EQ6 CheckIt

[AbbyANoble]

Reviewer: @kathypinzon

Instructions for tasks 1 and 3: Solve the following rational equation.

Three tasks in the problem. Answers can be whole numbers or fractions. Give fractional answers as fractions, not decimals/decimal approximations.

Task 1 should simplify to a linear equation. Pull from examples like:

• 3 = -4(x-4)
• 8/(x-3) = -6
• 3 = 1/4 - 5/x
• 10 - 8/(3x) = 5

Task 2 Instructions:

Suppose you manipulated $\frac{x+6}{x+2}-1=\frac{x+5}{x+8}$ into the equation $x^2+9x+8=0$. Which solutions of the quadratic equation are also solutions of the rational equation? Explain your reasoning.

• 4+1/(x+3) = -1/((x+2)(x+3)) (one solution!)
• (x-7)/(x-3) - 1 = (x-3)/(x-2) (one solution, variables in numerator)

Task 3 should simplify to a quadratic equation. Should have two solutions. Pull from examples like:

• 2 = x - 24/x
• 15 - 20/x = x + 24
• -6/((x+1)(x+6)) = -2 - 1/(y+6) (two solutions!)
• 6-5/(x+2) = -6/(x-2)
• (x+6)/(x+2) - 1 = (x+5)/(x+7) (two solutions, variables in numerator)

[StevenClontz]

Something I'm noticing here: your task 2 has a lot of special cases that wouldn't be asked on every question. Some require factoring, some require eliminating solutions from the denominator, etc.

I wonder if instead of having a lot of edge cases that may or may not pop up in task 2, we instead ask a few quicker yes/no questions about each possible special case (and prompt for a reason). Something like (I didn't check the actual math):

Suppose you manipulated $\frac{x+6}{x+2}-1=\frac{x+5}{x+8}$ into the equation $x^2+9x+8=0$, which factors to $(x+8)(x+1)=0$. Are the solutions $x=-1$ and $x=-8$ both valid for the original equation? Why or why not?

[AbbyANoble]

My only issue with those type of problems is that students don't actually have to solve a rational equation, which is the objective. They're just checking to see whether the solution is a solution, which is only a very small part of the objective. (And the easiest part if removed from the solving.)

We wouldn't necessarily have to use all of the types I listed. We could trim it down some. Or have two tasks and group these differently. Solve for one, and answer questions about the other.

[StevenClontz]

Sorry, I meant, have task 2 be a rational equation that solves to a quadratic without any edge cases, and introduce a task 3 that asks students to handle (perhaps a few) edge cases.

[AbbyANoble]

Ok. Yes, then I do like that. I generally have my students solve the "easier" ones of these on assessments.

[StevenClontz]

In general my CheckIt strategy is to make sure anything I might want assessed to appear on every generated exercise, balanced with making managable tasks for a single quiz. Having students fully work out some things, then quickly handle the edge cases is one way to balance this I think. (And add some randomization of order so task 4 isn't always the same edge case.)

[AbbyANoble]

Yeah. This is also why I try to make some of those harder edge cases show up in a homework set. So they are exposed to and held accountable for them, but not necessarily in a quiz/test environment.