Supporting tasks that involve change of visual representations with dynamic technology

Supporting tasks that involve change of visual representations with dynamic technology

Kevin A. [email protected]@kalawrence9 & @ipadmath

http://bit.ly/MCTM14KL

About me

• Taught high school math and physics for six years (IN).

• PhD student (third year) in the Program in Mathematics Education (PRIME) at MSU.

• Developmental Assistant on the Connected Mathematics Project (CMP).

• Research interests: Technology use in math classes, student transition from K12 to collegiate math classes.

Past presentations…

• Have mostly been on informing the audience of newer technologies (iPad Apps, Desmos, GeoGebra, GSP, etc.) that could be used in middle and secondary mathematics classrooms and how they could be used.

But…• I’m not sure I have made it a priority to show why it would be advantageous to use these technologies.

Goals for this session:

• Make you aware of why some static representations may (to some degree) promote student misconceptions in algebra and geometry.

• Show why using newer technologies with dynamic capabilities can help address student misconceptions in algebra and geometry that are generally supported with static representations.

Not Goals for this session:

• Teach you how to create on the software shown:• Desmos• GeoGebra• Geometer’s SketchPad

• NOTE: All the specific apps viewed in this session and the PowerPoint are available at http://bit.ly/MCTM14KL

High School: Functions » Building Functions » Build new functions from existing functions. CCSS.MATH.CONTENT.HSF.BF.B.3Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

Translation of Functions

• Many textbooks show translation of functions with two functions (one parent, one translated) on a single coordinate graph.

• The intent is show the net result, but does not show the actual movement from one to another.

Translation of Linear Functions

Holt

Holt

Vertical Translation of Linear Functions

Possible Student Misconception

Desmos – Online Graphing Calculator

• Desmos.com and iPad app• Similar display on any platform (computer, mobile, app)

• Save to Google Doc (not yet on iPad)• Pros:

• Enter either equation or data (tables)• View equation, table, and graph at the same time• Sliders for equations provide dynamic change in graphs and tables!

• Can import picture to sit behind the coordinate grid.

Desmos

• Given raw data as coordinates, find the function that matches• Test different types of function families with the use of sliders

• Allows the user to test different values

• See changes that occur when changing the value of parameters (how the graph moves/adjusts)

• https://www.desmos.com/calculator/nkio3nidn1

Translation of a Quadratic Function

Prentice Hall

Sample Student Work

Sample Student Work

Which point(s) is translated the most between these two functions? By how much?

What we see (hopefully).

Common student responses I received as to why the vertex point moves the most.

GeoGebra – Vertical Translation of Quadratic Function

• GeoGebra.org – Also a Google Chrome App• GeoGebraTube.org – The YouTube of GeoGebra• Similar to Desmos, we can graph the parent function and a function that moves by the use of a slider.

• The difference? Users can:• Create points on the translated graph that leave a vertical trail by using the “trace on” feature when the slider is moved.

• Create segments (geometric tool) between corresponding points that create vertical segments of equal length with measurements displayed.

Pythagorean Theorem 8th Grade Standards

• Understand and apply the Pythagorean Theorem.• CCSS.Math.Content.8.G.B.6Explain a proof of the Pythagorean Theorem and its converse.

• CCSS.Math.Content.8.G.B.7Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

• CCSS.Math.Content.8.G.B.8Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

Pythagorean Theorem HS Geometry Standards • Define trigonometric ratios and solve problems involving right triangles

• CCSS.Math.Content.HSG.SRT.C.8Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.*

HS Geometry Introduction

• “The definitions of sine, cosine, and tangent for acute angles are founded on right triangles and similarity, and, with the Pythagorean Theorem, are fundamental in many real-world and theoretical situations. The Pythagorean Theorem is generalized to non-right triangles by the Law of Cosines. Together, the Laws of Sines and Cosines embody the triangle congruence criteria for the cases where three pieces of information suffice to completely solve a triangle.”

So what exactly is the Pythagorean Theorem?• CCSSM doesn’t give a formal definition, but does explicitly state that it has a converse (conditional statement?).

• What do textbooks say (and show)?

UCSMP

UCSMP

Holt

Holt

Holt

Holt

Pearson

Pearson

Pearson

Is this a problem?

• Are textbooks defining the Pythagorean Theorem the same way? (conditional, converse, biconditional)• If the Pythagorean Theorem doesn’t hold true for a given scenario, what does that mean?

• Does there exist non-right triangles where the Pythagorean Theorem holds true?

• What opportunities do textbooks give students in learning about properties of right and non-right triangles? Are problem sets simply an exercise in computational skill or to develop a conceptual understanding of the Pythagorean Theorem?

• Do textbooks give visual representations for non-right triangle scenarios?

Grade 6 » Geometry » Solve real-world and mathematical problems involving area, surface area, and volume. » 1CCSS.Math.Content.6.G.A.1Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

Grade 7 » Geometry » Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. » 6CCSS.Math.Content.7.G.B.6

Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

Area of Triangles

• Generally, textbooks show finding the area of a right triangle first, followed by finding the area of non-right triangles (acute and obtuse).

• For right triangles, the altitude is generally one of the legs.

• The later involves altitudes that are not any particular side of the triangle. The altitude lies in the interior or exterior.

• Concentrating on just acute and obtuse triangles, students struggle with understanding:• why needing to know the altitude even matters.

• how to identify altitudes on the exterior of the triangle.

• triangles that have congruent bases and same altitudes have equal area, regardless of location of the altitude (interior, exterior, or on the triangle).

UCSMP

CMP3 – 6th Grade

CMP3 - 6th Grade

Holt

Pearson

Pearson

Link to GeoGebra, GSP files & Desmos

• http://bit.ly/MCTM14KL

Thank you!

Kevin A. [email protected]@kalawrence9 & @ipadmath