Modeling Irrational Approximations

To find rational approximations of irrational numbers students must first be able to differentiate between rational and irrational numbers (see the sorting activity highlighted in this previous Decimal Form of a Rational Number post). The geometric approach to evaluating square roots of perfect squares can be extended to approximating irrational numbers. This modeling activity provides a visual proof that square roots of non-perfect squares are indeed irrational (see the concept builder highlighted in this previous Square Roots and Cube Roots post). Consider scaffolding the geometric approach with this Stations Activity and record sheet based on square tiles.

8.NS.A.2 Resources

And then move students from the geometric model to a numeric approach with the number line (see this previous Irrational Approximations post). This will enable students to use rational approximations to compare the size of irrational numbers, locate them on a number line, and estimate the value of expressions.


This stations activity highlights Common Core State Standard 8.NS.A.2 included in MATH-8 and Accelerated MATH-7.

Grade 8 Benchmark

State testing is just around the corner! And all thoughts assessment can be found in the 'Tis the Season post. This Grade 8 activity set could be used with the Find Someone Who structure to conduct a general review of sample questions organized by Arkansas Frameworks strands. Each question is an adapted released item from Grade 8 Benchmark exams over the recent years. If you choose to use these in your classroom, remember to remind students to only pair with others who are not their teammates. This will allow students to return to their teams and wrap-up by using the RoundRobin structure to share solutions and discuss any questions that may arise.

Grade 8 Benchmark Resources

This structure requires students to coach each other as needed. Coach: Tip, Tip, Teach, Try again! Let me encourage you to use a coaching chart with these activities to strengthen the math vocabulary that is used during coaching. Perhaps let your students work with a classmate or two and then freeze the class to discuss a list of coaching tips. Review things they may say and things they may do while working collaboratively. For example, on the number and operations coaching chart you could list "Did you find the prime factorization?" or "GCF shows COMMON parts." under the say column. And then list these examples under the do column: "Draw a Venn diagram of prime factors." or "Make eye contact." or "Nod your head."

You may choose to accompany each Find Someone Who activity with a coordinating sample open response question. Details are not included in this post because everything I know about setting kids up for success with open response questions was learned from Rhonda Kobylinski. Without a doubt, she is your resident expert.

Happy testing to YOU!

Interactive Parallel Lines

Are your students struggling with the angle relationships formed when parallel lines are cut by a transversal? Check out this web-based interactive model. This prompts discovery of corresponding, alternate interior, and alternate exterior angles through measure and color coding. Angle notation is also supported in this model. Use for whole class demonstration or pair with a lab activity for 1:1 technology.

Interactive Discovery

Perhaps combine this interactive model with a reference foldable to summarize the angle relationships investigated in the AngLegs discovery.

Angle Relationships Reference Foldable

This activity set highlights Common Core Standard 8.G.A.5 included in MATH-8 and Accelerated MATH-7.

Volume of Cones, Cylinders, and Spheres

How do you address a standard that prompts students to "know" formulas? The word "know" is completely different from being prompted to "derive" formulas. 21st Century learning pushes us to ditch "know" being equated with "memorize" when a formula can quickly be obtained from Google. Perhaps a focus on the relationship within the structure of the formulas would best serve our students...thoughts?

The emphasis of volume as layered base area starts with cylinders. And since cylinders are critical for understanding volume of cones and spheres, be sure students are firmly rooted in this concept. The Oreo lab activity can confirm the level of student understanding. Then consider the following activities that use View-Thru Large Geometric Solids from Learning Resources for a hands-on investigation to determine the volume of cones and spheres from relational cylinders.

8.G.C.9 Resources

Short on time from the recent ice days? This animation is quick but powerful in showing the 3:1 volume ratio of cone to cylinder.



Similar extensions can be made to investigate the volume of a square pyramid from a relational cube. Note: Pyramids are included in Grade 7 standards (7.G.A.3) with respect to cross sections and High School Geometry standards (HSG-GMD.A.1 and HSG-GMD.A.3) with respect to volume.

These investigations highlight Common Core State Standard 8.G.C.9 included in MATH-8 and Accelerated MATH-7.

Square Roots and Cube Roots

How do you teach students to evaluate square roots of small perfect squares and cube roots of small perfect cubes? The foundational concept behind evaluating roots is the connection between side length of a square and its area or edge length of a cube and its volume. Do you stop there? No need. Start with the geometric connection and then extend to a visual proof that the square root of 2 is irrational. This investigation will pave the road to move approximating irrational numbers from concrete to abstract understanding...which is a natural prerequisite to the Pythagorean theorem.

8.EE.A.2 Resources

This investigation highlights Common Core State Standard 8.EE.A.2 included in MATH-8 and Accelerated MATH-7.

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.

Scientific Notation

Think back to your days in geometry. Do you remember the two-column proofs and the joy that would come with writing "by definition" as the reason to support a statement? There are just some things in mathematics that exist by declaration. By definition, the square root of negative one is represented with i. Why? Because the Italian mathematician Rafael Bombelli said so.

There is no "But why?" behind scientific notation either. It is a definition created to ease the cumbersome nature of incredibly large or microscopically small numbers. Scientific notation is a method of writing a number as a decimal multiplied by a power of ten. The definition restricts the decimal to a number greater than or equal to 1 and less than 10. It's still the same number...just written in a specified format.

Have you used place value and powers of ten to rewrite numbers according to the scientific notation format? Base ten blocks can model the decomposition of a number and serve as a beneficial launching pad for writing large numbers in scientific notation.

Scientific Notation with Base Ten Blocks

Once students grasp the definition, continue with the "Sort This!" activity to rewrite numbers from different forms including standard notation, product or quotient form, as a power of ten, or scientific notation. The "I have... Who has..." activity provides a quick informal assessment of converting numbers. Also the "RallyCoach" and "Stations" activities require students to solve problems involving numbers written in scientific notation.

Scientific Notation Resources

This activity set highlights Common Core State Standards 8.EE.A.3 and 8.EE.A.4 included in MATH-8 and Accelerated MATH-7.

The Distributive Property

How do you teach the distributive property? When planning lessons, be sure you are maintaining perspective. Ask: What have your students already experienced with the content? This is particularly critical when students have gaps and require support to bridge that gap. You will need to have the perspective of how the related content has developed over the years/courses so that you can replicate that progression for struggling students via individualized learning. Ask: What leaps will your students make in upcoming courses that stem from what you do with the content now? It is important to identify future learning so that you can ensure adequate depth in your course. Let's review the progression of the distributive property...

3.OA.B.5

In Grade 3, students use the distributive property as a strategy for multiplying. This is the concept builder for the distributive property. If students were unsuccessful when working strictly with numbers, then their efforts in applications with variables will be a stretch at best. Or worse yet...what if students memorized multiplication facts and never made sense of the concept through the distributive property?!?

6.EE.B.5

In Grade 6, students use the distributive property to produce equivalent expressions. Notice the example includes using the distributive property to toggle between factored form (the result from division) and expanded form (the result from multiplication).

7.EE.A.1

In Grade 7, students continue with the same concepts learned in Grade 6 with an extension to using rational numbers. Should 7th graders be solving equations that require using the distributive property? Continue reading.

8.EE.C.7b

In Grade 8 (and Accelerated MATH-7), students will use the distributive property to solve linear equations with rational coefficients. Notice the standard says "whose solutions REQUIRE expanding expressions using the distributive property..." in order to solve the linear equation.

Differentiate the necessity of the distributive property.

Perhaps MATH-7 students should be asked to solve equations that are similar to the equation in the left column above; however, this begs the question "How do we hope students solve that type of equation?". We also hope MATH-8 students recognize the need for the distributive property before they start the process of solving equations that are similar to the equation in the right column above.

This discussion highlights the Common Core State Standards 7.EE.A.1 and 8.EE.C.7b included in MATH-7, MATH-8 and Accelerated MATH-7.

Solving Systems of Equations

If MATH-8 teachers ever doubted that they teach algebra, diving into the Common Core State Standard 8.EE.C.8 would certainly end the debate. Students are required to solve and analyze pairs of simultaneous linear equations. The depth of this standard involves understanding the meaning of solution(s) of a system, solving a system graphically and algebraically, and using this knowledge to solve real-world problems that lead to a system of two linear equations.

The prerequisite for this content includes solving one-variable equations in Grade 7 and previously in Grade 8 as outlined in this Solutions of Linear Equations post. The tiling activity towards the end of the post would be a great review of types of solutions to set the stage for systems.

In a recent Genius Hour, we focused on how to teach 8th grade students to solve a system of linear equations algebraically. While it may be common to ask student to solve each equation for the variable "y" and set the two resulting expressions equal to solve for the variable "x", this process begs the question "Why?". Are we helping our students make connections so that they understand the origin of the algebraic method and know this process is valid? Do they understand the concept behind solving algebraically?

We looked at an introduction activity that extends from the "Balance This!" Activity in the Solutions of Linear Equations post. The visual of the balanced scale was used to discuss solutions and the adjustments necessary if the conditions are represented with two different scales. In this activity, students reason with a balance of shapes and a given value to determine the value of each shape.

Solving Systems of Linear Equations

We started with a school store scenario in which two different "store specials" were offered. One special involved a purchase of three items from the top shelf and two items from the bottom shelf for a total of $16. The second special involved a purchase of two items from the top shelf and three items from the bottom shelf for a total of $14.

• What are some possible values that keep the firsts scale balanced? How many different combinations can you find? How many combinations are possible?

• What about the second scale? How many total combinations are possible?

• What if you consider both scales simultaneously...meaning the possible values that keep the first scale balanced must also be the same values that keep the second scale balance? How many different combinations can you find? How many combinations are possible?

How much do the items on each shelf cost? Which shelf has the more expensive items? Justify your reasoning. Many students will use trial and error to begin solving the puzzle, but beware of students who verbalize a strategy that is actually substitution! The activity opens several paths, one of which is solving numerically and then making sense of how that cumbersome numeric approach connects to a systematic algebraic approach. The activity extends to all types of solutions and coordinates with several support activities that were reviewed.


These activities highlight the Common Core State Standard 8.EE.C.8 that is included in MATH-8 and Accelerated Algebra 1.