Drawing Triangles
Introduction
This group of lessons can be taught in first and second grades just after children learn how to measure in centimeters and millimeters and how to draw circles with a compass.
Equipment. Rulers, compasses and calculators. (Do not attempt this lesson if children do not have compasses that can keep a constant radius! The best are the more expensive ones, in which the radius is controlled by turning a small wheel.)
The purpose of this lesson is to teach the skill of drawing a triangle when the lengths of its three sides are given, and to teach when such a construction is possible.
(1) Given three positive numbers, is there a triangle whose sides have these lengths (measured in a given unit)?
The necessary (and sufficient) condition can be formulated in several ways.
Here are a few examples.
There is such a triangle when:
• The biggest of the numbers is smaller than the sum of the remaining two.
• The smallest number is bigger than the difference of the remaining two.
• Any number is bigger than the difference of the remaining two.
• Any number is smaller than the sum of the remaining two.
(2) Generic method of constructing the required triangle, given three numbers a, b, and c (and the unit of measurement).
1. Draw a straight line and mark on it two points forming a segment of length a. Use only the ruler. It is always better to draw the line too long, and later erase what is not needed, than to try to extend a line that is too short.
2. Choose on which side of the line you want your triangle, and to which end of your segment the second leg is to be attached.
Using the ruler for measuring, spread your compass to draw a circle with radius b. Put the sharp point of the compass on the chosen end, and draw at most half of a circle on the chosen side of the line.
3. Now spread your compass to draw a circle of radius c, and draw again at most half a circle on the same side of the line, but using the other end of the segment as the center point.
4. Connect the intersection of these half circles to the ends of the segment with straight lines, using a ruler.
5. Erase all unneeded portions of your picture, and MEASURE all three sides with the ruler, to check for errors.
Remark 1. If the parts of the two drawn circles do not intersect, it could be that:
• you drew arcs that were too small, or
• there is no triangle that has these three sides (see (1) above).
Remark 2. When children are drawing triangles, tell them what errors are acceptable. A reasonable approach at this level is to accept segments with lengths different by less than (+/-) 2 millimeters from the required one.
Lessons.
Part one.
Write three numbers on the blackboard. Explain and show children how to draw a triangle with sides of these lengths, following the procedure above. Repeat it until children are comfortable in doing this task.
This may take a long time (more than one class period). Therefore, vary the task by
• varying the sizes and shapes of the triangles.
• asking children to color and cut out the triangles they are drawing.
• providing such numbers for which triangles exist.
Children may help each other, but the work is individual.
(At the end, each child should have a nice collection of triangles.)
Example of a list of numbers left on the blackboard.
| 9 cm | 8 cm | 7 cm |
| 10 cm | 6 cm | 8 cm |
| 7 cm | 7 cm | 12 cm |
| 4.5 cm | 4.5 cm | 6 cm |
| 8.2 | 9.6 | 7.1 |
| 15.7 | 15.7 | 15.7 |
| 6.7 | 5.8 | 11.3 |
Part two.
Write again triplets of numbers, but now, for some of them, there is no triangle. (This can create consternation among children.)
Children will see that their parts of circles do not intersect.
Either the circles are disjoint, or one is inside the other. It is important that the children have already worked long enough with circles that they see this as a real phenomenon and not their "error." (You may tell the children that we will find out when three numbers lead to a triangle, and when they do not.)
Example of a list left on the blackboard.
| 12.3 | 7.6 | 8.9 | |
| 7 | 5 | 14 | no triangle |
| 7 | 5 | 10 cm | |
| 14.7 | 3.5 | 6.6 | no triangle |
| 14.7 | 13.5 | 6.6 | |
| ... | ... | ... |
Part 3.
When can you not draw a triangle?
• One of two numbers is too small.
• One number is too big.
Example. Suppose the three numbers are 7, 5, and 14. Then 7 (or 5) is too small, 14 is too big.
How to check it?
(Do not expect that children will discover it. Tell them, and let them check that it gives a correct answer with known examples.)
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* Add the two smaller numbers. If their sum is less than or equal to the biggest number, there is no triangle.
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Calculator procedure:
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* Add the two smaller numbers and subtract the biggest.
* If the answer is a negative number or zero, then there is no triangle.
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From this point on, children can check with a calculator if there is a triangle, before attempting a construction.
Examples.
7.7 5.6 11.2
[7.7][+][5.6][-][11.2][=] returns 2.1; there is a triangle.
>6.5 12.3 4.1
[6.5][+][4.1][-][12.3][=] returns -1.7; minus! No triangle.
Remark. An essential element of this part of the lesson is that children find the biggest number MENTALLY. If they have not learned it before, they should learn it at this point. We are stressing this point because an appropriate mixture of mental and mechanical calculations is important for efficient problem solving.