Which Variable to Solve for? (Part 2)

5 min

Narrative

In this Warm-up, students are given a simple equation in three variables and are prompted to rearrange it to solve for a particular variable. In each question, only the value of one variable is given, so students need to manipulate the equation even when some quantities are unknown. The work here prepares students to rearrange other variable equations later in the lesson.

As students work, look for those who substitute the given value before rearranging and those who first isolate the variable of interest before substituting. Invite them to share their approaches during class discussion.

Student Task

In an earlier lesson, you saw the equation V+F−2=EV + F - 2 = E, which relates the number of vertices, faces, and edges in a Platonic solid.

  1. Write an equation that makes it easier to find the number of vertices in each of the Platonic solids described:

    <p>Three dimensonal octahedron. Two triangular pyramids joined at rectangular bases.<br>  </p>

    1. An octahedron (shown here), which has 8 faces. 
    2. An icosahedron, which has 30 edges.

  2. A Buckminsterfullerene (also called a “Buckyball”) is a polyhedron with 60 vertices. It is not a Platonic solid, but the numbers of faces, edges, and vertices are related the same way as those in a Platonic solid.

    Write an equation that makes it easier to find the number of faces that a Buckyball has if we know how many edges it has.

Sample Response

    1. V=E−6V = E - 6 (or equivalent)
    2. V=32−FV = 32 - F (or equivalent)
  2. F=E−58F = E - 58 (or equivalent)

Synthesis

Select previously identified students to share their responses and strategies. Record and display for all to see the steps they take to rearrange the equations. Emphasize how each step constitutes an acceptable move and how it keeps the equation true.

Make sure students see that we can either substitute known values into the given equation before rearranging it, or we can rearrange the equation first before substituting known values. In the examples here, it doesn't matter which way it is done. Ultimately, we were solving for VV in the first question and for FF in the second question.

Explain that there will be times when one strategy might be more helpful than the other, as students will see in subsequent activities.

Anticipated Misconceptions

Some students may be unclear what it means to write an equation that "makes it easier to find the number of vertices (or faces)." Remind them of the work in an earlier activity. In the post-parade clean-up activity, for instance, they wrote the equation ℓ=2n\ell = \frac{2}{n} to quickly find the length of the road section that each volunteer would clean up, ℓ\ell, if there were nn volunteers. They wrote n=2ℓn=\frac{2}{\ell} to quickly find the number of volunteers, nn, if each volunteer were to clean up ℓ\ell miles.

Standards
Addressing
  • A-CED.4·Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. <em>For example, rearrange Ohm's law V = IR to highlight resistance R.</em>
  • A-CED.4·Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. <em>For example, rearrange Ohm's law V = IR to highlight resistance R.</em>
  • A-CED.4·Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. <em>For example, rearrange Ohm's law V = IR to highlight resistance R.</em>
  • A-CED.4·Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. <em>For example, rearrange Ohm's law V = IR to highlight resistance R.</em>
  • A-CED.4·Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. <em>For example, rearrange Ohm's law V = IR to highlight resistance R.</em>
  • A-CED.4·Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. <em>For example, rearrange Ohm's law V = IR to highlight resistance R.</em>
  • HSA-CED.A.4·Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. <span>For example, rearrange Ohm's law <span class="math">\(V = IR\)</span> to highlight resistance <span class="math">\(R\)</span>.</span>

20 min

10 min