What Do You Notice? What Do You Wonder?

Show students this video:

Then, ask students:

What do you notice?

What do you wonder?

Have students do a Think-Pair-Share routine:

1. Students have individual think time to jot down ideas on paper or whiteboard. 
2. Students share their initial ideas with a partner.
3. Students share as a whole group, either their own preference, or a meaningful observation they heard from their partner during the share (while giving credit to their partner). All contributions are acknowledged and recorded on an anchor chart on the board.

Spending time to acknowledge and address specific thoughts that students shared, whether a notice or a wonder, is crucial to building a culture in your classroom where students know that their voice is being valued and thus encourages them to continue sharing their thoughts and opinions later in this lesson and in future lessons.

Possible observations and questions that may come up include:

• I notice that all the photos are of tile work
• I notice that the tiles have radial patterns
• I notice that the patterns have symmetry
• I wonder how they make the tiles
• I wonder who makes them and how long it takes
• I wonder what they’re made of
• I wonder if the tiles are painted first, then assembled, or assembled then painted
• I wonder how they make the design of the tiles

Notice and Wonder 2

The second part of the Notice and Wonder routine focuses on answering some of the questions related to how the tile work is made, and hopefully spark some new questions. The video shows some of the tools and techniques a craftsman uses to create a small tile.

Show students the following video:

Then, ask students:

What more do you notice? What else do you wonder?

Share your thoughts.

Possible observations and questions that may come up include:

• I notice that he chips the tile twice (once in the front, once in the back)
• I notice the tiles are painted before he starts to cut them
• I wonder how the tiles all fit together
• I wonder if all the same tiles are exactly the same (or else the pattern doesn’t work/fit)
• I wonder how he gets all the tiles to be the same without a machine

Estimation: Prompt

Once students’ initial ideas have been acknowledged and noted, the class can settle on a question to explore: 

Are all the same colour pieces congruent? 

Make an estimate.

Facilitator’s note: Students have likely been introduced to the word “congruent” in earlier grades. Congruence is often introduced as meaning “same shape and size” which is an intuitive definition. In high school, congruence is often approached from a triangle congruence criteria perspective. Understanding congruence through transformations is a critical bridge between the two.

It may be worthwhile to consider beginning an anchor chart that records students’ ideas about congruence as they shift from the measurement definition (same shape and size) to the transformational definition (two polygons are congruent if it is possible to describe a series of transformations that maps one polygon onto the other). A simple prompt/title for the anchor chart might be: “Two shapes are congruent if…”

Students begin by creating a hypothesis before they are provided with all the information they need to answer the question, thus providing students who may be reluctant to share with a safe entry point. 

Be sure not to skip over asking students to make an estimate using only their spatial reasoning skills as this is a very important step in the Curiosity Path. Providing students an opportunity to make an estimate and try to articulate their thinking with their peers provides a very low floor opportunity for them to not only better understand the context, but to also begin nudging them to think about what will be important to make their estimate more precise as we continue through the lesson.

Students should be given an opportunity to share their ideas at this point, but refrain from sharing their rationale just yet in order to give everyone a chance to develop their own thinking.

Have students turn to a partner and generate questions they could ask that would provide them with information they could use to answer the question. Ask students how they would use the information to answer the question. 

Prompt:

What information might be helpful to answer this question? 

How might you use that information?

Estimation: Partial Reveal

Reveal to students the video showing how the tiles are made and assembled. Students may wish to update their predictions about which sets of tiles are congruent.

At this point, students will likely have an intuitive sense, or definition for congruence as being “same shape and size”. They may also suggest that corresponding angles and sides are congruent. Nonetheless, it is valuable for students to spend time exploring these ideas before further deepening their understanding to include transformation as an essential characterization of congruence.

Provide students with a copy of the tile pattern and prompt them to construct viable arguments in order to prove that pieces of similar color are congruent (or not).

Facilitator’s note:

It is worthwhile to consider giving students prompts that do not have a coordinate grid. It can help students who are just beginning to work with transformations get a sense of the general meanings for each type of transformation. Additionally, it is valuable in making explicit some of the relationships between figures. For example, connecting corresponding points on an image and a pre-image will highlight relationships that students may not consider on a coordinate grid. The same tile pattern is considered using a coordinate grid as part of the activities on Day 3 of this unit.

Some strategies that students may use include:

• cutting out different pieces and placing them on top of one another
• tracing different pieces on patty paper and placing them on top of one another
• measuring side lengths and comparing corresponding lengths on another figure
• measuring angles and comparing corresponding angles on another figure

Facilitator’s note: As students construct their arguments for or against the congruence of figures, this may be an opportunity to bring in some of the vocabulary associated with congruency, including:

• “distance preserving” (a transformation is distance preserving’ if the measures of corresponding distances on congruent figures are the same)
• “angle preserving” (a transformation is ‘angle preserving’ if the measures of corresponding angles on congruent figures are the same)