“THE DISTANCE FORMULA IS OFTEN
PRESENTED AS A ‘RULE’ TO MEMORIZE”
PURPOSE
• To help students to develop an understanding of
the meaning of the distance formula.
• A review of the distance formula.
• To present a real-world problem using the
distance formula.
• To encourage students to use reason to find a
solution.
TASK OVERVIEW
• Compute the distance between two locations in
a city.
• City streets are laid out on an evenly spaced
square grid.
• Define a coordinate system and think about how
to compute the distance.
REASONING HABITS / PROCESS STANDARDS
• Analyze a problem
using relationships and
structure
• Use connections across
different
representations
• Reflect on the solution
• Problem solving to build
new knowledge.
• Understand how math
ideas interconnect.
• Use representations to
model and interpret
physical, social, and
mathematical data.
STANDARDS FOR MATHEMATICAL PRACTICE
2. Reason abstractly and quantitatively.
4. Model with mathematics.
7. Look for and make sense of structure.
STANDARDS FOR MATHEMATICAL CONTENT
5.G
Graph points on the coordinate plane to solve realworld and mathematical problems.
8.G.7 Apply the Pythagorean Theorem to determine
unknown side lengths in right triangles in realworld math problems.
8.G.8 Apply the Pythagorean Theorem to find the distance
between two points in a coordinate system.
THE CLASS
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9th grade Transitional Algebra at LHS
23 students in class, 12 boys, 11 girls
21 students present for this lesson
20% are repeating this class
Class grades: 84 high, 76 average, 66 low
Teacher said they work at “5th grade level”
Students were encouraged to work in pairs.
Most did not.
TASK / PROBLEM LESSON
1. How many blocks
would you have to
drive to get from your
house to your friend’s
house? Draw a path
that you would drive,
and calculate the
distance.
“BEFORE WE BEGIN…”
Presentation and discussion reveals that the students:
•Have worked with the Pythagorean Theorem this year,
and in at least 3 earlier grades.
•Have worked with the coordinate plane.
•Understand the layout of a city grid.
•Understand what the problem is asking them to find.
•Require brief instruction to address misunderstandings
regarding helicopter flight.
QUESTIONS 1-2: PRE-CLASS DISCUSSION
Misconceptions:
• Students did not recognize that helicopter path
was different from car path.
• Comments included travelling same path as car,
but going faster.
• Some students recognized the flight path as on
the diagonal, but:
• Explained the shorter time in relation to speed.
QUESTIONS 1-2: PRE-CLASS DISCUSSION
Misconceptions (cont):
• Did not recognize the shape as a triangle, so did not
connect the solution to the Pythagorean Theorem.
• Explained with confidence that there was no way to
calculate an exact distance, because they couldn’t
count the boxes accurately.
• They attempted to solve the problem using ratios,
implying that they were trying to relate the task to a
more recent lesson on similar triangles.
STUDENT
EXAMPLE
1
STUDENT
EXAMPLE
2
DISCUSSION OUTCOMES
1. Students were directed to observe the
shape of the graph, eventually
recognizing a triangle. No one noted
the type of triangle.
2. When told it was a right triangle,
students needed much directed talk
to bring forth the Pythagorean
Theorem.
3. Once the Pythagorean Theorem was
in mind, the students worked the
problem quickly, finding the value of
“C.”
OBSERVATIONS
Reasoning Skills
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Students did not connect prior skills and knowledge.
Did not recognize city grid as coordinate plane.
Did not recognize the shape as a right triangle.
Students could place appropriate points on the grid,
once directed, but could not build a viable model of
the problem using mathematics.
OBSERVATIONS
Sense Making: Students Struggled To Make Sense
• Following the class discussion, students struggled to
connect the answer, “C=11.7” to the distance of travel.
• Students could be successful in each individual skills,
but struggled to make connections that led to
problem-solving.
OBSERVATIONS
Sense Making (cont)
•Students were unable to construct models for a follow-up
question, #5, which may imply that they did not engage in
sense making during the earlier task.
•Some students chose to measure the hypotenuse in
inches, and continued to believe the measurement was
correct even after discussion of the Pythagorean
Theorem.
STUDENT WORK: QUESTION 5
REFLECTIONS
• Students in this class seem to view math as disjointed skills with no realworld application.
• Many of these students have a history of poor performance and poor
work habits in math and have developed a “why bother” attitude,
which robs them of the ability to actively engage in problem solving.
• Some of the lower performing students showed strong potential to
succeed using this method, as two students from the bottom half of the
class attempted all 5 questions with success. It would be interesting to
see how the students would respond to lower level tasks, building slowly
toward more challenging current grade-level work.
• Students seemed to have a good toolbox of skills and knowledge, but
may lack the confidence to use their tools appropriately and at the right
times.
MORE SOLUTION ILLUSTRATIONS, Q #5
The End
“the crow has landed”