Modeling Functions

During the November 6th Genius Hour, we reviewed a variety of instructional practices that help build conceptual understanding. The following samples aligned to linear functions were provided.

The "Build This!" activity is the brilliant work of Nat Banting from his Relation Stations post over on his {Musing Mathematically} blog. The link was shared by Kathryn Freed during a recent #alg1chat on Twitter. Students view three patterns and determine what all three patterns have in common...a starting point and a constant change. Then students are asked to create a pattern of their own. Finally students rotated stations to investigate the models created by their classmates and build a table of values to model the pattern. The extension outlined in the blog post involves students using their record sheet to write equations for each pattern the following day. Starting with a concrete model of color tiles and eventually expanding to the abstract algebraic equation is powerful.

Nat mentioned in the #alg1chat that he has reversed the activity to provide students an equation and ask for the model. The possibilities are endless! You could have each team build two patterns with the same starting point...one with a constant change and one with a varying change to emphasize linear and nonlinear patterns. You could also add graphing to the table of values record sheet to make further connections.

Create a Model

The next activity is adapted from the Modeling with a Linear Function task on the Illustrative Mathematics website. (On a complete side note...we simply must chat about the word "task" and its use in today's math classroom. Care to define that one for me?) The activity poses a linear function and proceeds to outline a variety of models...some describe the linear function while others do not. However, the descriptions that do not match the linear function are common misconceptions. When structured as a Takeoff... Touchdown..., this activity is an engaging way to get students to differentiate between concept and misconceptions.

Differentiate Common Misconceptions

The following prompt (which would be a fabulous RoundTable...or, better yet, use three additional functions to create a Simultaneous RoundTable) is an adaptation of a released open response item from the Arkansas Department of Education. The original function was quadratic and the table didn't contain the same rational values. This prompt integrates prior knowledge of transformations and operations with rational numbers with the current knowledge of linear functions.

Integrate Prior Knowledge
All of these instructional practices help students build conceptual understanding and/or make connections between content. And that is the key to success in math...whether that's today, the next unit, the standardized test, or next year's math class.

These activities highlight Common Core State Standards 8.F.A.3 and 8.F.B.4 that are included in MATH-8 and Accelerated Algebra 1.


Definition of Function

Vocabulary development becomes essential in a mathematics classroom that focuses on the Common Core Standards for Mathematical Practice of attending to precision (SMP.6). In order for math students to communicate precisely to others, they must use clear definitions in their discussion and reasoning. And then subsequently, students can use their vocabulary in constructing arguments (SMP.3).

One strategy for vocabulary development involves the teacher telling a story that involves the new term. Use the Vending Machine Saga to help students determine if the machine is functioning properly for each transaction. (We are working under the assumption that if a product type is unavailable then the machine returns your dollar in exact change; therefore, no transaction has occurred for our review.)
Concrete Function Story Expands to Abstract
Another effective vocabulary strategy is engaging students in an activity that helps students add to their knowledge of the term. Present students with a two-column example/counterexample description. Provide think time for each student to generate facts or characteristics consistent with the term. Have students use the RoundRobin structure to share their pieces of a definition. With each rotation the team forms and refines a definition to present to the class. Once the teams reach consensus, the teacher and class discuss samples to develop a class definition. The final definition, examples, and counterexamples can be recorded in a Frayer Model template. Or the examples and counterexamples can be used as a card sort in an interactive student notebook.
Definition of Function

Although students investigate linear functions in MATH-8, the conceptual definition of a function is most critical for success with using function notation and finding an inverse function in Algebra 1. This investigation highlights Common Core State Standard 8.F.A.1 that is included in MATH-8 and Accelerated Algebra 1.


Integer Exponents

Middle school students generally enjoy working with patterns and find much success with number patterns. So discovering patterns with positive and negative exponents is a great way to conceptually teach integer exponents.
Integer Exponents Discovery Activity

The general laws of exponents can be developed from the number patterns discovered in the Integer Exponents RoundRobin activity and record sheet. And the discussion of base 10 will set the stage for scientific notation.
Integer Exponents Quiz-Quiz-Trade
Integer Exponents Quiz-Quiz-Trade
Integer Exponents Fan-N-Pick

This standard (CCSS 8.EE.A.1) will also bridge understanding to Algebra 1 as students explore geometric sequences and exponential functions.


Solutions of Linear Equations

Once students are fluent with the order of operations, they are ready to solve linear equations. When teaching the order of operations, I emphasize that we are simplifying numerical expressions. So when I provide the total value of a numerical expression BUT make one of the values in your expression a mystery, then my students have an equation to solve. And the process of solving for the mystery value requires students to use the order of operations in reverse.

2x + 5 = 17

What happened to the mystery value? It was doubled (or multiplied by 2) and then 5 was added for a total of 17. What happened last? The 5 was added to result in a total of 17. So take away (or subtract) 5. Now the total is 12...which is the result when I double the mystery value. So divide the total by 2 and you have revealed the mystery value of 6.

In middle school, students are moving from concrete thinking to abstract thinking...some a little more quickly than others. But our goal is to help all students find success dealing with abstract concepts. Solving linear equations may require the use of manipulatives to support concrete thinkers as they conceptualize this abstract concept.

And then when you add the possibility of multiple mysteries value that would reveal the same total result or no mystery value at all that would yield the total, the abstract becomes more abstract. Sarah over at Everybody is a Genius posted an excellent approach to Solving Special Case Equations. I love the introductory "puzzle" approach with balanced scales. I altered the format to a Showdown activity to provide structure for the discussion with the record sheet.
Balance This! Showdown Activity

Balance This! Record Sheet

Sarah also shared the notes her students created for their Interactive Notebooks to summarize their learning. I created the foldable in PowerPoint so it could be sent to the print shop if you prefer.
Types of Solutions Foldable (Student)

Types of Solutions Foldable (Teacher)

And this tiling activity is a great activity to informally assess student understanding of solution types.

Types of Solutions Tiling Activity
These activities highlight the Common Core State Standard 8.EE.C.7a included in MATH-8 and Accelerated MATH-7.

Decimal Form of a Rational Number

The progression of understanding rational numbers bridges between MATH-7 and MATH-8. (Note: This content is merged in Accelerated MATH-7.) And in the midst of discussing process teaching and concept teaching, we have highlighted the following standards.

Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.
Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
Hop over to the MATH-7 tab to see the Decimal Form of Rational Numbers post to jump start the conceptual understanding of these standards.

First ensure that students understand the classification of rational and irrational numbers. You could engage students in a sorting activity based on a few provided examples and counterexamples. (If you use all cards in the set, this activity will set the stage for Algebra 1 teachers to discuss the results of rational and irrational number operations.)

Sorting Activity

And then you could spin off of the MATH-7 activity of why some rational numbers terminate and others repeat to continue towards the concept of rewriting a repeating decimal as its equivalent fraction.

Algebraic Discovery

Would it be easier to teach the process for converting a repeating decimal to its equivalent fraction? Which benefits the students today, next month, next year? Process teaching or concept teaching?

Be sure to reflect on other related posts:

CCSS Call for Rigor

Process vs. Concept Teaching


Irrational Approximations

The Common Core State Standards are filled with number sense. Students engaged in the Common Core aligned curriculum will have a deep understanding of rational and irrational numbers...including a rational approximation of an irrational number (CCSS 8.NS.A.2).

How would you find the rational approximation of this number?

Ummm...without a calculator. ;)

Here's how we approach this one in my classroom...Well. Let's start with square roots that are rational. We know the square root of 20 is somewhere between the square root of 16 and the square root of 25...so something between 4 and 5. Can you be more specific? Hmmm...the list begins to form...
root 16 = 4
root 17 = ?
root 18 = ?
root 19 = ?
root 20 = ?
......................about 4.5
root 21 = ?
root 22 = ?
root 23 = ?
root 24 = ?
root 25 = 5

Since we have 9 numbers to cover an increase of 1 unit, each increase will be a little more than 0.1. Why? Because 1/9 > 1/10. So we can approximate that root 20 is approximately 4.4. {Ish! At this point we discuss if the "little more" is "enough more" to justify rounding.}

Find Someone Who
But what about simplest radical form?!? We've searched high and low. No sign of "simplest radical form" in our copy of the CCSS document. (If you find it...please put me out of my misery!)

Could we teach our students to simplify radicals and then in turn use that to find the rational approximation of an irrational number? In simplest radical form, the square root of 20 is exactly 2 times the square root of 5. Hmmm...the square root of 5 is some number between the square root of 4 and the square root of 9...between 2 and 3.

root 4 = 2
root 5 = ?
root 6 = ?
..................about 2.5
root 7 = ?
root 8 = ?
root 9 = 3

Since we have 5 numbers to cover an increase of 1 unit, each increase will be more than 0.1. Why? Because 1/5 > 1/10 (some students will automatically go to the fact that 1/5 = 0.2). So we can approximate that root 5 is approximately 2.2 and therefore 2(2.2) = 4.4.

Bottom line. We reach a valid approximation either way. I think it's important for our students to discover both paths...and even more important for them to critique which approach they prefer and under what circumstances they would select that particular method. I'm ready for a viable argument. How about you?!?

Cooperative practice would be crucial for students to have perspective to support their argument.

Quiz-Quiz-Trade or Inside-Outside Circle
This topic has a natural connection to Pythagorean Theorem application (CCSS 8.G.B.7). When students are determining the length of an unknown side in a right triangle, they should be able to state their solution as both an exact and approximate number. And how do we teach students to solve using the Pythagorean Theorem when they don't know the first thing about a quadratic equation? That will be for another post...after we finish investigating rational and irrational numbers...stay tuned!


Frayer Model Sample

The Frayer Model was developed by Dorothy Frayer and her colleagues at the University of Wisconsin. This graphic organizer will lead students to a thorough understanding of new words. The corners generally include definition, facts or characteristics, examples and non-examples.

This is the modified template used in my math classroom. Based on the content, the example corner may include non-examples as well.

In order for students to communicate mathematically, they need a deep understanding of the content. Summarizing critical vocabulary via the Frayer Model can jumpstart this process.

This vocabulary graphic organizer highlights the Common Core State Standard 8.G.A.3 that is included in MATH-8 and Accelerated MATH-7.


Thinking Transformations

In grade 4, Common Core standards include lines of symmetry based on folding two-dimensional figures...not reflections. In grade 7, Common Core standards include scale drawings based on proportional relationships...not dilations. So unless our 8th grade students remember slide, flip, and turn from their elementary days or they were in a 7th grade classroom that reviewed transformations for the Benchmark last Spring, the knowledge bank will likely appear significantly different than in the past years.

Let's begin with the end in mind. Our students must describe a sequence of transformations in a set of congruent two-dimensional figures. This implies that students need to
  • perform each type of transformation on a coordinate plane
  • recognize when conditions prove congruence vs. similarity
  • know the structure of coordinate notation to describe the sequence

Perhaps we can still draw from the elementary experiences with real-life images that display transformations and provide the backdrop for building vocabulary. Next we can transfer that knowledge to the coordinate plane; beginning with one point and then extending to polygons. In the past, students seem to struggle most with rotations. A set of question cards (with answers) can be used with the Kagan structures Inside-Outside Circle or Quiz-Quiz-Trade. Or perhaps 1-2 pages could be copied and used with the Kagan structure RallyCoach. This set is versatile to provide you with individual transformation practice or the freedom to create a mixed set of question cards. The set you create could be used strategically for small group meetings or Seminar study sessions.

When extending to polygons, let's work towards building the concept of congruence and similarity. The following lab activity uses color tiles and a work mat. Students build a figure and perform a variety of transformations to observe the properties. The record sheet is included and can be completed in pairs with the Kagan structure RallyCoach.

The key to every activity is the questioning that helps students gain conceptual understanding. The Properties of Transformations activity builds the conditions for congruence vs. similarity as summarized in the following mind map.

While building the mind map together, we can discuss rigid transformations...definitions of congruence and similarity...review proportional relationships within similarity...the possibilities are endless! And the understanding of these properties enable students to be successful in identifying a sequence of transformations from a pre-image to the subsequent image. A variety of approaches are included in the following set of questions and can be used with the Kagan structure Showdown.

These activities highlight the Common Core State Standards 8.G.A.1, 8.G.A.2, and 8.G.A.3 that are included in Accelerated MATH-7, MATH-8, and 7th Grade Accelerated Algebra 1.

Does that help? What else do you need? Chat with each other...some of our teachers tackled these standards last year...tap into their expertise. Remember to adjust your filter! And be sure to share your thoughts in the comments below. {All thoughts invited...the good, the bad, and the ugly!}


Coming soon...


This page will host content specific to the MATH-8 standards.