Grade 7

 

Geometry

 

 

Standard 7-4:   The student will demonstrate through the mathematical   processes an understanding of proportional reasoning, tessellations, the use of geometric properties to make deductive arguments, the results of the intersection of geometric shapes in a plane, and the relationships among angles formed when a transversal intersects two parallel lines.

 

The indicators for this standard are grouped by the following major concepts:

 

The indicators that support each of those major concepts and an explanation of the essential learning for each major concept follows.

 

 

Plane and Transformational- Plane

 

Indicators

7-4.2    Explain the results of the intersection of two or more geometric shapes in a plane.

7-4.3    Illustrate the cross section of a solid.

7-4.4    Translate between two- and three-dimensional representations of compound figures.

7-4.5    Analyze the congruent and supplementary relationships—specifically, alternate interior, alternate exterior, corresponding, and adjacent—of the angles formed by parallel lines and a transversal. 

 

          Seventh grade students are required to explain the results of the intersection of two or more geometric shapes in a plane.  For example, if a line intersects a circle, the result of the intersection is two points.  This can be modeled by drawing a picture of a circle and the line going through it.  Using models and pictures will aid students as they transition to attaining the ability to visualize the intersection of two or more geometric shapes.  An example of a hands-on model would be using a pen and a piece of paper to demonstrate what happens when a plane and line intersect, the intersection being a point. Poking the pen through the paper gives the students a visual demonstration of this concept.  Encourage students to think of everyday objects that can represent points, lines, planes, etc. in an effort to help them visualize the intersection.

          In 5th grade, students explored methods for translating between two-dimensional representations and three-dimensional objects. They learned to sketch the front, top, and side views of a three-dimensional object built with cubes. They also learned to draw a net for a given three-dimensional shape and construct and/or state the three-dimensional shape when given its two-dimensional representation (net).  In 7th grade students should acquire the ability to translate between two- and three-dimensional representations of compound (when two or more, two-dimensional or three-dimensional figures are joined together) figures and to illustrate the cross section of a solid.  Extensive modeling with concrete objects needs to be done in order for the students to develop a mental picture of compound three-dimensional shapes and the two-dimensional view points that give the figure it’s overall shape and vice versa. Student work with cross sections at seventh grade should be limited to “deconstructing” the layers of a three-dimensional object. For example, to illustrate the cross section of a rectangle, students may build a rectangle using interlocking cubes. Then the students might illustrate on isometric dot paper the original rectangle and a view of one of the “layers” or cross sections that make up the rectangle. The ability to do this will enable students to develop, justify, and understand formulas (such as area, surface area, volume, etc.) that are used in regards to two- and three-dimensional figures.  Because seventh grade is the first time students are introduced to the concept of volume, illustrating the cross section of three-dimensional shapes is an appropriate prerequisite to generating strategies for finding volume. It should be noted that in the example just cited, students should illustrate a horizontal and a vertical cross section. Also, it would be sound educational practice to do the same with a cube and discuss the relationship between the horizontal and vertical cross sections. When engaging in such a discussion, it is important for students to understand that the concrete models and pictorial illustrations are the result of the intersection of a plane in a segment of the geometric shape. (This stays true to the definition of a cross section – the intersection of a plane and a geometric solid.)     

          The general attributes and names of parallel, perpendicular, and intersecting

lines were discussed in prior grades. In 7th grade the specific angle

relationships formed between parallel lines and a transversal (a line that

intersects two or more lines in different points) are explored and identified. 

Students in 7th grade should analyze the congruent and supplementary

relationships of the angles formed by two parallel lines and a transversal. 

Students should have the ability to understand, identify, and use in

            mathematical communication the terms alternate exterior, alternate interior, corresponding, and adjacent in regards to the angles formed by parallel lines and a transversal. 

          Students need to formulate their own conclusions in regards to the angles formed by two parallel lines and a transversal through guided investigation before the formal definitions are introduced. They should have the opportunity to measure the angles formed by these lines, and from their findings, make conjectures and draw conclusions about which angles are congruent (a concept taught in 5th grade) and which angles are supplementary (a concept taught in 6th grade).  Once the students have a good understanding of which angles are congruent and/or supplementary, then the formal definitions can be introduced.

 

 

 

 

Plane and Transformational – Transformational

 

Indicators

7-4.9      Create tessellations with transformations.

7-4.10    Explain the relationship of the angle measurements among shapes   

              that tessellate.

7-4.1               Analyze geometric properties and the relationships among the properties of triangles, congruence, similarity, and transformations to make deductive arguments.

 

          Seventh grade students should use transformations to create tessellations, but more importantly should be able to explain the relationship of the angle measurements among shapes that tessellate. This is the students’ first introduction to tessellations. Tessellations (the covering of a plane without overlaps or gaps using congruent figures or a combination of congruent figures) are formed by transformations such as translations, reflections, and rotations (students studied transformations in both the 5th and 6th grades).  The emphasis should not be on creating a tessellation. The emphasis should be on the transformations used to create the tessellation and the relationship among the polygons that can be used to tessellate (or tile) a plane (piece of paper).

          Discovering the relationship of angle measures among shapes that tessellate

will require guidance by the teacher.  The objective is for students to

discover that in order for regular (all sides are the same length) polygons to

tessellate, the sum of the measures of the angles surrounding a point (at a

vertex) must be 360°.  This sum can be attained by using a combination of

different regular polygons or just several of the same polygon. Using regular

polygon manipulatives and the following chart may be helpful.

 

Shape

Triangle

Square

Hexagon

Octagon

 

Measure of one interior angle

 

60°

 

90°

 

120°

 

135°

Combinations that total 360° and therefore will tessellate.

Number

6

 

 

 

60 (6) = 360

of

4

 

1

 

60 (4) + 120 = 360

each

 

1

 

2

90 (1)+ 135 (2) = 360

shape

 

 

3

 

120 (3) = 360

used

1

2

1

 

60(1)+90(2) +120(1) = 360

 

          Prior to 7th grade, students have had experience with triangles and transformations. In fifth grade they classified shapes and congruent and in 6th grade they classified shapes as similar. Seventh grade students should build on those experiences by analyzing the geometric properties of congruent and similar triangles. After doing so, students should come to the conclusion that congruent triangles can be matched by simply placing one on top of the other. On the other hand, similar triangles have congruent corresponding angles and sides that match by a constant scale factor. Student experiences should enable them to come to those conclusions and be able to defend their thinking – not merely memorize the factual relationships. The goal is for students to begin to think more formally about the concepts of congruency and similarity and how transformations might be used when proving relationships. “Investigations into the properties of, and relationships among, similar “triangles” can afford students many opportunities to develop and evaluate conjectures inductively and deductively. For example, an investigation of the perimeters, areas, and side lengths of the similar and congruent triangles. . .could reveal relationships and lead to generalizations. Teachers might encourage students to formulate conjectures about the ratios of the side lengths, of the perimeters, and of the areas of the . . .triangles. (Students) might conjecture that the ratio of the perimeters is the same as the scale factor relating the side lengths and that the ratio of the atreas is the square of that scale factor.” (Principles and Standards for School Mathematics, 2000, page 234-234) All of this could then be linked to the Indicators listed in the following “Proportional Reasoning” section.

 

Proportional Reasoning

 

Indicators

7-4.6    Compare the areas of similar shapes and the areas of congruent shapes.

7-4.7    Explain the proportional relationship among attributes of similar shapes.

7-4.8    Apply proportional reasoning to find missing attributes of similar shapes.

 

          In sixth grade, the concept of similarity was introduced and students compared the angles, side lengths, and perimeters of similar shapes and also classified shapes as similar.  Fifth grade is when the concept of congruency was introduced and students compared the angles, side lengths, and perimeters of congruent shapes and also classified shapes as congruent.  In 7th grade, students will now compare the areas of similar shapes and the areas of congruent shapes.  Students should be given the opportunity to discover that the areas of congruent shapes are equal whereas the areas of similar shapes are not.  This indicator may be extended to allow students to discover and understand that the area of the larger of two similar shapes will be the area of the smaller shape multiplied by the square of the scale factor needed to create the larger similar shape.  This may be difficult for students to initially conclude; therefore numerous examples should be done to help students see this relationship. It is sound educational practice to start with Indicator 7-4.1 explained under the “Plane and Transformational - Transformational” section above. Once students are comfortable making deductive arguments with triangles, they are ready to move on to a variety of congruent or similar shapes.

          A natural link can be made to the major concept of ratio and proportion when students begin to compare attributes of similar and/or congruent figures such as ratios of angles to corresponding angles.  Seventh grade students should explore how similar shapes compare proportionally and be able to explain the proportional relationship among attributes of similar shapes.  For example, corresponding angle measures are congruent while corresponding sides are proportional.

          Seventh grade students should extend their understanding of ratio and proportion as they explore similarity at a more in-depth level.  This understanding will provide students the prerequisites needed to find missing attributes of similar shapes.  Once again, it is extremely important that students have made the connections in regards to similar figures that corresponding angle measures are congruent while corresponding sides are proportional.

          Seventh grade students must have an in-depth understanding of when and how to apply proportional reasoning to solve various types of problems. Teachers should provide problems that encourage students to use these concepts.  Students should be encouraged to not only find solutions to these problems, but also justify their solutions and explain the method they chose to derive their solution.

 

Connections to:

 

Other Seventh Grade Indicators

7-2.5 Apply ratios, rates, and proportions to discounts, taxes, tips, interest, unit costs, and similar shapes.

7-5.1  Use ratio and proportion to solve problems involving scale factors

          and rates.

 

Since the connections between these Indicators and the Indicators cited above were explained in the above essential learning information, no further explanation will be provided in this section. For specific information related to essential learning for these Indicators see the Number and Operations section for 7-2.5 and the Measurement section for 7-5.1