Download Math Grade 6: Unit 6 Rational Explorations

Math Grade 6: Unit 6
Rational Explorations
Learning Targets: I can…
• Explain the meaning of positive and negative
numbers and use positive and negative numbers in
real-world contexts. (6.NS.5)
• Place integers and other rational numbers in the
correct locations on a number line. (6.NS.6)
• Correctly order rational numbers on a number line.
(6.NS.7)
• Explain the meaning of and represent absolute
value. (6.NS.7)
• Solve real-world and mathematical problems by
graphing points in all four quadrants of the
coordinate plane. (6.NS.8)
Words to Know
absolute value: The distance between a number and
zero on the number line. The symbol for absolute
value is shown in the equation −8 = 8
integers: The set of whole numbers and their
opposites {… − 3, −2, −1, 0, 1, 2, 3, … }
negative numbers: the set of numbers less than zero.
opposite number: two different numbers that have the
same absolute value. Example: 4 and −4 are opposite
numbers because both have an absolute value of 4.
positive numbers: The set of numbers greater than
zero.
𝑎
rational number: A number that can be written as
𝑏
where a and b are integers, but b is not equal to 0.
sign: a symbol that indicates whether a number is
positive or negative. Example: in −4, the (−) sign
shows this number is read “negative four”.
Big Ideas
Every rational number has an opposite. -3 is
the opposite of 3. 3 is the opposite of
negative 3.
Any rational number above or to the right of
a number on a number line is greater.
-3 > -5 because -3 is two units to the right.
Likewise -5 < -3 because -5 is two units to the
left.
The absolute value of a rational number is its
distance from zero. Because distance is
positive, absolute value is always positive.
I-3I = 3 and I4I = 4
Every rational number has a unique place on
the number line. These positions help to
order numbers from least to greatest and
greatest to least. The number line to the
right has been used to order the numbers
listed at the top from least to greatest.
Every point in the coordinate plane has a
unique ordered pair with rational number
coordinates. One of the ordered pairs to the
right has the x-coordinate of -4 and a
y-coordinate of 2. If you were asked “Which
point would make the 4th corner of a
rectangle?” for the three points to the right,
it would be (2, -3).
Revised 01-19-14
Math Grade 6: Unit 6
Rational Explorations
How Can You Help Your Student?
Sample Problems
1.
Graph 3 and the opposite of 3 on a number line. Explain
how this proves that 3 > the opposite of 3.
Solution: 3 is greater than negative 3 or the opposite
of three because 3 is to the right of -3 on the number
line.
2.
A scuba diver is 30 ft. below sea level and a submarine is
75 ft. below sea level. Jim thinks the inequality for this
situation should be -30 ft. below sea level > -75 ft.
below sea level. Sally thinks the inequality should be -30
ft. below sea level < -75 ft. below sea level. Who is
correct? Why?
Solution: Jim is correct. If we drew a vertical number
line with zero representing sea level, -75 would be
below -30. This means -30 > -75.
3.
Given the point (2,3) in the coordinate plane where 2 is
x and 3 is y, graph the point (x, -y). How would you
describe the similarities and differences for these two
points?
Solution: Both points are two units to the right of
zero. However, (2,3) is three units up from 2 and (2,-3)
is three units down from 2.
Multiple problems, different solution paths, and teacher
commentary for the 6th grade standards in this unit.
http://www.illustrativemathematics.org/6.NS.C
• Before students can understand negative rational numbers, they must understand positive
rational numbers and all their equivalent forms – fraction, decimal, and percent. For
1
example 8 = 0.125 = 12.5%. The Fractions, Decimals, and Percents Stack-N-Pack game is
available for check out from your school’s Parent Center. There are Stack-N-Pack books for
math concepts from Kindergarten through High School. Here is the link for all the books:
Stack-N-Pack Mathematics Games for K-HS
• Use bills or money owed as an example of negative numbers. For example, the electric bill
is $67.50. That can be represented as -$67.50.
• Suppose below sea level is negative and above sea level is positive. Pose problems like “If a
plane is 200 ft. above sea level and a submarine is -150 ft. below sea level, how many feet
apart are they? (350 ft.)
• Math Goodies - Integers Introduction - Tutorial and Practice.
http://www.mathgoodies.com/lessons/vol5/intro_integers.html
• Starting with Set 5, a series of free online lesson sets related to this unit.
http://learnzillion.com/courses/44?collection_id=617#collection_617
• Fun Brain - What’s The Point?- Game/Practice where students must correctly locate or
name points on the coordinate plane. http://www.funbrain.com/cgibin/getskill.cgi?A1=choices&A2=co&A3=8&A4=0&A7=0&A8=math
Literature related to this unit to read together at home and discuss:
• The Fly on the Ceiling by Julie Glass - In this story, the author invents a way to keep track of
his possessions by using a grid and coordinates.
• Hottest, Coldest, Highest, Deepest by Steve Jenkins – The author describes various
wonders of the world such as the deepest lake, hottest/coldest temperatures, deepest
spot in the ocean, etc. Each natural wonder includes data, inset maps, diagrams, and
comparisons. Students can use this book to explore integers in the real world.
• Less Than Zero by Stuart Murphy - A penguin named Perry needs 9 clams to buy an ice
scooter, but he has difficulty saving. Negative numbers are explored as Perry earns,
spends, finds, loses, and borrows clams. Students can explore integers by making a line
graph to show how Perry’s clams increase and decrease as the story is read.
Revised 01-19-14