3.4 Ordering & Comparing Whole Numbers, Integers, Decimals & Fractions

3.4 Ordering & Comparing Whole Numbers, Integers, Decimals & Fractions

Quick Summary: Ordering and comparing whole numbers, integers, decimals and fractions is the first of the eleven fundamental knowledge and skills EQAO lists under Relationships and Proportional Reasoning — the dimension that carries 27 of the 50 mathematics questions. Comparison questions look easy and are quietly error-prone: candidates lose marks by comparing decimals digit-count-first, by assuming a bigger denominator means a bigger fraction, and by treating $-12$ as larger than $-3$. This section builds one dependable procedure for each number type and one procedure for mixed lists.

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Why Comparison Questions Appear Everywhere

Comparison is rarely the whole question on the MPT. It is embedded:

• Unit-rate items end with "which is the better buy?" — a comparison of two decimals.
• Data items ask you to order values before locating a median.
• Financial items compare a discount expressed as a fraction against one expressed as a percent.
• Estimation items in the no-calculator section ask which of four answers is closest — a comparison in disguise.

Because five mathematics questions are answered without a calculator, the ordering strategies below must work on paper.

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1. Whole Numbers: Compare by Place Value

For whole numbers written without leading zeros, more digits means a larger number. When the digit counts match, compare from the leftmost place and stop at the first difference.

$4{,}072{,}915 \;\text{vs.}\; 4{,}072{,}951 \;\Rightarrow\; \text{identical through the tens place; } 1 < 5 \;\Rightarrow\; 4{,}072{,}915 < 4{,}072{,}951$

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2. Integers: Use the Number Line, Not the Digits

The single most common integer error is comparing magnitude instead of value.

      -12   -8    -5    -3     0     3     5     8    12
 <-----|-----|-----|-----|-----|-----|-----|-----|-----|----->
        smaller  <-------------------------->  larger

• Every negative number is less than zero, and zero is less than every positive number.
• Among negatives, the number with the larger absolute value is the smaller number: $-12 < -8 < -3$, even though $|-12| > |-8| > |-3|$.
• A useful classroom framing: think of temperature. $-12^\circ\text{C}$ is colder — that is, less — than $-3^\circ\text{C}$.

Trap: "Which value is greatest: $-0.9$, $-\frac{1}{2}$, $-1.1$?" The answer is $-\frac{1}{2}$, because $-0.5$ sits furthest to the right.

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3. Decimals: Pad, Then Compare Left to Right

Never compare by length. $0.7$ is greater than $0.68$, even though $0.68$ has more digits.

Procedure:

1. Line up the decimal points.
2. Pad the shorter decimals with trailing zeros so every value has the same number of decimal places (appending zeros to the right of the last decimal digit does not change value).
3. Compare place by place from the left; the first place where the digits differ decides the comparison.

Worked comparison. Order $0.7$, $0.68$, $0.702$, $0.6$ from least to greatest.

$0.6 < 0.68 < 0.7 < 0.702$

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4. Fractions: Four Strategies, Ranked by Speed

Strategy A — Benchmarks $0$, $\frac{1}{2}$, $1$ (fastest)

Sort each fraction into a band before doing any arithmetic. A fraction is greater than $\frac{1}{2}$ when the numerator is more than half the denominator.

• $\frac{3}{8}$: half of 8 is 4, and $3 < 4$, so $\frac{3}{8} < \frac{1}{2}$.
• $\frac{5}{9}$: half of 9 is 4.5, and $5 > 4.5$, so $\frac{5}{9} > \frac{1}{2}$.
• Therefore $\frac{3}{8} < \frac{5}{9}$ with no common denominator required.

Strategy B — Same numerator: bigger denominator, smaller fraction

$\frac{3}{5} > \frac{3}{7} > \frac{3}{11}$. Cutting the same amount of pizza among more people gives each person less.

Strategy C — Same distance from 1

When each fraction is one unit fraction short of a whole, compare the size of the missing piece: $\frac{7}{8} = 1 - \frac{1}{8}, \qquad \frac{9}{10} = 1 - \frac{1}{10}$ Since $\frac{1}{10} < \frac{1}{8}$, less is missing from $\frac{9}{10}$, so $\frac{9}{10} > \frac{7}{8}$.

Strategy D — Cross-multiplication (two fractions only)

For positive $\frac{a}{b}$ and $\frac{c}{d}$, compare $ad$ against $bc$: $\frac{5}{8} \;\text{vs.}\; \frac{7}{11} \;\Rightarrow\; 5 \times 11 = 55, \quad 8 \times 7 = 56, \quad 55 < 56 \;\Rightarrow\; \frac{5}{8} < \frac{7}{11}$

Caution: cross-multiplication compares exactly two fractions. For a list of four, use benchmarks or convert to decimals.

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5. Mixed Lists: Convert Everything to Decimals

When a list mixes fractions, decimals, percents, and integers, one common format beats four separate tricks. Convert to decimals — the format that is easiest to order by place value.

Worked example. Order from least to greatest: $\frac{5}{8}$, $0.6$, $63%$, $\frac{2}{3}$, $0.58$.

1. Convert each value.

1. Order the decimals. $0.580 < 0.600 < 0.625 < 0.630 < 0.667$

2. Translate back to the original forms. $0.58 \;<\; 0.6 \;<\; \frac{5}{8} \;<\; 63\% \;<\; \frac{2}{3}$

Notice how close $\frac{5}{8} = 0.625$ and $63% = 0.63$ are. Multiple-choice distractors are built precisely on pairs like this, which is why memorizing the benchmark equivalents in Section 3.2 ($\frac{1}{8} = 0.125$, $\frac{3}{8} = 0.375$, $\frac{5}{8} = 0.625$, $\frac{7}{8} = 0.875$) saves both time and accuracy.

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6. Negative Rationals: Order the Magnitudes, Then Reverse

To order $-\frac{3}{4}$, $-0.7$, $-\frac{2}{3}$:

1. Compare magnitudes: $0.667 < 0.700 < 0.750$.
2. Reverse the order for negatives: $-\frac{3}{4} \;<\; -0.7 \;<\; -\frac{2}{3}$

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Classroom Connection

Ordering is one of the richest diagnostic tasks in the Grades 3–9 program. A student who writes $0.68 > 0.7$ is applying whole-number reasoning ("68 is bigger than 7") to decimals, and a student who writes $\frac{1}{3} > \frac{1}{2}$ is applying it to denominators. Both misconceptions are surfaced quickly by asking students to place cards on an open number line — a Representing task in Ontario's mathematical processes, and exactly the sort of diagnostic evidence Growing Success describes as assessment for learning.