Equivalent fractions: many names, one number, and lowest terms

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What to assume they know

Measuring a length by choosing a smaller unit: a fraction is a count of fractional units, and a strip can be folded into any number of equal parts
A fraction as an equal share, and why one-ninth is less than one-fifth: a fraction is what one person gets from an equal share
Factors and common factors, from Common factors: which jump sizes can land on a number
Multiplication and division facts up to 12, and division as the inverse of multiplication

What they should be able to do

Show that two fractions are equal by measuring one length with two different fractional units
Build a fraction wall and use it to read off equal lengths
Produce equivalent fractions for a given fraction, in both directions
Explain, in sharing terms, why the top and bottom may be multiplied by the same number
Write the division, addition and multiplication facts that go with a sharing picture
Decide whether a given fraction is in lowest terms, and say what test settles it
Reduce a fraction to lowest terms, both by repeated small divisions and by one division by the highest common factor
Recognise that a number has many equivalent fractions but only one lowest-terms name

Where it usually goes wrong

"Two fractions that look different must be different numbers." The whole section exists to break this, and the wall is the fastest demonstration. Vertical alignment, not arithmetic, is what should land first.
"Making the fraction simpler makes it smaller." Nothing about the amount changes. Keep the strip or the number-line point while the symbols change; the mark must not move.
"You can add the same number to the top and bottom." You cannot, and a student who has just learnt the multiplication rule will try it. Show a counter-example immediately: adding one to both numbers of one half moves the point on the line.
"Cancelling is a rule about crossing out digits." It is division by a common factor. The demonstration has to be a case where crossing digits fails: sixteen over sixty-two, where striking the sixes leaves one half while the fraction is nowhere near a half. Do not reach for sixteen over sixty-four, which is the trap — striking the sixes there happens to land on the right answer, so it demonstrates the opposite of what it is shown to demonstrate.
"There is one right way to reduce." The two routes on p.173 reach the same answer, and the book says so. Show both and let neither win.
"Every fraction has a simpler form." Only if the two numbers share a factor above 1. Four fifths does not, and that is the test.
"The fraction wall proves it for these numbers only." The wall stops at tenths for space. The argument does not stop there, and section 10 has to supply the reason that does not depend on a picture.

Questions to check understanding

Decide whether two given fractions are equivalent, and justify it
Write two or more fractions equivalent to a given one
Fill a missing number in an equality of two fractions
Given a sharing situation, write the fraction and the three facts that go with it
Reduce a fraction to lowest terms and state the factor used
Decide whether a given fraction is already in lowest terms
Find the second sharing situation that gives every child the same amount as a stated one
Use a fraction wall to answer how many of one fractional unit make another
The solutions appendix bound with this chapter answers the §7.6 question sets on its §7.6 pages

Examples worth working on the board

The three strips (§7.6, p.164). Checked against the printed page. Four bars are stacked down the left of the page: a plain one braced as one unit, then three bars in which the same left-hand stretch is shaded blue while the dashed divisions behind it get finer each time. A green callout to the right asks what the reader notices, and two bullet questions ask whether successive pairs of the shaded lengths are equal. Inputs: one shaded length, three different step sizes — two, four and eight parts to the unit.
The small fraction wall (§7.6, p.164). Checked against the printed page. A block of six rows, the top row a single cell headed as one unit, then rows of two, three, four, five and six equal cells, every cell carrying its fraction. Four questions follow it, all inputs: whether one half and three sixths are equal; whether two thirds and four sixths are equivalent and why; how many sixths make one half; how many sixths make one third.
The tall fraction wall (§7.6, p.165). Checked against the printed page. The same construction carried down to rows of sevenths, eighths, ninths and tenths. The book says a copy is bound at the end of the volume. Printed erratum: the right-hand cell of the sevenths row reads one seventh where the pattern requires seven sevenths; every other row closes correctly with its own denominator over itself. Checked against p.165.
The wall questions (§7.6, p.165). Inputs: are three sixths, four eighths and five tenths equivalent, and why; write two fractions equivalent to two sixths; fill a chain of boxes all equal to four sixths.
One roti among four (§7.6, pp.165–166). Checked against p.165: a quartered roti with four arrows running down to four stick-figure children, and a callout insisting the four shares match. The book then writes the same situation three ways — as a division, as a repeated sum, and as a product. Inputs: one roti, four children.
Three sharing questions (§7.6, p.166). Inputs only: three rotis among four children; two rotis among four children; two cakes among five children. Each asks for the picture and for the three facts as well as the fraction.
The scaling picture (§7.6, p.167). Checked against the printed page. Group 1 is drawn as two slabs above five children; Group 2 as two sets of two slabs above two sets of five children. A speech bubble reports the two shares as equal. Inputs: the two group compositions.
Draw and share (§7.6, p.167). Checked against the printed page. Three panels: one roti halved between two children, two rotis among four, three rotis among six, each child's share written under them. A ruled banner names the pattern. Inputs: the three situations.
The second scaling set (§7.6, p.168). Checked against the printed page. Three more panels — two rotis among three children, four among six, six among nine — with the first panel's shares filled in and the rest left blank, followed by a banner naming the first fraction as the simplest form of the other two, and a speech bubble asking what the reader notices about the two numbers of each fraction. Inputs: the three situations.
The missing-number questions (§7.6, pp.168–169). Inputs only: five glasses among four friends matched against eight friends; four kilograms in three bags matched against twelve kilograms; seven rotis among five children matched against an unstated larger pair, where more than one answer fits.
Lowest terms, worked (§7.6, p.172). The book takes sixteen twentieths, names 4 as a factor of both, divides, and checks that nothing divides the result. A speech bubble states the one-step method using the highest common factor. Inputs: the fraction and the common factor.
Lowest terms in stages (§7.6, p.173). Thirty-six sixtieths is reduced by halving twice and then dividing by three; the book then points out that both numbers are multiples of twelve and the same answer arrives in one move. Inputs: the fraction, and both routes. This pair is the whole content of section 12.
The reduction exercise (§7.6, p.173). Four fractions to put in lowest terms. Checked against the printed page: they are seventeen fifty-firsts, sixty-four one hundred and forty-fourths, one hundred and twenty-six one hundred and forty-sevenths, and five hundred and twenty-five one hundred and twelfths. The item labels are misprinted as a, b, e, d — there is no c, and e appears in third place. The last of the four is greater than 1, which is unusual for a reduction exercise and worth a remark.

Figures to have open

A fraction wall from one unit down to tenths, redrawn, with a movable vertical guide. Sections 2, 3, 4 and 12 all use it. It must be redrawn rather than reproduced, and it must correct the sevenths-row misprint described above.
Stacked strips in which one shaded length stays fixed while the divisions behind it change. Standard schematic; carries section 1.
Sharing panels — rotis above, children below, share written under each child — reusable for sections 5, 7, 8 and 9. Redraw; the printed rotis and stick figures on pp.165–168 are the book's own artwork.
A number line where a point stays put while its label is rewritten, for the first two misconceptions.
No photograph, table or data figure from the textbook is required.

Where this sits in the book

NCERT Ganita Prakash, Class 6, Chapter 7 "Fractions", §7.6 "Equivalent Fractions", pp.163–173 — the strips and the small wall (p.164); the tall wall and its questions (p.165); the quartered roti and the three facts (pp.165–166); the scaling groups (pp.166–167); the draw-and-share panels (p.167); the second scaling set and the simplest-form banner (p.168); the missing-number questions (pp.168–169); the lowest-terms definition and the worked reduction (p.172); the staged reduction and the exercise (p.173)
§7.3, p.156, for the folded strips this section reuses
§7.5, p.162, for the mixed form, which some of the reduction answers reach
Chapter 5 "Prime Time", §5.1, for factors and common factors
Summary, p.186, for the printed one-line statements of Equivalent Fractions and Lowest terms
Solutions appendix bound with this chapter file, §7.6 pages