Capítulo 4. Resultados y discusión
4.1. Artículos de investigación
4.2.4. Análisis Bivariado
4.2.4.3. Análisis Bivariado. Indicadores de sustentabilidad como variables explicativas
CCSS: Geometry: 3.G, 4.G
Mathematical Practices: 1, 2, 3
Visualization skills are called up on as students imagine shapes with 12 edges. The number 12 was stipulated because one of the most obvious shapes, the rect- angular prism, would be a possible response for students who cannot think of other things. The interesting question becomes whether or not there are other shapes and, indeed, there are.
The base of the prism could be any quadrilateral, for example, a kite or a trapezoid, instead of a rectangle.
But the shape could also be a pyramid with a 6-sided base.
Variations. Rather than indicating the number of edges, the number of vertices or faces could be given.
Th ere are at least three diff erent names you could use for a shape. What could that shape be?
CCSS: Geometry: 5.G
Mathematical Practices: 1, 2, 6
In this grade band, there is attention to the notion that some shapes are special cases of other shapes. For example, a square is also a parallelogram and a quadri- lateral. (It is also a rhombus and a rectangle.) An equilateral triangle is also a tri- angle, a polygon, and a regular shape.
A question like this one helps students focus on the fact that most shapes have many names.
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A shape has some pairs of perpendicular sides and some pairs of parallel sides. Th ere are more pairs of parallel sides. What could the shape be?
CCSS: Geometry: 4.G, 5.G
Mathematical Practices: 1, 2, 6
Addressing this question is likely to encourage students to think about all the shapes they know and the attributes of those shapes, making it a nice review sort of question. Students will realize that triangles cannot have parallel sides, so the shape could not be a triangle. Rectangles have two pairs of parallel sides, but four pairs of perpendicular sides, so they cannot be used. Parallelograms and trapezoids are possibilities, but not necessarily if they also have perpendicular sides. Regular octagons or regular hexagons would defi nitely meet the criteria.
Variations. The relationship between the number of pairs of perpendicular and parallel sides could be changed.
Create two shapes so that one of them has many more lines of symmetry than the other. Tell how many lines of symmetry each shape has.
CCSS: Geometry: 4.G
Mathematical Practices: 1, 2, 5
Students might choose very simple shapes to work with, but have an oppor- tunity to interpret the phrase many more in whatever way they wish. Some students will choose a circle, knowing it has many lines of symmetry, but they may be unsure when describing the number of lines of symmetry (since the number is infi nite). Students might then consider other shapes, such as hexagons, octagons, etc.
Students are likely to discover that regular shapes have more lines of symmetry than irregular shapes. In fact, it might be good to ask, How do you know that there have to be some equal side lengths if a shape has a line of symmetry?
Students should be expected to prove that a line is a line of symmetry by cut- ting out and folding.
Variations. Although students do not formally study planes of symmetry in three dimensions, they might explore linking cube structures that they believe have refl ective (or mirror) symmetry.
Draw two angles so that the obtuse angle is about three times as big as the acute angle.
CCSS: Measurement & Data: 4.MD Mathematical Practices: 1, 2, 5
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In this grade band, students begin to distinguish obtuse from acute and right angles. This question allows the student familiar with degrees to use that thinking to create angles. They would have to realize that if they use a 30° angle for the acute angle, the obtuse angle would need to be a bit more than 90°. Or they might realize that they could not use an acute angle of more than 60° if the angle three times the size is obtuse and not greater than 180°.
Other students could simply create an acute angle and lay it down three times to create the associated obtuse angle. If their acute angle is too small, the angle will not be obtuse and they will learn that they need a bigger, but not too big, acute angle.
Variations. The notion that the obtuse angle is three times the acute angle is simple to change; any other factor could be chosen instead. Another variant could stipulate that the larger angle must also be acute, forcing students to think about how small the initial angle would have to be.
You are required to construct a triangle where one side is 10 cm long, one side is a lot longer, and one angle is obtuse. Decide fi rst on the long side length and the angle measure. Th en construct the triangle.
CCSS: Geometry: 4.G
Mathematical Practices: 1, 2, 5, 6
Rather than instructing students as to what side and angle measures to use, there is value in letting students choose. One advantage is that students can include the angle between the two sides to make the task less diffi cult if they need to. That said, some students should probably be challenged to make the angle one of the non-included angles. Another advantage is that students will discover that certain combinations of side lengths may not be possible, for example, 10-30-10. Even more importantly, the diversity of triangles produced will allow for a rich class discussion.
If, for example, a student chooses a longer length of 20 cm and an angle of 120°, he or she might proceed as shown at the right: 120° 20 cm 10 cm 120° 20 cm 10 cm
If, though, the angle is not included, a possible triangle is this one: 120° 20 cm 10 cm 120° 10 cm
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3
BIG IDEA. Shapes can be located in space and relocated by using mathematical processes.An isosceles triangle is drawn on a coordinate grid. One of the vertices is at the point (4,5). Where might the other vertices be?
CCSS: Geometry: 5.G
Mathematical Practices: 1, 2, 5, 6
At this level, students begin to use the fi rst quadrant of a coordinate grid to locate objects in space. Asking students to create isosceles triangles on a grid is an interesting way to have them practice their coordinate graphing skills, while still supporting other geometric concepts.
There are many possible solutions with (4,5) as the vertex where the equal sides meet; one is shown below, but the base line could easily be extended the same amount on both sides of the vertex.
6 5 4 3 2 1 3 2 0 1 4 5 6 7 y x
However, (4, 5) need not be the place where equal sides meet, as shown below.
6 5 4 3 2 1 3 2 0 1 4 5 6 7 y x
Variations. Instead of an isosceles triangle, some other shape could easily be selected.
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OPEN QUESTIONS FOR GRADES 6–8
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BIG IDEA. Shapes of diff erent dimensions and their properties can be described mathematically.Choose an angle size and construct that angle. What could you do to create an angle of the same size and the angle’s supplement without using a protractor?
CCSS: Geometry: 7.G, 8.G
Mathematical Practices: 1, 2, 5
Although it may be necessary to ensure that students recognize that the supple- ment of an angle is the one with a measure equal to 180° less the original angle measure, students can take different directions to accomplish the task.
Some students might draw the angle and extend both rays to form both the supplement and an equal vertical angle.
supplement =
Other students will extend each ray and add a parallel line to create parallel lines cut by a transversal.
supplement =
Still others might choose to create a triangle with that angle, trace it, and duplicate it to create a congruent angle and put the other two angles in the triangle together to make the supplement.
Draw diff erent-looking triangles where one of the angles measures 100º and one of the sides is 3" long. What else do the triangles have in common?
CCSS: Geometry: 7.G
Mathematical Practices: 1, 2, 5
Students will, of course, draw only obtuse triangles because one of the angles must be 100°. Some will realize this right away, but others will need to actually do the drawings to see it.
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Because no other angle or side length is specifi ed, there will be great diversity in what the students draw. This experience will set them up for the realization that two pieces of information are never enough to defi ne a triangle and that three pieces of information may or may not defi ne a single triangle, depending on what pieces of information are offered.
A right triangle has two side lengths of 6" and 8". A mathematically similar triangle has one side length of 24". What could all three side lengths of each shape be?
CCSS: Geometry: 8.G
Mathematical Practices: 1, 2, 5, 6
What makes this question interesting is that there is no diagram, so the student does not know which of the original side lengths matches the new 24" side length. It could be the 6" that matches the 24" and all of the original side lengths are mul- tiplied by 4. But it could be the 8" that matches the 24" and all of the original side lengths are multiplied by 3. Or it could actually be the third side of the original right triangle, which is either 10" (√62+82) or √28" (√82–62) long, in which case the
scale factor is 24 divided by either amount. Thus, the question might be viewed as either simple or complicated.
It is because the triangle is a right triangle and the Pythagorean theorem can be applied that the third side is actually known.
Variations. Instead of using a common multiple of the two given side lengths in the smaller triangle, like 24, a number that is a multiple of only one of the provided length measures, or of neither measure, could be used.
Draw a right triangle. Build another shape on the hypotenuse so that one side of the shape is the hypotenuse of the original right triangle. Now build shapes mathematically similar to the one you created on the other sides of the original right triangle.
Calculate the areas of the three shapes you created. What do you notice?
CCSS: Geometry: 8.G
Mathematical Practices: 1, 3, 5, 7, 8
One of the interesting extensions of the Pythagorean theorem is that if similar shapes (not just squares) are built on the three sides of a right triangle, the largest area is the sum of the areas of the two smaller shapes built. By posing the question in this way, so that students can choose the shape to build, they can make the work more or less challenging, as is appropriate.
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Some students might build a rectangle or a simple isosceles right triangle on each side of the original right triangle. This makes the calculations for similarity as well as for determining area much simpler.
Other students might choose to put semicircles on each side of the original right triangle, and still others might choose much more challenging, unusual shapes.
You are given a shape and most of its dimensions. You need the Pythagorean theorem to fi gure out the area of the shape. What could the shape be and what dimensions would you already have?
CCSS: Geometry: 8.G
Mathematical Practices: 1, 2, 3
To respond to this question, students need to visualize what sort of shape to use. Initially, most will choose a right triangle, triggered by the phrase Pythagorean theorem, but if the values of the two legs of that triangle are given (rather than the value of one leg and the hypotenuse), the Pythagorean theorem is actually not required to determine the area.
Students who are ready should be encouraged to think of other shapes as well. One possibility is a trapezoid like the one shown at the right, where all four side lengths are given and the height is determined using the Pythagorean theorem.
Another possibility is a regular hexagon, with the side lengths given. Again, the Pythagorean theorem is useful for determining the height of each of the six triangles making up the hexagon. This is possible because the triangles are all equi- lateral, so the hypotenuse lengths are also known.
Students can be challenged to think of other shapes as well.
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BIG IDEA. Th ere are always many representations of a given shape.You draw a net of a 3-D shape in order to calculate its surface area. When you look at the net, there are only two face sizes. What shape might it be a net of?
CCSS: Geometry: 6.G
Mathematical Practices: 1, 2, 7
This question is designed to focus students on what sorts of nets they might see when they want to use a net to calculate surface area.
If, for example, the shape is a rectangular prism but not a cube, a student might see only two face sizes if the base is a square. Students might also see two face sizes if the shape is any pyramid where the base is a regular shape that is not an equilateral triangle. Students might also see only two face sizes for a non-rectangular prism.
A shape has a triangle for a cross-section. What could the shape be?
CCSS: Geometry: 7.G
Mathematical Practices: 1, 2
This question is suitable once students know what a cross-section is. Some students will assume that a pyramid is the logical shape to try and will fi nd that it is not as easy as they thought to get a triangle cross-section. It is possible, for example, if a triangle-based pyramid is cut parallel to a base or at a vertex.
Other students will recognize that if a cut is made whenever three faces meet at a vertex, a triangle can be formed. Thus, a cube could be cut to create a triangular cross-section, as shown at the right.
The question is suitable for a broader group of students if modeling clay is provided for students to make shapes and den- tal fl oss to allow them to cut cross-sections.
Choose a 3-D shape. Describe all the possible cross-sections you could create.
CCSS: Geometry: 7.G
Mathematical Practices: 1, 2
Allowing students a choice of the shape to cut can make the problem simple or more complex. Some students might choose shapes with curved surfaces, such as spheres, cylinders, or cones, whereas others might choose cubes, square pyra- mids, or triangular pyramids. If the shapes are made of materials that can be cut with dental fl oss, students can investigate the possibilities directly, which is much clearer to some students than using drawings or simply visualizing.
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ions for Grades 6–8
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BIG IDEA. New shapes can be created by either composing or decomposing existing shapes.Can you usually divide a shape up into one or more similar shapes?
CCSS: Geometry: 8.G
Mathematical Practices: 1, 2, 5, 6, 7, 8
To approach this problem, students will need to know that shapes that are similar are enlargements of or reductions of (or identical to) a given shape. A stu- dent is likely to begin with a simple shape such as a circle and try to divide it up. They will see that it is not possible to divide up a circle into only circles that are smaller.
If they try a square, they will see that it can be divided into more squares, in more than one way:
Some students will quit there, but others will investigate further, to see for which types of shapes this kind of division is possible and for which it is not. Because a simple shape like a square provides a solution, the problem should be accessible to all students in the class.
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BIG IDEA. Shapes can be located in space and relocated by using mathematical processes.A shape is completely in the fi rst quadrant. Where could it be after a translation?
CCSS: Geometry: 8.G
Mathematical Practices: 1, 2, 5
Some students will try one shape and one translation and come up with an answer. For example, a student might draw a square and move it up or to the right and correctly point out that the shape has stayed in the fi rst quadrant. Other stu- dents will realize that they could translate suffi ciently to the left to end up in the second quadrant, suffi ciently down to end up in the fourth quadrant, or suffi ciently left and down to end up in the third quadrant. Still other students will look for all the possible combinations, showing that the shape could end up in one, two, three, or even all four quadrants.
Variations. Similar questions can be created by changing the transformation (to rotations, refl ections, or even dilatations) or by stipulating the shape with which to start.
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ions for Grades 6–8
Draw several polygons on a coordinate grid to make a picture. Th e only conditions are that somewhere there is a side with a length of 3 coordinate units, somewhere a side with a length of 5 coordinate units, and somewhere a side with a length of 10 coordinate units.
CCSS: Geometry: 6.G, 8.G Mathematical Practices: 1, 5
This question is designed to provide an opportunity for students to show their understanding of how to measure horizontal and vertical distances on a coordinate grid, although some students who are ready might choose to measure diagonal distances, either using a ruler and relating the measures to the measures on the grid or using knowledge of the Pythagorean theorem. The question is posed in such a way that students can use simple shapes if they wish or more complex ones.
Making a picture is clearly not a critical part of the mathematics learning, but it does tend to make the activity more engaging for students.
One vertex of a triangle is at the point (1,2). After a refl ection, one vertex is at the point (5,8). Name all three vertices of the original and fi nal triangles.
CCSS: Geometry: 8.G
Mathematical Practices: 1, 2, 5
Some students will only be comfortable with either horizontal or vertical refl ection lines. The way the question is posed allows them success even with this limitation.
For example, the original triangle could have been positioned with vertices at (1,2), (1,8), and (3,8) and the new triangle with vertices at (5,2), (5,8), and (3,8):
(1,2)
(5,8)
Other students will assume that it was the point (1,2) that moved to (5,8). They will look for a “diagonal” refl ection line that accomplishes this task. In fact, they could use such a line and the original vertices (1,2), (3,5), and (6,3) move to (5,8), (3,5), and (6,3):
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ions for Grades 6–8
(1,2)
(5,8)
Still other students will recognize that there are even more possibilities than that. For example, the original triangle could have been positioned with vertices at (1,2), (0,3), and (−2,1) that move to (5,8), (6,7), and (7,10).
You draw a shape on a coordinate grid. After you rotate the shape, you draw the image. You remove evidence of the center of rotation, but someone else has to prove it was a rotation. What could he or she do?
CCSS: Geometry: 8.G
Mathematical Practices: 1, 2, 3
As students are introduced to transformations, the focus should be on the effects of those transformations. Students should understand that linear, area, and angle measures are unchanged, as are relationships of parallelism and perpen- dicularity.
In the case of a rotation, students can observe the turn, but they need to rec- ognize that there is a center of rotation that can be determined by locating the