# Meet Me in the Middle

## Discovering Midpoint Formula Through Estimations

K20 Center, Laura Halstied, Michell Eike, Samaya Williams, Laura Young, Lydia Baker, Kate Raymond | Published: August 15th, 2022 by K20 Center

• Subject Mathematics
• Course Geometry, Intermediate Algebra
• Time Frame 125 minutes
• Duration 3 class periods

### Summary

In this lesson, students will estimate population data leading them to discover the midpoint formula. They will then apply the formula they created to various other real-world applications.

### Essential Question(s)

How can we mathematically determine when something is "in the middle?"

### Snapshot

Engage

Students estimate the population of their school, town, state, country, and world.

Explore

Students use census data to estimate populations for several years and, in doing so, begin to construct a formula for a midpoint.

Explain

Students explain their method for estimation and discuss the different approaches used by their classmates to refine their definition and formula for midpoint.

Extend

Students brainstorm about the usefulness of a midpoint formula and apply their constructed formula to several real-life applications

Evaluate

Students use their midpoint formula to justify a decision about a road trip.

### Materials

• Lesson Slides (attached)

• Exploring Population handout (attached; one per student; printed front/back)

• Story Problem Cards handout (attached; one per group; printed front only)

• Story Problem Cards (Sample Responses) document (attached; for teacher use)

• Car Trouble handout (attached; one per student; printed front only)

• Car Trouble (Sample Responses) document (attached; for teacher use)

• Student devices with Internet access

• Calculator (one per student)

• Colored pencils (2 colors per student)

• Ruler (one per student)

• Graph paper

• Scissors (optional)

• Document camera (optional)

### Engage

15 Minute(s)

Use the attached Lesson Slides to guide the lesson. Review the essential question and objectives on slides 3 and 4 with students. Transition to slide 5 and ask students: "How much do we know about the population of our community?"

Use the Elbow Partner strategy and have students get out a piece of paper and work together to create an educated estimate for the following populations:

• our school

• our town

• our state

• our country

• our world

Provide time for students to estimate the populations.

Move to slide 6 and provide students with the real number of your school’s population and the actual population values of your town, state, country, and world.

Ask students to share their population amounts.

### Explore

30 Minute(s)

Move to slide 7 and pass out the attached Exploring Population handout to each student as well as two different colored pencils and a ruler.

Instruct pairs of students to use the following websites (shortened URLs are provided on slide 7) to complete Part A of their handout:

When they are finished, direct students to put their devices away.

Show slide 8 and direct students’ attention to Part B and have them plot their data from Part A. Then have students connect their dots, using their ruler, to create a line graph of the data.

Display slide 9 and have students use the data from Part A and Part B to estimate only the town’s and state’s populations for the years in the table for Part C. Guide students to not yet find the country's population.

### Explain

40 Minute(s)

Display slide 10 and ask students to explain how they found their estimates for Part C. Ask students to raise their hand based on the approach they used to find their estimate.

• How many used their graph?

• Who used a formula?

• Who found those values using a different approach? What was your approach?

Continue until all approaches have been shared. Ask for a group to volunteer their reasoning and display their data using a document camera if necessary.

After discussing the different approaches, move to slide 11. Ask students to consider the discussed approaches and use one of the shared methods that was different from their original approach to find their country’s population.

Display slide 12 and direct students’ attention to Part D of their handout. Use the Think-Pair-Share strategy and have students answer the question: "What do you notice about these points?"

Transition to slide 13 and direct students’ attention to Part E. Again, using the Think-Pair-Share strategy, ask students: "What must be true about the year you are estimating the population for in order to use your method from Part C?"

Next, have students finish Part E by writing a definition for midpoint (a word that hopefully came up during the earlier discussions) and writing a formula.

Make sure all students agree that the point they estimate must be in the middle of two known values.

Transition to slide 14. Provide the definition of midpoint to students and have students write it at the bottom of their handout.

Ask students to work in their group and determine if the estimates they found in Part C fit the definition of midpoint. Make students aware that they must be able to explain how it fits or does not fit the definition and defend their reasoning. After providing time for students to think this through, have each group share their reasoning.

Ask the class what they think the tick marks tell us about the location of the midpoint. Emphasize to students the relationship between the tick marks and the location of the midpoint. If the two segments are congruent, then that middle point is exactly in the middle and is therefore the midpoint.

Move to slide 15. Now that students have verified that their estimates are midpoints of line segments, tell them to continue to work with their partner to write a formula that allows them to find the coordinates (both x and y) of a midpoint using only the endpoints. Have them use the estimates they had already made in Part C to verify the formula they write.

As groups finish creating and verifying their formulas, have each group share the formula they created by going to the board and writing their formula. After all the formulas are recorded, ask students to discuss which formulas seem the most reasonable and which seem to be most user-friendly. Have the class vote on one formula to use as the midpoint formula.

If their chosen formula is written differently than the standard formula, introduce students to the standard formula for midpoint by displaying slide 16. Have students compare the class-created formula and the standard formula for validity and ease of use. If the class formula is invalid, discuss the reasons why. If the class formula is valid, but in a different form, discuss the differences but stress that either method is valid (and allow them to use either on future assessments).

### Extend

30 Minute(s)

Put students into groups of 2–3 and move to slide 17. Ask students to think about how the midpoint formula can be used in real life. Provide some time for students to discuss in their groups for a few minutes. Ask for volunteers from each group to share their ideas and record their responses on the board or somewhere else where all students can see them.

Show slide 18 and give each group of students a set of Story Problem Cards and a piece of graph paper and notebook paper. First, have students examine the six cards and only look for similarities and differences before solving. Ask for a few volunteers to share after students have looked at each card.

After providing time for students to examine the cards, tell students to begin solving the problem from the cards on a piece of notebook paper.

Every 10 minutes, or as needed, take a break from group work to call on each group to share an update of their progress. Ask each group to share a challenge they have discovered. As each group shares a challenge, ask the other groups if any of them have encountered the same challenge and, if so, how they solved it. Continue discussing the problems this way until all groups have shared a challenge. If a group says they have not yet encountered a challenge, have them share one of the solutions they have found instead. Encourage other groups to ask questions about the solutions presented. Continue in this manner until the groups have solved all six problems.

### Evaluate

10 Minute(s)

Transition to slide 19 and pass out the attached Car Trouble handout to each student to use with the Exit Ticket strategy. Be sure that they justify their answers using mathematical words and symbols. Collect the handout from students to assess understanding of the lesson content and inform future lessons.