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CBSE Class 8 · Mathematics

We Distribute, Yet Things Multiply

Official NCERT chapter from Ganita Prakash Part I (book code hegp1). ExamMaster notes are original teaching at CBSE Class 8 depth.

This lesson follows the official chapter “We Distribute, Yet Things Multiply” in Ganita Prakash Part I. The words below are ExamMaster’s teaching, not a paste from the book. Use the NCERT chapter for the classroom sequence; use these notes to hold the idea without copying exercises or figures.

  • CBSE Class 8
  • Easy level
  • 8 concepts

1Some Properties of Multiplication

Multiplication distributes over addition: a(b + c) = ab + ac. 6 × (10 + 3) = 60 + 18 = 78, the same as 6 × 13. The property is a rewrite, not a new kind of 6.

6 × (10 + 3) is not 6×10 + 3.

Figure. Six equal groups of 6 is the 6-by-6 array: 36. The inverse reading is 36 ÷ 6 = 6 with remainder 0. If the groups are not equal, it is not multiplication yet.

How it works

  1. See a multiply outside a suma(b+c).
  2. Multiply onto each partab + ac.
  3. Check by computing both ways if the numbers are smallSame value.

6 × 13 two ways

Compute 6 × (10 + 3) by distributing and by 6 × 13.

  • Distribute60 + 18
  • Sum78
  • 6 × 1378

Pro tip. Both ways land on 78. Dropping the 6 on the 3 is the miss.

2Special Cases of the Distributive Property

Special cases: a(b − c) = ab − ac; (a+b)c = ac + bc; and a(b+c+d) hits each term. Distributing across a minus keeps the minus.

7 × (20 − 2) is 140 − 14, not 140 − 2.

Figure. Distributive split of one 4-by-8 array: width 8 is 5 + 3, so 4 × 8 = 4 × 5 + 4 × 3 = 20 + 12 = 32. The two blocks share height 4 and sit in the 5:3 width ratio of the split.

How it works

  1. See + or − insideThe inside job.
  2. Hit each term with the outside factorKeep the + or −.
  3. A three-term sum gets three hitsNo leftover term.

3Mind the Mistake, Mend the Mistake

A mistake is a dropped hit (the 6 never reached the 3) or a sign slip on a minus. Mend means compute the other way: expand versus the original product. If they disagree, the rewrite failed.

Erasing the check because 78 “looked busy” is not mending.

Figure. The mistake is calling unequal piles a product. 5, 4, and 8 are not equal groups. Mend it by making equal groups of 6; then 7 × 6 = 42 is multiplication. The three sage boxes are three of the seven groups.

How it works

  1. Compute both writingsDistributed, and the original.
  2. If they differ, name the dropped term or signThe miss.
  3. Rewrite that termThe mend.

4This Way or That Way, All Ways Lead to the Bay

This way or that way: group first, or distribute first — the value is the same when the rewrite is legal. All ways lead to the bay means same number, not “any doodle”.

A third writing that drops a term is not a way to the bay.

Figure. This way or that way: an 8-by-5 array and a 5-by-8 array have the same product 40. The two rectangles are drawn with equal area and with side ratios 5:8 and 8:5.

How it works

  1. Write two legal routes6×13, or 60+18.
  2. Evaluate bothThe bay is 78.
  3. Refuse a route that skips a hitNot a way.

5A definition is a test you can run

Distribute is a test: every term inside gets the outside factor. If a term is left untouched, you do not yet have the property.

A pretty brackets drawing is not the test.

Figure. Multiplication is a test you can run: are the groups equal? Yes, then 7 × 6 = 42. No, then you do not yet have a product. The leftover after equal sharing is a remainder, not an error.

How it works

  1. List the inside termsb and c.
  2. Write a hit on eachab, ac.
  3. If one term has no hit, the definition failedRun it.

6Name the given before the unknown

The given is the product-with-a-sum. The unknown is the expanded form or the value. Copy every inside term before you multiply.

Inventing a +0 to “make it easier” without writing it is a silent given.

Figure. Name the given first: 4 rows and 7 in each row. The product 28 is last. Inverse: 28 ÷ 4 = 7 with remainder 0.

How it works

  1. Copy a, b, and cThe given.
  2. Name expand or valueThe unknown.
  3. Then hit each termGiven first.

7One worked case is enough at this class

One distributed product is enough. A page of 6×(10+3) clones does not add a new hit-rule.

Ten twins of 78 are still one idea.

Figure. One worked array is enough at this class: 6 rows of 7 is 42. Inverse: 42 ÷ 6 = 7 with remainder 0. You do not need a second array to hold the idea.

How it works

  1. Expand one caseThe case.
  2. A second only as a check (maybe a minus)Optional.
  3. StopThis class.

8A miss: swapping the school name for the picture

A bay-or-harbour story is a setting. The maths is a(b+c). Sailing a story-boat and skipping a hit is the swap-miss.

A coastline photo is not 78.

Figure. The miss is swapping the chapter name for the picture. The title is not the array. The picture is 7 equal groups of 5, product 35. The 7-by-5 rectangle is drawn in the 5:7 side ratio.

How it works

  1. Keep the bay as a storyA setting.
  2. Write the productThe maths.
  3. Hit each termDo not swap.
6 × (10 + 3) distributed is
  1. 60 + 18
  2. 60 + 3
  3. 6 + 13

Hit both 10 and 3.

Notes

  • The official chapter title is “We Distribute, Yet Things Multiply”. Teach the school test for that title, not a contest shortcut.
  • If a step needs a later class, stop. The next official chapter will pick it up.

Formulas

  • a(b + c) = ab + ac
  • a(b − c) = ab − ac

Recap

Hold these pegs from the official chapter “We Distribute, Yet Things Multiply”. The wording is ExamMaster’s teaching, not a textbook recap.

Some Properties of Multiplication
Multiplication distributes over addition: a(b + c) = ab + ac.
Special Cases of the Distributive Property
Special cases: a(b − c) = ab − ac; (a+b)c = ac + bc; and a(b+c+d) hits each term.
Mind the Mistake, Mend the Mistake
A mistake is a dropped hit (the 6 never reached the 3) or a sign slip on a minus.
This Way or That Way, All Ways Lead to the Bay
This way or that way: group first, or distribute first — the value is the same when the rewrite is legal.
A definition is a test you can run
Distribute is a test: every term inside gets the outside factor.
Name the given before the unknown
The given is the product-with-a-sum. The unknown is the expanded form or the value. Copy every inside term before you multiply.

Practise We Distribute, Yet Things Multiply

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