Fifth Grade Math Lesson Plan
Teacher: Hiroshi Tanaka, Elementary School, University of Tsukuba
Research Topic
Create Lessons That Helps Students Learn to Think Developmentally

1. Topic: The Secret to 5 cm2 Shapes (Estimation of area through Functional thinking whichmeans that the observer sees that the students are reasoning using the idea of a function
without explicitly saying that they know what a function is.)2. Preferred Approach to This Lesson
I give the students a piece of graph paper, and instruct them to draw a shape with an area of 5
cm2. The students try to draw various shapes byconnecting the dots on the paper. The students
・ ・ ・ ・ ・ ・spontaneously produce squares, rectangles, triangles, a・ ・ ・ ・ ・ ・variety of quadrilaterals, and other complex shapes. First・ ・ ・ ・ ・ ・they will try to find the area of the shape using theformula for finding area as well as by using the method of・ ・ ・ ・ ・ ・counting the number of 1 cm2 sections. Because the・ ・ ・ ・ ・ ・students are fifth graders, these activities are important・ ・ ・ ・ ・ ・themselves, but you should focus on the children who go astep further by trying to find the area using a function as・ ・ ・ ・ ・ ・shown below.
・ ・ ・ ・ ・ ・If you look closely at the figures drawn by the・ ・ ・ ・ ・ ・students, you will find that many of them are made byconnecting 12 points. You will also notice that none ofthese shapes contain points inside the shape. The students who notice this will try to figure out
if it holds true for the other shapes as well. Eventually they will discover a shape that contains
one point in the middle. At first they will think this is a special case, but after looking closely,
they will notice that this shape connects only 10 perimeter points.
Hopefully you will start to hear the students say things like, “Hey, that’s interesting.”
And also hopefully some of the students will start to notice that if 12 points means 0 interior
points, and 10 points means 1 interior point, then perhaps a shape connecting 8 points will
have 2 interior points. Following this clue, they will start to try to find other shapes.
The students will start to try to create shapes that connect eight perimeter points andcontain two interior points. Then they will confirm whether the shape they end up withactually has an area of 5 cm2. In the first part of the lesson, the students tried to draw shapeswith a certain area. In the second part, they will work from the opposite direction that is by
first creating a shape based on what they’ve learned and later checking the area. This lesson isdesigned to help students who have already learned about area to look at shapes from a new
and different perspective. A lot of students may think that studying area is all about using
formula, but I want to help them realize that they can make new discoveries if they look atthings using functions.
This experience can help students learn to look at everyday phenomena from a newperspective.
3. Development of the current lesson(1) Purpose
To use an activity in which students draw shapes of 5 cm2 on graph paper to make themrealize that when creating shapes of the same size, the number of perimeter points and thenumber of interior points vary in relationship to one another. And to teach them to try to find
new shapes based on their expectations regarding that variance.(2) Development plan
Lesson Activity
1. Students try to draw a 5 cm2 shape ongraph paper.

Teacher’s Role
- The teacher should let the students knowthat their shapes can be simple, but should
encourage them to make a variety of shapes
by suggesting that they try to come up withshapes that are different from those createdby their classmates.

2. They present their shapes to the class.
- Check whether the area is actually 5 cm2.
- Look at the completed drawings lined up
beside one another.
The teacher tries to suggest, based only on afew special cases, that “All of the shapes- The teacher should deliberately comment
seem to be made by connecting 12 points,on the number of perimeter points.
don’t they?”
Students respond・ “Wait, mine only connects 10 points.” Later the students will findcounter-examples, but this will help them・ “But here’s an exception.”・ “This has the same area, but the number realize that they cannot make judgments
based on a few special cases.
of points is different.”

3. The students compare the shapes
connecting 10 points with shapes connecting - Here the teacher creates a scenario in
which the area of the newly created figures
12 points and notice the difference that:- Even though they have a different number needs to be checked.
of perimeter points, they all have an area of 5 - The teacher should let the students find thesimilarities between the two groups of
cm2.
shapes.
- The 12-point shapes do not contain any- Sort the shapes by using the number ofinterior points, but the 10-point shapes
connecting points and the number of interior
contain a single interior point.
- In that case, maybe an 8-point shape would points and wait to see how the students start
to notice the pattern.
contain two interior points.
Connecting points 12 Interior points 0- And it would be interesting if a 6-point
shape contained three interior points. Let’s Connecting points 10 Interior points 1
Connecting points 8Interior points 2
find out.
4. Students try to find new shapes.
- They confirm whether the new shape has
the correct area.

- The level of difficulty of this activity willchange once the students make thisdiscovery. The students will conclude that afigure with 4 connecting points will have 4
interior points, and the teacher will have to
explain this.