Lesson 6

The purpose of this How Many Do You See is to allow students to use subitizing or grouping strategies to describe the images they see. Students may identify lines of symmetry within the dot arrangement and use this as a strategy to determine the total number of dots. The may also consider smaller arrays, or chunk the image into small groups and multiply or add.

In this activity, students have an opportunity to look for and make use of structure (MP7) because the arrangement contains smaller arrays and line symmetry.

• Groups of 2
• “How many do you see? How do you see them?”
• Display the image.
• 1 minute: quiet think time

• “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.

• “How does the arrangement of dots help you find the number?” (It has lines of symmetry. We can count the dots on one side of the line and double it, without counting one by one.)
• Consider asking:
  • “Who can restate the way _____ saw the dots in different words?”
  • “Did anyone see the dots the same way but would explain it differently?”
  • “Does anyone want to add an observation to the way _____ saw the dots?”

In this activity, students create their own figures that have certain symmetry-based attributes. Students are given a pair of parallel segments on a grid. They then add more segments to create figures with one, two, and zero lines of symmetry.

To create their own figures, students rely on their understanding of symmetry and parallel lines. They consider where possible lines of symmetry could be, how many segments to add, and where to place them. They may also experiment, rely on familiar shapes and their lines of symmetry, or imagine a line of symmetry and what it would tell them about the figure. As they do so, they look for and make use of structure (MP7).

• Groups of 2–4
• Provide access to straightedges

• 2–3 minutes: independent work on the first question
• 2 minutes: group discussion
• “Share your drawing for the first question. If needed, revise your thinking and drawing.”
• 5 minutes: independent work time on the remaining questions
• 3–5 minutes: group discussion
• “Be sure to discuss how you know that your drawings have the specified number of lines of symmetry.”
• Listen for attention to precision in students’ justifications. For instance, they may say the figure has line symmetry because:
  • the angles are the same
  • the sides have the same length and create the same angles when folded on top of each other
  • they measured all the parts and found them to be the same
  • they traced the figure and placed the copy on top of the original

If students create figures with more or fewer lines of symmetry than planned, consider asking:

• “How many lines of symmetry did you want your figure to have? How many lines of symmetry does it have?”
• “How could you change your figure so it has fewer (or more) lines of symmetry?”

• Invite students to share their drawings and how they determined how and where to add segments.
• “Which parts of the figure did you pay attention to when drawing? How did you make sure your drawing has line symmetry?“ (The segment lengths, angles, and distances from the line of symmetry all have to be the same on both sides of a line of symmetry.)
• “What was tricky about creating figures with line symmetry in this task?” (Sample response: It was hard to figure out what to draw to get the exact number of lines of symmetry.)

In this activity, students apply their understanding of symmetry, parallel and perpendicular lines, and types of quadrilaterals to create shapes with certain attributes on isometric dot paper. The arrangement and equal spacing of the dots give students structure for drawing parallel and perpendicular lines and to determine symmetry.

• Groups of 2
• Display the isometric dot image.
• “What do you notice?” (lots of dots forming lines, triangles, rhombuses)
• “What do you wonder?” (Are the dots spaced apart equally? Why is every other vertical stack of dots higher or lower than the stack next to it?)

• 10 minutes: group work time
• Monitor for students who:
  • use the equal distance between dots to draw parallel lines
  • notice that the dots in alternate columns form horizontal lines, which can be used to create right angles with vertical lines
  • use the dots to create equal lengths and angles

• Invite students to share their drawings. Highlight that many different drawings are possible for each description.
• “How did you make sure that the first two figures have line symmetry?” (Check that the figure has a line that splits it into two halves that mirror one another and match up when folded.)
• “How did you create parallel sides and know that they are indeed parallel?” (Use the spacing of the dots to draw segments that are the same distance apart.)
• “How did you create segments that are perpendicular?” (Connect dots that line up vertically and those that line up horizontally.)
• “What parts, specifically, did you need to draw to create a rectangle?” (2 sets of parallel segments—the same length for opposite sides—and 4 right angles)