Mathematics · 9th Grade · Exponent Laws and Polynomials · Weeks 19-27

Adding and Subtracting Polynomials

Performing addition and subtraction of polynomials by combining like terms.

• Common Core State Standards
• CCSS.Math.Content.HSA.APR.A.1
• CCSS.Math.Content.HSA.SSE.A.1

About This Topic

Scientific notation is a way to express very large or very small numbers using powers of ten. In 9th grade, students apply their knowledge of exponent laws to perform calculations with these numbers in the context of science. This topic is a key Common Core standard that connects 'Number and Quantity' to real-world applications in astronomy, biology, and chemistry.

Students learn that multiplying numbers in scientific notation involves multiplying the coefficients and adding the exponents. This makes it possible to compare the scale of a single cell to the scale of the entire Milky Way galaxy. This topic comes alive when students can engage in 'scale simulations' or collaborative investigations where they use scientific notation to solve 'Fermi problems', complex estimation challenges that require thinking in powers of ten.

Key Questions

1. Explain how combining like terms simplifies polynomial expressions.
2. Analyze common errors made when subtracting polynomials.
3. Construct a real-world problem that involves adding or subtracting polynomials.

Learning Objectives

• Calculate the sum of two polynomials by combining like terms.
• Determine the difference between two polynomials by distributing the negative sign and combining like terms.
• Identify common errors students make when subtracting polynomials, such as sign errors.
• Construct a word problem requiring the addition or subtraction of polynomials to model a real-world scenario.

Before You Start

Combining Like Terms

Why: Students must be able to identify and combine terms with the same variable and exponent to simplify expressions.

Distributive Property

Why: Understanding how to distribute a factor, especially a negative sign, is crucial for subtracting polynomials.

Key Vocabulary

Polynomial	An expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
Term	A single number or variable, or numbers and variables multiplied together. Terms are separated by addition or subtraction signs.
Like Terms	Terms that have the same variables raised to the same powers. For example, 3x^2 and -5x^2 are like terms.
Coefficient	The numerical factor of a term. For example, in the term 7x^3, the coefficient is 7.

Watch Out for These Misconceptions

Common Misconception

Students often think that a larger negative exponent means a larger number (e.g., thinking 10^-5 is bigger than 10^-2).

What to Teach Instead

Use the 'Scaling the Universe' activity. Peer discussion about 'place value' helps students realize that 10^-5 means the decimal is five places to the left, making it a much smaller fraction than 10^-2.

Common Misconception

Forgetting to adjust the coefficient after a calculation (e.g., leaving an answer as 15 x 10^4).

What to Teach Instead

Use 'Think-Pair-Share' to reinforce the 'standard form' rule. Students must check each other's work to ensure the coefficient is always between 1 and 10, adjusting the exponent accordingly (e.g., 1.5 x 10^5).

Active Learning Ideas

Simulation Game: Scaling the Universe

Groups are given objects ranging from an atom to the sun. They must research their sizes, write them in scientific notation, and then 'order' them on a giant classroom timeline of scale, explaining the massive jumps in powers of ten between them.

40 min·Small Groups

Inquiry Circle: Fermi Problems

Students work together to solve 'impossible' questions like 'How many grains of sand are on a beach?' They must use scientific notation to make reasonable estimates for each part of the problem and multiply them to find a final answer.

35 min·Small Groups

Think-Pair-Share: Significant Figures Check

Give students a calculation in scientific notation. One student performs the math, while the other checks that the final answer is written correctly (with only one digit before the decimal) and discusses how many 'significant figures' should be kept.

20 min·Pairs

Real-World Connections

• Urban planners use polynomials to model changes in population density or land use over time. Adding or subtracting these polynomials can help predict future city layouts or resource needs.
• In manufacturing, engineers might use polynomials to represent the cost of producing a certain number of items. Adding polynomial cost functions for different components allows for calculation of the total production cost.

Assessment Ideas

Exit Ticket

Provide students with two polynomials, for example, (3x^2 + 2x - 1) and (x^2 - 4x + 5). Ask them to find the sum and the difference of these two polynomials. Check their work for correct combination of like terms and accurate sign distribution during subtraction.

Quick Check

Present students with a subtraction problem like (5y - 2) - (3y + 7). Ask them to write down the first step they would take to solve this problem and explain why. This checks their understanding of distributing the negative sign.

Discussion Prompt

Pose the question: 'When subtracting polynomials, what is the most common mistake students make and why does this mistake occur?' Facilitate a brief class discussion where students share their observations and reasoning, reinforcing the importance of careful sign management.

Frequently Asked Questions

Why do scientists use scientific notation?

It makes it much easier to work with extreme numbers without having to count dozens of zeros. It also allows scientists to clearly show the 'precision' of their measurements using significant figures.

How can active learning help students understand scientific notation?

Active learning strategies like 'Fermi Problems' take scientific notation out of the textbook and turn it into a tool for exploration. When students have to estimate the number of atoms in a drop of water, they realize that standard numbers are useless for the task. This 'need' for a better system makes the rules of scientific notation feel like a powerful upgrade to their mathematical toolkit.

How do you add or subtract numbers in scientific notation?

To add or subtract, the exponents must be the same. You may need to rewrite one of the numbers so their powers of ten match before you can add or subtract the coefficients.

What is a 'power of ten'?

A power of ten is 10 raised to any exponent. Each increase in the exponent (from 10^2 to 10^3) represents a value that is ten times larger than the one before it.