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Teacher Guide: Converting from Scientific Notation

Learn to turn a number like 3.2 × 10⁻⁴ back into an ordinary decimal number by moving the decimal point and filling in zeros.

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All practice problems on paper, with a separate answer key.

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Class quiz

10 questions on Scientific Notation. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Convert numbers from scientific notation with positive exponents to ordinary numbers
  • Convert numbers from scientific notation with negative exponents to decimals, using placeholder zeros
  • Explain why 100=110^0 = 1 leaves the coefficient unchanged
  • Read calculator and spreadsheet E notation and real data given in scientific notation
Prerequisites
  • • Scientific Notation: the form a×10na \times 10^n with 1≤a<101 \le a < 10
  • • Writing Numbers in Scientific Notation (ordinary number to scientific notation)
  • • Multiplying and dividing decimals by 1010, 100100 and 10001000
Discussion Starters
  • 1. Why does a negative exponent make a number smaller but not negative?
  • 2. Is it easier to compare 0.000000120.00000012 and 0.00000120.0000012 or 1.2×10−71.2 \times 10^{-7} and 1.2×10−61.2 \times 10^{-6}? Why?
  • 3. Your calculator shows 6.02E23. How would you explain that number to someone who has never seen E notation?
  • 4. When would a scientist prefer the ordinary number, and when scientific notation?
Common Misconceptions

The exponent is the number of zeros to write

Remediation: Contrast 4×106=4,000,0004 \times 10^6 = 4{,}000{,}000 (6 zeros) with 4.5×106=4,500,0004.5 \times 10^6 = 4{,}500{,}000 (5 zeros). Have students circle each hop of the point instead of counting zeros.

A negative exponent means a negative number

Remediation: Build the pattern 102=10010^2 = 100, 101=1010^1 = 10, 100=110^0 = 1, 10−1=0.110^{-1} = 0.1, 10−2=0.0110^{-2} = 0.01: each step divides by 1010, and the values never drop below zero.

For 10−410^{-4} there are 4 zeros after the decimal point

Remediation: Move the point one hop at a time on a place value chart: 3.2→0.323.2 \to 0.32 adds no zero after the point. Then generalize: ∣n∣−1|n| - 1 zeros for a one-digit whole part.

Differentiation Ideas

For Struggling Students:

  • • Start with whole-number coefficients (5×1035 \times 10^3, 9×10−29 \times 10^{-2}) before decimals
  • • Use a place value chart and physically move a decimal point card one place per hop
  • • Write the direction rule on a card: positive = right = bigger, negative = left = smaller

For On-Level Students:

  • • Convert mixed sets with positive, negative and zero exponents
  • • Translate calculator E displays into ordinary numbers
  • • Convert real measurements and say them aloud in words

For Advanced Students:

  • • Convert non-normalized forms like 45×10−345 \times 10^{-3} and 0.6×1040.6 \times 10^{4}
  • • Convert and then change units (meters to millimeters, seconds to hours)
  • • Investigate how many digits a number has from its exponent alone
Standards Alignment
  • 8.EE.A.3 (CCSS.MATH.CONTENT.8.EE.A.3)

    Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities

  • 8.EE.A.4 (CCSS.MATH.CONTENT.8.EE.A.4)

    Interpret scientific notation that has been generated by technology

Lesson Resources
  • visualDecimal Point Mover

    Students hop the decimal point left or right, watch placeholder zeros appear and check the result against the exponent

  • activityCalculator Display Detective

    Students compute very large and very small products on a calculator, copy the E display and translate it into an ordinary number

  • worksheetScale of the Universe

    Convert ten real measurements (atom, cell, mountain, Sun, galaxy) to ordinary numbers and order them by size

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

In the previous lesson you wrote numbers in scientific notation. Now we go the other way: from a×10na \times 10^n back to an ordinary number written out in full, its decimal form. (Careful: "standard form" is another name for scientific notation in the UK, so in this lesson we say "ordinary number" or "decimal form".)
The exponent nn tells you how many places to move the decimal point of the coefficient aa:
  • n>0n > 0: move the point nn places to the right (the number gets bigger)
  • n<0n < 0: move the point ∣n∣|n| places to the left (the number gets smaller, but stays positive)
  • n=0n = 0: do not move it, because 100=110^0 = 1
When you run out of digits, fill the empty places with placeholder zeros:
4.5×106=4,500,0003.2×10−4=0.000324.5 \times 10^6 = 4{,}500{,}000 \qquad 3.2 \times 10^{-4} = 0.00032

Worked Examples

Write 4.5×1064.5 \times 10^6 as an ordinary number.
1

Read the exponent

n=6n = 6 is positive, so the point moves 66 places to the right → 6 places right
2

Use the digits you have

4.54.5 has one digit after the point, so the first hop uses the 55: 4545 → 1 place used, 5 left
3

Fill in placeholder zeros

The other 55 places become zeros: 45 00000‾45\,\underline{00000} → 4,500,0004{,}500{,}000
4

Check the size

10610^6 is one million, and 4.54.5 million is 4,500,0004{,}500{,}000 → Makes sense

Common Mistakes

Moving the point to the right for a negative exponent: 3.2×10−4=32,0003.2 \times 10^{-4} = 32{,}000

Why it's wrong: A negative exponent means dividing by 1010 again and again, so the number gets smaller, not bigger.

Correct: Negative exponent, move left: 3.2×10−4=0.000323.2 \times 10^{-4} = 0.00032

Adding nn zeros to the end: 4.5×106=45,000,0004.5 \times 10^6 = 45{,}000{,}000

Why it's wrong: The exponent counts places the point moves, not zeros to write. The digit 55 already uses one of the 66 places.

Correct: Only 6−1=56 - 1 = 5 zeros are needed: 4.5×106=4,500,0004.5 \times 10^6 = 4{,}500{,}000

Writing nn zeros after the point: 3.2×10−4=0.0000323.2 \times 10^{-4} = 0.000032

Why it's wrong: The first hop to the left puts the point in front of the 33 without adding a zero. That is one zero too many.

Correct: With a one-digit whole part, there are ∣n∣−1|n| - 1 zeros after the point: 0.000320.00032 (3 zeros)

Thinking a negative exponent makes the number negative: 5×10−3=−50005 \times 10^{-3} = -5000

Why it's wrong: The sign of the exponent controls the size. Only a minus sign in front of the coefficient makes the number negative.

Correct: 5×10−3=0.0055 \times 10^{-3} = 0.005 (positive), while −5×103=−5000-5 \times 10^{3} = -5000

Why It Matters

Scientific notation is compact, but people usually need the ordinary number to picture a size or to use it:
  • Space: the Sun is 1.496×1081.496 \times 10^8 km away, which is 149,600,000149{,}600{,}000 km
  • Biology: a red blood cell is 7×10−67 \times 10^{-6} m wide, which is 0.0000070.000007 m
  • Calculators and spreadsheets show very big or very small results like 6.02E23 or 2.5E-7. The E means "times ten to the power", so you must be able to read it back
  • Science reports and news mix both forms, and you will often need to compare a value in scientific notation with a value written out in full
Once you can convert in both directions, no number is too big or too small to understand.

Real World Applications

Astronomy

Astronomers write distances in scientific notation, but maps, books and news reports often give them in full.

Example:

Light travels about 3×1053 \times 10^5 km every second, which is 300,000300{,}000 km per second.

1Try It Yourself

The average distance from Earth to the Moon is 3.844×1053.844 \times 10^5 km.

Write this distance as an ordinary number of kilometers.

Step 1: Write the mathematical expression

Type the distance in km as an ordinary number:

Biology and Medicine

Cells, bacteria and viruses are measured in tiny parts of a meter, so lab results often use negative exponents.

Example:

A bacterium about 2×10−62 \times 10^{-6} m long is 0.0000020.000002 m, two millionths of a meter.

2Try It Yourself

A flu virus has a diameter of about 1.2×10−71.2 \times 10^{-7} m.

Write the diameter as an ordinary decimal number of meters.

Step 1: Write the mathematical expression

Type the diameter in meters:

Calculators and Spreadsheets

When a result has too many digits for the screen, calculators and spreadsheets switch to E notation: 4.54E9 means $4.54 \times 10^9$.

Example:

A spreadsheet cell showing 2.5E-4 holds the number 0.000250.00025.

3Try It Yourself

A calculator shows 4.54E9 for the age of the Earth in years.

How many years is that, written as an ordinary number?

Step 1: Write the mathematical expression

Type the number of years:

Key Takeaways

  • 1To convert a×10na \times 10^n to an ordinary number, move the decimal point of aa by ∣n∣|n| places
  • 2Positive exponent: move the point to the right, the number is 1010 or more
  • 3Negative exponent: move the point to the left, the number is between 00 and 11 (still positive)
  • 4Exponent 00: the number stays the same, because 100=110^0 = 1
  • 5Fill every empty place with a placeholder zero: 3.2×10−4=0.000323.2 \times 10^{-4} = 0.00032
  • 6Calculator E notation is scientific notation: 6.02E23 =6.02×1023= 6.02 \times 10^{23}

Frequently Asked Questions

How many zeros do I write for a positive exponent?

Subtract the number of digits after the point in the coefficient from the exponent. For 1.496×1081.496 \times 10^8: 8−3=58 - 3 = 5 zeros, so 149,600,000149{,}600{,}000.

How many zeros come after the point for a negative exponent?

When the coefficient is between 11 and 1010, there are ∣n∣−1|n| - 1 zeros between the point and the first digit. For 7×10−67 \times 10^{-6}: 55 zeros, so 0.0000070.000007.

Can a negative exponent ever give a negative number?

No. 10−n10^{-n} is always positive. A number is negative only if the coefficient has a minus sign, like −2.5×10−3=−0.0025-2.5 \times 10^{-3} = -0.0025.

What does the E on my calculator mean?

E stands for "exponent" and means ×10□\times 10^{\square}. So 3.1E-5 is 3.1×10−5=0.0000313.1 \times 10^{-5} = 0.000031. It has nothing to do with the number e≈2.718e \approx 2.718.

Glossary

Scientific notation
A number written as a×10na \times 10^n with 1≤a<101 \le a < 10 and nn a whole number
Ordinary (decimal) number
A number written out with all its digits, like 4,500,0004{,}500{,}000 or 0.000320.00032 (not "standard form", which in the UK means scientific notation)
Coefficient
The number aa in a×10na \times 10^n, between 11 and 1010
Exponent
The power nn in 10n10^n: how many places the decimal point moves, and in which direction
Placeholder zero
A zero written only to fill an empty place value, like the zeros in 0.000320.00032 or 4,500,0004{,}500{,}000
E notation
The calculator and spreadsheet way to show scientific notation: 2.5E-7 means 2.5×10−72.5 \times 10^{-7}

Formula Card

Positive exponent

a×10n, n>0a \times 10^{n},\ n > 0: move the point nn places right

Example: $2.75 \times 10^5 = 275{,}000$

Negative exponent

a×10−na \times 10^{-n}: move the point nn places left

Example: $8.4 \times 10^{-5} = 0.000084$ ($n - 1 = 4$ zeros after the point)

Zero exponent

a×100=aa \times 10^{0} = a

Because $10^0 = 1$, the point does not move

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