Powers of Ten

Overview

Today students will add and subtract numbers written in scientific notation, including cases where both scientific notation and decimals appear in the same problem. Students will use place value and exponent rules to justify their answers and interpret results in a reasonable way.

Learning intentions

• Students will be able to convert between decimal form and scientific notation.
• Students will be able to add and subtract numbers expressed in scientific notation.
• Students will be able to use properties of exponents to create equivalent expressions when needed.
• Students will be able to check whether a final answer makes sense by estimating magnitude.

Success criteria

• I can rewrite each number so the powers of 10 match before adding or subtracting.
• I can perform the arithmetic on the coefficients and keep the correct power of 10.
• I can convert the result back to scientific notation if it is not already in that form.
• I can estimate the size of the answer to verify it is reasonable.

Curriculum links

• Expressions and Equations: Know and apply properties of integer exponents to generate equivalent numerical expressions.
• Expressions and Equations: Perform operations with numbers expressed in scientific notation, including mixed decimal and scientific notation.
• Expressions and Equations: Use scientific notation to estimate very large or very small quantities and compare sizes.

Lesson structure (40 minutes)

1. 0–5 min · Warm-up: Quick magnitude check. Teacher writes two scientific-notation numbers on the board and asks which is larger and by about how many times; students respond with reasoning using powers of 10. Student task: Think-pair-share: “Which exponent is bigger and what does that mean for size?”

2. 5–12 min · Mini-direct teach: How to add/subtract scientific notation. Teacher models two examples:

• Example A: ((3.2 \times 10^5) + (7.4 \times 10^5))
• Example B: ((9.0 \times 10^3) - (2.5 \times 10^2)) with a conversion step so exponents match. Student task: Copy the key rule: “Match exponents first, then add/subtract coefficients.”

3. 12–20 min · Guided practice: Mixed formats. Teacher shows a problem that mixes decimal and scientific notation, such as ((1.6 \times 10^4) + 2500), and demonstrates converting 2500 to scientific notation (or rewriting (2500) as a multiple of (10^4)). Student task: Work individually for 3 minutes, then check with a partner for 5 minutes on the conversion and the final scientific-notation answer.

4. 20–30 min · Independent practice: 4 problems with teacher circulation. Teacher gives four problems (in increasing difficulty) and a checklist:

• Problem 1: ((4.5 \times 10^6) + (3.1 \times 10^6))
• Problem 2: ((8.0 \times 10^2) - (6.5 \times 10^1))
• Problem 3: ((2.7 \times 10^5) + 0.0032)
• Problem 4: ((9.6 \times 10^{-3}) - (4.0 \times 10^{-4})) Student task: Solve all four, convert to scientific notation, and underline where exponents were matched.

5. 30–36 min · Whole-class debrief: Reasonableness checks. Teacher selects two student methods (one correct, one with a common mistake such as not matching exponents or forgetting to adjust the coefficient after addition/subtraction) and asks the class to critique. Student task: Identify the error and propose the correction; justify using estimation based on powers of 10.

6. 36–40 min · Exit ticket (quick assessment). Teacher collects answers to one final mixed-format problem and one estimation question:

• Compute: ((3.4 \times 10^5) - 12,000) and express in scientific notation.
• Estimate: Is the answer closer to (10^5) or (10^6)? Student task: Provide the scientific-notation result and a one-sentence estimate justification.

Resources

• Board/markers or slides
• Student whiteboards or scrap paper
• Handout with guided steps (“Match exponents, combine coefficients, re-normalize”)
• Calculator only for checking arithmetic (not for doing the exponent matching)
• Example cards showing conversions between decimal and scientific notation (teacher-made or printed)

Assessment

• Observation during guided and independent practice: check that students match exponents before adding/subtracting.
• Quick checks during the debrief: listen for correct reasoning about magnitude and exponent meaning.
• Exit ticket: verifies correct operation in scientific notation and correct use of estimation for reasonableness.

Differentiation

• Support: Provide a “conversion helper” line: move the decimal the number of places indicated by the exponent, and write the matching scientific notation before combining.
• Support (sentence starters):
• “First I rewrote the numbers so the exponents match because…”
• “After I subtracted the coefficients, I adjusted the scientific notation since…”
• Extension: Give an extra challenge: ((5.2 \times 10^{-6}) + (7.5 \times 10^{-7})) and require a reasonableness estimate comparing to (10^{-6}).
• EAL/SEN: Use color-coding on the handout for coefficient and power-of-10 parts; allow step-by-step work space and a word bank for “exponent,” “coefficient,” “match,” and “estimate.”