Getting Ready to code Transformations: Unplugged

(Source: Ontario Association of Math Educators: Ontario Math Support)

Expectations:

Spatial Sense

• E1. describe and represent shape, location, and movement by applying geometric properties and spatial relationships in order to navigate the world around them

  • E1.3 plot and read coordinates in all four quadrants of a Cartesian plane, and describe the translations that move a point from one coordinate to another

  • E1.4 describe and perform combinations of translations, reflections, and rotations up to 360° on a grid, and predict the results of these transformations

Number

• B1. demonstrate an understanding of numbers and make connections to the way numbers are used in everyday life

  • B1.2 read and represent integers, using a variety of tools and strategies, including horizontal and vertical number lines

Social-Emotional Learning (SEL) Skills in Mathematics and the Mathematical Processes

• A1. Throughout this grade, in order to promote a positive identity as a math learner, to foster well-being and the ability to learn, build resilience, and thrive, students will apply, to the best of their ability, a variety of social-emotional learning skills to support their use of the mathematical processes and their learning in connection with the expectations in the other five strands of the mathematics curriculum.

Big Idea:

• In this lesson, to the best of their ability, students will learn to develop self-awareness and sense of identity and build relationships and communicate effectively as they apply the mathematical processes reasoning and proving (develop and apply reasoning skills (e.g., classification, recognition of relationships, use of counter-examples) to justify thinking, make and investigate conjectures, and construct and defend arguments and communicating (express and understand mathematical thinking, and engage in mathematical arguments using everyday language, language resources as necessary, appropriate mathematical terminology, a variety of representations, and mathematical conventions), so they can see themselves as capable math learners, and strengthen their sense of ownership of their learning, as part of their emerging sense of identity and belonging, and work collaboratively on math problems--expressing their thinking, listening to the thinking of others, and practising inclusivity -- and in that way fostering healthy relationships.

Students will be characterizing the location and movement of objects on the Cartesian plane, leading up to coding their own location and movement projects in subsequent lessons.

Activating Prior Knowledge:

Think: Begin by asking students what they think about when they hear the term graphing.

Note: A graph is a visual representation of data, and there are many types of graphs that students can describe. In this lesson, students are required to work with xy-graphs (i.e., plotting points on the Cartesian plane). If this type of graph is not mentioned, teachers can prompt students to think of other types of graphs that show the location (or position) of an object in space.

Pair: Have students share their thinking with a partner.

Share: Invite students to share their thinking with the larger group.

Record students’ ideas on a vertical surface for reference. With each lesson, you might choose to prepare an anchor chart.

Below is a sample of possible responses (illustrated as a word cloud).

To help students keep thinking, ask questions that further probe students' knowledge of the Cartesian plane and understanding of graphing.

You might consider asking:

• What do you notice? (Show Quadrant 1 of the Cartesian plane-- once with no grid; again with the same scales; and once more with different scales on both x- and y-axes; sample grids)

• What is similar about graphing a point (provide an example) on all of these grids? What’s different and why?

  • Teachers might consider displaying graphs, in real-time, using graphing technology like Desmos.

• How might grid-like systems, similar to this, be used every day? By others?

• What are you wondering?

Opportunity for Assessment

• Listen for key terms, gestures, explanations, as well as any questions that students are asking of their partner.

• The goal is not to tell; rather, it’s to better understand what students know, can do and what tasks they might need to experience to support their work towards the learning goal.

Scratch Project 1: Coordinate Graphing - Getting Started

Coordinate Graphing - Getting Started shows the following:

Graphing ordered pairs (x, y) in Quadrants 1 to 4

• Students can see the full animation (click green flag), hear the ordered pairs (pressing space-bar), and see a stamp left behind each ordered pair visited by the sprite (click on the cat).

• Invite students to ‘’look inside’’ to read the code that is there and have them converse about what they are reading.

• Exit Ticket 1 - Coordinate Graphing - Getting Started Doc

• Exit Ticket 1 - Coordinate Graphing - Getting Started Form

Scratch Project 2: Coordinate Graphing - Guess My Position!

Coordinate Graphing - Guess My Position! shows the graphing of ordered pairs (x, y), in Quadrants 1 to 4:

Activity 1A: Students can watch an animation (pressing the space-bar allows for multiple opportunities to view examples of a ball moving into position and labelled with its coordinates (x, y)) and/or

Activity 1B: Students can enter a guess, check their answer, and receive feedback about their choice of quadrant (1, 2, 3 or 4) on the Cartesian plane. Clicking on the green flag allows the student to continue trying.

Activity 2: Students can enter a guess, check their answer, and receive feedback about their choice of coordinates, (x, y). Clicking on the sprite allows the student to continue trying.

Invite students to ‘’look inside’’ to read the code that is there and have them converse about what they are reading.

• Exit Ticket 2 - Coordinate Graphing - Guess My Position! Doc

• Exit Ticket 2 - Coordinate Graphing - Guess My Position! Form

Tip: During consolidation, teachers might choose to project a few paths traced and have students offer up and discuss examples of “markers” (x, y) that lie along the path

Scratch Project 3: Translations & Reflections - What’s My Move?

Translations & Reflections - What’s My Move? depicts an arrow being drawn and then changed to a congruent, transformed image, on the Cartesian plane. The program is executed by clicking on the green flag.

The transformed image results from one reflection and two translations (one vertical; the other, horizontal). Each single transformation can be seen, as the arrow is re-drawn each time (the original image remains for students’ reference).

Invite students to ‘’look inside’’ to read the code that is there and have them converse about what they are reading.

Exit Ticket 3 - Translations and Reflections - What’s My Move? Doc

Exit Ticket 3 - Translations and Reflections - What’s My Move? Form

Tip: During consolidation, teachers might choose to project and discuss different lists of transformations collected (anonymously) by the form and model them by drawing them on Diagram - Translations and Reflections - What’s My Move? Students can also be asked to reflect on their own choices.

Learning Goal: To understand and code the movement of objects on the Cartesian plane

Success Criteria:

Describe the precise location of any point on the Cartesian plane using an ordered pair (x, y)

• Describe the precise location of any point on the Cartesian plane using an ordered pair (x, y)

• Recognize that the signs of both x- and y-coordinates depend on the quadrant where a point is located

• Use dynamic geometry (or other) software to understand how transformations behave.

Start with a math conversation, providing students with an opportunity to reflect.

Math Conversation:

Think: We began today’s lesson by sharing what we knew about graphing, and then we started to investigate transformations.

What else might we add to the list of information that was called “Graphing”?

If you were going to re-title the list we’ve been building, what would you call it? Why?

Pair: Have students share their thinking with a partner.

Share: Invite students to share their thinking with the class.

For online/hybrid options, consider creating a grid in Flipgrid to record these asynchronous conversations.

Next, to help develop a deeper conversation and highlight success criteria connected to the learning goal, facilitate a math congress with students.

Math Congress:

Using a small sample of student work, collected through one or more of the learning centres (and/or other means), focus on encouraging student sharing--explaining their thinking, listening to others, and asking questions of their peers.

Project 2:

From this learning centre’s exit tickets, choose a small number of paths students have traced from Exit Ticket - Project 2 and ask:

How can the precise location of an object on the Cartesian plane be described?

(If necessary, invite students to discuss among themselves and have them use the sample paths to illustrate their thinking.)

As students share their ideas, begin recording these as success criteria in a visible location in the classroom.

Scratch Project 3 - Translations and Reflections: What’s My Move?:

From this learning centre’s exit ticket, choose and display a small sample of lists of transformations alongside Diagram #2 and ask:

How can the movement of an object be described on the Cartesian plane?

(If necessary, invite students to discuss among themselves and have them use the sample lists as examples.)

Continue recording their ideas as success criteria, refining the criteria based on the conversation and teacher questioning.

Lastly, circle back to the list of terms started (math conversation, above), and with details shared by students during the math congress, work with students on developing or refining the learning goal for location and movement on the Cartesian plane.

Add this to the top of the list of success criteria charted.

Guided Learning Groups:

If your observations and conversations are pointing to students having difficulty in either expressing and showing precise locations, identifying, recognizing and applying transformations (reflections and translations), or both, then the challenge of the activities may have been greater than the skills students have already developed.

Finding some time between this lesson and the next planned activity to refer to the Scratch projects (animations) or different examples can further elicit student thinking and foster conversation (Opportunity for Assessment).

Challenge:

During the course of the lesson activities and debrief, some students may have arrived quickly and proficiently to the learning goal. In this case the challenge of the activities was either below or at the level of skills already developed by students. In this case, providing students with opportunities that lie just beyond their development can be productive in deepening their learning.

Teachers might choose to use a similar activity (unplugged) to Project #3 and ask different questions.

For example:

• What is the least number of transformations you can apply to produce the image arrow?

• Explore how you can incorporate rotations, and in combination with reflections and translations, to produce the same image arrow.

• If you were to explain this topic to someone else, what would you say to them? What would you have them try?

• If you had to create this type of activity, what improvements would you make? Why?

Extension:

For students who have some experience coding in Scratch (or in other block-coding environments) and are showing interest to “see inside”, they can re-mix the projects in this lesson or begin creating their own to demonstrate their ability to code computational representations of transformations.