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Chapter 7 · Fractions

Measuring a length by choosing a smaller unit

Teaching notesNCERT9 min

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9 min.

These teaching notes are for members

What the video covers, what to say before it, where a class usually goes wrong, and what to set afterwards. An account is free, and it opens every chapter of every book.

What to assume they know

What they should be able to do

  • Declare a chosen object to be one unit and measure other lengths against it
  • Fold a strip into halves, quarters and eighths and name each part
  • Build the run of multiples of a fractional unit and write each as a fraction
  • Explain why the last entry of such a run comes back to one whole
  • Read any fraction aloud as a count of fractional units, and say why that reading makes its size obvious
  • Name the two numbers of a fraction as the book names them
  • Make a thirds fold and use it to make sixths
  • Match a drawn piece to the fractional unit it represents

Where it usually goes wrong

  • "Three quarters means three, divided by four, and I must do the division." The book's preferred reading removes the division from the act of understanding the size. Both readings are legitimate; the point of section 11 is that only one of them makes the size visible.
  • "Folding in half twice gives thirds." It gives quarters. Say the count out loud each time the strip opens, and count creases as well as parts — three creases, four parts.
  • "You cannot fold a strip into three." You can, and p.158 asks for it. It is the first fold in the section that is not a halving, and it is what makes sixths reachable by one more halving.
  • "Eight eighths is a fraction, so it cannot be a whole number." The last row of the eighths run lands exactly on the whole strip. Show the strip, not the symbol, when this arrives.
  • "A fraction is always smaller than one." The half-roti run on p.157 breaks this on its second column. Pre-empt it here, and hand the naming problem to §7.5 rather than solving it now.
  • "The denominator counts what I have." It counts how many of these steps make one whole. The numerator counts what you have. Section 12 should state both halves of that sentence together.

Questions to check understanding

  • Fold or shade a strip to show a stated fraction, and say how many creases were needed
  • Write the run of multiples of a given fractional unit up to one whole
  • Rewrite a fraction as a count of fractional units, and rewrite a count of fractional units as a fraction
  • Say which is the numerator and which the denominator, and what each counts
  • Given a drawn piece, name the fractional unit it represents
  • Explain how a sixth can be made from a third by one more fold
  • Draw the picture and write the repeated-addition sentence for a stated number of quarters of a roti
  • The solutions appendix bound with this chapter answers the §7.3 Figure it Out on its §7.3 page

Examples worth working on the board

  • The strip and its folds (§7.3, p.156). Checked against the printed page. A plain grey bar is captioned as one strip of paper and braced underneath. It is then shown with one crease and two braces, and then with three creases and four braces. The explanation needs the physical action — fold, open, look at the crease — not just the finished picture.
  • The quarters run (§7.3, p.156). Checked against the printed page. Under the four-part strip the book stacks nested braces reading two of the quarter, then three of it, then four of it, each with its fraction beside it. Inputs: the fractional unit is one quarter, and the counts run 1, 2, 3, 4.
  • The eighths run (§7.3, p.157). Checked against the printed page. The same layout with eight parts, but most of the boxes are printed empty for the student to fill. The rows that are printed complete are the ones at counts 2, 4, 6 and 8, and the final row is closed off with the whole. Inputs: fractional unit one eighth, counts 1 to 8.
  • The boxed statement (§7.3, p.157). Checked against the printed page. A single ruled box states that a fractional amount is measurable once a fractional unit has been chosen. It is the thesis of the section printed as a banner, and it should get its own beat.
  • The half-roti run (§7.3, p.157). Checked against the printed page. A five-column table. Each column shows a growing collection of purple half-discs — one, then two, then three, four, five — with the matching repeated sum written under it and labelled as that many halves. Inputs: fractional unit one half, counts 1 to 5. Note: the second column is exactly one whole — two halves — and it is the third column, three halves, that first goes past it. That third column is the first amount beyond 1 the chapter writes as a fraction; amounts beyond 1 have appeared once already, on p.153, but only as spoken words. The book does not remark on the crossing here; §7.5 comes back for it.
  • Figure it Out (§7.3, p.158). Five tasks, all inputs: continue the half-run for two further steps; build the same run for the quarter; make a third by folding and then use it to make a sixth; draw and write the addition for five quarters of a roti and for nine quarters of a roti; and match four shaded circles to the fractional units one third, one fifth, one eighth and one sixth. The solutions appendix bound with this chapter gives the two extra half-run steps as six halves and seven halves, and reports the nine-quarter case as two wholes and a quarter left over.
  • Reading Fractions (§7.3, p.158). Checked against the printed page. A named panel with no section number of its own. It sets the ordinary spoken readings of three quarters against the reading as three copies of one quarter, and argues for the latter on the grounds that it exposes both the step size and the count. It then names the top and bottom numbers of a fraction using five sixths as the example. This panel is where numerator and denominator first appear in the chapter.
  • Teacher's Note (§7.3, p.158). A dashed panel asking teachers to repeat the fractional-unit exploration with circles, squares, rectangles and triangles. Useful as the closing beat: the argument does not depend on the strip.

Figures to have open

  • A foldable strip that can be creased and reopened, with braces that can be drawn under any run of parts. This is the workhorse figure; sections 1 to 6 all run on it. Standard schematic.
  • A run of half-discs that assemble into whole discs, for sections 7 and 8. The textbook's version on p.157 is drawn art; redraw it rather than reproducing it.
  • Four shaded circles to be matched with four fractional units, following the set printed on p.158, since the student will have that page open.
  • An annotated single fraction for section 12, with the two numbers labelled. Standard schematic.
  • No photograph or data table from the textbook is required.

Where this sits in the book

  • NCERT Ganita Prakash, Class 6, Chapter 7 "Fractions", §7.3 "Measuring Using Fractional Units", pp.156–158 — the strip and its folds and the quarters run (p.156); the eighths run, the boxed claim and the half-roti run (p.157); the Figure it Out set, the "Reading Fractions" panel and the Teacher's Note (p.158)
  • §7.1, pp.152–153, for the printed definition of fractional unit, and for the six spoken fraction words on p.153, three of which already name amounts beyond 1
  • §7.4, p.159, where the folded strip becomes a number line
  • §7.5, p.161, which names the amounts this section's runs pass through
  • Summary, p.186, for the printed one-line statement of Reading Fractions
  • Solutions appendix bound with this chapter file, §7.3 pages

The book

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