Quadrature of the kite-shape in " Area of the square and the triangle"(5th grade)
--Instruction of reconsideration for deepning pupils' thinking and cultivating task consciousness--
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1.Introduction
When I would image a lesson that pupils sudy enthusiastically mathematics, I might tell
to you that it can be seen in a lesson of Elementary School attached to Tsukuba University Tokyo. There, the teachers have made an effort toward continuing pupils' questions in their math lessons. To deepen pupils' thinking in math lessons, having many
questions one after another, finding out new problem, might deepen pupils' understanding contents of teaching material. And it might connect with fixing both basic and fundamental knowlege, understanding for them,too. The important matter of
instruction is a method how make pupils' thinking deepen and how make pupils find out
new problem(or task).
2.The method for continuing pupils' questions
The pupils have not any question by thinking about present contents, for continuing their
questions. Having some questions and finding some problems generate by thinking of
relation with contents taught yet and other contents. Then, generating parts not
understood wholy, new questions do. Therefore we find that we had better prepare some questions and directions, that is, have the pupils reconsiderate their learning to
have the pupils have any question.
3. From the lesson record
First I presented the pupils a paper written a kite shape(fig1.) and showed the task in the lesson-"Let's think of how to quadrate kite's area." It passed 3 minutes for the introduction.
Then I took a few minutes for the pupils to have an insight into problem solving, saying "Think of how you had better mesure the area." After then, I gave them problem cards that was written resized fig1. more little in order to themselve's problem solving, directing them to measure the kite's area. (fig2.) 
After the pupils thought of the problem for about ten minutes, I had four pupils present their solving method how to measure the kite's area with presentstion boards(small white board). I had the pupils write their method in about 5 mibnutes after
self-problem solving began.
Fisrt, Sakura(girl) presented as follows. (fig3.)
She divided the shape into two triangles and quadrated every tringle's area, the method
of whom is aorthodox one that doubles congruent triangles.
Next Mayumi(girl) presented her idea, that is, the same area-changing. (fig.4)
She made four triangles by two diagonal lines and moved two right triangles into left.
And she explained about the rectangle made, adding the length of the rectangle.
Then Makoto(boy) presented his idea, the double area-changing. It is one that double area rectangle was made as the kite-shape was surrounded by it.(fig.5)
Therefore, the kite shape's area is a half of it.
When Makoto was presenting his idea, as Yousuke(boy) muttered that Makoto's formulas can be reformed into only one formula, I asked Yousuke how he would reforme it. And
he answered "7~62" , I wrote it down the whiteboard.
Finally, Takahiro(boy) presented rather atrange method. In the fig.2, he cut it by a vertical diagonal line, making two triangles, turning over one triangle, made a parallelogram. This method was unexpectedly to us. I had guessed that an idea turning over the shape would not generated because of the problem card printed sections. It is
very original and unique method.
After four pupils showed their ideas, we verified that every method is useful to measure the area. Next I asked them whose method is easier and they answered that
they are Mayumi's and Takahiro's and Makoto's.
As we could quadrate the area of the kite shape, I announced them that we thought
of the formula of the kite shape's area. And we made sure of using mathematical words
for the formula as rectangle's one is "multiply height by width" or trapezoid's one is "(upper side+lower side)~height2".
"How will we make the formula of the kite shape's area?", I asked them. And for the
question Mayumi answered "height ~width2". I asked her to explain of it more conversantly and she told that she thought the kite shape is like the rectangle. (fig.7)
Her saying means the sides of height and width in the bigger rectangle which Makoto had written. Just then, Yousuke came to the blackboard, explaing that their sides means
diagonal lines of the kite shape. And we verified that two explanations showed the same
length and I had them made sure that its area can be quadrated by multiplying two lengths of diagonal lines and dividing it by 2.
After that, I directed them to write the formula on their notebooks.
But many pupils could not make the formula because of not understanding the learning. I had better have instructed them that its area could be quadrated by "diagonal line(length)~diagonal line2". Or the pupils might have a little resistance for using the
diagonal line as tha formula of the kite shape's area. Anyway I found that it is rather difficult for them to make the formula of the area beyond expectaion.
Next I concluded the formula of the kite shape's area is "one diagonal line ~
another diagonal line 2", and we read the formula, making sure of multiplying two diagonal lines and dividing them by 2.
And I had the pupils quadrate two kites' area which were printed as an exercise.(fig.8 & fig.9) The pupils all could quadrate of @,
but half of them of A , because of inexactness in the figures. As the length of two
diagonal lines were a few millimeters shorte than the integral number, the pupils might waver in measuring the length. But I found that they understood how to quadrate of the
areas by reading their impressions after lesson.
By the way, we conclude the learning in case of the ordinary lesson then, but I had
them reconsiderate the formula of kite shape' area. I asked them whether the formule
of kite shape's area could be used in case of quadrating other square's areas. And then the pupils said, "trapezoid, parallelogram-----", but I asked them what shape is
like the kite shape. This time the pupils said, "rhombus", and I had them tried whether
the area of the rhombus learned the last time(the former lesson) could be quadrated
by using the formula of kite shape's area. Then we found that we can quadrate the area of the rhombus by the formula, "one diagonal line ~another diagonal line 2".
And moreover I asked them again how two shapes (kite shape and rhombus) are like
as two shapes can be quadrated by the same formula. Three pupils presented their
thinkings for the question.
Koushi(boy): crossing the sides----
Yousuke(boy): the number of the sides is the same--
(but other pupils disagreed with it, saying
"It shows the whole squares")
Motoki(boy): (in the kite shape)how to cross this diagonal
line and another diagonal line, I think, right angle,
that shape also(rhombus) gets right angle.
Motoki noticed that both the kite shape and the rhombus cross the diagonal lines perpendiculary. When Motoki was presenting his thinking, Koushi murmured that
the diagonal lines cross perpendiculary in the square too. I asked them whether the square can be quadrated by using the formula if we can quadrate the area of the quadrangles crossing the diagonal lines perpendiculary. The pupils answered," it
could do---". Just then we saw the figure(fig.10)
of the square in the textbook and I had them measure both one side line and the diagonal line, quadrating the area.
And then they found one side has just 3cm but diagonal line has about 4.2cm that shows subtle error. The result indicated that the area in case of "one side ~ one side"
shows 3~3=9(cu), and the area in case of the formula "one diagonal line ~another
diagonal line
2" shows 4.2~4.22=8.82(cu). We found that two areas are different. The pupils
saying, " We cannot quadrate-----, different------". For the fact I asked them
whether its difference of tha areas should be the same or whether this formula still
cannot be used, how they did think about it. But when we were searching this problem,
we have not yet spare time and I told them that we would think whether this formula could be applied to the square's area, too.
Lastly I had them write the impression about the lesson and had two pupils announce them. The lesson was over.
4. Conclusion
I selected a developing teaching material,that is, quadrature of the kite shape's area,
for searching how to instruct of reconsideration for deepning pupils' thinking and cultivating task consciousness. Though I heard at the study session that there had been the practise that used the kite shape, I think that my lesson program and basic
idea are different from them. I created an original teaching material and practised as
original lesson.
In Japan, they propose of "New learning ability"which emphasize bringing up pupils' creativity and independence. Then I think that I would like to construct my lessons creatively and independently. In such sense, emphasizing in learning is not new but using
to instruct reconsiderately is , I think , one of creative acts.
[This article was carried on "The Elementary Mathematics Education"(Meiji-Tosyo publishing co.,) published Feb. 1996.]
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