EDCP 342A Unit planning: Rationale and
overview for planning a 3 to 4 week unit of work in secondary school
mathematics
Your name: Shan Huang
School, grade & course: Churchill Secondary, Grade 8
School, grade & course: Churchill Secondary, Grade 8
Topic of unit: Chap. 6: Square Roots
and the Pythagorean Theorem
Preplanning questions:
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(1)
Why do we teach this unit to
secondary school students? Research and talk about the following: Why is
this topic included in the curriculum? Why is it important that students
learn it? What learning do you hope they will take with them from this? What
is intrinsically interesting, useful, beautiful about this topic? (150 words)
In Grade 8 mathematics, after learning
integers, fractions, percents/ratios/proportions, as well as decimal numbers,
students start to understand different ways of expressing numbers. These
numbers are accurately expressed and internally transferable. However, not
all the numbers can be expressed accurately (for example, π, e, etc.). When we define “the square of a
number” as the product of the number multiply itself, we are also interested
in the reversal operation, i.e., if we have a number, what is the number
whose square is the given number? Are there any constraints about the given
number? Are there any interesting properties about the results? These ideas
can be expanded to 3rd power, 4th power, etc.
The idea of exponent will be introduced in
later course, but in this Chapter, the Pythagorean Theorem will be taught as
an application of using square/square roots. Pythagorean Theorem is such an
important bridging property between algebra and geometry that it connects
these two as a whole and provides useful insight in thinking geometry with an
algebraic perspective.
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(2)
What is the history of the
mathematics you will be teaching, and how will you introduce this history as
part of your unit? Research the history of your topic through resources like
Berlinghof & Gouvea’s (2002) Math through the ages: A gentle history for
teachers and others and Joseph’s
(2010) Crest of the peacock: Non-european roots of mathematics, or equivalent
websites. (100 words)
Square is a shortcut of expressing the product
of a number multiplies itself, while square root is the reversal operation of
square. The concepts of them are straight forward but the sign of square root
comes from the “r” in the word “root”.
The history of Pythagorean Theorem is
interesting. Ancient people in Egypt, Babylon, and China at different times
somehow figured out a triangle with side lengths of 3 units, 4 units, and 5
units has a “perfect angle”----now we know that is a 90-degree angle, or a
right angle. In general, any right triangle will satisfy the following
relationship: a^2 + b^2 = c^2.
A wonderful YouTube video will be used to
demonstrate the history of finding Pythagorean Theorem (https://www.youtube.com/watch?v=PrjTkWGLk2Q ).
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(3)
The pedagogy of the unit: How to offer this unit
of work in ways that encourage students’ active participation? How to offer
multiple entry points to the topic? How to engage students with different
kinds of backgrounds and learning preferences? How to engage students’ sense
of logic and imagination? How to make connections with other school subjects and
other areas of life? (150 words)
To encourage students’ active participation, on-hand activity is
necessary to give students opportunities to think about the meaning of the
theorem. By that, students may realize the theorem is not merely a formula,
but a bridge that connects geometry and algebra. A shape may have algebraic
meanings, while an algebraic expression may some geometrical explanations.
Students should also
prepare for the standardized exams. Emphasizing on calculation, explanation,
and problem solving skills is never too much for students in this age. Various
problems within different scenarios will be posed and students will be
encouraged to solve them.
Scenarios should be
diversified to make sense to all ethnic background students. In order to help
students understand more about the topic, various subjects like physics,
geography, and tech education topics that are relevant to Pythagorean Theorem
will be used as demonstrations. All in all, mathematics should be applicable
in people’s daily lives.
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(4)
A mathematics project connected to
this unit: Plan and describe a student mathematics project that will form
part of this unit. Describe the topic, aims, process and timing, and what the
students will be asked to produce. (100 words)
One
interesting project that encourages students to discover the meaning of
Pythagorean Theorem is to do a shape-matching game. This is a game developed
by Blake Peterson (2009).
(a) Students in pairs will make two squares. These squares can be with any
dimensions, but for convenience, they had better have different and drawable
sizes
(b) In each pair, assume the smaller square is with side length a, and the
larger is with side length b.
(c) When two squares are put adjacent to each other, the total area of the
two squares are
(d) Cut the two squares as shown on figure 1 (see lesson plan). Then each pair will have
five shapes.
(e) Move these shapes to make a larger square.
(f) The result will be similar as figure 2 (see lesson plan).
Then it is illustrated that the larger square
with side length c has the same area as the two previous squares, or a^2 + b^2 = c^2.
Also, students can also see that this relationship is independent of the
values of a and b.
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(5) Assessment
and evaluation: How will you build a fair and well-rounded
assessment and evaluation plan for this unit? Include formative and
summative, informal/ observational and more formal assessment modes. (100
words)
Formative assessment:
(a)
Communicate with students during class time;
(b)
Peers’ solution sharing;
(c)
Quizzes every week (one quiz every two classes);
(d)
Workbook from page 211 to 240;
(e)
Mathematical project explanation;
(f)
Explanations on different ways of proving Pythagorean
Theorem.
Summative assessment:
(a)
Basically all the formative assessments can be
treated as summative assessments, and vice versa;
(b)
Various in-class problems based on different
scenarios;
(c)
The mathematical project itself can be
considered as a summative assessment as even though students can make the
large square, s/he may not be able to explain how they see it.
(d)
Unit exam focus on the applications of the theorem,
as well as some different proofs of the theorem.
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Elements of your unit plan:
a) Give a numbered list of the topics of the
10-12 lessons in this unit in the order you would teach them.
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Lesson
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Topic
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1
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Square roots
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2
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Modelling
square roots
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3
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Non-perfect
square roots
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4
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The Pythagorean
Theorem
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5
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Various proofs
of Pythagorean Theorem
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6
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Various applications
of Pythagorean Theorem 1
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7
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Various applications
of Pythagorean Theorem 2
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8
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Extension: irrational
numbers and nun-perfect square roots
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9
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Chapter review
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10
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Summative unit
test
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