Transformations Continued

Do you have a classroom theme? A motto of sorts...something that drives everything that you and your students do in your classroom...and hopefully beyond those walls as well?!?

Do you see the same idea within your course curriculum? With the implementation of the CCSS, there has been speculation as to the theme (or emphasis) for each grade level. Educators present arguments for a particular theme that acts as the ongoing thread in that grade level of math. For Grade 8, educators seem to fall into different camps as two threads have risen to the top of discussions in the past few years. Some say transformations. Some say functions. What are your thoughts? Would you say that transformations or functions drive everything you do in MATH-8? Is it a difference between results and relationships?

That's a big discussion.
And it's a discussion that needs to occur sooner rather than later.
Because it really is different to spy all standards with a lens of transformations versus a lens of functions.

As for the MATH-8 standards that involve developing the concept of transformations, what seems to be the most difficult for students to grasp? Historically, my students have struggled with coordinate notation and the process of rotations. Review a previous post on Thinking Transformations that discusses the transformations standards that are expected for students in MATH-8 and Accelerated MATH-7.

Transformations Resources

The following resource includes record sheets to coordinate with the transformations activities adapted from the Hands-On Standards Grades 7-8 book. Create four stations, one per transformation, for students to rotate through as a team. Consider duplicating to set two stations for each transformation in an effort to manage manipulatives setup.

Transformations Stations Activity Record Sheets

These activities highlight Common Core State Standards 8.G.A.1, 8.G.A.2, 8.G.A.3, and 8.G.A.4 included in MATH-8 and Accelerated MATH-7.

Similarity

There are four types of transformations or changes that can be made to the position of a geometric figure. When a geometric figure undergoes a transformation, the line segments still result in line segments and angles still result in angles. Translations, reflections, and rotations produce congruent figures meaning the shape is identical in that the corresponding line segments maintain their same length and the corresponding angles maintain their same measure...the exact figure is simply in a new position. (CPCTC=corresponding parts of congruent triangles are congruent!) The location changes, and sometimes the orientation changes as well. Dilations produce similar figures meaning the figure is the same shape and the corresponding angles maintain their same measure but the corresponding line segments are proportional in length. (CCSS 8.G.A.4)

In CCSS 8.G.A.5 students investigate angle-angle criterion for similarity of triangles. What happened to the unmentioned third angle? Why not angle-angle-angle criterion for similarity? Ask your students. Prompt them to realize that the third angle is implied by interior angle sum of a triangle. In the lab activity below, students discover that two angle measures in a triangle do not yield congruence between the figures. This sets the stage to discuss the definition of similarity. The AngLegs Simultaneous RoundTable puts the definition to work and makes connections to dilation and percent increase. This is prerequisite knowledge for CCSS 8.EE.B.6 standard that connects slope and similar triangles including the derivation of y=mx+b.

8.G.A.5 Similarity Resources

These discovery activities highlight Common Core State Standards 8.G.A.5 and 8.EE.B.6 included in MATH-8 and Accelerated MATH-7.

Angle Relationships

Prior to Common Core implementation, angle measure was a frequently repeated topic. Now, basic angle rules begin in Grade 7 and extend to angle relationships in Grade 8. These geometry standards prompt visual investigations with the use of manipulatives. The standards call students to "use informal arguments to establish facts..." about angle relationships. Grab your AngLegs and patty paper to accompany the activities outlined below.

8.G.A.5 Resources

Are your students struggling with basic angle rules and angle relationships formed by parallel lines cut by a transversal? This game focuses on these angle relationships and as the levels increase the content extends to angle relationships with polygons and circles for enrichment.

Game Description:
Help Itzi the spider climb the clock to rescue his family! Solving cunning angle puzzles to reveal a path through each level's maze of tangled webs and reach the goal. It's sure to make your head spin!

Manga High Math Games

These concept builders highlight Common Core State Standard 8.G.A.5 included in MATH-8 and Accelerated MATH-7.

Formative and Summative Assessment

With the implementation of Common Core State Standards, it is no surprise that assessment has been a topic of discussion from the beginning. There are proponents for almost every possibility who claim a particular assessment option is required for CCSS. And the terms are certainly flying...rigor, depth of knowledge, task, inquiry learning, PBL (does that mean project-based learning, problem-based learning, or performance-based learning?!?). Perhaps if everyone had a common meaning for each "buzz" word then significant analysis could occur.

Meanwhile, let's not get overwhelmed by one right way. Each student is different; fair isn't always equal...and equal isn't always fair. Focus on the big picture first. Categorize the types of assessment used in a classroom as formative or summative. Confused by those terms? Consider the road test that is required to receive a driver's license as an analogy...

What if, before getting your driver's license, you received a grade every time you sat behind the wheel to practice driving? What if your final grade for the driving test was the average of all of the grades you received while practicing? Because of the initial low grades you received during the process of learning to drive, your final grade would not accurately reflect your ability to drive a car. In the beginning of learning to drive, how confident or motivated to learn would you feel? Would any of the grades you received provide you with guidance on what you needed to do next to improve your driving skills? Your final driving test, or summative assessment, would be the accountability measure that establishes whether or not you have the driving skills necessary for a driver's license—not a reflection of all the driving practice that leads to it. --Catherine Garrison and Michael Ehringhaus in Formative and Summative Assessments in the Classroom 

Students need the driving practice [formative assessment]. And the detailed feedback provided before, during, and after those practice sessions is critical to (1) influence the design of the next practice session and (2) enable the student to realize the key skills they are using effectively and the ones that need refining for continued growth. The final driving test [summative assessment] definitely paints a picture of how well students combine a variety of skills. Caution: Do you know anyone who has a driver's license and cannot parallel park? Students can understand a concept, make connections between concepts, and still lack depth in skill. Find a balanced approach to assessment in the classroom that checks both skills and connections.

With that being said, there is a difference between students demonstrating their mathematical understanding via tasks that have one correct solution and tasks in which multiple solutions can be accurately justified. When using tasks with one correct solution, let's strive for multiple paths to arrive at that solution. For example, review the following fundraiser problem. There is only one correct answer; however, students are not required to follow a particular path to reach the solution. A student could begin problem solving by using a graph while another student could generate data to initially solve the problem. Furthermore, the detail with which they communicate their reasoning will also reflect comprehension.

Sample Assessment Resource

Thoughts? What do you find most effective in gauging student understanding? Try the sample assessment above and provide feedback. What adjustments needed to be made for your students?

This assessment document highlights Common Core State Standard 8.EE.C.8 included in MATH-8 and Accelerated Algebra 1.