Thermodynamics and Kinetics · Weeks 19-27

Ozone Depletion and CFCs

Students will investigate the chemistry of the ozone layer, its depletion by CFCs, and international efforts for recovery.

About This Topic

Optimization problems use the features of quadratic functions to find the 'best' possible outcome in a given scenario. In 9th grade, this usually involves finding the maximum area of a fenced region or the minimum cost of a production run. This is a high-level Common Core standard that demonstrates the practical utility of the vertex in business and engineering.

Students learn that the vertex of a quadratic model represents the optimal point, either the peak of a profit curve or the bottom of a cost curve. This topic comes alive when students can engage in 'design challenges' where they must use a fixed amount of 'fencing' (string) to create the largest possible area. Collaborative investigations help students discover that for a rectangular area, the 'optimal' shape is always a square.

Active Learning Ideas

Inquiry Circle: The Fencing Challenge

Groups are given a fixed length of string (the 'fence'). They must create different rectangles, record the width and area of each in a table, and then find the quadratic equation that models the relationship. They must identify the width that produces the maximum area.

⏱ 45 min·Small Groups

Generate mission

Think-Pair-Share: Max or Min?

Give students two scenarios: 'Maximizing the height of a rocket' and 'Minimizing the cost of a factory.' Pairs must discuss whether the vertex in each quadratic model represents a 'high point' or a 'low point' and how the 'a' value of the equation tells them which one it is.

⏱ 20 min·Pairs

Generate mission

Simulation Game: The Price Optimizer

Students act as business owners. They are given a model showing how raising prices reduces the number of customers. They must write a quadratic revenue function (Price x Customers) and find the 'perfect' price that maximizes their total income.

⏱ 40 min·Small Groups

Generate mission

Watch Out for These Misconceptions

Common Misconception

Students often think that 'more' of something (like a longer width) always leads to a better result.

What to Teach Instead

Use 'The Fencing Challenge.' Peer discussion helps students see that as the width gets too long, the 'length' must shrink to stay within the perimeter, eventually making the area smaller. This 'trade-off' is why an optimal middle point exists.

Common Misconception

Confusing the 'optimal input' (x-value) with the 'optimal result' (y-value).

What to Teach Instead

Use 'The Price Optimizer' activity. Collaborative analysis helps students clarify that the x-value is the 'price they should set,' while the y-value is the 'maximum profit they will make.' Keeping these separate is key to answering optimization questions correctly.

Frequently Asked Questions

What does 'optimization' mean in math?

Optimization is the process of finding the best possible solution to a problem given certain constraints. In 9th grade, this usually means finding the maximum or minimum value of a quadratic function by locating its vertex.

How can active learning help students understand optimization?

Active learning strategies like 'The Fencing Challenge' turn an abstract maximum into a physical discovery. When students see that their area starts small, grows, and then shrinks again as they change the dimensions, the 'parabola' shape of the data makes perfect sense. This hands-on experience helps them understand why the vertex is the 'answer' to the problem, rather than just a point they were told to find.

Why is the square the 'optimal' rectangle for area?

Mathematically, the area of a rectangle with a fixed perimeter is a quadratic function. The vertex of that function always occurs when the length and width are equal, which is the definition of a square.

How do I know if a quadratic has a maximum or a minimum?

Look at the 'a' value (the coefficient of x^2). If 'a' is negative, the parabola opens down and has a maximum. If 'a' is positive, the parabola opens up and has a minimum.

Browse curriculum by country

• Americas
• US
• CA
• MX
• CL
• CO
• BR

• Europe
• UK
• IE
• FR
• DE
• ES
• IT
• NL
• PT
• SE

• Asia & Pacific
• IN
• SG
• AU