SEVENTH GRADE > 7.G.1 TEACHER GUIDE

## TEACHER GUIDE TO CLARIFICATION

###### 7.G.1

Draw, construct, and describe geometrical figures and describe the relationships between them.

7.G.1   Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.

###### Scale Drawings

In this Standard students will understand scale drawings of geometric figures, including:

• Computing lengths from a scale

• Computing area from a scale drawing

• Reproducing a scale drawing at a different scale

This cluster focuses on the importance of visualization in the understanding of Geometry. Being able to visualize and then represent geometric figures on paper is essential to solving geometric problems.

Scale drawings of geometric figures connect understandings of proportionality to geometry and lead to future work in similarity and congruence. As an introduction to scale drawings in geometry, students should be given the opportunity to explore scale factor as the number of time you multiple the measure of one object to obtain the measure of a similar object. It is important that students first experience this concept concretely progressing to abstract contextual situations. Pattern blocks (not the hexagon) provide a convenient means of developing the foundation of scale. Choosing one of the pattern blocks as an original shape, students can then create the next-size shape using only those same-shaped blocks. Questions about the relationship of the original block to the created shape should be asked and recorded. A sample of a recording sheet is shown

Use Pattern Blocks to explore scale

Shape

Square

Triangle

Rhombus

Original Side Length

1 unit

1 unit

1 unit

Created Side Length

Scale Relationship of Created to Original

This can be repeated for multiple iterations of each shape by comparing each side length to the original’s side length. An extension would be for students to compare the later iterations to the previous. Students should also be expected to use side lengths equal to fractional and decimal parts. In other words, if the original side can be stated to represent 2.5 inches, what would be the new lengths and what would be the scale.

Shape

Square

Parallelogram

Rhombus

Original Side Length

2.5 inches

3.25 centimeters

(Actual measurements)

Created Side Length

Scale Relationship of Created to Original

Length 1

Length 2

Provide opportunities for students to use scale drawings of geometric figures with a given scale that requires them to draw and label the dimensions of the new shape. Initially, measurements should be in whole numbers, progressing to measurements expressed with rational numbers. This will challenge students to apply their understanding of fractions and decimals.

Note the scaffolding involved in this standard –

moving through conceptual understanding to application of knowledge

After students have explored multiple iterations with a couple of shapes, ask them to choose and replicate a shape with given scales to find the new side lengths, as well as both the perimeters and areas. Starting with simple shapes and whole-number side lengths allows all students access to discover and understand the relationships. An interesting discovery is the relationship of the scale of the side lengths to the scale of the respective perimeters (same scale) and areas (scale squared). A sample recording sheet is shown.

Students should move on to drawing scaled figures on grid paper with proper figure labels, scale and dimensions. Provide word problems that require finding missing side lengths, perimeters or areas. For example, if a 4 by 4.5 cm rectangle is enlarged by a scale of 3, what will be the new perimeter? What is the new area? If the scale is 6, what will the new side length look like? Suppose the area of one triangle is 16 sq units and the scale factor between this triangle and a new triangle is 2.5. What is the area of the new triangle?

Great

Idea!

Reading scales on maps and determining the actual distance (length) is an appropriate contextual situation.

Constructions facilitate understanding of geometry. Provide opportunities for students to physically construct triangles with straws, sticks, or geometry apps prior to using rulers and protractors to discover and justify the side and angle conditions that will form triangles.

Explorations should involve giving students: three side measures, three angle measures, two side measures and an included angle measure, and two angles and an included side measure to determine if a unique triangle, no triangle or an infinite set of triangles results. Through discussion of their exploration results, students should conclude that triangles cannot be formed by any three arbitrary side or angle measures. They may realize that for a triangle to result the sum of any two side lengths must be greater than the third side length, or the sum of the three angles must equal 180 degrees. Students should be able to transfer from these explorations to reviewing measures of three side lengths or three angle measures and determining if they are from a triangle justifying their conclusions with both sketches and reasoning.

This cluster is related to the following Grade 7 cluster “Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.” Further construction work can be replicated with quadrilaterals, determining the angle sum, noticing the variety of polygons that can be created with the same side lengths but different angle measures, and ultimately generalizing a method for finding the angle sums for regular polygons and the measures of individual angles. For example, subdividing a polygon into triangles using a vertex (N-2)180° or subdividing a polygons into triangles using an interior point 180°N - 360° where N = the number of sides in the polygon. An extension would be to realize that the two equations are equal.

Slicing three-dimensional figures helps develop three-dimensional visualization skills. Students should have the opportunity to physically create some of the three-dimensional figures, slice them in different ways, and describe in pictures and words what has been found. For example, use clay to form a cube, then pull string through it in different angles and record the shape of the slices found. Challenges can also be given: “See how many different two-dimensional figures can be found by slicing a cube” or “What three-dimensional figure can produce a hexagon slice?” This can be repeated with other three-dimensional figures using a chart to record and sketch the figure, slices and resulting two-dimensional figures.

7.G.1

Students determine the dimensions of figures when given a scale and identify the impact of a scale on actual length (one-dimension) and area (two-dimensions). Students identify the scale factor given two figures.  Using a given scale drawing, students reproduce the drawing at a different scale.

Students understand that the lengths will change by a factor equal to the product of the magnitude of the two size transformations.

Example:

Julie showed you the scale drawing of her room. If each 2 cm on the scale drawing equals 5 ft, what are the actual dimensions of Julie’s room? Reproduce the drawing at 3 times its current size.

Kansas Association of Teachers of Mathematics (KATM) Flipbooks. Questions or to send feedback: melisa@ksu.edu.

PARCC Grade 7 Computer-Based EOY Practice Test
http://parcc.pearson.com/practice-tests/math/

The scale on a map shows that 5 centimeters = 2 kilometers.

Part A

What number of centimeters on the map represents an actual distance of 5 kilometers?

centimeter(s)

Part B

What is the actual number of kilometers that is represented by 2 centimeters on the map?

kilometer(s)

###### Coherence and Connections: Need to Know

6.G.1

7.G.1

7.RP.2

8.EE.6

This cluster is connected to the Grade 7 Critical Area of Focus #3, Solving problems involving scale drawings and informal geometric constructions, and working with two- and three- dimensional shapes to solve problems involving area, surface area, and volume.

Connections should be made between this cluster and the Grade 7 Geometry Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. (7.G.4-6).  Grades 6 and 7: Ratios and Proportional Relationships.  This cluster leads to the development of the triangle congruence criteria in Grade 8.

Kansas Association of Teachers of Mathematics (KATM) Flipbooks. Questions or to send feedback: melisa@ksu.edu.

Students grow in their ability to analyze proportional relationships. They decide whether two quantities are in a proportional relationship (7.RP.2a); they work with percents, including simple interest, percent increase and decrease, tax, markups and markdowns, gratuities and commission, and percent error (7.RP.3); they analyze proportional relationships and solve problems involving unit rates associated with ratios of fractions (e.g., if a person walks 1/2 mile in each 1/4 hour, the unit rate is the complex fraction ½ / ¼ miles per hour or 2 miles per hour) (7.RP.1); and they analyze proportional relationships in geometric figures (7.G.1).

Examples of Opportunities for Connections among Standards, Clusters or Domains
Students use proportional reasoning when they analyze scale drawings (7.G.1).

PARCC Model Content Frameworks: Mathematics Grades 3-11 (version 5). (2017, November).

Illinois Assessment of Readiness Mathematics Evidence Tables

Evidence

Statement Key

Evidence Statement Text

Clarifications

MP

7.G.1

Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.

2,5

Calculator

yes

i). Tasks may or may not have context.

###### Classroom Resources

7.G.1 Daily Discourse

###### HOT Questions

1.  I created a rectangular dog pen for my puppy using 36 feet of fence.  He is now three times bigger.  How many feet of

fencing will I need to create a space that is 3 times bigger?

15 feet

3 feet

3 feet

15 feet

15 + 15 + 3 + 3 = 36 feet         perimeter = 36 feet

15 x 3 = 45                              area = 45 sq. ft

In order to make the space bigger I just need to multiply one side length by three to create a space that is 3 times bigger.

15 x 9 will be my new pen

15 + 15 + 9 + 9 = I will need 48 feet of fencing

And the area will increase from 45 sq. feet to 135 sq. feet which is 3 times bigger.

2.  Mariko has an 80:1 scale drawing of the floor plan of her house. On the floor plan, the dimensions of her rectangular          living room are      inches by 2 ½ inches.

What is the area of her real living room in square feet?

3.  I would like to make a gravel path to my garden.  The path is 10 ft long by 2 ft wide.  The gravel should be about 2 inches

thick.  Gravel costs \$15.00 per cubic foot.  How much will it cost to create my gravel path way?

Solution

10 x 2 x 2 = 40ft3

\$15.00 x 40 ft3 = \$600.00

4.   The scale on a map shows 2 centimeters = 30 feet.  What is the actual number of centimeters that represents a distance of       10 feet?

2/3 cm