Mathematics 2015-16
Grade 8 The Number System
Month: September – Week 2
Critical Areas: The Number System
Content Emphases for Grade 8
Major Cluster
Supporting Cluster
Additional Cluster
70% of time
20% of Time
10% of Time
8.EE.A
8.NS.A
8.G.C
8.EE.B
8.SP.A
8.EE.C
8.F.A
8.F.B
8.G.A
8.G.B
Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning
Domains:
The Number System
Clusters:
Know that there are numbers that are not rational, and approximate them by rational numbers.
Standards:
8.NS.A.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.A.2 Use rational approximations of irrational numbers to compare size of irrational numbers, locate them approximately on a number line diagram,
and estimate the value of expressions (e.g., π2).
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Mathematics 2015-16
Grade 8 The Number System
Month: September – Week 2
Key Student Understandings
8.NS
 Students understand the concept of rational and irrational numbers.
 Students understand properties of square roots, powers of ten, base ten, and
exponents.
Assessments

Formative Assessment Strategies

Evidence for Standards-based Grading
Common Misconceptions/Challenges
8. NS Know that there are numbers that are not rational, and approximate them by rational numbers.
 Since we historically in math class name so few irrational numbers (e.g. pi) students sometimes conclude that irrational numbers are unusual
or rare; in fact, the square root of a whole number is either a whole number itself or is irrational. Through an in-depth study of irrational
numbers, this misconception will be addressed.
 Challenge: Students need to realize as they use calculators that calculators can only display a finite number of digits; they can’t display the
exact value of an irrational number.
Instructional Practices
8. NS.1 Students can use graphic organizers to show the relationship between the subsets of the real number system. The diagram should show
that all real numbers (numbers on the number line) are either rational or irrational.
Students distinguish between rational and irrational numbers. Any number that can be expressed as a fraction is a rational number. Students
recognize that the decimal equivalent of a fraction will either terminate or repeat. Fractions that terminate will have denominators containing
only prime factors of 2 and/or 5. This understanding builds on work in 7th grade when students used long division to distinguish between
repeating and terminating decimals.
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Mathematics 2015-16
Grade 8 The Number System
Month: September – Week 2
8.NS.2 Students locate rational and irrational numbers on the number line and compare and order rational and irrational numbers:
Example 1: Compare √2 and √3 by estimating their values, plotting them on a number line, and making comparative statements.
Solution: Statements for the comparison could include:
√2 is approximately 0.3 less than √3
√2 is between the whole numbers 1 and 2
√3 is between 1.7 and 1.8
During exploration, each group could have a number line and an envelope filled with cards, each card with a different irrational number. Each
person takes a card, places it on the number line, and justifies its placement, comparing the size of irrational numbers.
Example 2: Approximate the value of
5 to the nearest hundredth.
5 falls between 2 and 3 because 5 falls between 22 = 4 and 32 = 9.
The value will be closer to 2 than to 3. Students continue this “ iterative” process with the tenths place value. 5 falls between 2.2 and 2.3
because 5 falls between 2.22 = 4.84 and 2.32 = 5.29. The value is closer to 2.2. Further iteration shows that the value of 5 is between 2.23 and
Solution: Students start with a rough estimate based upon perfect squares.
2.24 since 2.232 is 4.9729 and 2.242 is 5.0176.
Students explore the idea that the value of a square root can be approximated between integers and that non-perfect square roots are
irrational. Students also recognize that square roots may be negative. Students recognize the existence of negative square roots, and
understand that, for example, the √2 always represents the positive square root of two, so that the negative square root must be written as
-√2.
Differentiation
8. NS Know that there are numbers that are not rational, and approximate them by rational
numbers.
 given a number (e.g. square roots, fractions, decimals), determine if it is rational or irrational
and justify;
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Literacy Connections
 Academic vocabulary terms

Vocabulary Strategies
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Mathematics 2015-16
Grade 8 The Number System
Month: September – Week 2
 approximate a given irrational number by rational numbers;
 Given two distinct numbers on a number line, find both a rational and an irrational number
between them.

Literacy Strategies
Instructional Resources
Instructional Pacing Guide
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