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Application of all of the above, and can be supplemented.

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Conceptual lenses create connections between factual informations and conceptual knowledge. They can be referred to as "key concepts" and encourage a focus on transferrable ideas.
Generalizations are in each strand, and are enduring, essential understandings.

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Skills fall under general processes, and include problem-solving, making connections, proofs, and reasoning.

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There are three types of guiding questions that should allow students to make connections and draw out conceptual understanding:
- factual
- conceptual
- provocative/debatable

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Begin with:
1. Identify results
2. Determine acceptable evidence
3. Plan learning experiences/instruction
Remember that conceptual understandings must be at the forefront of planning.

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A tool for planning concept-based units, with knowledge strands and process strands.

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Graphic organizer
Unit webbing tool
Time of reflection
Hint jar

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Guiding students to discover generalizations and principles through inquiry as opposed to telling students the generalizations and principles and then having them study and focus on specific examples. It involves having students write in math.

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Pythagorean Theorem Generalization: Level 1

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Pythagorean Theorem Generalization: Level 1

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Students will understand that...
In a right angle, the lengths of the two sides affect the length of the third.

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Generalizations that have risen to the level of theorem or law, based on facts and truths, no longer qualified.

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Pythagorean Theorem Generalization: Level 3

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Pythagorean Theorem Generalization: Level 3

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What is the significance? Why is this important?
Knowing the area of the square drawn from the hypotenuse or of one side of a right-angled triangle can help solve/calculate for an unknown side.

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Pythagorean Theorem Generalization: Level 2

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Pythagorean Theorem Generalization: Level 2

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Why and how?
The area of the squares drawn from the two sides of a right-angled triangle represent the area of the square drawn from the hypotenuse.

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Two types of generalizations are overarching ones and topical ones, meaning that there are generalizations that are topic-specific and some that are not.
Ex: logarithms as a process generalization including problems-solving and making connections.

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Intro to Logarithms (article) | Logarithms | Khan Academy

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Learn what logarithms are and how to evaluate them.

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Think of the following when considering generalizations: two or more statements, use words like "may" and "can", provides goals and student understanding, endure beyond specific topics.
Think of the following when considering principles: factual truths, theorems, hold the generalizations together, do not contain words like "may" or "can".

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Strategies for solving problems are composed of skills as outlined in this link.

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Problem Solving Strategies

Look for a pattern

Find the sum of the first 100 even positive numbers.

The sum of the first 1 even positive numbers is 2 or 1( 1+1) = 1(2).
The sum of the first 2 even positive numbers is 2 + 4 = 6 or 2( 2+1) = 2(3).
The sum of the first 3 even positive numbers is 2 + 4 + 6 = 12 or 3( 3+1) = 3(4).
The sum of the first 4 even positive numbers is 2 + 4 + 6 + 8 = 20 or 4( 4+1) = 4(5).

Look for a pattern:
The sum of the first 100 even positive numbers is 2 + 4 + 6 + ... = ? or 100( 100+1) = 100(101) or 10,100.

Make an organized list

Find the median of the following test scores: 73, 65, 82, 78, and 93.

Make a list from smallest to largest:
65
73
78 Since 78 is the middle number, the median is 78.
82
93

Guess and check

Which of the numbers 4, 5, or 6 is a solution to (n + 3)(n - 2) = 36?

Substitute each number for "n" in the equation. Six is the solution since (6 + 3)(6 - 2) = 36.

Make a table

How many diagonals does a 13-gon have?

Look for a pattern. Hint: If n is the number of sides, then
n(n-3)/2 is the number of diagonals. Explain in words why this works. A 13-gon would have 13(13-3)/2 = 65 diagonals.

Work backwards

The pig received $2000.
1/3 of ? = $2000
? = $6000
6/7 of ? = $6000
? = $7000
1/2 of ? = $7000
? = $14,000

Use logical reasoning

At the Keep in Shape Club, 35 people swim, 24 play tennis, and 27 jog. Of these people, 12 swim and play tennis, 19 play tennis and jog, and 13 jog and swim. Nine people do all three activities. How many members are there altogether?

Hint: Draw a Venn Diagram with 3 intersecting circles.

Draw a diagram

Fortune Problem: a man died and left the following instructions for his fortune, half to his wife; 1/7 of what was left went to his son; 2/3 of what was left went to his butler; the man's pet pig got the remaining $2000. How much money did the man leave behind altogether?

Solve a simpler problem

Try a simpler problem. How much would you pay if a half pound of sliced roast beef costs $2 and the person slices 3 pounds? If a half pound costs $2, then one pound would cost 2 x $2 or $4. Multiply by the number of pounds needed to get the total: 3 x $4 = 12.
Now try the original problem: If a half pound costs $2.49, then one pound would cost 2 x $2.49 or $4.98. Multiply by the number of pounds needed to get the total: .53 x $4.98 = $2.6394 or $2.64.

Read the problem carefully
Know the meaning of all words and symbols in the problem.

List the ten smallest positive composite numbers.

Sort out information that is not needed.

Last year the Williams family joined a reading club. Mrs. Williams read 20 books. Their son Jed read 12 books. Their daughter Josie read 14 books and their daughter Julie read 7 books. How many books did the children of Mr. and Mrs. Williams read altogether?

Determine if there is enough information to solve the problem.

Create problem solving journals
Students record written responses to open-ended items such as those tested on FCAT in mathematics. Student identifies problem solving strategies.

Copyright Statement for this Assessment and Evaluation Services Publication

Authorization for reproduction of this document is hereby granted to persons acting in an official capacity within the State System of Public Education as defined in Section 228.041(1), Florida Statutes. The copyright notice at the bottom of this page must be included in all copies.

The Administrator
Assessment and Evaluation Services
Florida Department of Education
Turlington Building, Room 414
325 West Gaines Street
Tallahassee, Florida 32399-0400

Copyright © 2000
State of Florida
Department of State

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Problem Solving Strategies Look for a pattern Find the sum of the first 100 even positive numbers. The sum of the first 1 even positive numbers is 2 or 1( 1+1) = 1(2). The sum of the first 2 even positive numbers is 2 + 4 = 6 or 2( 2+1) = 2(3).

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Plotting a graph is an example of a skill. 

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You can often learn a lot about a function by sketching its graph. The problems in this feature lend themselves to graph sketching, or exploration using software such as GeoGebra or Desmos. We hope that by working through them, you will have a clearer understanding of functions and their graphs.

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You can often learn a lot about a function by sketching its graph. The problems in this feature lend themselves to graph sketching, or exploration using software such as GeoGebra or Desmos. We hope that by working through them, you will have a clearer understanding of functions and their graphs.

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It looks trivial but it keeps going viral. What answer do you get when you calculate [maths]$6\div 2(1+2)$? This question has reached every corner of social media, and has had millions of people respond with two common answers: $1$ and $9$[/maths]. You might think one half of those people are right…

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These all make up the mathematical processes. Skills are small actions used in strategies that assist in obtaining a solution to a problem. Multiple skills make up a strategy. Algorithms are specific to math.

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These are broad techniques that can be applied to different disciplines, such as science or math. Each can also be broken down into skills, algorithms, and strategies that are or can be specific to mathematics.
1. Problem Solving
2. Reasoning and Proof
3. Communicating
4. Making Connections
5. Creating Representations
6. Investigating

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The structure of process includes the six mathematical processes, algorithms, skills, and strategies.

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Theorems are proven statements, connecting explanations of concepts.

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A formula that is represented by facts.

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While different, both fall under the structure of knowledge.

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The Fundamental Theorem of Calculus is an example of a concept that would fall under this level of the structure of knowledge.

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This level allows for connections between topics/concepts because of the statements of conceptual understanding (theorems).

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Page ID

2513

1. The Mean Value Theorem for Integrals

2. Fundamental Theorem of Calculus Part 1: Integrals and Antiderivatives

3. Fundamental Theorem of Calculus, Part 2: The Evaluation Theorem

4. Key Concepts

5. Key Equations

6. Glossary

7. Contributors and Attributions

Learning Objectives

Describe the meaning of the Mean Value Theorem for Integrals.

State the meaning of the Fundamental Theorem of Calculus, Part 1.

Use the Fundamental Theorem of Calculus, Part 1, to evaluate derivatives of integrals.

State the meaning of the Fundamental Theorem of Calculus, Part 2.

Use the Fundamental Theorem of Calculus, Part 2, to evaluate definite integrals.

Explain the relationship between differentiation and integration.

In the previous two sections, we looked at the definite integral and its relationship to the area under the curve of a function. Unfortunately, so far, the only tools we have available to calculate the value of a definite integral are geometric area formulas and limits of Riemann sums, and both approaches are extremely cumbersome. In this section we look at some more powerful and useful techniques for evaluating definite integrals.

These new techniques rely on the relationship between differentiation and integration. This relationship was discovered and explored by both Sir Isaac Newton and Gottfried Wilhelm Leibniz (among others) during the late 1600s and early 1700s, and it is codified in what we now call the Fundamental Theorem of Calculus, which has two parts that we examine in this section. Its very name indicates how central this theorem is to the entire development of calculus.

Isaac Newton's contributions to mathematics and physics changed the way we look at the world. The relationships he discovered, codified as Newton's laws and the law of universal gravitation, are still taught as foundational material in physics today, and his calculus has spawned entire fields of mathematics.

Before we get to this crucial theorem, however, let's examine another important theorem, the Mean Value Theorem for Integrals, which is needed to prove the Fundamental Theorem of Calculus.

The Mean Value Theorem for Integrals

The Mean Value Theorem for Integrals states that a continuous function on a closed interval takes on its average value at the same point in that interval. The theorem guarantees that if \(f(x)\) is continuous, a point \(c\) exists in an interval \([a,b]\) such that the value of the function at \(c\) is equal to the average value of \(f(x)\) over \([a,b]\). We state this theorem mathematically with the help of the formula for the average value of a function that we presented at the end of the preceding section.

\[f(c)=\dfrac{1}{b−a}∫^b_af(x)\,dx.\]

Proof

\[ m(b−a)≤∫^b_af(x)\,dx≤M(b−a).\]

Dividing by \(b−a\) gives us

\[ m≤\frac{1}{b−a}∫^b_af(x)\,dx≤M.\]

\[ f(c)=\frac{1}{b−a}∫^b_a f(x)\,dx,\]

and the proof is complete.

□

Solution

\[A=\dfrac{1}{2}(4)(8)=16. \nonumber\]

\[\dfrac{1}{4}(16)=4 \nonumber\]

At \(c=2,f(2)=4\).

Figure \(\PageIndex{1}\): By the Mean Value Theorem, the continuous function \(f(x)\) takes on its average value at \(c\) at least once over a closed interval.

Solution

Figure \(\PageIndex{2}\): Over the interval \([0,3]\), the function \(f(x)=x^2\) takes on its average value at \(c=\sqrt{3}\).

Exercise \(\PageIndex{2}\)

Fundamental Theorem of Calculus Part 1: Integrals and Antiderivatives

As mentioned earlier, the Fundamental Theorem of Calculus is an extremely powerful theorem that establishes the relationship between differentiation and integration, and gives us a way to evaluate definite integrals without using Riemann sums or calculating areas. The theorem is comprised of two parts, the first of which, the Fundamental Theorem of Calculus, Part 1, is stated here. Part 1 establishes the relationship between differentiation and integration.

\[F(x)=∫^x_af(t)\,dt,\]

then \(F′(x)=f(x)\) over \([a,b]\).

Before we delve into the proof, a couple of subtleties are worth mentioning here. First, a comment on the notation. Note that we have defined a function, \(F(x)\), as the definite integral of another function, \(f(t)\), from the point a to the point \(x\). At first glance, this is confusing, because we have said several times that a definite integral is a number, and here it looks like it's a function. The key here is to notice that for any particular value of \(x\), the definite integral is a number. So the function \(F(x)\) returns a number (the value of the definite integral) for each value of \(x\).

Second, it is worth commenting on some of the key implications of this theorem. There is a reason it is called the Fundamental Theorem of Calculus. Not only does it establish a relationship between integration and differentiation, but also it guarantees that any integrable function has an antiderivative. Specifically, it guarantees that any continuous function has an antiderivative.

\[ \frac{1}{h}∫^{x+h}_x f(t)\,dt=f(c). \nonumber\]

\[ \lim_{h→0}f(c)=\lim_{c→x}f(c)=f(x) \nonumber\]

and the proof is complete.

□

\[g(x)=∫^x_1\frac{1}{t^3+1}\,dt. \nonumber\]
Solution
\[g′(x)=\frac{1}{x^3+1}. \nonumber\]

Solution

Solution
Thus,

Fundamental Theorem of Calculus, Part 2: The Evaluation Theorem

The Fundamental Theorem of Calculus, Part 2, is perhaps the most important theorem in calculus. After tireless efforts by mathematicians for approximately 500 years, new techniques emerged that provided scientists with the necessary tools to explain many phenomena. Using calculus, astronomers could finally determine distances in space and map planetary orbits. Everyday financial problems such as calculating marginal costs or predicting total profit could now be handled with simplicity and accuracy. Engineers could calculate the bending strength of materials or the three-dimensional motion of objects. Our view of the world was forever changed with calculus.

After finding approximate areas by adding the areas of n rectangles, the application of this theorem is straightforward by comparison. It almost seems too simple that the area of an entire curved region can be calculated by just evaluating an antiderivative at the first and last endpoints of an interval.

\[ ∫^b_af(x)\,dx=F(b)−F(a). \label{FTC2}\]

We often see the notation \(\displaystyle F(x)|^b_a\) to denote the expression \(F(b)−F(a)\). We use this vertical bar and associated limits \(a\) and \(b\) to indicate that we should evaluate the function \(F(x)\) at the upper limit (in this case, \(b\)), and subtract the value of the function \(F(x)\) evaluated at the lower limit (in this case, \(a\)).

The Fundamental Theorem of Calculus, Part 2 (also known as the evaluation theorem) states that if we can find an antiderivative for the integrand, then we can evaluate the definite integral by evaluating the antiderivative at the endpoints of the interval and subtracting.

Proof

\[F(x_i)−F(x_{i−1})=F′(c_i)(x_i−x_{i−1})=f(c_i)\,Δx.\]

\[ F(b)−F(a)=\sum_{i=1}^nf(c_i)\,Δx.\]

\[ F(b)−F(a)=\lim_{n→∞}\sum_{i=1}^nf(c_i)Δx=∫^b_af(x)\,dx.\]

□

Use Equation \ref{FTC2} to evaluate

\[ ∫^2_{−2}(t^2−4)\,dt. \nonumber\]
Solution
If \(y=x^n\),

\[∫x^n\,dx=\frac{x^{n+1}}{n+1}+C. \nonumber\]
Analysis

Figure \(\PageIndex{3}\): The evaluation of a definite integral can produce a negative value, even though area is always positive.

\[ ∫^9_1\frac{x−1}{\sqrt{x}}dx. \nonumber\]
Solution
See Figure \(\PageIndex{4}\).

Figure \(\PageIndex{4}\): The area under the curve from \(x=1\) to \(x=9\) can be calculated by evaluating a definite integral.

Solution
\[ ∫^5_0(5+2t)\,dt. \nonumber\]

A Parachutist in Free Fall

Figure \(\PageIndex{5}\): Skydivers can adjust the velocity of their dive by changing the position of their body during the free fall. (credit: Jeremy T. Lock)

Figure \(\PageIndex{6}\): The fabric panels on the arms and legs of a wingsuit work to reduce the vertical velocity of a skydiver's fall. (credit: Richard Schneider)

Key Concepts

The Mean Value Theorem for Integrals states that for a continuous function over a closed interval, there is a value c such that \(f(c)\) equals the average value of the function.

The Fundamental Theorem of Calculus, Part 1 shows the relationship between the derivative and the integral.

The Fundamental Theorem of Calculus, Part 2 is a formula for evaluating a definite integral in terms of an antiderivative of its integrand. The total area under a curve can be found using this formula.

Key Equations

Mean Value Theorem for Integrals

If \(f(x)\) is continuous over an interval \([a,b]\), then there is at least one point \(c∈[a,b]\) such that \[f(c)=\frac{1}{b−a}∫^b_af(x)\,dx.\nonumber\]

Fundamental Theorem of Calculus, Part 1

If \(f(x)\) is continuous over an interval \([a,b]\), and the function \(F(x)\) is defined by \[ F(x)=∫^x_af(t)\,dt,\nonumber\]

then \[F′(x)=f(x).\nonumber\]

Fundamental Theorem of Calculus, Part 2

If \(f\) is continuous over the interval \([a,b]\) and \(F(x)\) is any antiderivative of \(f(x)\), then \[∫^b_af(x)\,dx=F(b)−F(a).\nonumber\]

Glossary

fundamental theorem of calculus

the theorem, central to the entire development of calculus, that establishes the relationship between differentiation and integration

fundamental theorem of calculus, part 1

uses a definite integral to define an antiderivative of a function

fundamental theorem of calculus, part 2

(also, evaluation theorem) we can evaluate a definite integral by evaluating the antiderivative of the integrand at the endpoints of the interval and subtracting

mean value theorem for integrals

guarantees that a point \(c\) exists such that \(f(c)\) is equal to the average value of the function

Contributors and Attributions

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5.3: The Fundamental Theorem of Calculus

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Page ID 2513 The Mean Value Theorem for Integrals Fundamental Theorem of Calculus Part 1: Integrals and Antiderivatives Fundamental Theorem of Calculus, Part 2: The Evaluation Theorem Key Concepts Key Equations Glossary Contributors and Attributions Learning Objectives Describe the meaning of the Mean Value Theorem for Integrals.

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                    STRUCTURE OF KNOWLEDGE

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                    STRUCTURE OF KNOWLEDGE

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The structure of knowledge is comprised of two levels: the factual level and the generalizations and principles level.

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The factual level is comprised of facts, memorization, and information that is not transferable to other concepts/topics.
Ex: Pythagorean Theorem, names of triangles.