Create Transversal Problems
Lesson 6 of 16
Objective: SWBAT create a problem about the angles surrounding a transversal.
I suggest some teaching techniques around this lesson in this video:
The idea is for students to create a transversal problem involving algebraic thinking, specifically around balancing an equation. The goal is to create a problem that looks something like this: Transversal algebra sample
The thinking of this project is around questions like, "how do I create expressions that fit within the relationship between alternate interior, exterior and other angles formed between parallel lines and a transversal?"
The starting point is to have students use graph paper to sketch two parallel lines and then intersect them with some transversal. Then they can use a protractor to measure the angles formed at the intersection. This is the start of the project. One misconception to look for is to watch and see how students round their angles, since the sums of the angles must match the complementary and supplementary pairs.
I like to pull an example on my document camera and review the rounding process.
Once students are confident that they have the correct angle measures, we move onto the algebraic thinking. The goal of the project is to have students create a transversal diagram so that the expressions accurately represent the measures of the angles in their diagrams. One partnership will create the diagram and problem and then later we will swap and have another partnership find the angle measures using the algebraic expressions we create.
Again we use an example from the class to help students understand the purpose of the project.
In one example, we might have two supplementary angles 30 and 150. One approach is to let x = 30 and then decompose 150 into some addend pair, like 30 + 120. Then students can rewrite this in terms of x. 30 + 120 = x + 3x = 4x. However, we encourage students to mix it up. They might choose to write 30 + 120 as 30 + 4x.
To the person solving the they would only know that the two angles are x and 30 + 4x. Then to solve this problem (we share them later in class), a students would have to solve for x in the equation x + 30 + 4x = 180
Another (more advanced) approach is to set x to some value. For example, a group might have two alternate interior angles, measuring 40 degrees each. We could set x equal to a factor of 40. We could really pick anything for x and still create a fun problem. It might make sense to use a factor when first introducing the idea.
For example, we could set x = 5. Then 40 = 8x. However, we could also write 40 in terms of 5 using any combination of terms. for example if we start with 11x and we know that x = 5, we could finish the expressions by asking, "if 11x = 55, 55 minus what equals 40?" Then we could write the expression 11x -15. In this case the expressions would then be 8x and 11x - 15. This is a tougher problem since students have to solve for x and then substitute x to find the correct angle measure.
Throughout this project, I conference with students as they work. It might take the entire class and some work at home for them to finalize their expressions. Once the expressions are finished, I conference with students by asking them for their expressions and solving the angles with them. We often have fun here and have conversations around tracing the source of an error or about why this process of decomposition works.
Highlight and reflect upon a few key student problems. A common misconception is that the value of x doesn't match the correct angle measure. Another more common problem is that students will leave one angle measure intact and then only write another angle as an algebraic expression.
For example, one partnership gave the supplement angle measures of 120 and 6x. This problem is far too easy. Here x = 10 and the other angle measure is 60. The question I then get is, "how could I make this tougher?"
To scale up this work, I revisit our initial discussion and suggest that x = some number. We might pick 4. Then 4 + 116 = 120 or 8 + 112 = 120 and we could write 2x + 112 = 120 for one of the expressions and the other angle of 60 could be written as 1/2x + 58. If they want to make the problem even more difficult, I encourage them to factor. For example they could write 1/2(x - 29) for one of the angles.
The goal is to swap and share project, so we reserve time to swap and solve other project problems. I give students post its solve for the angle values. Then they check their work with the project they are reviewing and discuss the experience with each other.