MBots on the Coordinate Plane

Engineers design, build, program, and test robotic systems that use coordinate systems and other spatial information to determine position and navigate through an environment. Precise measurements, directions, and movement commands enable robots to travel to specific locations and perform designated tasks. Similar concepts are used in warehouse and manufacturing robots, autonomous vehicles navigating between locations, NASA rovers exploring the surface of Mars, and robotic systems assisting with surgical procedures. Understanding coordinates, positive and negative directions, spatial relationships, and transformations provide a mathematical foundation for designing and controlling robotic systems. The mBot provides a simplified model of this engineering application as it is controlled or programmed to navigate to specified locations on a coordinate plane.

After this activity, students should be able to:

• Accurately label and describe the parts of a coordinate plane, including the x-axis, y-axis, origin, quadrants, and ordered pairs.
• Graph, interpret, and navigate to ordered pairs in all four quadrants using positive and negative integers.
• Determine and apply reflection rules across the x-axis and y-axis to identify and locate image coordinates.

Students should have:

• An understanding of positive and negative integers.
• The ability to read and write ordered pairs in the form (x, y).
• A basic understanding of horizontal and vertical movements.
• Familiarity with identifying the x-axis, y-axis, and origin on a coordinate plane.
• Experience counting units on a grid accurately and consistently.
• Introductory familiarity with using a Chromebook or tablet.

Today, you are not just math students… you are robotics engineers! Robotics engineers control and program robots to move to exact locations in warehouses, hospitals, and even on Mars. If a robot is off by just a few centimeters, it can crash, damage equipment, or miss its target. Precision matters.

(Point to the coordinate plane.) Engineers don’t just guess where things are. They use coordinate systems just like this one. Who has ever used GPS on a phone? (Allow hands.)

What happens if the GPS location is slightly wrong? (Possible answers: You get lost, wrong address, wrong house, wrong turn.) Exactly. Coordinates guide movement in the real world.

(Walk to the origin and stand on it.) This point is (0,0). Engineers call this the origin. It’s the starting reference point. If I tell a robot to move to (4,3), what information am I giving it? (Possible answers: Go right 4, up 3; x first, then y.)

Today your challenge is to program and drive an mBot to precise coordinates — and later, to its reflection. That means you’ll need to think like engineers and test, adjust, and improve.

(Write on board: Precision. Direction. Coordinates. Reflection.) Before we begin, I want you to think about this: Where else in real life do we use grids or coordinates? (Possible answers: video games, maps, city blocks, football plays, Minecraft, architecture plans.)

When have you experienced something that required exact positioning? (Give students 30 seconds to talk with their shoulder partner. Circulate and listen. Call on 2–3 students to share.)

As engineers, your job today is to solve a design problem: How do we move a robot accurately in a mapped space? What questions do we need to investigate before we start? (Possible student questions: Does order matter? What happens if we mix up x and y? How do negative numbers change movement? What if the robot overshoots?)

Excellent questions. Keep those in mind. Engineers don’t just follow directions — they ask why, test ideas, and refine their thinking.

Let’s begin at the origin.

Background  

The Coordinate Plane

The coordinate plane consists of a horizontal x-axis and a vertical y-axis that intersect at the origin (0, 0). The axes divide the coordinate plane into four regions called quadrants, numbered I through IV. Ordered pairs are written in the form (x, y). The x-coordinate indicates horizontal position, with negative values to the left of the origin and positive values to the right. The y-coordinate indicates vertical position, with negative values below the origin and positive values above the origin. When plotting an ordered pair, the x-coordinate is addressed first, followed by the y-coordinate.

Movement on the Coordinate Plane

Horizontal movement on the coordinate plane changes the x-coordinate while the y-coordinate remains the same. Vertical movement changes the y-coordinate while the x-coordinate remains the same. A point located on the x-axis always has a y-coordinate of 0, while a point located on the y-axis always has an x-coordinate of 0.

Reflections

A reflection is a geometric transformation that produces a mirror image of a point or figure across a specified line. When a point is reflected across the x-axis, the x-coordinate remains the same and the y-coordinate changes sign. For example, (3, 4) becomes (3, -4). When a point is reflected across the y-axis, the y-coordinate remains the same and the x-coordinate changes sign. For example, (3, 4) becomes (-3, 4).

Coordinate Systems and Robotics

Coordinate systems can be used in robotics to describe position and movement in physical space. Robotic movement relies on precise instructions, measurements, direction, and spatial relationships. Errors in direction, distance, sign, or sequence can result in a robot reaching a different location than intended. Testing movement, observing the resulting position, identifying sources of error, and adjusting instructions are examples of the iterative troubleshooting processes used in robotics and engineering.

mBot Setup and Control

The mBot can be controlled manually or through programmed commands, depending on the configuration used for the activity. The mBot Setup Sheet (PDF) provides information about setting up and controlling the mBot.

Before the Activity

• Gather materials for each group: charged mBot, tablet/Chromebook with mBlock installed, laminated coordinate plane (≈ 60 cm × 60 cm), dry-erase markers, eraser, pencils, and directions sheet.
• Print copies of the Activity Exploration Sheet (PDF) (1 per student), the Coordinate Plane Printing Sheet (PDF) (1 per student), the Reflection Exit Ticket (PDF) (1 per student), and the mBot Warmup Activity (PDF) (1 per group).
• Fully charge and test all mBots and device connections (Bluetooth) before class.
• Calibrate robots to ensure they drive straight and measure approximately 5 cm per grid unit.  If not moving in a straight line, inspect the robot wheels and align by adjusting the tires or screwing them more tightly to the chassis.
• Prepare a large demonstration coordinate plane for modeling (see Coordinate Plane Printing Sheet (PDF) Suggestions for instructions regarding the preparation of this coordinate plane).
• Review coordinate plane vocabulary and reflection rules.
• Arrange classroom space to allow at least 1 m × 1 m (~3 ft. × 3 ft.) per group for safe movement.

During the Activity

Part 1: Introduction & Concept Review (10 minutes)

1. Divide the class into groups of three or four students.
2. Distribute materials to each group (e.g., mBot, device, coordinate mat, markers, mBot Warmup Activity (PDF)). 
3. Ask students to complete the mBot Warmup Activity (PDF) as their first task.
4. Collect warmup activity sheets when complete.
5. Give one Activity Exploration Sheet (PDF) and one Coordinate Plane Printing Sheet (PDF) to each student.
6. Direct students to label the coordinate plane sheet with the x-axis, y-axis, origin, Quadrants I–IV, and positive/negative directions.
7. Have students then write down the definitions of each in Part 1: Vocabulary and Labeling of their Activity Exploration Sheet (PDF).
8. Review ordered pair format (x, y).

• • Write the ordered pair (3, -2) on the board.
  • Ask: “Which direction do we move first?” (Possible answer: x first, then y.)

9. Model how horizontal movement affects only the x-coordinate and vertical movement affects only the y-coordinate. (See Teacher Script (PDF) for one way to introduce these concepts.)

Part 2: Axis Navigation (10 minutes)

10. Give one laminated coordinate plane mat to each group.
11. Instruct groups to place their mBot at the origin (0,0).
12. Have students drive the mBot along the positive x-axis, then return to the origin. (Note: Details on how to set up the mBot for manual control or programmed control are in the mBot Setup Sheet (PDF).) 
13. Optional: If students are unfamiliar with the mBots, give them a few minutes to become comfortable with them and to get practice controlling in their groups.
14. Repeat along the positive and negative y-axis and along the positive and negative x-axis.
15. Instruct students to complete Part 2: Exploring Movement of their Activity Exploration Sheet (PDF).
16. Ask guiding questions: “What value stays the same when moving along the x-axis?” (Possible answer: y = 0.)

How can you move the robot forward? Backward? Turn left? Turn right?CopyrightCopyright Original photo from project team

Part 3: Ordered Pair Challenge (15 minutes)

17. Provide a list of ordered pairs (e.g., (4,3), (-2,5), (-3,-4)).
18. On the group’s coordinate plane, have students plot each point with a marker before moving the robot.
19. Give students time to program or manually drive the mBot to each coordinate. (Note: Details on how to set up the mBot for manual control or programmed control are in the mBot Setup Sheet (PDF).)
20. Encourage students to each take a turn controlling the driving of the mBot to a coordinate.
21. Have students complete Part 3 of their Activity Exploration Sheet (PDF).
22. Circulate and check students' plotted coordinates and mBot movements. If students get incorrect answers, instruct groups to troubleshoot and adjust movement distances.

How can you move the robot from (5,4) to (-3, -3)?CopyrightCopyright Original photo from project team

Part 4: Reflection and Challenge (10 minutes)

23.   Provide an original coordinate (e.g., (3,2)).

24. Instruct students to determine its reflection over the x-axis and drive to that location. (Answer: (3,-2))
25. Instruct students to repeat with reflection over the y-axis (-3, 2). 
26. Have students complete Part 4: Reflections of their Activity Exploration Sheet (PDF).
27. Discuss patterns in how coordinates change. Samples of patterns of change for going over the x-axis and y-axis are below:

• • Over the x-axis: The y-coordinate changes sign, but the x-coordinate stays the same.
     Example: (3, 2) → (3, -2)
  • Over the y-axis: The x-coordinate changes sign, but the y-coordinate stays the same.
     Example: (3, 2) → (-3, 2)

28. Have students complete the first part of Part 5: Challenge of their Activity Exploration Sheet (PDF).

Can you program the mBot to move automatically from Quadrant I to Quadrant IV?CopyrightCopyright Original photo from project team

Part 5: Exit Reflection (5 minutes) worksheet.

29. Have students complete the second part of Part 5: Reflection of their Activity Exploration Sheet (PDF).
30. Distribute one Reflection Exit Ticket (PDF) to each student.
31. Have students complete the Reflection Exit Ticket (PDF) explaining reflection rules and quadrant identification.
32. Facilitate a brief whole-class discussion to summarize key learning.

Pre-Activity Assessment

Warmup Activity: Before beginning the activity, have students complete the three-question mBot Warmup Activity (PDF) to assess their prior knowledge of ordered pairs, coordinate axes, and quadrants. Questions include plotting the point (-3, 4) and identifying its quadrant (Quadrant II), explaining what the x-coordinate represents (horizontal position; left or right), and identifying the x-coordinate of a point located on the y-axis (0). Review responses to identify misconceptions, such as reversing the x- and y-coordinates, before proceeding with the activity.

Activity Embedded (Formative) Assessment

Activity Exploration Sheet: As students complete the mBot navigation challenges, have them complete the Activity Exploration Sheet (PDF) to demonstrate their understanding of coordinate-plane vocabulary, movement along the axes, ordered pairs, and reflections.

Work Questions: Circulate among groups and observe whether students correctly label the coordinate plane, accurately plot coordinates before moving the mBot, navigate along the correct axes and directions, and correctly apply reflection rules. Ask questions such as, “Why did you move horizontally first?” (Possible answer: Because the x-coordinate comes first in an ordered pair.) and “What changes when reflecting a point over the x-axis?” (Possible answer: The y-coordinate changes sign while the x-coordinate remains the same.) Use student responses and observations to identify misconceptions and provide feedback as students troubleshoot and revise their mBot movements.

Post-Activity (Summative) Assessment

Reflection Exit Ticket: At the conclusion of the activity, have students independently complete the Reflection Exit Ticket (PDF) to demonstrate their understanding of quadrants, ordered pairs, and reflections. Questions include identifying the quadrant containing (-4, -2) (Quadrant III), reflecting (5, -3) over the y-axis (-5, -3), and explaining why the order of coordinates matters in an ordered pair (Possible answer: Reversing the coordinates changes the location of the point). Collect and review the exit tickets to assess student understanding and identify concepts that may require additional instruction or review.

• Clear at least 1 m × 1 m (~3 ft × 3 ft) space per group.
• Secure cords to prevent tripping.
• Keep liquids away from devices.
• Do not allow students to pick up moving robots.
• Place robots on the ground, not desks, so that unexpected movements do not result in mBots falling off desks.

• Robot does not drive straight or overshoots.
  Solution: Calibrate before class (e.g., inspect wheels, align tires, screw motors securely to chassis) and lower the speed of the robot for more precise control.
• Bluetooth/connectivity issues.
  Solution: Test devices beforehand. Ensure tablet and mBot are fully charged. Reboot mBot (by toggling power switch off and rthen on) to put device into pairing mode. If mBot is turned on for a few minutes before pairing is attempted, it can often fail (but is fixed by restarting the mBot).

For Lower Grades (5th–early 6th)

• Limit work to Quadrant I only (positive numbers only).
• Provide a pre-labeled coordinate plane.
• Use whole numbers within ±5 to reduce cognitive load.
• Focus only on plotting and axis movement (omit reflections initially).
• Allow students to physically walk the grid before programming the robot.
• Provide step-by-step driving cards (e.g., “Move right 3, move up 2”).

Dictionary.com. Lexico Publishing Group, LLC. Accessed April 30, 2026. (Source of some vocabulary definitions, with some adaptation) http://www.dictionary.com.

mBot2. Makeblock.  Accessed April 30, 2026. (Source of details about mBot2 with some adaptation) https://www.makeblock.com/pages/mBot2-coding-robot.