MADISON COUNTY PUBLIC SCHOOLS

District Curriculum Map for Mathematics: Grade 8

Unit Description / Unit 1
Rational / Irrational Numbers
Suggested Length: 2 weeks
*Students should NOT use calculators for this unit.
Big Idea(s)
What enduring understandings are essential for application to new situations within or beyond this content? / Enduring Understandings
Develop an understanding of expressions and equations.
Square roots can be rational or irrational.
An irrational number is a real number that cannot be written as a ratio of two integers.
Every number has a decimal expansion, for rational numbers it repeats eventually, and can be converted into a rational number.
All real numbers can be plotted on a number line.
Rational approximations of irrational numbers can be used to compare the size or irrational numbers, locate them approximately on a number line, and estimate the value of expressions.
Enduring Skills Rubric measures competency of the following skills:
Find square roots and cube roots of perfect squares and perfect cubes.
Explain the difference between a rational and an irrational number.
Convert either repeating or terminating decimals into a fraction.
Write a decimal approximation for an irrational number to a given decimal place.
Place rational and irrational numbers on a number line.
Essential Question(s)
What questions will provoke and sustain student engagement while focusing learning? / What is the difference between rational and irrational numbers?
When are rational approximations appropriate?
Why do we approximate irrational numbers?
How do you locate the approximate location on a number line and estimate the value of irrational numbers?
Standards / Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them.
In grade 8, students solve real world problems through the application of algebraic and geometric concepts. Students seek the meaning of a problem and look for efficient ways to represent and solve it. They may check their thinking by asking themselves, “What is the most efficient way to solve the problem?”, “Does this make sense?”, and “Can I solve the problem in a different way?”
2. Reason abstractly and quantitatively.
In grade 8, students represent a wide variety of real world contexts through the use of real numbers and variables in mathematical expressions, equations, and inequalities. They examine patterns in data and assess the degree of linearity of functions. Students contextualize to understand the meaning of the number or variable as related to the problem and decontextualize to manipulate symbolic representations by applying properties of operations.
6. Attend to precision.
In grade 8, students continue to refine their mathematical communication skills by using clear and precise language in their discussions with others and in their own reasoning. Students use appropriate terminology when referring to the number system, functions, geometric figures, and data displays.
7. Look for and make use of structure.
Students routinely seek patterns or structures to model and solve problems. In grade 8, students apply properties to generate equivalent expressions and solve equations. Students examine patterns in tables and graphs to generate equations and describe relationships. Additionally, students experimentally verify the effects of transformations and describe them in terms of congruence and similarity.
8. Look for and express regularity in repeated reasoning.
In grade, students use repeated reasoning to understand algorithms and make generalizations about patterns. Students use iterative processes to determine more precise rational approximations for irrational numbers. During multiple opportunities to solve and model problems, they notice that the slope of a line and rate of change are the same value. Students flexibly make connections between covariance, rates, and representations showing the relationships between quantities.
Standards for Mathematical Content
8.NS.1 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.
8.NS.2 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π2). For example, by truncating the decimal expansion of √2 (square root of 2), show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
8.EE.2 Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
Supporting Standard(s)
Which related standards will be incorporated to support and enhance the enduring standards? / 6.NS.6abc Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
a. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., -(-3) = 3 and that 0 is its own opposite.
b. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
c. Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.
7.NS.2d Apply and extend previous understandings of multiplication and division of fractions to multiply and divide rational numbers.
d. Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in zeroes or eventually repeats.
Instructional Outcomes
What must students learn and be able to do by the end of the unit to demonstrate mastery? / I am learning to…
·  Convert any number to decimal form and show (informally) that every number has a decimal expansion.
·  Classify all real numbers as rational or irrational.
·  Recognize decimal expansions as rational or irrational numbers.
·  Evaluate small perfect squares up to 225 and small perfect cubes up to 125.
·  Categorize the square root of a number as irrational or rational.
·  Approximately locate irrational numbers on a number line.
·  Approximate irrational numbers using rational numbers.
·  Compare the size of irrational numbers using rational numbers.
·  Apply square/square roots and cube/cube roots within an equation and identify that they are inverse operations.
·  Justify that the square root of 2 is irrational because it is not a perfect square.
Essential Vocabulary
What vocabulary must students know to understand and communicate effectively about this content? / Essential Vocabulary
Rational numbers, Irrational numbers, Terminating decimals, Repeating decimals, Bar notation, Decimal expansion, Square root, Cube roots, Perfect Square, Perfect Cube, Truncating, , and π(Pi)
Supporting Vocabulary
Estimation and Approximation
Resources/Activities
What resources could we use to best teach this unit? / Resources/Activities
www.estimation180.com
www.visualpatterns.org
101Questions
http://www.101qs.com/
Dan Meyer’s Website
http://blog.mrmeyer.com/
Dan Meyer has created many problem-based learning tasks. The tasks have great hooks for the students and are aligned to the standards.
Andrew Stadel
https://docs.google.com/spreadsheet/ccc?key=0AkLk45wwjYBudG9LeXRad0lHM0E0VFRyOEtRckVvM1E#gid=0
Andrew Stadel has created many problem-based learning tasks using the same format as Dan Meyer.
Robert Palinsky
http://robertkaplinsky.com/lessons/
Robert Palinsky has created many tasks that engage students with real life situations.
Geoff Krall’s Emergent Math
http://emergentmath.com/my-problem-based-curriculum-maps/
Geoff Krall has created a curriculum map structured around problem-based learning tasks.
Mathematics in Movies
http://www.math.harvard.edu/~knill/mathmovies/
Short movie clips related to a variety of math topics.
Mathematical Fiction
http://kasmana.people.cofc.edu/MATHFICT/browse.php
Plays, short stories, comic books and novels dealing with math.
The Shodor Educational Foundation
http://www.shodor.org/interactivate/lessons/byAudience/
This website has extensive notes, lesson plans and applets aligned with the standards.
NEA Portal Arkansas Video Lessons on-line
http://neaportal.k12.ar.us/index.php/9th-12th-grades-mathematics/
The NEA portal has short videos aligned to each standard. This resource may be very helpful for students who need review at home.
Learnzillion
http://learnzillion.com/common_core/math/hs
This is another good resource for parents and students who need a refresher on topics.
Math Words
http://www.mathwords.com/
This is a good reference for math terms.
National Library of Virtual Manipulatives
http://nlvm.usu.edu/en/nav/vlibrary.html
Java must be enabled for this applet to run. This website has a wealth of virtual manipulatives helpful for use in presentation. Listed by domain.
Geogebra Download
http://www.geogebra.org/cms/download
Free software similar to Geometer’s Sketchpad. This program has applications for algebra, geometry & statistics.
Remember there are other sources in your school that may not be listed on this common resources list due to variation in each individual school. Examples of other great resources your school may have access to include: Everyday Math Games, Investigations, Everyday Partner Games, AVMR file folders, Ongoing Assessment Project, etc. The Kentucky Numeracy Project is also a great resource that can be searched by AVMR strand, CCSS, and grade level. Find this resource at http://knp.kentuckymathematics.org/#!/page_knphome. Kentucky teachers can use it for free. Just put in your school email address and the username “mathfun”, and password is “859”.

Curriculum and Instruction 2015-2016 Page 1 of 6

Curriculum and Instruction 2015-2016 Page 1 of 6