MathBored

Virginia SOL Mathematics Textbook

Grade 7 Workbook — Chapter 1: Powers of Ten and Scientific Notation

SOL 7.NS.1 · Companion to Textbook Chapter 1

Each page below is one Canva page. Headings are sized for direct paste: page title as H1, section labels as H2. Figures referenced by filename live in ../figures/. Item numbers match the textbook exactly, so one answer key serves both books.

PAGE 1 — Chapter opener

Chapter 1 · Powers of Ten and Scientific Notation

Standard 7.NS.1

In this chapter you will:

Words to know: exponent · base · power · negative exponent · fraction form · decimal form · scientific notation · coefficient · ascending order · descending order

Calculator note: this chapter is fully pencil-and-paper. No calculator needed, and none allowed on the state test for this standard.

PAGE 2 — Follow the pattern down

1.1 Powers of 10 with Negative Exponents

FIGURE: fig1-powers-of-ten-pattern.png (full width)

Complete the ladder. Each step down divides by 10.

Power Value
10310^{3}
10210^{2}
10110^{1}
10010^{0}
10−110^{-1}
10−210^{-2}
10−310^{-3}

Fill in the blanks.

Each time the exponent goes down by 1, the value is ____________ by 10.

Each time the exponent goes up by 1, the value is ____________ by 10.

A negative exponent does NOT make the number ____________. Every power of ten is greater than ______.

Guided practice.

  1. 103=10^3 = ______ 102=10^2 = ______ 101=10^1 = ______ 100=10^0 = ______

  2. 10−1=10^{-1} = ______ 10−2=10^{-2} = ______

  3. What operation moves you from 10−310^{-3} to 10−410^{-4}? _______________

  4. Write 10−410^{-4} as a decimal. ______

  5. Which power of ten equals 0.0010.001? ______

PAGE 3 — Powers of ten in place value

The Place-Value Chart

FIGURE: fig2-place-value-chart.png (full width)

Label each column with its power of ten.

Place thousands hundreds tens ones tenths hundredths thousandths
Power of ten

Independent practice.

  1. Evaluate as a decimal: a) 10−110^{-1} ______ b) 10−310^{-3} ______ c) 10−510^{-5} ______ d) 10010^{0} ______

  2. Write the power of ten: a) 0.010.01 ______ b) 0.00010.0001 ______ c) 0.0000010.000001 ______

  3. Complete: 10−2=10−1÷10^{-2} = 10^{-1} \div ______

  4. Order from least to greatest: 10−110^{-1}, 10−410^{-4}, 10010^{0}, 10−210^{-2}

  5. In the decimal form of 10−610^{-6}, how many zeros stand between the decimal point and the 11? ______

PAGE 4 — Apply and explain

Thinking About Negative Exponents

  1. Application. One sheet of printer paper is about 10−410^{-4} meters thick.

    Thickness as a decimal: ______________ meters

    Sheets needed to stack up to 1 meter: ______________

    How did the exponent give you that second answer without long division?

  2. Reasoning. Jonah says 10−310^{-3} must be a negative number because the exponent is negative.

    What is Jonah confusing?

    What does 10−310^{-3} actually equal? ______________

    Show the pattern that proves it:

    100=10^{0} = ______ → 10−1=10^{-1} = ______ → 10−2=10^{-2} = ______ → 10−3=10^{-3} = ______

PAGE 5 — Exit ticket 1.1

Exit Ticket · Lesson 1.1

Name: ________________________ Date: ____________

  1. Evaluate 10−210^{-2}. ______

  2. Which power of ten equals 0.000010.00001? ______

  3. What happens to the value each time the exponent decreases by 1?

  1. Explain why 100=110^0 = 1, using the pattern instead of a memorized rule.

PAGE 6 — Three forms of the same number

1.2 Fraction Form and Decimal Form

10−n=110n10^{-n} = \frac{1}{10^{n}}

FIGURE: fig3-three-forms-table.png (full width)

The exponent counts the zeros in the denominator AND the decimal places.

Guided practice.

  1. 10−1=1    ‾=10^{-1} = \dfrac{1}{\underline{\ \ \ \ }} = ______ (decimal)

  2. 10−4=1    ‾=10^{-4} = \dfrac{1}{\underline{\ \ \ \ }} = ______ (decimal)

  3. Write 1100\dfrac{1}{100} as a power of ten. ______

  4. Write 0.0010.001 in fraction form. ______

  5. Complete: 10−6=110  ‾10^{-6} = \dfrac{1}{10^{\underline{\ \ }}}

PAGE 7 — Build the table

All Three Forms

  1. Complete the table.
Exponent form Fraction form Decimal form
10−110^{-1}
10−210^{-2}
10−310^{-3}
10−510^{-5}

  1. Write each as a power of ten: a) 110\dfrac{1}{10} ______ b) 110 000\dfrac{1}{10\,000} ______ c) 11 000 000\dfrac{1}{1\,000\,000} ______

  2. Write each in fraction form: a) 0.10.1 ______ b) 0.00010.0001 ______ c) 0.000010.00001 ______

  3. True or false: 10−2=12010^{-2} = \dfrac{1}{20}. ______

    Reason: _______________________________________________

  4. Which is greater, 11000\dfrac{1}{1000} or 10−210^{-2}? ______

    How you decided: _______________________________________________

PAGE 8 — Apply and explain

Small Numbers in the Real World

  1. Application. A milligram is 11000\tfrac{1}{1000} of a gram.

    As a power of ten: ______ As a decimal: ______ gram

    Mass of a 250 mg tablet, in grams: ______

    Work space:

     

     

  2. Reasoning. Explain why the exponent in 10−n10^{-n} counts BOTH the zeros in the denominator of the fraction form AND the decimal places in the decimal form.

    Why must those two counts always agree?

PAGE 9 — Exit ticket 1.2

Exit Ticket · Lesson 1.2

Name: ________________________ Date: ____________

  1. Write 10−310^{-3} in fraction form ______ and decimal form ______

  2. Write 1100 000\dfrac{1}{100\,000} as a power of ten. ______

  3. Write 0.010.01 in fraction form. ______

  4. In your own words, how is 10−n10^{-n} related to 10n10^{n}?

PAGE 10 — What scientific notation is

1.3 Scientific Notation and Decimal Form

FIGURE: fig4-scientific-notation-anatomy.png (full width)

A positive number is in scientific notation when it looks like a×10na \times 10^{n} with 1≤a<101 \le a < 10.

Label the two parts.

In 6.3×10−56.3 \times 10^{-5}, the coefficient is ______ and the power of ten is ______.

Circle YES or NO: is it in scientific notation?

4.8×1034.8 \times 10^{3} YES / NO 0.7×1020.7 \times 10^{2} YES / NO
12.5×10412.5 \times 10^{4} YES / NO 9.99×10−69.99 \times 10^{-6} YES / NO

PAGE 11 — Moving the decimal point

Which Way, and How Far?

FIGURE: fig5-decimal-point-shift.png (full width)

Positive exponent → move RIGHT → the number grows.

Negative exponent → move LEFT → the number shrinks.

Guided practice.

  1. 2.6×103=2.6 \times 10^{3} = ______________

  2. 9.1×10−2=9.1 \times 10^{-2} = ______________

  3. 54 000=54\,000 = ______________ in scientific notation

  4. 0.0072=0.0072 = ______________ in scientific notation

  5. Is 12.5×10212.5 \times 10^{2} in scientific notation? ______

    Why or why not? _______________________________________________

    Written correctly: ______________

PAGE 12 — Convert both directions

Conversion Practice

  1. Write in decimal form.
a) 3.2×1053.2 \times 10^{5} b) 7.05×10−47.05 \times 10^{-4}
c) 1.9×10−11.9 \times 10^{-1} d) 8×1068 \times 10^{6}
  1. Write in scientific notation.
a) 6 300 0006\,300\,000 b) 0.000580.00058
c) 470470 d) 0.00000910.0000091
  1. Which are correctly written in scientific notation? Circle them, and fix the ones that are not.
Expression Correct? Fix
0.9×1040.9 \times 10^{4}
3.05×10−23.05 \times 10^{-2}
10×10510 \times 10^{5}
1×10−31 \times 10^{-3}

  1. 2.05×10−3=2.05 \times 10^{-3} = ______________

  2. 5.6×10n=0.0565.6 \times 10^{n} = 0.056. Find nn. ______

PAGE 13 — Apply and explain

Very Big and Very Small

  1. Application. A red blood cell is about 7×10−67 \times 10^{-6} meters wide. A sold-out arena holds about 6.5×1046.5 \times 10^{4} people.

    Red blood cell in decimal form: ______________ meters

    Arena capacity in decimal form: ______________ people

    Which number required moving the decimal point farther? ______________

  2. Reasoning. Why does moving the decimal point RIGHT always go with a positive exponent, and moving it LEFT always go with a negative one?

    Connect your answer to what multiplying and dividing by 10 actually do.

PAGE 14 — Exit ticket 1.3

Exit Ticket · Lesson 1.3

Name: ________________________ Date: ____________

  1. 4.3×10−3=4.3 \times 10^{-3} = ______________

  2. 91 000=91\,000 = ______________ in scientific notation

  3. 0.00026=0.00026 = ______________ in scientific notation

  4. How do you decide whether the exponent should be positive or negative?

PAGE 15 — The two-step comparison rule

1.4 Comparing Numbers in Scientific Notation

FIGURE: fig6-magnitude-number-line.png (full width)

Step 1: Compare the EXPONENTS. The greater exponent wins. Step 2: Only if the exponents are equal, compare the COEFFICIENTS.

Remember: the more negative the exponent, the smaller the number.

Guided practice. Fill in <<, >>, or ==.

  1. 7.4×105 □ 7.4×1037.4 \times 10^{5} \ \square \ 7.4 \times 10^{3}

  2. 2.8×10−4 □ 6.1×10−42.8 \times 10^{-4} \ \square \ 6.1 \times 10^{-4}

  3. 5×102 □ 5.0×1025 \times 10^{2} \ \square \ 5.0 \times 10^{2}

  4. Which is greater, 8.3×10−28.3 \times 10^{-2} or 8.3×10−58.3 \times 10^{-5}? ______________

  5. Ascending order: 4.2×1014.2 \times 10^{1}, 4.2×10−14.2 \times 10^{-1}, 4.2×1034.2 \times 10^{3}

PAGE 16 — Ordering practice

Ordering Up to Four Numbers

FIGURE: fig7-ordering-scientific-notation.png (full width)

  1. Fill in << or >>.
a) 3.6×104 □ 9.1×1033.6 \times 10^{4} \ \square \ 9.1 \times 10^{3} b) 1.5×10−2 □ 1.5×10−11.5 \times 10^{-2} \ \square \ 1.5 \times 10^{-1}
c) 6.7×10−5 □ 6.07×10−56.7 \times 10^{-5} \ \square \ 6.07 \times 10^{-5} d) 2.4×106 □ 9.9×1052.4 \times 10^{6} \ \square \ 9.9 \times 10^{5}

  1. Ascending order: 3.1×10−23.1 \times 10^{-2}, 3.1×10−43.1 \times 10^{-4}, 9.9×10−39.9 \times 10^{-3}, 1.2×10−21.2 \times 10^{-2}

  2. Descending order: 8.4×1038.4 \times 10^{3}, 8.04×1038.04 \times 10^{3}, 8.4×1028.4 \times 10^{2}, 8.44×1038.44 \times 10^{3}

  3. Which is least: 5×10−35 \times 10^{-3}, 4.9×10−24.9 \times 10^{-2}, 5.1×10−35.1 \times 10^{-3}, 9×10−49 \times 10^{-4}? ______________

  4. Which is greater, 0.000720.00072 or 7.9×10−47.9 \times 10^{-4}? ______________

    Conversion used: _______________________________________________

PAGE 17 — Apply and explain

Measuring Dust and Catching Errors

  1. Application. Four dust particles measured in a lab:
Particle Diameter
A 2.5×10−62.5 \times 10^{-6} m
B 8×10−78 \times 10^{-7} m
C 1.4×10−51.4 \times 10^{-5} m
D 9.2×10−69.2 \times 10^{-6} m 
Smallest to largest: _______________________________________________

Diameter of the largest particle in decimal form: ______________ m

  1. Reasoning. Priya says 9.6×10−3>1.1×10−29.6 \times 10^{-3} > 1.1 \times 10^{-2} because 9.6>1.19.6 > 1.1.

    Her error: _______________________________________________

    Correct comparison: _______________________________________________

    Decimal forms that support it: ______________ and ______________

PAGE 18 — Exit ticket 1.4

Exit Ticket · Lesson 1.4

Name: ________________________ Date: ____________

  1. Fill in: 6.2×10−3 □ 6.2×10−26.2 \times 10^{-3} \ \square \ 6.2 \times 10^{-2}

  2. Ascending order: 7.5×1027.5 \times 10^{2}, 7.5×10−27.5 \times 10^{-2}, 2.9×1022.9 \times 10^{2}

  3. Which is greater, 4.8×1054.8 \times 10^{5} or 5.1×1045.1 \times 10^{4}? ______________

  4. State the two-step rule for comparing numbers in scientific notation.

PAGE 19 — Chapter 1 review, part 1

Chapter 1 Review

Part A · Powers of 10 with negative exponents

  1. Evaluate: a) 10−210^{-2} ______ b) 10−510^{-5} ______ c) 10010^{0} ______

  2. Which power of ten equals 0.00010.0001? ______

  3. Continue the pattern: 10110^{1}, 10010^{0}, 10−110^{-1}, ______, ______

  4. What happens to the value each time the exponent decreases by 1? Why does that make 10−110^{-1} smaller than 10010^{0}?

Part B · Fraction form and decimal form

  1. 10−310^{-3} in fraction form ______ and decimal form ______

  2. 1100 000\dfrac{1}{100\,000} as a power of ten: ______

  3. Decimal form: a) 10−110^{-1} ______ b) 10−610^{-6} ______

  4. Which is greater, 10−410^{-4} or 11000\dfrac{1}{1000}? ______

    Explain: _______________________________________________

PAGE 20 — Chapter 1 review, part 2

Chapter 1 Review (continued)

Part C · Scientific notation and decimals

  1. Decimal form: a) 5.4×1045.4 \times 10^{4} ______________ b) 3.06×10−33.06 \times 10^{-3} ______________

  2. Scientific notation: a) 720 000720\,000 ______________ b) 0.000450.00045 ______________

  3. Why is 62×10362 \times 10^{3} not in scientific notation? _______________________________________________

    Written correctly: ______________

  4. 8.1×10n=0.0818.1 \times 10^{n} = 0.081. Find nn. ______

  5. A grain of fine sand is about 5×10−45 \times 10^{-4} m across. In decimal form: ______________ m

Part D · Comparing and ordering

  1. Fill in << or >>: a) 2.7×104 □ 2.7×1052.7 \times 10^{4} \ \square \ 2.7 \times 10^{5} b) 6.3×10−3 □ 6.03×10−36.3 \times 10^{-3} \ \square \ 6.03 \times 10^{-3}

  2. Ascending order: 4.5×10−24.5 \times 10^{-2}, 4.5×10−34.5 \times 10^{-3}, 1.9×10−21.9 \times 10^{-2}, 8.8×10−48.8 \times 10^{-4}

  3. Descending order: 3.3×1063.3 \times 10^{6}, 9.7×1059.7 \times 10^{5}, 3.03×1063.03 \times 10^{6}, 1.1×1061.1 \times 10^{6}

PAGE 21 — Chapter 1 review, part 3

Chapter 1 Review (continued)

Part E · Application and reasoning

  1. Application. Four wavelengths of visible light: 4.7×10−74.7 \times 10^{-7} m, 6.2×10−76.2 \times 10^{-7} m, 5.1×10−75.1 \times 10^{-7} m, 4.07×10−74.07 \times 10^{-7} m

    Ascending order: _______________________________________________

    Which step of the comparison rule did all the work here? _______________________________________________

  2. Application. Write 0.0000940.000094 in scientific notation: ______________

    Is it greater or less than 9.4×10−49.4 \times 10^{-4}? ______________

    By how many powers of ten do they differ? ______

  3. Reasoning. Why is 10−610^{-6} less than 10−210^{-2}, even though 6>26 > 2? Use repeated division by 10.

  4. Reasoning. Why must the coefficient be at least 1 and less than 10? Name one specific problem that would show up without that rule.

Canva production notes