A Quadratic Trinomial/Factoring Investigation for Algebra I/II

In my Christmas post, I raised the issue of how much time should be spent on factoring quadratic trinomials over the integers in light of the new ADP Standards for Algebra I and II. Hopefully, some of you will provide us with the benefit of your knowledge and experience. I may even make this into a poll or survey to be voted on but, in this post, I will appear to contradict myself and propose an investigation of this topic which requires some effort and time on the part of the student. The target audience would be the regular or accelerated Algebra I/II student.  

We all need to become more creative in the strategic use of time in our classrooms (I still think of myself as being in the classroom!). What are some alternatives to using class time for this? I'll suggest one approach and I'm hoping others will offer their suggestions:


Assign the following as an extra credit or "long-term" project to be due in a week or two. I would not even take classroom time to discuss it. Just hand it out or post it on your website or the department's website if it is to be given to all the Algebra classes. Students can easily download it or print directly if they wish. After they are collected, graded and returned, you may choose to discuss it briefly for about 10 minutes using an overhead transparency, opaque projector or via your computer and a projector. You can also post some  student solutions on the website.


THE INVESTIGATION/PROJECT

[OPTIONAL HINT OR CUE]
The following may require an application of the ac-method learned in class.

(1) Factor the following over the integers and show all steps used in your method of factoring:

(a) 12x² + 27x + 15

(b) 12x² + 28x + 15

(c) 12x² + 29x + 15

(2)
(i) List all positive integers values of b, including the ones from part (a), for which 12x² + bx + 15 is factorable over the integers.
(ii) For each value of b, factor the resulting trinomial.
(ii) How many of these trinomials produce a gcf ≠ 1 for 12, b and 15?

(3) If we knew in advance that 180 has 18 positive integer factors, explain how it follows that there are 9 values for b in part (2).

(4)
(i) If the "12" and "15" were interchanged, explain why this would not change the possible values for b in part (2)?
(ii)For each resulting trinomial such as 15x² + 28x +12, determine its factors and explain how they are related to the factors of the original trinomial (i.e., before interchanging the 12 and 15).


QUESTION FOR OUR DISCUSSION (No, these are not rhetorical! Some are quite knotty)

(1)  What do you see as the benefits of this investigation, if any?

(2) Do the new standards and assessments discourage us from investing time into this type of in-depth problem-solving?

(3) Do you believe this type of assignment should be reserved for the accelerated/honors Algebra I student in 7th or 8th grade or even for the stronger Algebra II student?

(4) With the new ADP Algebra standards, do you believe this type of investigation is reasonable, particularly since it is unlikely that any variation of this would appear on an End of Course Test?

(5) If you were to give this problem, how would you edit the investigation? Parts you would delete or change? Parts you would add?

(6) My goal for this blog has always been to provide you with useful and engaging examples of in-depth problems for your students that require going beyond the mechanical aspects of the course. These problems are developed for this blog -- they do not come from my notes from 30 years ago! Would you be interested in a supplementary resource of such problems for each course you teach? Do you already have one from the publisher or from another source which you really enjoy? Share it!

How Much Factoring In 1st Year Algebra?

SEASON'S GREETINGS

Math Notations 3rd Birthday- Thank You!

The American Diploma Project is and will be impacting on what is being taught in both Algebra I and II in the 15 states who have joined the ADP Consortium. The classic flow from Standards to Assessments to Course Content is leading to the type of content standardization in our schools which I envisioned decades ago. A natural part of this process is deciding what topics in our traditional courses need to be deemphasized or eliminated to allow more time for the study of linear and non-linear function models, one of the central themes of the new Algebra standards.This leads to curriculum questions like...

How much time should be spent on factoring quadratic trinomials in Algebra I?

My assumption is that factoring ax²+bx+c where a ≠ 1 is still taught in Algebra I. Please challenge that assumption if wrong! If we also assume there is sufficient justification for teaching this, then we move on to the issue of how much time should be devoted to instruction. Two days? More? Time for assessment?

Here are some arguments pro and con...


PRO

(1) It is required by the ADP Standards (see below).

(2) Learning only simple trinomial factoring of the form x²+bx+c is not sufficient for solving more complex application problems.

(3) The various algorithms, such as the "ac-method", which have been developed for factoring quadratic trinomials, are of value in their own right; further, the "ac-method" introduces or reinforces the important idea of factoring by grouping.

(4) Students gain technical proficiency by tackling more complicated trinomials.

(5) Students should be given the option of more than one method, not just the quadratic formula.



CON 

(1) The AP Calculus exam generally avoids messy quadratics in their problems. If such occur, students normally go directly to the Quadratic Formula.

(2) The SATs generally avoid asking students to factor such quadratics directly, particularly since it is easy to "beat the question" by working backwards from the choices. Instead, they ask the student to demonstrate an understanding of the process.

Here's a typical question they might ask:

If 6x² + bx + 6 = (3x + m)(nx + 3) for all values of x, what is the value of b?


(3)The ADP standards for Algebra I do include this topic but it does not appear to be stressed. The following are taken from the ADP Algebra I standards and practice test:

(3) Do other nations teach our traditional methods of factoring or are students told to go directly to the quadratic formula?

(4) Current Alg I texts seem to have deemphasized factoring in general and some have moved this topic to later in the book.


So I am opening the floor for your input here!


(a) How much time is spent on factoring quadratic trinomials in Algebra I in your school?
(b) Do you teach the "ac-method"? If yes, do you motivate it or teach it mechanically?
(c) Do you believe factoring quadratic trinomials is essential or should it be deemphasized?


By the way, here is an example of the ac-method:

Factor completely over the integers:   6x² + 13x + 6

Step 1: Find a pair of factors of ac = (6)(6) = 36 which sum to b = 13.
Hopefully, students think of 9 and 4 without a calculator!

Step 2:  Rewrite the middle term 13x as 9x + 4x (works in either order)
Then 6x² + 13x + 6  =   6x² + 9x + 4x + 6

Step 3: Group in pairs and factor out greatest common monomial factor from each pair:
3x(2x + 3) + 2(2x + 3)

Step 4: Factor out the common binomial factor 2x + 3:
(2x + 3) (3x + 2)

Step 5: Check carefully by distributing.


Here is a "proof" of this method (some details omitted like the meaning of h and k):

Divisibility, Counting, Strategies, Reasoning -- Just Another Warmup

Most of my readers know that my philosophy is to challenge ALL of our students more than we do at present. The following problem should not be viewed therefore as a math contest problem for middle schoolers; rather a problem for all middle schoolers and on into high school


List all 5-digit palindromes which have zero as their middle digit and are divisible by 9.

Comments:
(1) Should you include a definition or example of a palindrome as is normally done on assessments or have students "look it up!"

(2) Is it necessary to clarify that we are only considering positive integers when we refer to a 5-digit number?

 (3) What is the content knowledge needed? Skills? Strategies? Logic? Reasoning? Do these questions develop the mind while reviewing the mathematics? In other words, are they worth the time? 

(4) BTW, there are ten numbers in the list. Sorry to ruin the surprise!

(5) How would this question be worded if it were an SAT problem? Multiple-choice vs. grid-in? 

Demo For Building An Investigation In Geometry For All Levels

Note: Diagram has been modified from original.

 For Figures 1 and 2, the following is given:

AD + AC = BD + BC

Perimeter of triangle = 36

AC = 15

Show that the length of AD = 3.

In other words, demonstrate that the length of AD is independent of sides AB and BC.

Instead of imposing or suggesting my way of using this question to build an investigation, how would you do it?

If you're new to this blog, I have published dozens of examples of investigations which are intended to develop process, conceptual understanding, generalization  and a different view of what mathematics is for our students. An investigation allows students to explore particular cases before attempting to generalize and abstract. Some might call this scaffolding. I see it as creating an experimental environment in the classroom, encouraging our students to become mathematical researchers! I know every argument against this approach but, remember, I'm suggesting that this type of activity only be used perhaps once a month...

The question above can be given as is to some groups of students but may not be appropriate in its present form for many others. The question can be reworded or changed completely.

What would you do?

Using WarmUps in Middle School/HS to Develop Thinking and Review/Apply Skills

My 500 or so subscribers may not have seen the following anagrams which have been in the right sidebar of my home page  for the past month or so. No one has yet taken the time to solve them. They're not that hard! Pls email me at dmarain at gmail dot com with your answers.


VORTEC SCAPE


(1) Hidden Steps OR


(2) General Arrows



The following problems are similar to ones I posted recently...



Mental Math and No Calculator! 


1)  The following sum has a trillion terms:


0.01 + 0.01 + 0.01 + ... + 0.01 = 1000...0
How many zeros will there be in the sum?




2)  The following product has a trillion factors:


(0.01) (0.01) (0.01) ... (0.01) = 0.000...1
How many zeros after the decimal point will there be in the product?



A Few Comments...
(a) You may want to adjust the "trillion" for your own groups but I'm intentionally using this number for a few reasons, not the least of which is to review large powers of 10 (Will most think: "A million has 6 zeros, a billion has 9 zeros, so a trillion has..."?).


(b) The second one is more challenging and intended for Prealgebra students and above but, using the "Make it simper" and "Look for a pattern" strategies, make it possible for younger students.


(c)  How many of you are reacting something like: "Is Dave out of his mind? My students don't know their basic facts up to 10 and he wants mental math with a trillion!" I have found that large numbers engage students since they know there is a way of doing these without a lot of work if you know the "secrets"! Besides, we either push our students or we don't. You decide...


(d) These questions review several important concepts and skills. You may want to use these to introduce or review the importance of exponents and their properties. 

INSTRUCTIONAL STRATEGIES SERIES: Teaching for Meaning - More Than Just A Geometry/Algebra Problem

HAPPY THANKSGIVING!

 

Alright, you're teaching about the rule for slopes of perpendicular lines in Algebra or Geometry. 

Here are some of the instructional strategies or approaches you may have used...

(1) State the theorem without explanation followed by 3-4 demo examples of how it's used
(2) Motivate the theorem using the lines y = (3/4)x and y = (-4/3)x, choosing the points (4,3) and (-3,4) to demonstrate why these lines are perpendicular
(3) A more abstract approach using the following diagram

NOTE: Q(-b,a) is the point on line M in quadrant II. The label is too far from the dot!

FROM THE GIVEN INFORMATION IN THE DIAGRAM PROVE THAT ∠QOP IS A RIGHT ANGLE, THAT IS, LINES L AND M ARE PERPENDICULAR.

Comments

(a) If your group was advanced, would you omit the perpendiculars QR and PS?

(b) Would you draw the diagram to scale to prevent confusion for most students?

(c) Would you even consider Option (3) with a regular or weaker group of students? Would Option 2 be more than enough to get at the main idea?

(d) To more strongly suggest the use of slopes and/or similar triangles, would it be better to use the points (4,3) and (-6,8) on the lines? I personally would prefer this (and not give the equations of the lines). What do you imagine most students would do with this problem a few weeks (or even days!) later? Would they make the connection to slopes immediately if they had moved on to another unit or if this appeared on an assessment?

(e) Would some students need more than one example to suggest a generalization? Exactly what questions would you ask to promote a generalization?

(f) What have you done with this topic and/or how would you modify the above ideas??? The floor is open..

By the way, do you believe it is likely or unlikely that some version of this problem might appear on a standardized test like ADP's Algebra 2 End of Course Test or the SATs?

The Return of the WarmUp Challenges!

Just when you thought that MathNotations is on permanent hiatus or in hibernation, here are a couple of WarmUps/Problems of the Day/Test Prep/Challenges/// to consider for your students. 

Actually, I'm embarking on a new venture - an online tutoring website with live audio and video for OneOnOne math tutoring for Grades 6-14 (through Calculus II). In addition, I'm also working on setting up a small group (5-10 students) online SAT or ACT Course grouped by ability (a 600-800 SAT group, a 450-600 group, etc.).  If you're interested in getting more information about these before the official launch just contact me at dmarain at gmail dot com.


Update: Answers/comments are at the bottom...

1.   NOTE: ANGLE B IS A RIGHT ANGLE IN DIAGRAM BELOW - THANKS TO JONATHAN FOR CATCHING THAT OVERSIGHT!

2.   If 10^-1000 - 10^-997 is written as a decimal, answer the following:


(a) How many decimal places are there, i.e., how many digits to the right of the decimal point?
(b) One can show that the decimal digits end in a string of 9's. How many 9's?
(c) How many zeros are to the right of the decimal point and to the left of the string of 9's?

Notes:
(1) If we write the negative exponent expressions as rational numbers, this is perfectly appropriate for middle schoolers and, in fact, I think they need more of these experiences!
(2) The "Make It Simpler - Look for a Pattern" Strategy should be second nature to our youngsters, but when they see questions like these on the SATs, how many of our students really think of it!
(3) The fact that some calculators return a value of zero for the expression in the problem is a teachable moment - seize it!!
(4) See below for an algebraic approach.



--------------------------------------------------------------------------------------------


ANSWERS


1. 9√3


2. (a) 1000   (b) 3   (c) 997


An Algebraic Approach to #2:
First, students need to be familiar with the basic pattern:
10^-1 = 1/10 = .1 Note that there is one decimal digit.

10^-2 = 1/10² = 1/100 = .01  Note that there are two decimal places, etc.


10^-1000 - 10^-997 = 1/10¹⁰⁰⁰ - 1/10⁹⁹⁷
Using 10¹⁰⁰⁰ as the common denominator, we obtain
1/10¹⁰⁰⁰ - 10³/10¹⁰⁰⁰ =
-999/10¹⁰⁰⁰ from which the results follow (with some additional reasoning)...

Note: I could have worked directly with the exponent form by factoring out 10^-1000 but I chose rational form for the younger student.

THE OPEN-ENDED CONTEST PROBLEM AND SOLUTIONS

As promised, here is the open-ended, rubric-based, holistically scored, performance-assessed, student-constructed first problem from MathNotation's Third Contest:

1. A primitive Pythagorean triple is defined as an ordered triple of positive integers (a,b,c) in which a² + b² = c² and the greatest common factor (divisor) of a, b and c is 1. If (a,b,c) form such a triple, explain why c cannot be an even integer.

Comments
(a) The content here is number theory. Is some of this covered in your district's middle school curriculum or beyond? More importantly, at what point do students begin to formulate and write valid mathematical arguments?

(b) The immediate reaction of most students was that this seemed like a fairly simple problem. However, only a couple of teams scored any points. Perhaps the challenge here was the construction of a deductive argument, although as you will see below, there is one challenging part.

(c) There were two successful approaches used by the teams. Both involved indirect reasoning. Do your students begin to do these in middle school or are "proofs" first introduced in geometry?

(d) I allowed students to assume without proof the following:

(i) The general rules of parity of the sum of two integers
(ii) The square of a positive integer has the same parity as the integer

(e) Interestingly, none of the teams considered an algebraic approach to the one challenging case, i.e., demonstrating that the sum of the squares of two odd integers is not divisible by 4.

If a and b are odd, they can be represented as
a = 2m+1 and b = 2n+1, where m and n are integers.
Then a² + b² = (2m+1)² + (2n+1)² =
(4m² + 4m + 1) + (4n² + 4n + 1) =
4(m² + n²) + 4(m + n) + 2, which leaves a remainder of 2 when divided by 4.
BUT, if c is even, say c = 2k, then c² = 4k², which is divisible by 4.

(f) The two best solutions came from our first and second place teams, Chiles HS in FL and Hanover Park Middle School in CA. Both used the ideas of congruence modulo 4.

Here is the indirect method used by Chiles:

Let's assume that c can be an even integer. We'll prove by contradiction. An even integer can be summed in two ways:
1. with two even integers or
2. two odd integers
If it is the latter case, then looking at the residuals of modulo 4, the two odd integers summed will be equal to 2, but this is not the case as 2 is not a modulo of 4 residue. If it is the former case, then it does not satisfy the problem as then a, b, and c have common factor of 2. Therefore c must be an odd integer. Q.E.D.


Here is the indirect method used by Hanover Park:

Suppose, for the sake of contradiction, that there is a PPT (primitive Pythagorean Triple) s.t. c is even. Then c² ≡ 0 (mod 4).

We break this into cases based on the parity of a,b.

Case I: Both a and b are even; gcd(a,b,c) ≥ 2 because a,b,c are even, a contradiction.

Case 2: One of a and b is even. Then, a² + b² ≡ 0 + 1 ≡ 1
not ≡ 0 (mod 4), a contradiction.

Case 3: Both of a, b are odd. Then a² + b² ≡ 1 + 1 ≡ 2
not ≡ 0 (mod 4), a contradiction.
We have covered all cases for a, b with no valid cases. Thus, in a PPT, c cannot be even.

Both of these arguments represent a more sophisticated understanding of mathematics and the methods of proof. Clearly, these students are quite advanced and exceptional, however, I feel many middle school teachers begin early on to encourage their students to explain their thought processes both orally and in writing. Am I right? I would like to hear your thoughts on this...