Warm-up: How Many Do You See: Dot after Dot (10 minutes)
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 many do you see? How do you see them?
- “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?”
Activity 1: You’re Gonna Draw It: It’s Symmetric (20 minutes)
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).
Materials to Gather
- 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
Here is a pair of parallel segments that have the same length.
Add one or more segments to create a figure with only 1 line of symmetry.
Here are two more pairs of parallel segments. Add more segments to make:
a figure with 2 lines of symmetry
a figure with no lines of symmetry
If you have time: Here are some other pairs of parallel lines. Add more segments to create a figure with 1 line of symmetry.
Advancing Student Thinking
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.)
Activity 2: Hidden Shapes (10 minutes)
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.
Advances: Conversing, Reading
- 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
Here is a field of dots.
Can you connect the dots to create each of the following shapes? If so, draw the shapes. If not, be prepared to explain your reasoning.
- A triangle with only one line of symmetry
- A quadrilateral with only one line of symmetry
- A quadrilateral with two pairs of parallel sides
- A quadrilateral with one pair of perpendicular sides
- A rectangle
- A six-sided shape with only one line of symmetry
- 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)
“Today we used our understanding of the attributes of figures to draw figures with varying lines of symmetry and varying numbers of parallel or perpendicular sides.”
Display the images:
“When figures are shown on a line grid or dotted grid, we can often learn a lot about their attributes. Here are two figures, one on a square grid and the other on a dotted triangular grid.”
“How might we use grids to see if:
- two segments have the same length?” (On a grid with lines, we can count the units. On dot paper, we can use the distance between dots to see segments are the same length.)
- two segments are parallel?” (On a square grid, the horizontal lines are parallel, and so are the vertical lines. On dot paper, any two rows or columns of dots that are always the same distance apart are parallel.)
- two segments are perpendicular?” (On a square grid, the vertical and horizontal lines are perpendicular. On dot paper, there are vertical stacks of dots and horizontal rows.)
- a figure has line symmetry?” (Use the grid or the dots to check if the figure has two parts that are the same size and mirror each other across a line.)
Cool-down: Can You See It? (5 minutes)
Student Section Summary
In this section, we looked at different attributes of shapes, such as the number and length of sides, the measurements of sides and angles, and whether the shapes had parallel and perpendicular sides.
We then used these attributes to classify quadrilaterals and triangles.
Triangles with a right angle are right triangles.
Quadrilaterals with two pairs of parallel sides are parallelograms.
Quadrilaterals with two pairs of parallel sides and four right angles are rectangles.
Quadrilaterals with four equal sides are rhombuses.
Quadrilaterals with four equal sides and four right angles are squares.
We also learned about lines of symmetry. A figure that has a line of symmetry can be folded along that line to create two halves that match up exactly.