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.