place larger. This connects well with NF.5.b, “Explaining why multiplying a given number by a fraction greater
than 1 results in a product greater than the given number
…; explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b b.1) There are some great examples of multiplying with larger numbers beginning on page 23 of this document:
Click to access NCSMJournal_ST_Algorithms_Fuson_Beckmann.pdf
In my opinion, we need to continue to give examples of area models for smaller numbers with each block shown to maintain that connection.
To clarify, I’m wondering if, even at the high school level, we need to be reducing expressions like that.
I’ve wondered how far we will go with this as well. It takes a lot of practice to be able to reduce fractions in the form of (((3xy^-4z^5)^-3(4y^-2z^-3))/((7x^-2)^3)(y^6)^-7) and it is a challenge to answer the question, “When will we ever…”
I think I like the word “reduce” at least as much as “cancel” if not more, but there may be something I’m not considering. I didn’t see “reduce” in the standards.
Now that you mention it, the “functions based approach” is nebulous in some aspects to me as well. If anyone finds a complete, succinct explanation, please post.
Thank you for your quick response. In the interest of narrowing curriculum, it makes sense to have the expectation that students solve 5-4|2x-3| = -11 as a system of equations. After all, we would not teach a separate method for finding solutions for something like cos x – 3 tan x = 7.
For 5-4|2x-3| = -11, our high school students struggle with having to first use order of operations to get the absolute value alone: |2x-3| = 4 , then take the bars off and write 2x-3 =4 or 2x-3 = -4, and then solve for x twice. It all becomes a confused, memorized procedure. On the other hand, if students know 6.NS.7c (the absolute value of a rational number is its distance from zer0 on the number line), the solutions to |x| = 3 can be visualized by writing x= above both the 3 and the -3 on a number line, then it would not be such a leap for them to write 2x-3 on the number line twice, once above 4 and once above negative 4.
