In my last post, I was writing about students who are not following the standard algorithms as presented by our textbook, but still trying to solve forces problems in ways that make sense to them. One pattern of their work I talked about in that post was how these students don’t typically write out component using Fy = F sin(θ) and Fx = F cos(θ).

Here are some examples of how students solve problems without doing this to the following problem.

**Approach #1**

Here is an approach by a student who began with the standard approach, summing the forces in both the x and y directions.

- After summing the forces in the y-direction, you can see the student concluding that T_By = 4500, which is correct. The vertical component of Tb has to hold the weight.
- For the vertical direction, the student uses
**tangent** to relate 4,500 N Vertical component of tension, to the horizontal component of tension.
- Then to find the magnitude of the tension, they use the Pythogorean Theorem.

Note: The “standard” solution would have written T_By and T_Bx in terms of cosine and sine and directly solved for magnitude of the Tension, and only considered the value of components implicitly. This student solved for each of the components explicitly and combined them to find the magnitude.

**Another approach using a combination of trig and Pythagorean theorem. **

This approach also uses a combination of cosine and sine, but they never explicitly write an algebraic sum of forces statement. Because of this, you might be tempted to think that this student is just cobbling together random math in the hopes that it will work out. That was certainly my first response. But these concerns are largely gone once you see the image and how the student checks their work to show that the forces do in fact sum to zero.

In first picture,

- It seems like implicitly said vertical component of tension B = weight (4,410)
- Then, it used cosine to solve for the magnitude of tension

In the picture, below

- They implicitly say that horizontal component of Tb = Ta
- Used Pythagorean Theorem to solve for Ta

In this last picture,

- Student checked their answer using cosine and sine to make sure their vector triangle worked out
- Checked their answer by drawing a component only FBD to conclude that Fnet = 0

I actually think the work they did to check their work shows sophistication, and is better evidence of sense-making them someone who just writes down the standard algorithm.

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