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A short class with Sarah and her students. The other ways are under More.
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Today we're going to talk about how to identify linear relationships in word problems. Can anyone tell me what a linear function is?
Is it something that graphs as a straight line?
Exactly! A linear function graphs as a straight line and can be expressed in the form y = mx + c. Now, what about some real-life examples where we might see these linear functions?
Like how a taxi charges a flat fee and then per mile?
Great example! If the taxi charges $5 for the first mile and $2 for each additional mile, we can express this as a linear function. Can you represent this mathematically?
It would be C(x) = 2x + 5 where x is the number of miles.
Perfect! Always remember to identify the constant and variable in such scenarios. Now, let's summarize: identifying keywords in problems can give us clues about the variables involved.
Now that we've identified linear functions, let's practice translating word problems into equations. For example, 'A plant grows 3 cm per day. At day 0, it’s 5 cm tall.' How can we turn this into a linear function?
I think we would express height as H(x) = 3x + 5?
That's correct! Here, x represents days, and the plant starts at 5 cm, growing 3 cm each day. Any questions about how to identify the variables here?
What if there were more variables, like multiple plants growing at different rates?
Excellent question! In that case, you’d set equations for each plant separately but still follow the same principle. Now, let’s recap: identifying key phrases is crucial to understanding and translating the problems correctly.
Now we will solve some word problems! Here’s one: 'A mobile plan costs $10 per month plus $0.05 per text. Can we write a function for total cost?'
I think it’s C(x) = 0.05x + 10, where x is the number of texts.
Exactly! Now, if a user sends 200 texts in a month, what is the total cost?
So, we calculate it as C(200) = 0.05 * 200 + 10, which equals $20.
What if someone sends more texts? Does the cost change linearly?
Yes! That’s the beauty of linear functions—they model these kinds of relationships consistently. Let’s summarize: writing down equations helps us systematically find solutions.
Let's review everything we've learned! We talked about how to identify, translate, and solve word problems using linear functions. Can anyone provide an example or practice problem?
How about a situation like, 'A car rental company charges a base fee plus extra per mile.'
Great! We can represent that with C(x) = base fee + rate * x. What is the base fee and rate in your scenario?
Let’s say it’s $50 per day plus $0.25 per mile.
Exactly! So, C(x) = 0.25x + 50. Now, let’s recap what we’ve learned: identifying scenarios, creating equations, and solving them step by step!
Understand how to identify and express situations in the format of linear functions, such as y = mx + c.
Familiarize yourself with common occurrences where linear functions apply, including growth rates and costs tied to usage, as exemplified by plant growth rates and mobile plans.
Learn techniques for converting text-based descriptions into mathematical representations. This section will enhance your ability to think mathematically when faced with day-to-day challenges.
For cost and growth we need to know, / Just write the function and let it flow.
Imagine a gardener keeping track of his plants; each day, they grow taller. If he starts at 5 cm and each day adds more, he can easily predict their height using a simple equation.
F.L.O.W. – Find the linear function, Label the variables, Organize the equation, Write the solution.
C.A.R.E. – Costs And Rates Equate; helps us remember how to build equations from costs.
Term
What is a linear function?
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What does the intercept in a linear equation represent?
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How do you find the slope in a linear function?
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A function whose graph is a straight line, expressed in the form y = mx + c.
Symbols used to represent numbers in equations, typically x for independent and y for dependent.
Points where the line crosses the axes; y-intercept is where x=0, and x-intercept is where y=0.
A type of linear function that models the total cost related to a specific activity or usage.
The speed at which a quantity, such as a plant's height, increases over time.
This section focuses on constructing and solving word problems using linear functions.
The Word Problems section covers the identification of linear relationships in real-life scenarios and how to translate these relationships into mathematical expressions, leading to problem-solving through linear equations.
In this section, we delve into the concept of word problems that can be modeled with linear functions. A word problem often describes a real-life scenario where relationships between variables can be expressed linearly. By transforming these scenarios into equations, we can solve for unknowns systematically.
y = mx + c.