PrepShorts · Teaching notes · Class 7 Mathematics · Chapter 8, Working with FractionsPrepShorts

Chapter 8 · Working with Fractions

Why multiplying can make a number smaller

Teaching notesNCERT9 min

LearnTeach

This video could not be loaded. Reload the page to try again.

Sign in with Google

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.

Create a free accountSign in

What to assume they know

What they should be able to do

  • Predict, before computing, whether a product will land above or below each of its factors, from where those factors sit relative to 1
  • Produce an example of each of the three printed situations and check the prediction against the computed product
  • Compare a product with a factor by rewriting both over a common denominator
  • State the chapter's conclusion in its own careful form — a comparison with the other factor, one factor at a time
  • Say exactly when a product falls below both of the numbers multiplied, and give a counter-example to the claim that it always does
  • Explain why interchanging the two factors cannot change the product, using the rectangle picture
  • Recognise that 1 is the hinge of the whole discussion, and why the chapter sets it aside before starting

Where it usually goes wrong

  • "Multiplication makes things bigger." The chapter's whole block exists to break this. Do not correct it by replacing it with "multiplying by a fraction makes things smaller", which is the same mistake with a different sign.
  • "If one factor is a fraction, the product is below both numbers." 1/4 × 8 = 2 is printed on p.184 precisely to stop this. The product is above the 1/4.
  • "Fraction means below 1." 4/3 is a fraction and sits above 1, and the chapter puts it in Situation 1 for that reason. The hinge is 1, not the presence of a fraction bar.
  • "The conclusion is that a product always falls below both factors." The chapter's conclusion is per factor — a factor below 1 pushes the product below the other number. Below both is the special case where both factors are below 1, which is Situation 2 and only Situation 2. The chapter reaches that conclusion for one worked pair on p.185; it does not generalise it.
  • "You can compare 6/20 with 3/4 by looking at them." You cannot, reliably, and the chapter does the rewriting to twentieths rather than asking the reader to see it. Show that rewriting as a step, not as a formality.
  • "1 belongs in the table too." The block removes 1 in its opening sentence, because a factor of 1 leaves the other number exactly where it was and would make every row need an "or equal to".
  • "Swapping the factors gives a different answer because you cut differently." You do cut differently — the two pictures on p.186 are genuinely different pictures — and you land on the same area. That is the point worth slowing down on.

Questions to check understanding

  • Given two factors, say without computing whether the product will be above or below each of them, and justify from where the factors sit relative to 1
  • Complete the two statements the chapter leaves blank on p.185
  • Produce your own example for each of the three situations
  • Decide the truth of a claim such as "this product is above 1" for a product of two fractions given only as fractions — the printed item on p.197 asking which of six comparisons hold for 565/465 × 707/676 is exactly this, and is answered by placing each factor against 1 rather than by multiplying
  • Compare a product with one of its factors by rewriting both over a common denominator
  • Explain why the order of the two factors does not matter, using a rectangle
  • Spot the error in a stated rule that says "less than both" without a condition

Examples worth working on the board

  • 3 × 5 (Part I, §8.1, p.184). Inputs: two counting numbers above 1. The chapter notes the product exceeds each of them. This is the baseline the rest of the block is measured against.
  • 1/4 × 8 (Part I, §8.1, p.184). Inputs: one factor below 1, one above. The chapter states the product 2, and observes that it sits above the 1/4 and below the 8. This single line is the counter-example to "a fraction always makes it smaller than both", and it is printed before any rule.
  • 3/4 × 2/5 (Part I, §8.1, pp.184–185). Inputs: both factors below 1. The chapter computes 6/20, then — and this step is the mathematical content — rewrites 3/4 as 15/20 and 2/5 as 8/20 so that all three numbers are twentieths and can be compared by their numerators alone.
  • The three-situation table (Part I, §8.1, p.185). Checked against the printed page. A three-column table headed Situation, Multiplication, Relationship, with a pink header band and red rules:
    • Situation 1 — both factors above 1, worked with 4/3 × 4, product 16/3, described as above both.
    • Situation 2 — both factors below 1, worked with 3/4 × 2/5, product 3/10, described as smaller than either of them.
    • Situation 3 — one factor below 1 and one above, worked with 3/4 × 5, product 15/4, described as below the larger factor and above the smaller one. Note that Situation 1's example is 4/3 × 4, not 4 × 4/3, and that 4/3 is a fraction sitting above 1 — the table is deliberately not "whole numbers versus fractions".
  • The printed hint (Part I, §8.1, p.185). The chapter tells the reader outright that the relationship turns on whether each number lies between 0 and 1 or above 1, and asks for more examples of each situation. Treat the hint as the chapter naming its own variable.
  • The two blanks (Part I, §8.1, p.185). Checked against the printed page. The block ends with two fill-in-the-blank sentences, each offering greater or less, one for a factor between 0 and 1 and one for a factor above 1. The chapter does not print the answers here; the completed statements appear in the SUMMARY on p.198.
  • Order of multiplication (Part I, §8.1, p.186). Inputs: 1/2 × 1/4 and 1/4 × 1/2, both 1/8. Checked against p.186: two unit squares are drawn. In the upper one the rectangle is a half wide and a quarter tall, with the brace for 1/2 along the top and 1/4 down the left; in the lower one the rectangle is a quarter wide and a half tall, with the braces swapped. Both hatched rectangles are labelled 1/8. The chapter gives two reasons the order is irrelevant: the area of a rectangle is unchanged when length and breadth are interchanged, and Brahmagupta's formula shows the same thing symbolically.

Figures to have open

  • A number line marked with 0, 1 and a stretch beyond, on which two factors and their product can be placed. Every comparison in sections 3 to 7 runs on this one axis, and re-using it is what makes 1 visible as the hinge. Standard schematic.
  • The three-situation table, redrawn rather than lifted, with its three columns.
  • Two unit squares carrying congruent rectangles in the two orientations, with side braces that can swap labels, for section 10.
  • No photograph or data figure from the textbook is needed.

Where this sits in the book

  • NCERT Ganita Prakash, Class 7, Part I, printed Chapter 8 "Working with Fractions", §8.1 — the unnumbered block headed "Is the Product Always Greater…?", pp.184–185, including the three-situation table and the two closing blanks
  • Same part, §8.1, the unnumbered block "Order of Multiplication", p.186
  • Same part, SUMMARY, p.198, third bullet, where the completed statements appear
  • Same part, §8.1, Figure it Out, p.197, item 9, for the assessment shape this block is examined in
  • Same part, §8.2, pp.189–190, for the matching question about division, handled in Why dividing can make a number bigger

The book

Open in a new tab