Gifted Issues Discussion homepage Forums General Discussion Hitting a brick wall with algebra

IMO, bluemagic's graph idea is a very good one. You get to show a 'linear equation' describes a line and that the solution to a set of them is going to at their common intersection, i.e. where the x-values AND the y-values coincide.

It will also build on somethng that your DD has done before - providing a 'seed crystal' for further understanding to bind onto.

I'd agree. This is, btw, the approach taken by the secondary math stream of the Singapore Math series I've used with some of my children (Discovering Math/Dimensions Math CC; I prefer the sequence of Discovering over Dimensions). Plenty of graphing prior to introducing substitution or elimination, so that students understand that the solution of simultaneous equations is the coordinates of the point of intersection between lines, not just some procedural manipulation. Find a free equation graphing app (our iMac appears to have come with one) and graph them that way, if hand-drawing them becomes overly laborious.

I'm going to disagree and agree at the same time. Yes, in the current situation, graphing the lines will help. But IMO, it doesn't address the fundamental problem, which is that current teaching doesn't create a proper foundation.

Here's how the ancient pre-algebra book handles the idea of functions in chapter 1 (set theory):

Section 1.3 focuses on 1:1 correspondence between sets. Section 1.4 introduces a function in terms of 1:1 pairs between sets. It uses diagrams of sets to depict definitions for domain and range. Then it gets into the idea of one input ---> one output = function. It next brings up the subject in the final chapter (intro to functions and basic graphing).

Knowing this stuff is essential to getting what it means to solve a pair of equations. The intersecting graphs help IF the student has a solid understanding of function and intersecting sets, which I suspect is extremely unlikely in pre- or early algebra, especially as taught today.

I could be wrong; the OP could ask her daughter to explain functions and see what she says. She could also ask how the intersecting lines relate to set theory. If the answer is something like, "the point where the lines meet is the intersection of the sets of the two ranges," she understands. I suspect that very few kids get this. Yet it is so fundamental....

you guys are all over my heads, I'm afraid--I've had to relearn systems of equations to help her, and the graphing stuff is just gone for me, though I'm sure I could relearn. My sense is that her fundamentals of ALL things algebraic are shaky indeed. She is quick and picks up most things "procedurally," but some of this stuff is a bit less procedural. I had her here in Khan beacuse that's where they are:



and she was struggling, getting maybe 50% of the problems right. She actually had the basic concept down but was shaky enough that she kept fudging things up.

I took her back to here: and she walked through the first few videos and examples and got everything right and the lights seemed to be going on. I feel confident she could go through the KA systems of equations lessons and get this down. But what else doesn't she have down that she should have down? I'm kind of threatening/sweet talking her into doing this extra work...she's used to breezing through everything...

Techncally, she is in prealgebra. (How systems of equations is prealgebra, I don't know.) Next year, she will take Algebra 1. I'm still confused about how this all works.

Agree with various posters above that the current commercial textbooks tend to be a mess.

I'll observe that the Common Core standards for grade 8 do call for doing systems of two linear equations. From:

http://www.corestandards.org/Math/Content/8/introduction/

Students solve systems of two linear equations in two variables and relate the systems to pairs of lines in the plane; these intersect, are parallel, or are the same line. Students use linear equations, systems of linear equations, linear functions, and their understanding of slope of a line to analyze situations and solve problems.

which is presumably why your DD is encountering them in her 8th grade equivalent pre-algebra class.

I personally have inclinations towards the sort of theoretical approach Val recommended.

However, I'll mention another approach that I've seen work well, which is the cpm.org approach. I would describe the CPM approach as "Common Core done right". CPM Is a non-profit consortium, staffed by middle and high school math teachers, that developed, pre-CC, a math curriculum based on collaboration, communicating math ideas to others, and problem solving, with spaced-repetition/spiraling rather than drill-and-forget. CPM was able to do a quick-and-natural transition to CC alignment, rather than the sort of garbled hack jobs coming out from the commercial textbook publishers.

I'm not personally familiar with their "Course 3" textbook (which would tend to be used as the 3rd year of a middle school curriculum, that is, grade 8, prior to an Algebra class in H.S.), but looking at the table of contents:

http://cpm.org/cc3

Chapter 1: Problem Solving
Chapter 2: Simplifying with Variables
Chapter 3: Graphs and Equations
Chapter 4: Multiple Representations
Chapter 5: Systems of Equations
Chapter 6: Transformations and Similarity
Chapter 7: Slope and Association
Chapter 8: Exponents and Functions
Chapter 9: Angles and the Pythagorean Theorem
Chapter 10: Surface Area and Volume

it does appear to be following a sensible enough order of working with equations in one variable, then graphing, then making sure students can go between graphs/functions/tabular sorts of representations, and only then moving on to systems of equations: the sort of topic ordering various posters recommended.

However, CPM books can be challenging to use as a supplement for an individual student since they do expect (and are built around) students working in groups and bouncing ideas off each other, and explaining their work to others: it's discovery-style learning, based on problem-solving and collaboration. So a CPM book may not be much help to the original poster looking for a supplement and way to fill in gaps for a single student -- but in a situation with another kid or two to work together through the material, I'd look at CPM (and then supplement as the kids seemed ready with additional "pure math" discussions of fundamental principles).

consider also a video + book option that my son was able to get through fairly easily (age 14 at the time), and does cover systems of equations towards the end. the presenter/flow is very good at keeping a person's attention so it doesn't end up feeling like a lot of work.
http://www.thegreatcourses.com/courses/algebra-i.html

In the decent schools with which I am familiar, Pre-Algebra is not your parents' Pre-algebra. Perhaps partly because of all the exposures to "elementary algebra" and "elementary geometry" in the standard elementary curriculum, Pre-algebra is half review of elementary math and half beginning Algebra and beginning Geometry. In her GT Algebra I course, DD12 (7th) finished systems of linear equations and systems of linear inequalities a couple of months ago.

Systems of linear equations may be a bit early for Pre-Algebra and DD's GT Pre-Algebra course (using a standard textbook) did not cover it last year. However, three years ago, DS12 used a pre-2000 Pre-Algebra book in his extra-accelerated program that was specifically marketed for GT only and did cover systems of equations during the second half of the year.

Anyhow, here are my advice as related to your original concern although I must confess that I haven't dealt with brick walls or re-teaching as some PP have. First, I cannot overemphasize how critically important it is for your DD to graph graph graph graph graph. It is the key to conceptual understanding. Ultimately, she needs to look at an equation or an inequality (rather it is linear or non-linear) or a system of equations/inequalities and visualize what it looks like and therefore what it means. In fact, I think this is in part why the current curriculum order looks so jumbled and spirally to many - the objective is to provide a big picture structure/frame to be filled in rather than a linear approach of one subject followed by another and within each subject brick by brick.

Another advice is to put the system of equations in context - this is a standard requirement in our district even pre CC when my oldest took Algebra I 5 years ago. Make sure she can set up two linear equations based on a real life problem or in the beginning at least understand what two linear equations can represent in real life. For example, constraints on spending/purchases are relevant to a tween/teen. You don't want your DD to mechanically follow a procedure to solve for X and Y. You want her to see why she would bother.

Finally, use AOPS alcumus to check her foundation in Pre-Algebra. If she has mastered Pre-algebra, it doesn't take very long to pass all the Pre-algebra topics plus in the process she wil deepen her understanding. DD12 who is strong in math but not mathy by inclination/thinking breezed through school Pre-Algebra last year but I was concerned with the adequacy of her school curriculum so I had her go through alcumus Pre-Algebra. She passed all the alcumus Pre-algebra topics pretty quickly during the summer but did learn some new stuff/approaches, which I think probably contributed to her finding Algebra I really easy this year. Anyhow, my point is that a solid foundation in Pre-Algebra is critical before you can build the structure of Algebra, Geometry, Trigonometry on top.

Without reading carefully all replies (so with the risk of repeating someone else's point), I wonder whether the pace might contribute to the issue.

My DD is doing Algebra through an online course arranged by our district. What I noticed was that there were a few topics that took her quite a while to truly understand, even though the rest was straight-forward. With the online course this was not a problem. She simply spent more time on the difficult topics (much more time, actually, on these topics). But in a classroom kids might not have this kind of flexibility. So if they feel stuck on something, and don't have time to *unstuck*, and are forced into the next topic and the next topic, that might be a source of stress and frustration.

Just a thought... My son actually has similar moments. He is in college-level math, and he tells his teacher that he often needs a Eureka moment for a new concept to truly make sense. Before this moment comes, his homework and classroom discussion won't be of high quality but all he needs is a bit more time. And his teacher totally gets it.

Maybe talking with the teacher and see what they suggest?

I'm confused about that too, and think now that I was wrong about how wonderful the CC math standards are. A major point about them was supposed to be "deeper understanding without rushing." That doesn't seem to be the case here at all.

My understanding of the high amount of algebra in pre-algebra is this:

Lots of students are struggling with algebra 1 and not doing well on the standardized tests.

The problem must be that they need two years to learn the material. Therefore, we will put lots of algebra in pre-algebra and give them that extra time.

The same is happening with calculus, though to a lesser degree: Calculus with precalculus.

OP, I understand the problem better now, and have realized that your DD is in precisely the same situation my younger DS was in. He was doing CC 8th grade math and getting As, and then the very end of the first semester hit, and he was back to being completely lost. He was getting stuck on precisely the same thing your DD is on. I had a meeting at school about it, and he ended up doing that pre-algebra course I mentioned.