Download Unit 1

Unit #: 1
# of weeks: 5
Subject(s): AMPS Math
Grade(s): 7
Designer(s): Julie Felix, Erin Lee
STAGE 1 – DESIRED RESULTS
Unit Title: Operations with Rational Numbers
Transfer Goal(s):
Students will be able to independently use their learning to solve a variety of real world problems to find the most appropriate solution(s).
Enduring Understandings:
Students will understand that…
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Real numbers (rational and irrational) can represent quantity,
position, location and relationships.
• Mathematical properties guide operations with real numbers.
• Real numbers can be expressed in a variety of formats
(exponential expressions, decimals, scientific notation and
rational & irrational numbers).
• Many rules in mathematics (laws of exponents, scientific
notation and radicals) are derived from patterns.
Students will know:
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The difference between rational and irrational numbers
and they are both subsets of the real number system.
(8.NS.1)
Numbers can be written in different forms. (8.NS.1, 8.NS.2)
How to execute the laws of exponents. (8.EE.1)
The inverse operations needed to solve a variety of
problems (exponents, square root, and cube root). (8.EE.2)
Scientific notation will be used to represent relatively large
and relatively small numbers. (8.EE.3 and 8.EE.4)
Operations using scientific notation rely on laws of
exponents. (8.EE.1, 8.EE.2, 8.EE.3 and 8.EE.4)
Essential Questions:
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How can I make rational and irrational numbers more manageable?
How do I know my answer is reasonable?
Why do we approximate irrational numbers?
Under what situations is it appropriate to give approximate answers?
Students will be able to:
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Apply the distributive property (and factoring) to negative numbers,
to include negative fractions and negative decimals. (7.NS.2)
Evaluate problems using all operations with real numbers, to
include negative fractions, decimals, exponents, and roots.
(7.NS.2, 8.NS.1, 8.NS.2, 8.EE.1 and 8.EE.2)
Identify different forms of terminating and repeating decimals to
include fractions and irrational numbers. (7.NS.2, 8.NS.1 and
8.NS.2)
Apply the properties of integer exponents, square roots and cube
roots to solve problems. (8.NS.1, 8.NS.2, 8.EE.1 and 8.EE.2)
Compare order and graph real numbers on a number line.
(7.NS.1 and 8.NS.2)
Represent and perform operations with scientific notation (8.EE.3
and 8.EE.4)
Adapted from Understanding by Design, Unit Design Planning Template (Wiggins/McTighe 2005)
*Bold print indicates 8th grade material.
Last revision 9/8/16
1
Unit #: 1
# of weeks: 5
Subject(s): AMPS Math
Grade(s): 7
Designer(s): Julie Felix, Erin Lee
STAGE 1– STANDARDS
Cluster
Know that there are numbers that are not rational, and approximate
them by rational numbers.
Work with radicals and integer exponents.
Standards
Know that numbers that are not rational are called irrational.
Understand informally that every number has a decimal expansion;
8.NS.1 for rational numbers show that the decimal expansion repeats
eventually, and convert a decimal expansion which repeats
eventually into a rational number.
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
8.NS.2
example, by truncating the decimal expansion 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.
Know and apply the properties of exponents to generate equivalent
8.EE.1
numerical expressions. For example, 32 x 3-5 = 3-3 = 1/33 = 1/27.
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
8.EE.2
rational number. Evaluate square roots of small perfect squares and
cube roots of small perfect cubes. Know that √2 is irrational.
Use numbers expressed in the form of a single digit times an integer
power of 10 to estimate very large or very small quantities, and to
express how many times as much one is than the other. For example,
8.EE.3
estimate the population of the United States as 3 × 108 and the
population of the world as 7 × 109, and determine that the world
population is more than 20 times larger.
Perform operations with numbers expressed in scientific notation,
including problems where both decimal and scientific notation are
used. Use scientific notation and choose units of appropriate size for
measurements of very large or very small quantities (e.g., use
8.EE.4
millimeters per year for seafloor spreading). Interpret scientific
notation that has been generated by technology.
Apply and extend previous understandings of addition and
subtraction to add and subtract rational numbers; represent addition
Apply and extend previous understandings of operations with fractions
7.NS.1 and subtraction on a horizontal or vertical number line diagram.
to add, subtract, multiply, and divide rational numbers.
a) Describe situations in which opposite quantities combine to
make 0.
Adapted from Understanding by Design, Unit Design Planning Template (Wiggins/McTighe 2005)
Last revision 9/8/16
2
*Bold print indicates 8th grade material.
Unit #: 1
# of weeks: 5
Subject(s): AMPS Math
Grade(s): 7
7.NS.2
Designer(s): Julie Felix, Erin Lee
b) Understand p + q as the number located a distance |q| from
p, in the positive or negative direction depending on whether
q is positive or negative. Show that a number and its
opposite have a sum of 0 (are additive inverses). Interpret
sums of rational numbers by describing real-world contexts.
c) Understand subtraction of rational numbers as adding the
additive inverse, p – q = p + (-q). Show that the distance
between two rational numbers on the number line is the
absolute value of their difference, and apply this principle in
real-world contexts.
d) Apply properties of operations as strategies to add and
subtract rational numbers.
Apply and extend previous understandings of multiplication and
division and of fractions to multiply and divide rational numbers.
a) Understand that multiplication is extended from fractions to
rational numbers by requiring that operations continue to
satisfy the properties of operations, particularly the
distributive property, leading to products such as (– 1)(– 1)=
1 and the rules for multiplying signed numbers. Interpret
products of rational numbers by describing real-world
contexts.
b) Understand that integers can be divided, provided that the
divisor is not zero, and every quotient of integers (with nonzero divisor) is a rational number. If p and q are integers
then – (p/q) = (-p)/q = p/(-q). Interpret quotients of rational
numbers by describing real-world contexts.
c) Apply properties of operations as strategies 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 0s or eventually repeats.
Adapted from Understanding by Design, Unit Design Planning Template (Wiggins/McTighe 2005)
*Bold print indicates 8th grade material.
Last revision 9/8/16
3