##
* *Reflection: Quizzes
Distances in the Coordinate Plane - Section 3: Concept Development

I noticed this year that my students love formulas with a passion. To them, formulas are great because they can memorize them, plug numbers into them, and get correct answers if they calculate correctly.

I this the first lesson of the year requiring students to ** derive** a formula, I noticed that many students seemed to be tuning me out as I was teaching the deriving of the formula. These same students perked up when I revealed the actual formula. They copied the formula into their notes and eagerly began to attempt the practice problems that would require them to use the formula.

My main goal, though, was to have them be able to derive the formula and understand its connection to the Pythagorean Theorem. So I had to stop and explain in very clear terms that the main learning objective was for students to be able to derive the distance formula on their own and that this learning objective would be assessed on a quiz. This announcement definitely upped the level of engagement and concern about how to derive the formula. All of the sudden, students had questions about the PowerPoint. For example, students weren't clear on where x2-x1 and y2-y1 came from. These and other questions gave me an opportunity to go more into depth with my explanations and made the lesson more rich.

I encountered this type of situation several times during the year. These were times when I had a goal for students to derive a formula or prove a theorem and also be able to apply the formula/theorem. I would spend lots of time and energy teaching students how to do the derivations but sometimes I would not require them to do the derivations on the unit tests. It got to the point where students would just ask me, as I was teaching a derivation or proof, "Do we have to know this for the test?"

Ideally, students would be intrinsically motivated and I would not include these types of items on a unit test. These items require lots of time for students to complete and for me to grade. In the real world, where many of my students are motivated by grades, I have to find ways to quiz them on these derivations and proofs and assess them on unit tests. For example, I might ask on a multiple choice test, "In the distance formula, which of the following is the length of a leg of a right triangle?"...

This experience taught me that I need to strategically convey to students that this is a high school geometry course that is different in nature than their middle school geometry course. Whereas the focus in middle school was learning the formulas and applying them, the focus in the high school course is formally deriving the formulas and explaining why they work.

*Putting the focus on deriving*

*Quizzes: Putting the focus on deriving*

# Distances in the Coordinate Plane

Lesson 3 of 14

## Objective: SWBAT find the midpoint and length of a segment given coordinates of its endpoints

## Big Idea: May the formula be with you. In this lesson students wield the power of the distance formula.

#### Activating Prior Knowledge

*10 min*

**Where we've been: **In the past two lessons, students have dealt with segments lengths abstractly (segments expressed as algebraic expressions) and concretely (using rulers to measure).

**Where we're going:** In this lesson, students will determine segment lengths and midpoints using coordinate formulas.

So in this section of the lesson, we want to make sure that students have a command of the notation involved in measuring segment lengths, and that they have a conceptual understanding of segment length and midpoint of a segment.

Materials: Ruler with centimeter markings; Calculator (Optional); Paper

**Task**:

1. Create segment AB with length 13 cm. Be precise.

2. Find the exact midpoint of segment AB and label it S.

3. Measure to verify that S is the midpoint of segment AB. If it is, mark the diagram appropriately to communicate this.

4. Write a concise paragraph, using geometric notation where appropriate, explaining how you verified that S is the midpoint of segment AB.

See a Sample Paragraph here.

#### Resources

*expand content*

#### Concept Development

*30 min*

In this lesson there are some essential understandings that we want students to develop. First is the idea that ** coordinate planes do not exist in and of themselves; we define them in order to suit our purposes.** Next is the idea that in a two-dimensional coordinate system,

**Finally, the idea that**

*horizontal and vertical distances are simpler to determine than diagonal distances.***is important.**

*diagonal distances can be analyzed in terms of their horizontal and vertical components*

We’ll also be reinforcing some basic notation that students have been learning.

So to begin this section of the lesson, I give each student a sheet of grid paper. Working from the document camera, I give the directions contained in the resource, Distance Formula Concept Development: Student Instructions. [I intentionally do not give students a copy of the directions, nor do I show all of the directions at once. When students receive a list of instructions all at once, they tend to approach it as a checklist rather than a learning activity.]

After #14 on Distance Formula Concept Development Student Instructions, we pause for a mini-lesson on Deriving the Distance Formula.

*expand content*

#### Skill Development

*20 min*

Once students have a good feel for the distance formula, and a good set of worked examples for reference, it's time for them to work on some examples on their own.

The goal is for them to develop fluency and automaticity with the formula. Ideally, they will not have to draw out a right triangle every time they get a distance problem.

Instead, they should start to see the coordinates and have a good feel for the value of (x2 - x1) and (y2 - y1) just by asking themselves how far apart x1 and x2 are on the number line. Same for y1 and y2. And they should quickly be able to find the square root of the sum of the squares of these two quantities.

Which reminds me, make sure that students are skilled at squaring numbers (as opposed to multiplying them by two), that they know their basic multiplication facts, and that they can estimate irrational roots so that they know when their answers are unreasonable.

Finally, this is definitely a lesson suited for using calculators. As much as students need to learn math facts and number sense, they also need to know how to use basic technology and this skill needs to be developed too.

*expand content*

#### Check for Understanding

*25 min*

To check for understanding, let's see if students can transfer what they've learned about the distance formula to a 'not so novel' context: Perimeter and Area of Triangles, Rectangles and Squares. At this point, students may only be comfortable using the distance formula when instructed to do so. The goal of this section is to have students practice selecting the formula as a tool when appropriate. Having this type of command of the formula will be useful, for example, later when students will have to derive equations of parabolas and circles based on their geometric definitions (CCS HSG-GPE)

The Area and Perimeter in the Coordinate Plane resource is intended to be used as part of an in-class activity.

At this point in the lesson, I am intentional about fostering student independence. Students are all too willing to hand over the reins of their cognitive architecture to the teacher. For example, as they look at (as opposed to read) the directions, students will say things like “What is it you want me to do here?” I clarify that it is not ME that *wants* them to do anything. The DIRECTIONS are calling for them to do something and it’s their job to figure out what that is. So, the bottom line: we have to get students to read and interpret the directions.

The next type of question I get: “So should I use the distance formula to find the perimeter (or area)?” I recognize this, again, as an attempt to put the onus on the teacher. So I redirect with questions like:

Why does it make sense to you to use the distance formula?

What does perimeter mean?

What information would you need in order to calculate its perimeter?

What is the distance formula for?

What specifically could the distance formula tell you about the figure?

Another issue that arises is that students are unsure which coordinates to “plug into” the distance formula when presented with the multiple ordered pairs that form the vertices of a figure. Students will say things like, “Which points do I use?” The gut feeling seems to be that they should just throw all of the points into the formula and out would come the perimeter. Directing their attention to the graph, I ask, “What are you trying to find?” Some might say "The Perimeter"...."Ok then, if you had a ruler, what would you do first?" Then, based on their response I might say “Oh, if you want to find the length of segment AB, then which points would be appropriate to use in the formula?”

So the basic idea while enacting this section of the lesson is to place the onus for learning with the student and force them to grapple with question: When do I apply the distance formula?

*expand content*

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- UNIT 1: Community Building, Norms, and Expectations
- UNIT 2: Geometry Foundations
- UNIT 3: Developing Logic and Proof
- UNIT 4: Defining Transformations
- UNIT 5: Quadrilaterals
- UNIT 6: Similarity
- UNIT 7: Right Triangles and Trigonometry
- UNIT 8: Circles
- UNIT 9: Analytic Geometry
- UNIT 10: Areas of Plane Figures
- UNIT 11: Measurement and Dimension
- UNIT 12: Unit Circle Trigonmetry
- UNIT 13: Extras

- LESSON 1: Origins of the Geometric Universe
- LESSON 2: Line Segments
- LESSON 3: Distances in the Coordinate Plane
- LESSON 4: Exploring Midpoint Quadrilaterals
- LESSON 5: Investigating Points, Segments, Rays, and Lines
- LESSON 6: Formative Assessment Day 1
- LESSON 7: Introducing Angles
- LESSON 8: Angle Measurements
- LESSON 9: Basic Constructions
- LESSON 10: Measuring to find Perimeter and Area
- LESSON 11: Finding Perimeter and Area in the Coordinate Plane
- LESSON 12: Geometry Foundations Summative Assessment Practice Day 1 of 2
- LESSON 13: Geometry Foundations Summative Assessment Practice Day 2 of 2
- LESSON 14: Introduction to Transformations