Placing a rational number on the line, and distance as |a − b|

What to assume they know

• What "rational" means, and why the denominator cannot be zero — what a rational number is, equivalent writings, and the co-prime representative
• Debts and fortunes: negative numbers close subtraction — negative numbers, and the number line extended both ways
• Dividing a segment into a given number of equal parts
• Improper fractions and mixed numbers, and converting between them
• Subtraction of rationals, from §3.4

What they should be able to do

• Set up a number line with an origin and equally spaced integers
• Place a proper fraction on the line by the chapter's construction, stating the number of parts and the number of steps
• Place a fraction greater than 1 by first identifying the two integers it lies between
• Place a negative rational, and say which direction the sign sends you
• Define absolute value as distance from the origin and evaluate it for positive, negative and zero inputs
• Explain why absolute value is never negative
• Compute the distance between two rationals as the absolute value of their difference, and show the result does not depend on the order
• Read a mixed number line carrying both integers and fractions, above and below the axis
• Locate a decimal on the line by first reading it as a fraction

Where it usually goes wrong

• "You place a fraction by estimating where it looks about right." The rule is a construction with two numbers of instructions in it. Estimation is what the section replaces.
• "The numerator tells you how fine the cuts are." It is the other way round, and this swap is the commonest error in the topic. Say the rule aloud in the right order every time: the lower number cuts, the upper number walks.
• "An improper fraction cannot go on the line because it is bigger than one." Fig. 3.6 exists to answer this. Convert to a mixed number first, which identifies the interval, and then apply the same construction inside it.
• "−3/4 means go three quarters right and then flip." It means walk three quarter-steps leftwards. There is one motion, not two.
• "Absolute value means removing the minus sign." That description gives the right answer for every case in the chapter and the wrong idea. Absolute value is a distance; that is why it is never negative, why |0| is 0, and why it appears in the distance formula at all.
• "|a − b| and a − b are the same when a is bigger." They agree in that case, which is exactly why students stop distinguishing them and then get a negative distance when the order flips.
• "Equivalent fractions sit near each other on the line." They sit on the identical point. Fig. 3.7 and the previous topic together make this concrete.
• "1.15 with a bar is a bit more than 1.15." It is 1.1555… and placing it requires converting the repeating decimal to a fraction first.

Questions to check understanding

• Place a given proper fraction on a number line, stating the construction
• Place a fraction greater than 1, showing the mixed-number step
• Place a negative rational
• Represent several rationals with unlike denominators on one line
• Evaluate absolute values, including of a negative and of zero
• Compute the distance between two given rationals, and verify the answer is the same in both orders
• Read positions off a supplied number line and name the numbers marked
• Locate a terminating decimal on the line by converting it to a fraction first
• State why the absolute value of any rational is never negative

Examples worth working on the board

Values marked verified are worked out here on the chapter's stated inputs. The chapter prints no answers.

• Fig. 3.3 (p. 50). A plain integer line with arrowheads both ends, ticked and labelled from −7 to 7, illustrating the equal spacing the section sets up.
• Fig. 3.4 (p. 50). The same line shortened to run from −2 to 2, with two fractions marked: 1/2 sitting midway between 0 and 1, and −3/4 sitting between −1 and 0. Labels for the fractions are set beneath the axis. Read from the printed page.
• The construction rule (§3.4.1, p. 50). Printed: cut the unit interval into q equal parts, then move p of those parts from zero, rightwards if the number is positive and leftwards if it is negative.
• Fig. 3.5 (p. 51). The interval from 0 to 1 quartered, with ticks labelled 0, 1/4, 1/2, 3/4 and 1 — the worked instance for 3/4. Read from the printed page.
• Fig. 3.6 (p. 51). A line from 0 to 3 divided into quarters throughout, with every quarter labelled: 0, 1/4, 1/2, 3/4, 1, 5/4, 3/2, 7/4, 2, 9/4, 5/2, 11/4, 3. The chapter's worked instance is 9/4, rewritten as 2¼ so that the correct unit interval — between 2 and 3 — is identified before any cutting happens. Verified: 9/4 = 2¼, so one quarter-step right of 2.
• Think and Reflect (p. 51). Asks the student to place 8/5 and −7/4. Verified: 8/5 = 1⅗, so three fifths right of 1; −7/4 = −1¾, so three quarters left of −1.
• Fig. 3.7 (p. 51), read from the printed page. A mixed line running −5 to 5 with labels split above and below the axis. Above the axis: −18/5, −9/4, −7/8, 7/8, 9/4, 26/7, 32/7. Below the axis: −5, −22/5, −4, −3, −2, −1, 0, 1, 3/2, 2, 3, 4, 5. The caption describes the marks as a mixture of integers and rationals sitting on one line. Verified: −18/5 = −3.6, −9/4 = −2.25, −7/8 = −0.875, 26/7 ≈ 3.714, 32/7 ≈ 4.571, −22/5 = −4.4 — all consistent with the drawn positions. This is the richest single figure in the section and worth redrawing in full.
• Absolute value, Example 1 (p. 51). Printed: |5/3| is 5/3; |−5/3| is also 5/3; |0| is 0. The section then states that absolute value is never negative, written |x| ≥ 0.
• Fig. 3.8 (p. 52), read from the printed page. A line from −5 to 4 with b marked at −4 and a marked at 3, both labels above the axis, and a double-headed arrow spanning the two points labelled |a − b| = 7. Verified: 3 − (−4) = 7, and (−4) − 3 = −7, whose absolute value is also 7. The figure is drawn on integers, which makes it a clean template to re-use on fractions.
• Order-independence, worked (not in the book). Take a = 3 and b = −4 from Fig. 3.8 and compute both differences. Verified: the two differences are opposites, and the absolute value collapses them to one number. Say why this is required rather than convenient: a gap has one length.
• Exercise Set 3.4, Q1 (p. 52). Place 2/3, −5/4 and 1½ on a single line. Verified: thirds and quarters do not share a grid, so this exercise forces a twelfth-part scale — the first place a student must choose a common cutting for several fractions at once. Flag it: the chapter's construction rule handles one fraction at a time and does not mention this.
• End-of-chapter Q4 (p. 64). Locate 0.532 and 1.15 with a bar over the final digit, so 1.1555…, on the number line. Verified: 0.532 = 532/1000 = 133/250; the second is a repeating decimal and needs Converting a terminating or repeating decimal back to p/q before it can be placed exactly. The overline on the second item was read from p. 64.

Figures to have open

• Fig. 3.7 (p. 51) redrawn in full, with labels above and below the axis exactly as the printed figure splits them. This is the chapter's own figure, it carries more data than any other in the section, and the above/below split is what lets it hold twenty labels without collision — the seven above and thirteen below listed in the bullet above.
• Fig. 3.8 (p. 52) redrawn: a line with two marked points, a span arrow, and the distance formula attached. The chapter's own figure; the argument of sections 9 and 10 sits on it.
• Fig. 3.5 and Fig. 3.6 (p. 51) as a build sequence rather than two static images — the first is contained in the second, and showing the extension from one unit to three is cheaper than drawing both.
• A twelfth-part grid for Exercise Set 3.4, Q1, showing thirds and quarters landing on a shared scale. The chapter prints no such figure and the exercise needs one.
• No photograph is needed.

Where this sits in the book

• NCERT Ganita Manjari, Class 9 Mathematics, printed Chapter 3 on the world of numbers. Section §3.4.1, whose printed heading begins "Representation of Rational Numbers" and continues onto a second line naming the number line, runs pp. 50–52.
• Figs. 3.3 and 3.4, p. 50; Figs. 3.5, 3.6 and 3.7, p. 51; Fig. 3.8, p. 52.
• The absolute value material sits under an unnumbered bold subheading on p. 51, which can be named but not cited by number; Example 1 is printed there.
• Think and Reflect, p. 51.
• Exercise Set 3.4, Q1, p. 52. End-of-chapter exercises, Q4, p. 64.