Mathematics
Grade 10
15 min
Compare decimals and fractions on number lines
Compare decimals and fractions on number lines
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1
Introduction & Learning Objectives
Learning Objectives
Convert any rational number between fraction and decimal form to facilitate logical comparison.
Accurately plot fractions and decimals on a number line with appropriate scaling and partitioning.
Justify the relative position of two or more numbers on a number line using logical arguments and inequality symbols.
Formulate a logical proof to demonstrate why one rational number is greater than, less than, or equal to another using their number line positions.
Analyze and interpret number line models to deduce relationships between sets of fractions and decimals.
Apply the concept of density of rational numbers to locate and compare values within infinitesimally small intervals on a number line.
Is 0.667 closer to 2/3 or 3/5? How can you prove your conclusion...
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Key Concepts & Vocabulary
TermDefinitionExample
Rational NumberA number that can be expressed as a ratio p/q of two integers, where q ≠ 0. All terminating and repeating decimals are rational numbers.3/4, -5, 0.25, and 0.3̅ (which is 1/3) are all rational numbers.
Number LineA geometric representation of numbers where points on a line correspond to real numbers. The line is ordered, with values increasing from left to right.A horizontal line with 0 in the center, negative numbers like -2 and -1 to the left, and positive numbers like 1 and 2 to the right.
Ordering PrincipleFor any two distinct numbers 'a' and 'b' on a number line, if 'a' is to the left of 'b', then it is logically necessary that a < b.Since -1.5 is to the left of -1 on the number line, we can conclude that...
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Core Formulas
Conversion: Fraction to Decimal
\frac{a}{b} = a \div b
To convert a fraction to a decimal, divide the numerator by the denominator. This is the first step to comparing a fraction with a number already in decimal form.
Conversion: Terminating Decimal to Fraction
0.d_1d_2...d_n = \frac{d_1d_2...d_n}{10^n}
To convert a terminating decimal to a fraction, use the digits as the numerator and a power of 10 (corresponding to the number of decimal places) as the denominator, then simplify. This allows for comparison using fractional methods.
Logical Comparison by Cross-Multiplication
For b > 0 and d > 0, \frac{a}{b} > \frac{c}{d} \iff ad > bc
A logical test to compare two fractions without converting them to decimals or finding a common denominator. This is a form...
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Challenging
To prove that a rational number exists between 7/11 and 8/11, a student must apply the Density Property. What is a valid logical first step in constructing this proof without finding the average?
A.Assert that since 7 and 8 are consecutive, no integer exists between them, so no fraction can exist.
B.Find an equivalent fraction for both, such as 14/22 and 16/22, which makes the intermediate fraction 15/22 immediately evident.
C.Convert both to repeating decimals and show that the decimal expansions are different.
D.Partition a number line into 11 parts and show there is no mark between the 7th and 8th mark.
Challenging
Let p/q be any positive rational number. Formulate a logical proof that there always exists another rational number, r, such that 0 < r < p/q. Which statement is the key to this proof?
A.The number r can be constructed as p/(q+1). Since the denominator is larger, the fraction is smaller.
B.The number r can be constructed as (p-1)/q. This is always smaller than p/q.
C.The number r can be constructed as p/(2q). This value is equivalent to (1/2) * (p/q), which is always positive and half the original value, thus placing it between 0 and p/q.
D.The Density Property only applies between two *distinct* rational numbers, not between a number and zero.
Challenging
On a number line, point M is located at x/y and point N is at z/w, with x/y < z/w. Point P is the midpoint of the segment MN. A logical proof for the value of P would conclude that P is located at:
A.(x+z)/(y+w)
B.(xz)/(yw)
C.The average of the two numbers, which is 1/2 * (xw + zy)/(yw)
D.The average of the two numbers, which is 1/2 * (x/y + z/w)
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