After playing with the Contemplate then Calculate routine a
little bit at the beginning of the year, a workshop with Grace Kelemanik and Amy Lucenta inspired
me to really commit to planning a series of CthenC tasks as part of my Systems
of Equations unit. I started by looking at New Visions’ Algebra I systems tasks, which made me decide that my focus would be on using structure to solve
systems represented visually, with the hope that they would translate those
structural moves to solving systems algebraically. So I sought out a whole
bunch of visual systems and thought about how I would solve them, which led me
to the following list of big ideas.

## Big Ideas:

- Any time you can get an equation with one variable, you can find the value of the variable. You can make an equation with one variable…
- through substitution of either a value or another variable
- through elimination when you have two equations that have all but one of the variables the same
- The equations given to you don’t always lend themselves to creating equations with one variable. You can create new equivalent equations by…
- scaling an equation
- adding two equations together
- finding the difference between two equations
- adding/subtracting the same thing from each side of the equation

## Phase 1 of the Unit

I have spent the first two weeks of my unit
on understanding what a system of linear equations is, understanding what a
solution is, solving systems graphically, introducing standard form linear
equations, and revisiting slope-intercept form equations. During this time, I have been using the CthenC routine with a variety of algebraic systems represented
visually, with the goal ofing prompt most (if not all) of the big ideas above. I wanted each task to be able to be solved more than one way, so I intentionally gave more
equations than the bare minimum to solve. Here's the progression of tasks I have made so far:

## Task #1 (Most basic substitution and elimination, see what they do):

###
**Potential noticings that lead to structural thinking:**

- We noticed that the second and third equations were almost the same, but the second equation had one more star and a greater value. Therefore, we found the difference between the two equations and since 20 – 16 = 4, the red star has to equal 4.
- We noticed that two stars
were equal to a square, so we replaced the square in the 3
^{rd}equation with two stars. Then we had an equation with all stars, so we could divide the 16 equally between the 4 stars. -
We noticed that two stars
were equal to a square, so we replaced the square in the 2
^{rd}equation with two stars. Then we had an equation with all stars, so we could divide the 20 equally between the 5 stars. -
We noticed that two squares
equal a star, so we replaced the two stars in the 3
^{rd}equation with a square. Then we had an equation with all squares, so we could divide the 16 equally between the two squares.

###
**Annotations of actual student strategies:**

## Task #2 (have a couple of different elimination options):

### Potential noticings that lead to structural thinking:

- We noticed that the last equation had one more triangle than the first equation/We noticed that the first and last equations both have a chunk of two hearts and a square
- We noticed that second equation has two more triangles than the first equation/We noticed that the first and second equations both have a chunk of two hearts and a square
- We noticed that the second equation has one more triangle than the third equation/We noticed that the second and third equations both have a chunk of two hearts and a square

###
**Annotations of actual
student strategies:**

## Task #3 (keep an elimination possibilities, prompt substitution again):

### Potential noticings that lead to structural thinking:

- We noticed that a pen costs
$3 more than a notebook, so in the 2
^{nd}equation we can replace the pen with $3 and a notebook - We noticed that the third equation has one notebook and one more pen than the second equation/We noticed that the second and third equations both have a chunk of two notebooks and a pen

###
**Annotations of actual student strategies:**

## Task #4 (substitution hopefully more appealing than elimination):

### Potential noticings that lead to structural thinking:

- We noticed that a slice of
cake costs $2 more than an ice cream cone, so we can replace the slice of cake
in the 1
^{st}equation with $2 and an ice cream cone. - We noticed that the 2
^{nd}equation has 2 more pieces of cake than the first equation and that the items in the second equation cost $7 more than the ones in the first equation/ We noticed that the first and second equations both have 3 ice cream cones and a piece of cake

## Phase 2 of the Unit (up next):

After the first two weeks, we
go into solving systems algebraically using elimination and then using
substitution. I am planning for my CthenC tasks to have only one method for
solving and for that method to directly link to the algebraic manipulations
students will be doing that day or the next day.

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