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CBSE Class 7 · Mathematics

Another Peek Beyond the Point

Official NCERT chapter from Ganita Prakash Part II (book code gegp2). ExamMaster notes are original teaching at CBSE Class 7 depth.

This lesson follows the official chapter “Another Peek Beyond the Point” in Ganita Prakash Part II. 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 7
  • Easy level
  • 8 concepts

1A Quick Recap of Decimals

A recap: the point is a fence; tenths then hundredths; compare from the left; add by lining up points. This peek starts from that and then multiplies and divides.

Skipping the fence and treating 3.2 as 32 is a recap you dropped.

Figure. 1.35 is 1 one, 3 tenths, 5 hundredths. The point starts the tenths house. The next digit is already a new house, not a spare ones digit.

How it works

  1. Place a decimal by housesThe recap.
  2. Compare or add one pair if you need to warm upStill the recap.
  3. Then get ready to × and ÷This peek’s new job.

2Decimal Multiplication

To multiply decimals, multiply as whole numbers, then count the total decimal houses in the factors and place the point. 0.2 × 0.3 = 0.06 (two houses).

0.2 × 0.3 is not 0.6 — you owed two houses.

Figure. 1.5 × 3 is three equal groups of 1.5. The product 4.5 is the joined span, not a new kind of number. 1.5 + 1.5 + 1.5 = 4.5.

How it works

  1. Multiply ignoring the points2 × 3 = 6.
  2. Count houses after the points1 + 1 = 2.
  3. Place the point so the product has that many houses0.06.

Houses in a product

Compute 1.2 × 0.4.

  • As wholes12 × 4 = 48
  • Houses1 + 1 = 2
  • Product0.48

Pro tip. Two houses in the factors, two in the product.

3Decimal Division

To divide decimals, you may make the divisor a whole by moving both points the same number of houses, then divide. 3.6 ÷ 0.12 → 360 ÷ 12 = 30.

Moving only one point is a house-slide.

Figure. 4.8 ÷ 3 asks for three equal shares of 4.8. Each share is 1.6. The cut is equal parts first; the 1.6 is what each part holds.

How it works

  1. Move both points equally until the divisor is wholeFair move.
  2. Divide the new whole numbersThe quotient.
  3. Check by multiplying back if you canQuotient × divisor ≈ dividend.

4Look Before You Leap!

Look before you leap: estimate first. 1.9 × 3.1 is near 2 × 3 = 6, so 58.9 is a leap that failed. A wild point-placement should lose to the estimate.

Trusting a calculator-shaped guess without an estimate is the leap.

Figure. 2.9 × 4 is near 3 × 4 = 12, not 29 × 4 = 116. Look at the point before you leap. The product is about 12 (exactly 11.6).

How it works

  1. Round each factor to a nearby one-digit or wholeThe look.
  2. Multiply thoseA landing zone.
  3. Then compute; if you are far from the zone, mend the pointDo not leap.

5A definition is a test you can run

Decimal × here is a test: house-count after a whole-number multiply. If you cannot say how many houses you owe, you do not yet have the definition.

A point dropped “in the middle” is not a test.

Figure. The test for 4.5: the digit just after the point must sit in the tenths house. If it does, the writing is a decimal, not a nickname.

How it works

  1. Multiply the digitsThe size.
  2. Count houses owedThe test.
  3. Place the pointThe definition ran.

6Name the given before the unknown

The given are the two decimals and × or ÷. The unknown is the product or quotient. Count houses (or move both points) only after you copy the givens.

Inventing extra houses “to be safe” is a silent given.

Figure. Name the given first: 1.7 and 4. The product is the unknown. Do not chase the product before both givens have a house.

How it works

  1. Copy both decimalsThe given.
  2. Name × or ÷The unknown’s job.
  3. Then count or shift housesGiven first.

7One worked case is enough at this class

One product and, if needed, one quotient is enough. A page of 0.2×0.3 clones does not add a new house-count.

Ten twins of 0.06 are still one idea.

Figure. One worked product is enough: four groups of 2.5 make 10. Hold the method; do not pile extra cases that repeat the same join.

How it works

  1. Do one × (and one ÷ if that is the ask)The case.
  2. A second only as a checkOptional.
  3. StopThis class.

8A miss: swapping the school name for the picture

A shop price in rupees-and-paise is a setting. The maths is 1.25 × 4, not the photo of the till. Swapping the till for the house-count is the miss.

A price tag does not place the point for you.

Figure. The school word “decimal” is not the picture. 8.6 is 8 ones and 6 tenths. Swap the name for the houses and the point has a job.

How it works

  1. Keep the shop as a storyA setting.
  2. Write the decimal jobThe maths.
  3. Run house-count or fair shiftDo not swap.
0.2 × 0.3 equals
  1. 0.06
  2. 0.6
  3. 6

Two decimal houses in the factors.

Notes

  • The official chapter title is “Another Peek Beyond the Point”. 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

  • decimal houses in a product = houses in the factors, added
  • shift both points equally to make the divisor whole

Recap

Hold these pegs from the official chapter “Another Peek Beyond the Point”. The wording is ExamMaster’s teaching, not a textbook recap.

A Quick Recap of Decimals
A recap: the point is a fence; tenths then hundredths; compare from the left; add by lining up points.
Decimal Multiplication
To multiply decimals, multiply as whole numbers, then count the total decimal houses in the factors and place the point.
Decimal Division
To divide decimals, you may make the divisor a whole by moving both points the same number of houses, then divide.
Look Before You Leap!
Look before you leap: estimate first. 1.9 × 3.1 is near 2 × 3 = 6, so 58.9 is a leap that failed. A wild point-placement should lose to the estimate.
A definition is a test you can run
Decimal × here is a test: house-count after a whole-number multiply.
Name the given before the unknown
The given are the two decimals and × or ÷.

Practise Another Peek Beyond the Point

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