standard_description: Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
problem_activity: Activity
The purpose of this activity is to extend what students know about constructing a perpendicular line through a point on the given line to a new situation in which the constructed perpendicular line goes through a point
not
on the given line.
Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).
Digital
Launch
Arrange students in groups of 2. Display the image for all to see:
[IMG]
Remind students that in the previous lesson, they used straightedge and compass moves to create a line perpendicular to
\(\ell\)
that goes through point
\(B\)
, which was on line
\(\ell\)
. Now display an image of the construction of a perpendicular line through
\(B\)
for all to see throughout the activity:
[IMG]
Display a list of constructions students already know. This display should be posted in the classroom for the remaining lessons within this unit. It should look something like (only include the first 6 constructions now):
circles of a certain radius
lines and line segments through two points
regular hexagons
equilateral triangles
a perpendicular bisector of a given segment
a perpendicular line through a point on the given line
a perpendicular line through a point
not
on the given line (added in this lesson)
a parallel line through a point
not
on the given line (added in this lesson)
After quiet work time, ask students to compare their responses to their partner’s and decide whether they are both correct, even if they are different. Follow with a whole-class discussion.
Action and Expression: Provide Access for Physical Action.
Provide access to tools and assistive technologies such as dynamic geometry software.
Supports accessibility for: Visual-spatial processing;Conceptual processing; Organization
Student Facing
Here is a line
\(m\)
and a point
\(C\)
not
on the line. Use straightedge and compass tools to construct a line perpendicular to line
\(m\)
that goes through point
\(C\)
. Be prepared to share your reasoning.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
The line segment
\(AB\)
has a length of 1 unit. Construct its perpendicular bisector and draw the point where this line intersects our original segment
\(AB\)
. How far is this new point from
\(A\)
?
We now have 3 points drawn. Use a pair of points to construct a new perpendicular bisector that has not been drawn yet and label its intersection with segment
\(AB\)
. How far is this new point from
\(A\)
?
If you repeat this process of drawing new perpendicular bisectors and considering how far your point is from A, what can you say about all the distances?
[IMG]
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Print
Launch
Arrange students in groups of 2. Display the image for all to see:
[IMG]
Remind students that in the previous lesson, they used straightedge and compass moves to create a line perpendicular to
\(\ell\)
that goes through point
\(B\)
, which was on line
\(\ell\)
. Now display an image of the construction of a perpendicular line through
\(B\)
for all to see throughout the activity:
[IMG]
Display a list of constructions students already know. This display should be posted in the classroom for the remaining lessons within this unit. It should look something like (only include the first 6 constructions now):
circles of a certain radius
lines and line segments through two points
regular hexagons
equilateral triangles
a perpendicular bisector of a given segment
a perpendicular line through a point on the given line
a perpendicular line through a point
not
on the given line (added in this lesson)
a parallel line through a point
not
on the given line (added in this lesson)
After quiet work time, ask students to compare their responses to their partner’s and decide whether they are both correct, even if they are different. Follow with a whole-class discussion.
Action and Expression: Provide Access for Physical Action.
Provide access to tools and assistive technologies such as dynamic geometry software.
Supports accessibility for: Visual-spatial processing;Conceptual processing; Organization
Student Facing
Here is a line
\(m\)
and a point
\(C\)
not
on the line. Use straightedge and compass moves to construct a line perpendicular to line
\(m\)
that goes through point
\(C\)
. Be prepared to share your reasoning.
[IMG]
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
The line segment
\(AB\)
has a length of 1 unit. Construct its perpendicular bisector and draw the point where this line intersects our original segment
\(AB\)
. How far is this new point from
\(A\)
?
We now have 3 points drawn. Use a pair of points to construct a new perpendicular bisector that has not been drawn yet and label its intersection with segment
\(AB\)
. How far is this new point from
\(A\)
?
If you repeat this process of drawing new perpendicular bisectors and considering how far your point is from A, what can you say about all the distances?
[IMG]
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Some students may struggle more than is productive. Ask these students to draw a line segment and construct the perpendicular bisector of it. In that construction, the perpendicular bisector will go through an intersection point of two circles. Ask, “What happens if you create a circle centered at that intersection point that goes through an endpoint of the segment? Why does that happen? How can you use this idea in this new activity?”
Activity Synthesis
Focus on the process of using a previous construction to generate new constructions. Here are some questions for discussion:
“How was this construction different from perpendicular line constructions you have done before? How did thinking about the differences help you plan what to do?” (I didn't have a segment to bisect, because the point wasn't on the line. I realized I could still make a segment using the new point.)
“How does knowing some constructions help you do other, more complicated constructions?” (I can use the same moves, but just change them a little bit.)
Writing, Speaking, Conversing: MLR 1 Stronger and Clearer Each Time
. Use this routine to give students a structured opportunity to revise and refine the reasoning used to create their construction. Ask each student to meet with 2–3 other partners in a row for feedback. Provide students with prompts for feedback that will help individuals strengthen their ideas and clarify their language. For example, “Can you explain how…?”, “Why did you choose to construct a circle there?”, “What do you know about radii that helps here?”, or “What do you mean by…?”. Students can borrow ideas and language from each partner to strengthen their final explanation.
Design Principle(s): Optimize output (for justification); Support sense-making


aligns: no