Transforming Geometric Instruction

Part 1

Understanding the Instructional Shifts in the Iowa Core Mathematics

Standards Grades 6-8

Session Outcomes

In order to effectively implement Iowa Core Geometry Standards, teachers will:• become familiar with the content shifts in middle school

geometry (not all geometry content represents a shift).• experience those shifts through activity based learning.• identify connections between content and the

Standards for Mathematical Practice. • understand the importance of the van Hiele levels of

geometric learning.

Standards for Mathematical Practice

“The Standards for Mathematical Practice describe varieties of expertise that

mathematics educators at all levels should seek to develop in their students.”

IC Mathematics page 8

https://www.youtube.com/watch?v=c8MPXdgq1i0

Standards for Mathematical Practice 1. Make sense of problems and persevere in solving

them.2. Reason abstractly and quantitatively.3. Construct viable arguments & critique the reasoning

of others.4. Model with mathematics.5. Use appropriate tools strategically.6. Attend to precision.7. Look for and make use of structure.8. Look for and express regularity in repeated reasoning.

4

Refer to pages 8-10

in IC Mathematics

Read through the Standards

for Mathematical Practice* for one of the

three middle school grade levels.

How do you think these will be exhibited in

geometry?

*adapted from KATM Flip Books, grades 6 - 8

As you continue through the different activities, use this template to record student actions (or activities) that

encompass each SMP.

Van de Walle, John A. (2004), Elementary and Middle School Mathematics: Teaching Developmentally, Boston: Pearson, pp. 349.

Levels of Geometry

Read “Understanding the van Hiele Levels”.

Be prepared to share what level you are and what level your students are.

Van Hiele Levels of Reasoning about Shapes

Visualization

Analysis

Informal Deduction

Deduction

Rigor

Characteristics of the Levels

The levels are sequentialThe levels are not age-dependentGeometric experience is the greatest

single factor influencing advancement through the levels.

When instruction or language is at a level higher than that of the student, there will be a lack of communication.

Where Have We Been? Elementary

• Identify and describe shapes (K)• Analyze, compare, create, and compose shapes (K)• Reason with shapes and their attributes (1-3)• Measure lengths (2nd)• Understand concepts of area and perimeter (3rd)• Draw and identify lines and angles, measure angles, and

classify shapes by properties of lines and angles (4th)• Graph points on the coordinate plane to solve problems (5th)• Classify two-dimensional figures based on their properties

(5th)• Understand concepts of volume (5th)

Where Are We Going?Middle School

• Solve problems involving area, surface area, and volume (6th)• Draw, construct, and describe geometrical figures and

describe the relationships between them (7th)• Solve problems involving angle measure, area, surface

area, and volume (7th)• Understand congruence and similarity using physical

models, transparencies, or geometry software (8th)• Understand and apply the Pythagorean Theorem (8th)• Solve problems involving cylinders, cones, and

spheres (8th)

On a sheet of paper, sketch the table shown below. Fill in the missing information.

Find the volume for the rectangular prism when each cube measures…

Unit Cube Measure

Length Width Height Volume

1 x 1 x 1

x x

x x

At your table, discuss any patterns you notice. Why do these patterns occur?

Cube-shaped boxes will be loaded into the cargo hold of a truck. The cargo hold of the truck is in the shape of a rectangular prism. The edges of each box measure 2.5 feet, and the dimensions of the cargo hold are 7.50 feet by 15.00 feet by 7.50 feet, as shown below.

A. What is the volume, in cubic feet, of each box?B. Determine the number of boxes that will completely fill the

cargo hold of the truck. adapted from Smarted Balanced Assessment

Going Bananas!

Leo has another pan that is 9 in. x 9 in. x 3 in., and from past experience he thinks he needs about an inch between the top of the batter and the rim of the pan.

Should he use this pan? Explain.

Leo’s recipe for banana bread won’t fit in his favorite pan. The batter fills the 8 in. x 11 in. x 1 in. pan to the very top, but when it bakes it spills over the side.

adapted from Illustrative Mathematics

Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l x w x h and V= bh to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

6.G.2

from Illustrative Mathematics

Select 4 of the polygons listed on the handout. • Construct your four polygons on the coordinate graph by

plotting the given points.• Label each polygon with its corresponding letter.• Determine the area of each polygon.• In your group, discuss the method(s) you used to determine

the area.

Secret Chamber ProblemAn archaeologist discovers a secret treasure chamber and maps the chamber on a coordinate graph. The coordinates he plots are:

(2 , 5)(2 , 1)(-4 , 1)(-4 , 5)

Plot the points on the coordinate grid. What are the dimensions of the secret chamber?

adapted from Big Ideas Math (2014)

Secret Chamber Problem

The archaeologist finds a gold coin at (-1 , 4), a silver coin at (-4 , 2) and pottery at (-4 , 4).

How much closer is the pottery to the silver coin than to the gold coin?

adapted from Big Ideas Math (2014)

Going to the Moviesadapted from Big Ideas Math (2014)

Going to the Movies

Which Standards for Mathematical Practice are evident when solving this problem? Use your template to record any notes.Make sense of problems and persevere in solving them.Reason abstractly and quantitatively.Construct viable arguments and critique the reasoning of others.Model with mathematics.Use appropriately tools strategically.Attend to precision.Look for and make use structure. Look for and express regularity in repeated reasoning.

6.G.3 Draw polygons in the

coordinate plane given coordinates for the vertices;

use coordinates to find the length of a side joining points

with the same first coordinate or the same second

coordinate. Apply these techniques in the context of

solving real-world and mathematical problems.

Where Are We Going?

Middle School

• Solve problems involving area, surface area, and volume (6th)• Draw, construct, and describe geometrical figures and

describe the relationships between them (7th)• Solve problems involving angle measure, area, surface

area, and volume (7th)• Understand congruence and similarity using physical

models, transparencies, or geometry software (8th)• Understand and apply the Pythagorean Theorem (8th)• Solve problems involving cylinders, cones, and

spheres (8th)

Focusing on Triangles

LaunchSketch a shape with the following conditions:A quadrilateral with at least one right angle and a side of 4 units.

Now, use a ruler and protractor to construct the shape.

Compare your creation with a partner. Was the figureyou made “unique”?

LaunchSketch a shape with the following conditions:A parallelogram with a 40 degree angle and a side of 5 units.

Now, use a ruler and protractor to construct the shape.

Compare your creation with a partner. Was the figureyou made “unique”?

LaunchSketch a shape with the following conditions:A square with a perimeter of 24 units.

Now, use a ruler and protractor to construct the shape.

Compare your creation with a partner. Was the figureyou made “unique”?

ExploreWith your partner, work through situations 1 – 3 on the student handout. You will be sketching the figure and then creating it with Geogebra and/or a ruler and protractor.If you have time, continue onto situations 4 – 6.

Summarize

Were all the triangle situations possible? Explain any conclusions or conjectures you made.Were there any unique triangle situations? Explain any conjectures you made.Are there other situations with other types of figures that would be unique? Why?

Iowa Core:7.G.2. Draw (freehand, with

ruler and protractor, and with technology) geometric shapes

with given conditions. Focus on constructing triangles from three measures of angles or

sides, noticing when the conditions determine a unique

triangle, more than one triangle, or no triangle.

“2-D from 3-D”Adapted from “A Plane Intersecting a Cube” and “A Plane

Intersecting a Triangular Pyramid”, Jack Gittenger

LaunchLook at the 3-d shapes. Describe the attributes of:• Right rectangular prism• Right rectangular pyramid• Triangular Pyramid• Cube

If a knife were used to slice one of these and cut it into 2 parts, what 2-d shape would we see on the two parts that used to be connected?

Would it make a difference the way the cut was made?

Explore

Use Geogebra and the student activity handout “2-D from 3-D” to explore a cube and a triangular pyramid being cut by a plane.

Summarize

• Using the “Cube” program, describe some of the 2-d figures you found. Can you describe how the plane intersected the cube?

• What other, if any, 2-d figures could you have created with a right rectangular prism that is not a cube? Explain the cut you would make.

• Using the “Pyramid” program, describe some of the 2-d figures you found. Can you describe how the plane intersected the pyramid?

• What other, if any, 2-d figures could you have created with a right rectangular pyramid? Explain the cut you would make.

• There are other 3-d figures that we could slice. What figure can you suggest to the class and what 2-d figure do you think the slice would make?

Iowa Core

7.G.3. Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.