Download What is a rational number?

Math 7 Warm-Up↑ 11/29/16
Divide each of the following problems. Be sure to place the appropriate sign on
each problem.
1)
−54
−6
=
2) If s = -4 and t = -6, then
𝑠+𝑡
5
=
3) The hot air balloon Midnight Express changes -2,700 feet of altitude in 135
minutes. What is the rate of change in feet per minute?
4) During the past week, Mrs. Thorne recorded the following amounts in her
checkbook: $150, -$75, -$15, and -$32. Write and evaluate an expression to
find the average of these amounts.
Math 7 The Number System:
7.NS.2 Apply and extend previous understandings of multiplication and division of
fractions to multiply and divide rational numbers.
7.NS.2b Understand that integers can be divided, provided that the divisor is not
zero, and every quotient of integers (with non-zero divisor) is a rational number.
7.EE.3 Apply properties of operations to calculate with numbers in any form;
convert between forms as appropriate.
Mathematical Practices:
1 – Make sense of problems and persevere in solving them.
3 – Construct viable arguments and critique the reasoning of others.
5 – Use appropriate tools strategically.
6- Attending to precision.
7 – Look for and make use of structure.
Objective: Know what is a rational number and be able to graph rational numbers
on a number line.
A. What is a rational number?
A rational number is any number that can be expressed as a division of
two integers as long as the denominator of the fraction is not equal to zero.
𝑎
A rational is of the form
Examples:
1= ,
9=
,
𝑏
, 𝑏 ≠ 0.
2
, − 2.3 =
9
√25 =
,
Other examples of rational numbers:
Examples of numbers that are not rational:
,
5
14 8 =
−8=
,
B. Placing Rational Number on the Number Line
To place a rational number follow these steps:
1) Find the two whole numbers the rational is in between.
2) Divide the space in between the two whole numbers by the denominator
of the rational number. (Need 1 fewer lines than the denominator).
3) Count over the numerator and place a dot at the location where the
rational number should be placed and label it correctly.
Example 1:
Example 2:
3
Graph the rational 1 on the given number line.
4
3
Between which two whole numbers does 1 lie?
4
How many sections do we need?
How many lines is that?
3
Draw a dot at 1 .
4
Example 3:
5
Graph the rational −1 on the given number line.
8
(*Remember that negative numbers decrease in value as the digits get larger).
5
Between which two whole numbers does −1 lie?
8
How many sections do we need?
How many lines is that?
5
Draw a dot at −1 .
8
You Try!
Example 4:
2
Graph the rational −4 on the given number line.
5
2
Between which two whole numbers does −4 lie?
5
How many sections do we need?
How many lines is that?
2
Draw a dot at −4 .
5
Example 5:
4
Graph the rational 6 on the given number line.
7
4
Between which two whole numbers does 6 lie?
7
How many sections do we need?
How many lines is that?
4
Draw a dot at 6 .
7
Homework p. 167 Review Problems 1 – 6 & p. 168 Reflection
Closure: What is a rational number?
Which of the following are rationals?
Explain your reasoning:
0.2, 𝜋, − 4,
4
√36 , √13 , 2 5
0. 6̅,
-7.2,
Math 7 Warm-Up↑ 11/30/16
3
1) Graph the rational −1 on the given number line.
8
3
Between which two whole numbers does −1 lie?
8
How many sections do we need?
How many lines is that?
3
Draw a dot at −1 .
8
2) What is the definition of a rational number?
3) Give three examples of numbers which are rational using your definition
above in question 2.
1)
2)
3)
Math 7 The Number System:
7.NS.2 Apply and extend previous understandings of multiplication and division of
fractions to multiply and divide rational numbers.
7.NS.2b Understand that integers can be divided, provided that the divisor is not
zero, and every quotient of integers (with non-zero divisor) is a rational number.
7.NS.2d Converting rational numbers to decimals using long division and
knowing that a rational number either terminates or repeats.
7.EE.3 Apply properties of operations to calculate with numbers in any form;
convert between forms as appropriate.
Mathematical Practices:
1 – Make sense of problems and persevere in solving them.
3 – Construct viable arguments and critique the reasoning of others.
5 – Use appropriate tools strategically.
6- Attending to precision.
7 – Look for and make use of structure.
8 – Look for and express regularity in repeated reasoning
Objective: Know that a rational number can be expressed as a repeating or
terminating decimal and be able to accurately represent a rational as a decimal
using long division.
Writing Rationals As Terminating or Repeating Decimals
All rational numbers can either be expressed as a terminating or repeating
decimals. Terminating decimals are decimals that stop or which have a zero that
repeats. Repeating decimals have one or more digits that repeat over and over
again. Repeating decimals can be represented using bar notation over the first
digit or set of digits that repeat.
First divide the numerator by the denominator.
Example 1:
1
3
=
Example 2:
2
9
=
Example 3: 2
3
8=
Example 4: 5
4
6=
Example 5:
74
100
Example 6: −3
=
7
20 =
You Try!
Example 7: −6
Example 8:
Example 9:
7
9
1
2=
=
−1
40
=
Which denominators do you think cause decimals to repeat? Why?
Which denominators do you think cause decimals to terminate? Why?
Closure: All rationals can be expressed as what two kinds of numbers?
Homework: p. 179 – 180 (2, 6, 8, 10, 11, 15, 16, 18, 19, 22)
Math 7 Warm-Up↑ 11/16/15
Express each rational number as a terminating or repeating decimal without a
calculator.
1)
1
8
=
8
2) −2 =
9
If you finish #1 and #2, try these two problems with your calculator.
3) 5
3
16
4) −7
=
8
11
=
Math 7 The Number System:
7.NS.2 Apply and extend previous understandings of multiplication and division of
fractions to multiply and divide rational numbers.
7.NS.2b Understand that integers can be divided, provided that the divisor is not
zero, and every quotient of integers (with non-zero divisor) is a rational number.
7.NS.2d Converting rational numbers to decimals using long division and
knowing that a rational number either terminates or repeats.
7.EE.3 Apply properties of operations to calculate with numbers in any form;
convert between forms as appropriate.
Mathematical Practices:
1 – Make sense of problems and persevere in solving them.
3 – Construct viable arguments and critique the reasoning of others.
5 – Use appropriate tools strategically.
6- Attending to precision.
7 – Look for and make use of structure.
8 – Look for and express regularity in repeated reasoning
Objective: Be able to represent any terminating or repeating decimal as a fraction
in simplest form using the concept of place value.
A. Writing Terminating Decimals as Rational numbers.
By using the last place value a number terminates in we can write all
terminating decimals as rational numbers.
0
0
2
-3
.
.
.
.
7
1 9
1 0 5
0
4
Complete the table below. Write the fractions in simplest form.
Words
Decimal
Fraction
0.7
seven tenths
0.19
Two and one hundred five
thousandths
2.105
-3.04
Example 1: -0.2 =
Example 2: 0.04 =
Example 3: -0.875 =
B. Writing Terminating Decimals as Rationals
Every repeating number can also be written as a fraction with a denominator
of 9, 99, 999, 9999, etc. Use the place value of the final digit that repeats
the first time minus 1 as the denominator. Then simplify your answer if
possible.
Example 4: 0. 5̅ =
̅̅̅̅ =
Example 5: 0. 09
̅̅̅̅ =
Example 6: −1. 27
̅̅̅̅̅ =
Example 7: 3. 123
You Try! Turn each of these mixed examples of both terminating and repeating
decimals into fractions (rationals) in simplest form:
Do at least examples 8 & 9.
Example 8: 0.8 =
Example 9: 0. 6̅ =
Example 10: 0.34 =
̅̅̅̅ =
Example 12: 0. 72
Homework: p. 181 - 182 (24, 25, 26, 27, 28, 29, 30, 35, 36, 37 )
Name ________________________________________________ 11/30/16
Ticket out the Door Closure:
What is the main math concept that you can use to change a decimal into a
fraction?
_____________________________________________________________
_____________________________________________________________
Give an example of a repeating or terminating decimals show how you would
convert each into a fraction.
Terminating Example:
______________ =
Repeating Example:
______________ =