Mixed fractions: naming what lies past 1

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What to assume they know

Every fraction has one place on the number line: fractions occupy definite places on the number line, and some of those places lie past 1
Measuring a length by choosing a smaller unit: a fraction is a count of fractional units, and a whole is a complete set of them
Repeated addition and its short form, from Every number sequence is a rule, not a list
Division with a remainder, informally

What they should be able to do

Sort a list of fractions into those under one whole and those over it
State the size test for a fraction that exceeds one whole, and explain why it must hold
Count how many whole units a given fraction contains
Rewrite a fraction greater than 1 in the mixed form, and name its two parts as the book names them
Rewrite a mixed form back as a single fraction, and explain the step that makes the two equal
Decide whether every fraction greater than 1 can be written in the mixed form, and say what happens at exact wholes
Place a mixed form on the number line

Where it usually goes wrong

"The mixed form is a different, bigger number." It is the same number regrouped. Show both forms landing on the same point of the number line before any arithmetic.
"Two and two thirds means two multiplied by two thirds." The compact notation hides a plus sign, and that is genuinely confusing. Say the plus out loud every time the form appears, at least until section 10.
"To convert back, multiply the whole part by the denominator and add — that's the rule." It is the rule, but on p.163 the book earns it by replacing each whole with a full set of fractional units and counting. Show the counting first; the rule is what the counting collapses into.
"A fraction bigger than one is wrong and must be tidied into mixed form." Neither form is more correct. §7.8 will add fractions and leave answers in both forms on the same page.
"If the top is bigger the fraction is bigger." True only when the bottom numbers match. Nothing in this section compares two different denominators.
"Every fraction over 1 has a leftover." Four halves does not. The chapter's own eighths run in §7.3 ended exactly on a whole, and §7.5 puts the question to the reader directly on p.162. Section 9 should answer it, and answer it the way the book does, rather than leaving a student to meet the answer only in the solutions appendix.

Questions to check understanding

Sort a list of fractions into those below and those above one whole
State how many whole units a given fraction contains
Convert a fraction greater than 1 into the mixed form
Convert a mixed form into a single fraction
Mark a mixed form on the number line
Given a mixed form, name its whole part and its fractional part
Explain, without using the rule, why a fraction whose top number exceeds its bottom number must be more than one whole
The solutions appendix bound with this chapter answers the §7.5 Figure it Out sets on its §7.5 pages

Examples worth working on the board

The sorting table (§7.5, p.161). Checked against the printed page. A two-column table is printed empty, headed for lengths under one unit and lengths over one unit, and the student is sent back to the fractions marked in §7.4 to fill it. Inputs: the fractions of pp.159–160, which include one half, two thirds, two fifths, four fifths, three halves and the run of fifths beyond 1.
The size test (§7.5, p.161). The book states the pattern in terms of which of the two numbers is the larger. Do not present it as a definition handed down: section 3 should derive it by asking how many steps one whole needs.
Three halves and five halves (§7.5, p.161). Printed as repeated sums of one half that are then regrouped. Inputs: the fractional unit is one half; the counts are 3 and 5.
The thirds speech bubble (§7.5, p.161). Checked against the printed page. A character observes that three thirds close up into one whole, so a fourth third must carry the total past one. This is the cleanest available argument for the size test and deserves its own section.
Whole units inside a fraction (§7.5, pp.161–162). Two question sets, both inputs only. The first asks how many complete units sit inside seven halves, then inside four thirds and seven thirds. The second asks the same of eight thirds, eleven fifths and nine quarters.
The worked mixed form (§7.5, p.162). Eight thirds is regrouped as two whole units with two thirds left, and the book then writes it in the compact mixed notation and gives it a spoken name. Inputs: the fraction, and the fact that a whole here takes three thirds.
Jaya's conversion (§7.5, p.163). Checked against the printed page. A named character takes a mixed form with whole part 3 and fractional part three quarters, replaces each whole by four quarters, and collects the lot. She writes the collection out twice — once as a long sum of quarters bracketed into groups, and once with each bracket compressed to a multiplication — before reading off the single fraction. Inputs: whole part 3, fractional part three quarters, and the fact that one whole is four quarters. The two-line presentation is the point: the multiplication line is not a new rule, it is the addition line shortened.
Both practice sets (§7.5, pp.162–163). Inputs only. One set asks for six fractions to be rewritten in the mixed form: nine halves, nine fifths, twenty-one nineteenths, forty-seven ninths, twelve elevenths and nineteen sixths, with nine halves worked as the printed example. The other asks for six mixed forms to be rewritten as single fractions: three and a quarter, seven and two thirds, nine and four ninths, three and one sixth, two and three elevenths, and three and nine tenths.
The open question (§7.5, p.162). The book asks whether every fraction greater than 1 can be written in the mixed form and prints no answer beside it. The bound solutions answer no, and give as the counter-case a fraction that reduces to an exact whole number. The p.162 definition is what settles it: the book requires a mixed number to carry a fractional part that is less than 1, and a fraction like four halves or eight quarters leaves nothing to put there. Stage the question, let the class try it, and then give the book's answer — an explanation that answers yes contradicts the marking scheme the student is holding.

Figures to have open

A number line running 0 to 4 on which both spellings of the same number can be marked at the same point. This is the figure that carries the thesis and should return in sections 1, 9 and 11. Standard schematic.
A tray of identical fractional-unit tiles that can be grouped into wholes and regrouped, for sections 3, 4, 6 and 10. Standard schematic.
An annotated mixed form showing the suppressed plus sign. The textbook does not print this annotation; it is added here and should be labelled so.
No photograph, table or data figure from the textbook is required. The two-column sorting table on p.161 is printed empty, so it is a worksheet frame rather than a figure to reproduce.

Where this sits in the book

NCERT Ganita Prakash, Class 6, Chapter 7 "Fractions", §7.5 "Mixed Fractions", pp.161–163 — the sorting table, the size test, the halves regrouped and the thirds speech bubble (p.161); the whole-unit counts, the worked eight-thirds case, the naming of the form, the open question and the first practice set (p.162); Jaya's conversion and the second practice set (p.163)
§7.4, pp.159–160, which supplies the fractions the sorting table asks for
§7.3, p.158, for the printed wording of numerator and denominator
§7.8, p.177, where a sum is left in both spellings on the same page
Summary, p.186, for the printed one-line statement of Mixed fractions
Solutions appendix bound with this chapter file, §7.5 pages