Unit 7, Lesson 1
Connecting Distance and Circles
Integrated Math 2
1.1
Activity
Instructional Routines
Poll the ClassMaterials
To Gather
Scientific calculatorsActivity Narrative
The purpose of this activity is for students to recall how to plot points a given distance away from a particular point. Students may also make the observation that a set of points a given distance from a single point makes a circle. Students will continue building toward this understanding in subsequent activities, so it is not necessary for students to make that observation at this time.
Making scientific calculators available for this activity gives students the opportunity to choose appropriate tools strategically (MP5).
Launch
Provide access to scientific calculators.
Activity
NonePlot as many points as you can that are a distance of 5 away from the point .
Activity Synthesis
The purpose of this discussion is for students to explain how they found points a given distance away from a particular point, and to make observations about their results. Display the coordinate plane with the point . Poll the class for the points they found and annotate the graph with those points.
Here are some questions for discussion:
- “What were the first points you found?” (I found the points directly in line vertically and horizontally with the given point without much calculation.)
- “How did you find other points that ended up on a grid line?” (Since the distance from the set point was 5, I used a right triangle with legs of length 3 and 4 to find more points. Some of these were , , and .)
- “What do you notice or wonder about the points that are a distance of 5 from the point ?” (I notice that you can use the Pythagorean Theorem to find points. It appears to make a circle centered at . I wonder whether the circle has a radius of 5.)
If no students make use of a right triangle with legs of length 3 and 4 to find additional points, invite students to do so now. Recognizing that will be helpful in later activities.
Lesson Synthesis
The purpose of this discussion is for students to connect distance with the radius of a circle. Display this image of the points from the Warm-up with a circle drawn through all the points.
Draw a new point along the circle that does not lie on a grid line. Ask students, "How far away is this new point away from the center? How do you know?" (It must be 5 because the radius of the circle is 5, so the distance from any point on the circle to the center is 5.)
Ask students, "How does the distance between a point on the circle and the center of the circle connect to the radius of a circle?" (If you plot all of the points that are the same distance from a central point, then you get a circle, and the distance from the center point to any point on the circle is the radius of the circle.)
Then ask, "How could you decide whether a point lies on the circle, inside the circle, or outside the circle?" (If it is on the circle, its distance to the center will be the same as the radius. If it is inside the circle, its distance to the center will be less than the radius, and if it lies outside the circle, its distance to the center will be greater than the radius.)
Student Lesson Summary
Let's find some points that are a distance of 5 away from the point and plot them on a grid. We can use gridlines to find 4 points:
We can use the Pythagorean Theorem to help us find some more points. Recall that for a right triangle with a hypotenuse of 5, we can make a right triangle with legs 3 and 4. This means that if the horizontal and vertical distances between the points are 3 and 4, then the distance between them will be 5. We can find 8 more points this way!
As we fill in more points, it looks like the points are forming a circle around the point , and in fact, they are. The circle has a radius of 5, which makes sense since we know that a circle is the set of all points a given distance away from a central point. In this case, that given distance is 5.
How can we check whether a point lies on this circle if we are not sure? Let's check the point . We can use the Pythagorean Theorem to test the distance between the center of the circle and the point: , which is not equal to 5. That means the point is close to the circle, but does not lie on the circle.
We can say that for a circle of radius and center , if a point is a distance of away from , it will lie on the circle. If point is not a distance of away from , then it will not lie on the circle. Another way to say this is that for a circle of radius and center , will lie on the circle if and only if it is a distance of away from .