##
* *Reflection: Intervention and Extension
Problem Solving with Quadrilaterals - Section 2: Independent Problem Solving

I sometimes use the Quadrilateral Practice problem set for scaffolding and extensions within a class. If students are grouped homogeneously, all of the students in the classroom can work on exactly the same problems, but with different expectations for different groups. For struggling students, you might ask for less detail in their answers, while for those students who are ready to be challenged, you might make it clear that “all possible information” includes all angle and segment measures whenever possible.

As an example of this, for Example_1 in the rectangle section students could fill in their diagram so that the opposite sides are congruent and all four angles of the rectangle are right angles. They could go further and use the Pythagorean Theorem or remember their Pythagorean triples to fill in the length of the diagonal. Or, as a further extension, students could go even further and find the angle measures created by the diagonals, by using trigonometry. I suspect that asking students to solve each and every one of the problems to this latter level of detail could become excruciating for the students. This is where planning ahead and giving careful thought to the problems comes in. I might, before making copies, for example, write a star next to all the problems on which I would like the more able students (or all of the students) to use trig. Or I might ask groups of students needing a challenge to select one problem in each section, when possible, to use trig.

There are many possibilities – you just need to be clear in your own mind, going into this problem set, where you want your students to go with it.

*Intervention and Extension: Grouping Students to Challenge Them*

# Problem Solving with Quadrilaterals

Lesson 3 of 5

## Objective: Students will be able to apply their knowledge of quadrilaterals to numeric and algebraic problem solving.

#### Guided Practice

*25 min*

To begin today's class, I give my students a problem set entitled Guided Practice. This handout begins with a chart that asks the students to recall what they have learned about the sides, angles, and diagonals of the members of the parallelogram family. I expect my students will refer to their Family Tree of Quadrilaterals as they fill in the chart.

The problem set then includes one or two numerical problems for each type of the quadrilaterals These problems are not particularly challenging. They are a warmup designed to reinforce the students’ basic understanding of these figures. One of the problems (#5b) offers the possibility of applying knowledge with regard to the sides of an isosceles right triangle. My students may choose to pursue this strategy, and if so, it will be a point of discussion to prepare for what follows in the lesson. Overall, I want this Guided Practice to go quickly.

To help students work precisely, I ask that they work in groups, one problem at a time. I check in with the whole class, one problem at a time, when it appears each group has completed a problem. As we go over each problem, I am careful to ask “How do you know…*?”.* I expect my students will start to follow this practice in their group work:

**In problem #2a, it is important to ask, “How do you know a = 8 and b = 10?” and I expect my students to answer, “Opposite sides of a parallelogram are congruent.” **

I think it is important to ask students to justify over and over again in this unit – it helps them with their work on quadrilateral proofs and, by frequently writing and speaking the justifications, also helps them to remember the different features of all the different quadrilaterals.

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#### Independent Problem Solving

*18 min*

Now it’s time to allow the students to work at their own pace with the handout entitled Quadrilateral Practice. As students work I circulate around the room, watching for any students who might struggle, and also continue to challenge individual students with that question, “How do you know?”

**Teacher's Note**: Before using this problem set, it is **really** important that you, the teacher, spend some time looking at it beforehand. I have indeed made the mistake of just grabbing it out of my file and not looking at it ahead of time. It created lots of confusion!

Many of the problems read, “Fill in all possible information.” These problems can be approached from a very basic level or, more ambitiously, using Pythagorean triples, Special Right Triangles, and trigonometry. Sides can be found in either simplest radical form, or rounded to the nearest tenth. These are all decisions that you need to make before setting the students loose on the problems.

Each section of the problem set includes one problem in which the students must draw their own diagram. I have found that it is very important to include this skill, and I will address this skill in a future lesson, as well. But it seems like each and every year that I teach this unit, I am caught off guard at least once by a student who becomes stumped when faced with having to draw his or her own diagram. This year, for example, a young lady raised her hand *during the unit test* and asked if I could show her what a trapezoid looks like. Yet there were two diagrams of isosceles trapezoids on the very next page of the test. Guess I need to keep working at it!

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#### Wrap-Up + Homework

*2 min*

It is typical that many of my students are not finished with the Quadrilateral Practice problems at the end of this lesson. Instead of reviewing the answers, I ask them to stop what they are doing to ask for feedback:

- How are the problems going?
- Are there any concepts in particular with which they are struggling?
- Which kinds of problems are hard and which are easy?

I let them know that we will continue with this problem set in the next class meeting, and I hand out tonight's Homework Proof.

*expand content*

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