##
* *Reflection: Gradual Release
Exploring Midpoint Quadrilaterals - Section 2: Independent Practice

When students were asked to determine whether HJKL had any congruent or parallel sides, several of them were unclear on how they should go about doing that. These students tended to call me over to their desk and say something like, "What do you want us to do here?"

My response was typically, "*I* don't *want* you to do anything...what is it that the *directions* are asking you to do?" In that way, I was trying to put the onus on the students to interpret the directions rather than me doing it for them.

The typical student response to my question would be "They want me to see if any of the sides are congruent."

My response: "What does it mean for two sides to be congruent?"

Student: "It means they're the same."

Me: "What about them is the same?....They're color?, Their smell? What?"

Student: "Oh their lengths are the same."

Me: "So how could you determine whether two segments are congruent?"

Student: "I could use a ruler and measure."

Me: "Yes, but in this exercise you will not have a ruler available to you; all you have is the coordinates of the vertices of HJKL. What could you do?"

So after this type of exchange, the student usually gets the idea that they should use the distance formula to determine if there are any congruent sides. Before I leave, though, I emphasize to the student that the type of questioning I put them through is the type of internal dialogue they need to be having with themselves as they make sense of the instructions and find a way to approach the task. I might, then, ask them to replay the logical sequence for me. That should go something like:

"I'm trying to determine if HJKL has any congruent sides. If there are congruent sides, then their lengths must be equal. I can use the distance formula to find the lengths of the sides and determine if any of them are equal to each other."

This experience reminds me that some students are very hesitant to take initiative when they have not been given explicit directions on what to do. Even though we practiced the distance formula in the Activating Prior Knowledge section, and even though it was not that much of a stretch to apply the formula to determine if there were congruent sides, the sheer fact of not being told explicitly what to do threw some students for a loop. I have no doubt that they understand the concepts; I just think they need to get more comfortable taking that first step to think about what is being asked and take the risk of trying to respond appropriately.

*Getting students to take initiative*

*Gradual Release: Getting students to take initiative*

# Exploring Midpoint Quadrilaterals

Lesson 4 of 14

## Objective: SWBAT find midpoints of segments; SWBAT use distance formula to verify that segments are congruent

#### Activating Prior Knowledge

*15 min*

**Where We've Been:** Students have just learned the distance and midpoint formula. Or, should I say (given this is an 8th grade standard) students have just had their memories refreshed on these formulas.

**Where We're Going:** Soon we'll be getting into analytic geometry and proof in the coordinate plane. Students will need to have full command of these formulas.

In this section I give students Activating Prior Knowledge_Coordinate Plane Formulas, which has three problems involving the relevant formulas for the day.

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#### Independent Practice

*45 min*

In this section students will be following directions (on their own) to complete the Independent Practice: Midpoint Quadrilaterals activity. As students are working, I walk around. I do my best not to answer questions. Instead I pose open ended questions that force students to read and interpret directions and/or confront their misconceptions. I also check to see that students are meeting the specifications that were laid out for quadrilateral ABCD. If they've failed to meet one or more of the specs, I'll ask them to explain to me how they met ALL of the specifications. As they do, I'll pop an open-ended question that forces them to confront their misconception...if they don't see it for themselves first.

*expand content*

#### Cooperative Activity

*20 min*

As we know, the quadrilateral formed by joining successive midpoints of the sides of any quadrilateral is a parallelogram. In this section of the lesson, I get students together in groups of three or four to compare the work they have done with midpoint quadrilaterals and try to arrive at some general conclusions.

Each will be given the Cooperative Activity: Midpoint Quadrilaterals handout to complete.

Note: If any students in a group have congruent quadrilaterals ABCD, it's a good idea to put them into different groups.

When the time has elapsed for the activity, I randomly call students to share their paragraphs with the class. As students read them, I provide feedback.

By the end, I want everyone to walk away having a basic understanding that it seems whenever we join midpoints of the sides of a quadrilateral, two pairs of opposite sides will be both parallel and congruent.

*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