# 4.2E: Exercises - Mathematics

## Exercise (PageIndex{1})

1. What is "total signed area"?

2. What is "displacement"?

3. What is (int_3^3 sin x,dx)

4. Give a single definite integral that has the same value as (int_0^1 (2x+3),dx +int_1^2 (2x+3),dx).

Under Construction

## Exercise (PageIndex{2})

A graph of a function (f(x)) is given. Using the geometry of the graph, evaluate the definite integrals.

1.

(a) (int_0^1 (-2x+4),dx)
(b) (int_0^2 (-2x+4),dx)
(c) (int_0^3 (-2x+4),dx)
(d) (int_1^3 (-2x+4),dx)
(e) (int_2^4 (-2x+4),dx)
(f) (int_0^1 (-6x+12),dx)

2.

(a) (int_0^2 f(x),dx)
(b) (int_0^3 f(x),dx)
(c) (int_0^5 f(x),dx)
(d) (int_2^5 f(x),dx)
(e) (int_5^3 f(x),dx)
(f) (int_0^3 f(x),dx)

3.

(a) (int_0^2 f(x),dx)
(b) (int_2^4 f(x),dx)
(c) (int_2^4 2f(x),dx)
(d) (int_0^1 4x,dx)
(e) (int_2^3 (2x-4),dx)
(f) (int_2^3 (4x-8),dx)

4.

(a) (int_0^1 (x-1),dx)
(b) (int_0^2 (x-1),dx)
(c) (int_0^3 (x-1),dx)
(d) (int_2^3 (x-1),dx)
(e) (int_1^4 (x-1),dx)
(f) (int_1^4 left ((x-1)+1 ight ),dx)

5.

(a) (int_0^2 f(x),dx)
(b) (int_2^4 f(x),dx)
(c) (int_0^4 f(x),dx)
(d) (int_0^4 5f(x),dx)

Under Construction

## Exercise (PageIndex{3})

A graph of a function (f(x)) is given; the numbers inside the shaded regions give the area of that region. Evaluate the definite integrals using this area information.

1.

(a) (int_0^1 f(x),dx)
(b) (int_0^2 f(x),dx)
(c) (int_0^3 f(x),dx)
(d) (int_1^2 -3f(x),dx)

2.

(a) (int_0^2 f(x), dx)
(b) (int_2^4 f(x), dx)
(c) (int_0^4 f(x), dx)
(d) (int_0^1 f(x), dx)

3.

(a) (int_{-2}^{-1}f(x),dx)
(b) (int_{1}^{2}f(x),dx)
(c) (int_{-1}^{1}f(x),dx)
(d) (int_{0}^{1}f(x),dx)

4.

(a) (int_0^2 5x^2,dx)
(b) (int_0^2 (x^2+1),dx)
(c) (int_1^3 (x-1)^2,dx)
(d) (int_2^4 left ( (x-2)+5 ight ),dx)

Under Construction

## Exercise (PageIndex{4})

A graph of the velocity function of an object moving in a straight line is given. Answer the questions based on that graph.

1.

(a) What is the object's maximum velocity?
(b) What is the object's maximum displacement?
(c) What is the object's total displacement on [0,3]?

2.

(a) What is the object's maximum velocity?
(b) What is the object's maximum displacement?
(c) What is the object's total displacement on [0,5]?

Under Construction

## Exercise (PageIndex{5})

An object is thrown straight up with a velocity, in ft/s, given by (v(t) = -32t+64), where (t) is in seconds, from a height of 48 feet.
(a) What is the object's maximum velocity?
(b) What is the object's maximum displacement?
(c) When does the maximum displacement occur?
(d) When will the object reach a height of 0? (Hint: find when the displacement is -48ft.)

Under Construction

## Exercise (PageIndex{6})

An object is thrown straight up with a velocity, in ft/s, given by (v(t)=-32t+96), where (t) is seconds, from a height of 64 feet.
(a) What is the object's initial velocity?
(b) What is the object's displacement 0?
(c) How long does it take for the object to return to its initial height?
(d) When will the object reach a height of 210ft?

Under Construction

## Exercise (PageIndex{7})

Use these values to evaluate the given definite integrals.

• (int_0^2 f(x) ,dx=5),
• (int_0^3 f(x) ,dx=7),
• (int_0^2 g(x) ,dx=-3), and
• (int_2^3 g(x) ,dx=5).

1. (int_0^2 left ( f(x)+g(x) ight ),dx)

2. (int_0^3 left ( f(x)-g(x) ight ),dx)

3. (int_2^3 left ( 3f(x)+2g(x) ight ),dx)

4. Find values for (a) and (b) such that
(int_0^3 left ( af(x)+bg(x) ight ),dx=0)

Under Construction

## Exercise (PageIndex{8})

Use these values to evaluate the given definite integrals.

• (int_0^3 s(t),dt =10),
• (int_3^5 s(t),dt =8),
• (int_3^5 r(t),dt =-1), and
• (int_0^5 r(t),dt =11).

1. (int_0^3 left ( s(t)+r(t) ight ),dt)

2. (int_5^0 left ( s(t)-r(t) ight ),dt)

3. (int_3^3 left ( pi s(t)-7r(t) ight ),dt)

4. Find values for a and b such that
(int_0^5 left ( ar(t)+bs(t) ight ),dt=0)

Under Construction

## Exercise (PageIndex{9})

Evaluate the given indefinite integral:

1. (int (x^3-2x^2+7x-9),dx)

2. (int (sin x -cos x +sec^2 x),dx)

3. (int (sqrt[3]{t}+frac{1}{t^2}+2^t),dt)

4. (int left ( frac{1}{x} -csc x cot x ight ),dx)

Under Construction

## 4.2(E) Math TEKS 4.2E KIT – Representing decimals- tenths & hundredths with models & money

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## Larson Algebra 2 Solutions Chapter 13 Trigonometric Ratios and Functions Exercise 13.4

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## Expanded Notation

Expanded form or expanded notation is a helpful way to rewrite numbers in order to show case the place value of each digit.

There are basically two acceptable ways to show numbers in expanded notation. Here are some examples.

Method #1: 4,000 + 900 + 80 + 1

In this method, we can see that 4 is in the thousands place because we have rewritten it as 4,000. The 9 is in the hundreds place, so we wrote it as 900.

To write a number in the form, you just replace all the numbers that came after the digit with zeros.

Here is another example using method 1.

Method #1:: 10,000 + 5,000 + 800 + 7

Again, we can see the place value of each digit in the number. Notice that there were no tens in this number.

The second method takes the work from method #1 a step further.

Method 2: (4 x 1,000) + (9 x 100) + (8 x 10) + (1 x 1)

This method shows the place value as a power of ten. We could have even taken this another step and used exponents to show the powers of ten.

The answer would now be: (4 x 10 3 ) + (9 x 10 2 ) + (8 x 10 1 ) + (1 x 10 0 )

Method 2: (1 x 10,000) + (5 x 1,000) + (8 x 100) + (7 x 1)

Expanded Notation with Decimals

Using the powers of ten, we can also write numbers with decimals in expanded notation.

We will start by using the first method. This will help us build to method #2.

(8x 10) + (9 x 1) + (3 x + (4 x )

The expanded notation helps us to see that the 3 is in the tenths place and the 4 is in the hundredths place.

(7 x 100) + (1 x 10) + (3 x 1) + (5 x ) + (2 x )

You might note that the number of zeros in the denominator for the decimals is the same as the number of places from the decimal point to that digit. For example, the 2 is three digits from the decimal point, so there are three zeros in the denominator.

## Binomial Theorem, Exponential and Logarithmic Series

The binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power (a + x) n into a sum involving terms of the form C(n,r) a n- r x r .

Application of binomial theorem

Binomial theorem is used to find the sum of infinite series and also for determining the approximate value s of certain algebraic and arithmetical quantities.

Exponential and logarithmic series

Let us consider the function y = f(x) = a x , a > 0 where,a is a base and x is a variable, is called an exponential function. The inverse of the logarithmic is exponential, which is denoted by loga x

The series of the exponential function is called exponential series and l the series of logarithmic function is called logarithmic series

Expansion e x

This expansion is known as the exponential series.

When this series is replaced by &ndashx wehave,

When x = 1 and -1 respectively

The value of series expansion for e is between 2 and 3

e = $mathop sum limits_<< m> = 0>^ <12>frac<1><<< m>!>>$ = 2.718281828

Expansion of a x

Logarithmic series

Since it is an identity, so the coefficient of x n of the LHS of (iii) should be equal to the coefficient of x n of the RHS of (iii)

Coeff. of x n in the expantion of (1 + x) 2n = C(2n,n) &hellip(iv)

Find the middle term in expansion of $> + frac<1><<3<< m>^2>>>> ight)^9>$

Here, n = 17 (odd number) therefore there are two middle terms.

i.e. 9 th term and 10 th term.

We know that, (1 + x) n = 1 + nx + $frac<<< m>left( << m> - 1> ight)>><<2!>>$x 2 + $frac<<< m>left( << m> - 1> ight)left( << m> - 2> ight)>><<3!>>$x 3 + &hellip..

Here are three solutions -- see the example.

I find that using environments like equation , equation* , align , align* help make LaTeX look simpler. Using qquad< ext<>> to add comments next to specific lines in an align environment, for example, helps transition code from something cluttered to something more pleasant to read. So, left-align will be taken care of by either align or align* for longer equations, or equation and equation* for single lines, and commenting from the aforementioned quad or qquad spacing modifier.

## 4.2E: Exercises - Mathematics

Solution of the singular Sturm-Liouville Problem: vibration of the closed ring string (4.6.6 p.333)

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Lecture Notes 2.1-2.5
(modified May 25, 2005)

Lecture Notes 2.6 Systems of ODEs
(modified May 20, 2005)

2nd order linear ODE with constant coefficients

3rd order linear ODE with constant coefficients

Euler-Cauchy Equation, example 4 (p.149)

Power Series Solution, example 2.11 (p.164)

Example of case 2 of Frobenius Theorem (TEST 1 #5c)

Systems of ODEs, example 4

Systems of ODEs with Maple, example 1

example 2 (autonomous system)

Half and Quater Range Expansions

Lecture Notes (modified March 5, 2005)

4.5.0 Banach and Hilbert Spaces
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4.5.5 Sturm-Liouville problem for equation X''-mu*X=0, case 2) Robin-Diriclet b.c.'s

4.6.2 Laplace's Equation
Diriclet problem, basic case

4.6.2 8. Poisson's's Equation
Diriclet problem

4.6.3 1) 1-d homogeneous equation and boundary conditions (Neumann-Neumann)

4.6.3 2) 1-d non-homogeneous equation and boundary conditions (Dirichlet-Dirichlet)

4.6.3 3) 1-d homogeneous equation and boundary conditions (Dirichlet-Robin) - application of Sturm-Liouville Theorem

4.6.3 4) 2-D temperature field in rectangular domain

Example of Diffusive Process:
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4.6.4 1) 1-d homogeneous equation and boundary conditions (Dirichlet-Robin)

4.6.4. 2) vibration of circular membrane (standing waves)

4.6.6. Singular SLP: vibration of the ring

Movies:

Orthogonal Sets in Annular Domain:

1 Dirichlet-Dirichlet:
Maple: nu=0 nu=1

2 Neumann-Dirichlet:
Maple: nu=0 nu=1

3 Dirichlet-Neumann:
Maple: nu=0 nu=1

Lecture Notes (modified Feb 15, 2005: with Banach Fixed Point Theorem)

Fourier Integral Representation(p.98)

Laplace's Equation (Poisson integral formula, p.107)

Laplace Equation (p.116)

8.2 The Laplace Transform:

Solution of Heat Equation(p.121)

Solution of Wave Equation(p.123)

8.3 The Hankel Transform:

Solution of Heat Equation(p.129)

Solution of Wave Equation(p.134)

8.4 Finite Fourier Transform:

8.4.2 Heat Equation in the Finite Layer(p.147)

8.4.3 Heat Equation in the Sphere(p.151)

8.6 Generalization of the Finite Integral Transform Method:

8.6.5 Transient Heat Transfer in the Fin(p.176)

## 4.2E: Exercises - Mathematics

The materials concentrate on the development of the primary focal areas for the grade level. The materials spend the majority of concept development on the focal areas as outlined in the Texas Essential Knowledge and Skills (TEKS), and they strategically and systematically develop students’ content knowledge. There are practice opportunities for students to master the content.

Evidence includes but is not limited to:

Throughout all modules, the materials contain planning documents such as unit plans and the TEKS for Mathematics Correlations that clearly state the focal areas of a unit these focal areas align with the grade-level TEKS. The materials clearly and consistently showcase focal areas of the curriculum that are aligned to the grade-level TEKS the Process Standards are combined with the content strands in the majority of the modules. For example, in the Teacher Edition, at the beginning of each unit, there is an outline that shows teachers a broad scope and sequence as well as all of the modules that fall under that focal area. The “Engage, Explore, Explain, Elaborate, Evaluate” (5E) lesson plan design in the materials informs the teaching and learning of math concepts.

The lessons begin with activating prior knowledge and progress to higher-order critical thinking and problem solving throughout the units, ensuring students master the full concept. The materials include inserts that outline the primary focal areas for instructional emphasis with a narrative and a graphic depicting vertical alignment. The lesson design permits instruction in each grade to focus on skills in greater depth while simultaneously building a foundation for the next grade, establishing an effective learning progression. In addition, the materials provide various practice opportunities through the use of additional resources such as “Response-to-Intervention (RTI) Tiered Lessons,” “Enrich” lessons, STEM activities, and the “Grab-and-Go” activities found in the digital resources. The materials explain that students will be able to apply mathematical skills in a variety of ways in order to be mathematicians. The introduction explains that through using manipulatives, models, and rigorous questions, students are able to move beyond a basic level of learning to develop deep conceptual understanding and then practice, apply, and discuss what they know.

Modules 1–8 focus on representing, counting, writing, and comparing whole numbers. Modules 9–14 focus on composing and decomposing numbers, addition, and subtraction. Module 17 identifies 2D shapes, while Module 18 identifies 3D shapes. These key focal areas are spiraled in most of the modules after they are taught for example, Module 1 teaches “Count, Write, and Represent numbers through 4.” This is scaffolded through to “Counting, Writing, and Representing the whole numbers through 20” in Module 8.

In Module 9, the center activities consist of a bus stop game that could be played at an actual bus stop, a literature piece, and a subtraction activity involving various objects that extend beyond the classroom walls. The lesson cycle for each module contains opportunities for students to practice the concept, utilizing manipulatives prior to moving to the abstract.

In Module 11, lessons provide scaffolding to activate prior knowledge and continue to build on the concept being taught. The lessons begin with activating prior knowledge about terms used when adding. The teacher then uses hands-on activities to introduce the concept of joining. Each lesson introduces a different strategy for addition up to 5. Additionally, the opportunities to practice in the lessons as well as the additional materials available in the resources promote rigor and guide the students to build their “toolbox” with varying strategies for problem-solving and critical thinking. In Module 11, students learn to use at least three strategies to solve addition problems.

In Module 17, students distinguish triangles from non-triangles and explain their selections using evidence-based justifications. The “Explain, Elaborate, and Evaluate” portion of this lesson illustrates how students utilize two-dimensional shapes and two-color counters in counting sides and vertices and communicate mathematical ideas within whole-group and small-group discussions, center activities, and through independent practice.

### Evaluation for 2.2 Materials strategically develop students’ conceptual understanding by following a progression of learning from concrete to representational to abstract (CRA) as is appropriate for the grade-level and content.

The provided materials include concepts sequenced from concrete to representational to abstract (CRA) as is grade-level appropriate. Materials include a variety of appropriate content and grade-level types of concrete models and manipulatives, pictorial representations, and abstract representations. However, materials do not clearly support teachers in understanding and developing students’ progression along the CRA continuum.

Evidence includes but is not limited to:

Throughout the materials, in order to understand whole numbers, Modules 1 and 2 include two-color counters, MathBoards, stones, connecting cubes, and five-frames Module 3 includes two-color counters, connecting cubes, number and symbol tiles, number cards, sets of blocks, five-frames, MathBoards, and dot cards. For addition and subtraction, Module 9 uses MathBoards, two-color counters, connecting cubes, and paper bags. The lessons all include step-by-step instructions for how to administer the concrete models and manipulatives. Although many concrete models and manipulatives repeat themselves over the course of the entire curriculum, there is plenty of variety. Because these manipulatives repeat themselves throughout the curriculum, students have previous experience with the concrete models and manipulatives. Additionally, there are paper-and-pencil opportunities for students in each lesson. As skills are spiraled in and built upon, students are using these manipulatives in more complex ways. Students are using the counters for example, students represent numbers in a ten-frame. Next, they build numbers with counting cubes and then write numbers. The use of the ten-frame is spiraled back in as students begin exploring addition and subtraction and represent equations with the connecting cubes. However, the materials do not explicitly state how these manipulatives should be used, and kindergarteners have likely not had prior experience using these tools.

Throughout the modules, the materials do not explicitly provide guidance about what tools should be available to students at different points in their development and how to push students to use increasingly sophisticated tools as appropriate along the CRA continuum. There is some evidence that the materials do support teachers with instructional suggestions to help students to progress through learning new concepts, knowledge, and skills however, there is a lack of direct instruction on how or when to move students along the CRA continuum. For example, every model in each grade level includes a “Show What You Know” section to assess students’ understanding of the concept. This section directs teachers to provide intervention using the “Response to Intervention” materials or to provide enrichment through the enrichment materials based on the students’ responses, but there is no explicit reference to utilizing the CRA continuum.

In Module 4, in the “Making Connections” section, students begin by counting up to 5 by using their fingers, and then think of other ways they can show the number 5. As a whole class, students use five-frames and counters to demonstrate the number 5 and one more. This is continued in the “Explore” section. In the “Elaborate” section, students manipulate two-color counters to identify different ways to represent 6 with two different numbers, such as 2 and 4. In the “Homework and Practice” section, students draw counters on a ten-frame as they count each rocket ship, and then select the right number to match the number of counters in three different ten-frames. The lessons provide instructions for RTI, for students who need additional support, and enrichment for students who understand the concepts however, there is no direction or instruction for teachers in reference to moving along the CRA continuum.

In Module 9, the lesson invites students to share what they know about modeling numbers teachers ask, “What number is one more than 1?” and “What number is one more than 2?” The lesson recommends using the “Interactive Digital Lesson,” where students are guided through the process of modeling numbers using items such as cubes or their fingers. The “Explore” section scaffolds through modeling numbers to composing numbers, using both counters and five-frames, up to the number 3. Materials instruct students on how to use the counters and five-frames teachers tell them how to place the red and yellow counters, asking questions such as, “How many yellow counters are there? Write the number. How many red counters are there? Write the number. What is 0 and 3? 3. Point to the number. There are 3 counters in all.” As the lesson progresses, the teacher digs deeper by asking, “Two children are sitting on a rug. One more child joins them. How many children are sitting on the rug now? How can you solve this problem?” The lesson contains “Dig Deeper” recommendations, such as asking students how they can solve this problem and guiding students to suggest acting out the problem or using manipulatives. The lesson ends with a higher-order thinking skills (HOTS) question: “Sasha wants to have 3 counters. She has 2 yellow counters. How many red counters does she need to have 3?”

In Modules 9–14, the introduction recommends that students complete the “Show What You Know” independently as a pre-assessment test. The materials then recommend a “Diagnostic Interview Task,” using two-colored counters, to have students demonstrate what they understand. The “Vocabulary Builder” lists strategies to help build students’ vocabulary, such as visualizing and drawing representations for the vocabulary words. In Module 14, “Differences within 7,” teachers instruct students on how to use pictorial models to solve subtraction problems by crossing out the pictures to show subtraction.

### Evaluation for 2.3 Materials support coherence and connections between and within content at the grade-level and across grade levels.

The provided materials support coherence and connections between and within content at the grade level and across grade levels. The resources include supports for students to build vertical content knowledge by accessing prior knowledge and understanding of concept progression. The included tasks connect two or more concepts, and they provide opportunities for students to explore relationships and patterns within and across concepts. The materials mostly support teachers in understanding horizontal and vertical alignment however, there are not always sufficient explicit details or directions to indicate why the materials build, especially for teachers less familiar with the TEKS for the grade level or grades above.

Evidence includes but is not limited to:

At the beginning of the year, in Module 5, focusing on modeling, counting, and writing numbers to 10, students use their knowledge of representing numbers they use counters in a ten-frame. In the middle of the year, in Module 10, students again connect their knowledge of numbers they compose and decompose numbers to ten. At the end of the year, in Modules 11–14, they once again connect their sense of numbers they add and subtract.

In Module 8, Lesson 8, students solve a word problem to find out how many apples Kaelin has. Through the step-by-step process, they find out that Kaelin has 18, and Chase has 16. Materials then prompt teachers to ask students which set is larger and to clarify that this also means which set has more. Students discuss how they know which set has more. This word problem alone connects addition, subtraction, greater than, less than, and abstract concepts. Also in this module, a lesson titled “Count and Write 18 and 19” includes an Essential Question: “How can you count and write 18 and 19 with words and numbers?” Students have modeled, counted, and written numbers 1–17 using counters and ten-frames in a consistent manner. During the “Making Connections” piece of the “Lesson Opener,” students tell the teacher what they know about counting and writing up to 17 then, they count the number of legs on an octopus out loud, up to 17, starting with 10 finally, they write 18 in number form. Students have done this in previous “Lesson Openers,” beginning in Unit 1, with the corresponding numbers of study. Students utilize the ten-frame models as part of the “Explore,” “Explain,” or “Evaluate” portions of the lesson just as they have throughout Modules 4, 5, 6, 7, and 8.

In Module 14, in the “Go Deeper” section, materials prompt students to find the similarities and differences in number sentences. Through this, students can begin to understand the relationship between addition and subtraction, which correlates with the following first grade TEKS: “Students use properties of operations and the relationship between addition and subtraction to solve problems.”

In Module 15, the materials use real-life problems that require students to recognize and apply mathematics in contexts outside of mathematics. For example, in an “Enrich” activity, students draw an art piece to sell. They put a price on it between 1 and 5 pennies. The teacher gives each student 5 pennies and has them buy and sell one drawing each. Students exchange pennies for the drawing and then count how many pennies they have left.

In Module 15, students use their sorting skills when sorting coins they continue to sort, using shapes, in Modules 17 and 18 in Module 20, students sort objects. Building on prior practice in sorting is essential in helping to solidify students’ understanding of building, creating, reading, and interpreting picture graphs.

In Module 16, the teacher asks students to share stories about a time when they saw or used eggs students draw up to 20 eggs on their paper and then count and write down the number of eggs they drew. Later in the lesson, students play a game to increase their familiarity with numbers from 1 to 30 they repeatedly count forward to move along the game path. Students complete an activity card by showing sets of 10 objects using the ready-made “Grab-and-Go” differentiated centers kit.

In Module 20, students sort objects and create a graph. Teachers ask students to read the graph, asking questions such as “How many cubes are red?” and “Which color has few cubes?” This activity builds on the previous eight modules, where students learned to count, write, and represent numbers.

### Evaluation for 2.4 Materials are built around quality tasks that address content at the appropriate level of rigor and complexity.

The provided materials are built around quality tasks that address content at the appropriate level of rigor and complexity. Tasks are designed to engage students in the appropriate level of rigor as identified in the Texas Essential Knowledge and Skills (TEKS), as appropriate for the development of the content and skills, and they clearly outline for the teacher the mathematical concepts and goals behind each task. Materials integrate contextualized problems throughout, providing students chances to apply knowledge and skills in varied situations. The resources provide teachers guidance on anticipating student responses and strategies however, the materials do guide teachers on ways to revise content to be relevant to their specific students, the students’ backgrounds, and the students’ interests. In addition, there are embedded opportunities for discourse, but there are no rubrics for teachers to evaluate the quality of the discussions.

Evidence includes but is not limited to:

Throughout all modules, the materials begin with concrete models, allowing students to use tools and manipulatives to represent numbers. The materials guide students through CRA (concrete to representational to abstract) tools, models, and understandings, with increasing depth and complexity materials provide increasing rigor throughout a given unit and across units over the course of the year. Each unit (which includes multiple modules) includes an introduction called “Introduce the Unit” it describes and explains the overall concepts and the goals of the unit. This is exemplified in the “TEKS for Mathematics” section in the Teacher Edition (TE) as well as in the teacher professional development videos the student edition lists the TEKS for each lesson in the top right-hand corner of the first page. Materials note multiple goals behind a task, emphasizing that the process is just as important for student learning as the product they guide teachers to facilitate discussion on how differences in strategy relate to efficiency and how well strategies work for the problem type. The unit page explains each component of the lesson unit pages also include “Essential Questions” that teachers focus on throughout each unit. Lessons follow a “5E” format (“Engage, Explore, Explain, Elaborate, and Evaluate”). Each lesson starts out with accessing prior knowledge and a “Making Connections” section in the Explore section, the lesson dives into the on-level material. During the Explain portion, materials encourage moving students to Elaborate only if they master the previous teaching. There are enrichment questions in the higher-order thinking (HOT) problems, “Go Deeper,” and in independent practice, labeled “Homework and Practice.” The unit page also informs teachers that the “Diagnostic Interview Task” may be used for intervention on prerequisite skills. For each grade level, several sections throughout the TE unit pages direct teachers to students’ prerequisite skills, such as “Show What You Know,” “Quick Checks,” and the “Vocabulary Builder.” Each lesson includes TEKS and learning objectives to address the “Common Errors” section lists possible misconceptions for each lesson in the TE.

In Module 1, students progress to finding all the ways to create 10 using counters and ten-frames this culminates in Module 6. As they extend this work to 20 in Module 8, students are expected to understand and verbalize the conceptual understanding behind these skills, explaining and demonstrating that numbers can be composed and decomposed in different ways. This is a precursor to understanding the relationship between addition and subtraction as well as algebraic relationships in writing expressions.

In Modules 1–8, “Numbers and Operations,” teachers guide students to use connecting cubes or two-color counters to model the numbers 1–20. As the unit and modules progress, students use picture representations to identify and count numbers 1–20. These may be pictures that they draw themselves or pictures that are provided for them in their student edition. Specifically, in Module 5, students see pictures of ten objects. Later in that same module, students draw their own ten objects and write the numbers 1–10. At the end of the unit, in Module 8, teachers guide students to think abstractly as they order and compare numbers up to 20.

In Module 3, students begin a lesson by reviewing counting up to 5. Students then make cube towers in the Explore section of the lesson. They make cube towers in order, using one cube, then two cubes, and so on up to five cubes. The Explain section has students “Share and Show” what they’ve learned and their reasoning behind how they know the cube trains are in order. In the Elaborate section, students move on to filling in blanks, using numbers 1 to 5. Finally, in the “Problem Solving” section, students must figure out which set has one more than a set of three blocks. This is much more advanced than the beginning of the lesson, which is just rote counting from 1 to 5. Tasks increase throughout any given module and unit. For example, in Module 3, students compare numbers through 5 they begin with counting and ordering to 5, then move on to greater than and less than, and finish with comparing by matching and counting sets of 5.

In Module 5, a lesson titled “Represent, Count, and Write through 10” begins with activating prior knowledge through the use of the “Lesson Opener” activities. For example, materials state: “Review counting objects and counting to 8 with children. What numbers do you say when you count to 8? Would you use counters and a five-frame to count 8? Why not?” The lesson then moves to Explore materials introduce the Essential Question: “How can you show and count 9 objects?” Students use manipulatives or other strategies previously taught to work through a problem. The teacher reads the problem: “Daniel wins eight prizes at the fair. Then he wins one more prize. How many prizes does he win?” The teacher guides the students through the steps to solve the problem the teacher gives each child nine counters, models counting to 9, and has children count to make sure they have nine counters. If needed, the teacher rereads the problem and directs students, “Place eight counters on the page. Now place one more counter. How many prizes did Daniel win? Count to be sure that you have nine. Now draw the counters to model the prizes. Is 9 greater than or less than 8? When you are counting, is a number always greater than or less than the one before it?” The next step in the lesson is Explain teachers use the “Model and Draw” strategy to model their thinking process and how to solve a problem. The lesson progresses from conceptual understanding to procedural fluency in the Elaborate, Share and Show, and Problem Solving sections of the lesson, where students repeat practices building on rigor.

In Module 10, “Compose and Decompose Numbers Up to 10,” a discussion question embedded within the Elaborate portion asks, “There are six bananas on the snack table. Two bananas are green. How many bananas are yellow? How many bananas are on the table? How many are green?” Students count out the two bananas and mark them with an x. Teachers then instruct students to count the remaining bananas and determine if they put together or took apart (added or subtracted) for this problem. “Go Deeper” guides the teacher on how to proceed should a student exhibit mastery “Differentiated Instruction” guides the teacher on how to proceed should a student experience difficulty. Also in Module 10, in the “Springboard for Learning” section, the materials remind teachers that children must ask themselves what the problem asks them to do. If the problem is a “put together” problem, children will add to solve it, thus leading to further application in problem-solving requiring addition.

In Module 16, materials direct the teacher to ask children to share stories about a time when they saw or used eggs students draw up to 20 eggs on their paper and then count and write down the number of eggs they drew. Later in the lesson, students play a game where they increase their familiarity with numbers from 1 to 30, repeatedly counting forward to move along a game path. Students complete an activity card by showing sets of 10 objects, using the ready-made “Grab-and-Go” differentiated centers kit.

Crystal Hede, Kate Russell, Ron Weatherby, Monique Brennan, Wayne Gore, Ben Williams

### Description

The new Queensland Senior Physical Education syllabus affects all aspects of teaching and learning, featuring new teaching content, new course structure and a new approach to assessment.

As EPAA Secondary Publisher of the Year 2017, 2018 and 2019, Oxford University Press is committed to helping teachers and students in Queensland reach their full potential.

Physical Education for Queensland provides in-depth and complete coverage of the new syllabus in a format that offers complete support for teachers and their students. This comprehensively updated edition now has two-volumes covering Units 1 & 2 (Book 1), and Units 3 & 4 (Book 2).

Key features include:

• The Physical Education toolkit: a stand-alone reference section that explains the structure of the syllabus, supports the acquisition of key skills and provides practical tips for success in Physical Education
• Learning pathways mapped clearly and directly to the syllabus to ensure complete coverage
• Engaging content, including media articles, case studies and practicals, brings the syllabus to life
• Assessment support and resources, including exam preparation and practice
• Key subject mattercoverage presented using clear, concise language, supported by engaging visual elements and sequenced to scaffold student learning
• Differentiated teaching supported by a range of appropriately levelled questions and activities included for each section
• Inquiry learning and critical thinkingapproaches clearly modelled throughout
• Additional digital learning resources included to support both teachers and students.

### Contents

Chapter 1: Physical Education toolkit
1.1 Course overview for QCE Physical Education
1.2 Assessment overview for QCE Physical Education Units 3 & 4
1.2A Tips for success on the Project &ndash folio
1.2B Tips for success on the Investigation &ndash report
1.2C Tips for success on the Examination &ndash combination response
1.2D Understanding cognitive verbs
1.3 The importance of data in QCE Physical Education
1.3A Putting data to use in Physical Education
1.4 Careers in physical education, sport and fitness

UNIT 3: TACTICAL AWARENESS, ETHICS AND INTEGRITY AND PHYSICAL ACTIVITY

Chapter 2: Tactical awareness
2.1 Introduction to tactical awareness
2.2 Approaches to motor learning and the development of tactical awareness
2.3 The dynamic systems approach and dynamic models of learning
2.4 Introduction to a constraints-led approach to teaching and learning
2.5 Implementing a constraints-led approach
2.6 Assessment support &ndash Summative internal assessment 1: Project &ndash folio
2.7 Developing tactical awareness in &lsquoinvasion&rsquo physical activities [ONLINE ONLY]
2.7A Australian football
2.7C Futsal
2.7D Netball
2.7E Soccer
2.7F Touch football
2.7G Water polo
2.8 Developing tactical awareness in &lsquonet and court&rsquo physical activities [ONLINE ONLY]
2.8B Tennis
2.8C Volleyball
Chapter 2 review

Chapter 3: Ethics and integrity
3.1 Introduction to ethics and integrity
3.2 Ethics and integrity in sport and physical activity
3.3 Fair play
3.4 Developing personal values and ethical behaviours
3.5 The influence of ethical values and ethical strategies on fair play and integrity
3.6 The influence of globalisation and mass media coverage on ethical values and behaviours
3.7 Ethical dilemmas
3.8 The ethical decision-making framework
3.9 Assessment support for summative internal assessment 2
Chapter 3 review

UNIT 4: ENERGY, FITNESS AND TRAINING AND PHYSICAL ACTIVITY

Chapter 4: Energy, fitness and training
4.1 Introduction to energy, fitness and training
4.2 Energy requirements for physical activity
4.3 Energy systems used in physical activity
4.4 Fitness requirements for physical activity
4.5 The role of oxygen in performance
4.6 Training requirements for physical activity
4.7 Training zones
4.8 Principles of training
4.9 Training methods
4.9A Continuous training
4.9B Fartlek training
4.9C Resistance training
4.9D Interval training
4.9E Flexibility training
4.9F Circuit training
4.10 Fatigue and recovery in training
4.11 The theory of periodisation
4.12 Developing a training program
4.13 Developing a training session plan
4.14 Assessment support &ndash Summative internal assessment 3: Project &ndash folio
Chapter 4 review

Chapter 5: Unit 4 revision and examination preparation
5.1 Energy requirements for physical activity including energy systems
5.2 Fitness requirements for physical activity
5.3 The role of oxygen in performance and training zones
5.4 Principles of training
5.5 Training methods, fatigue and recovery in training
5.6 The theory of periodisation
5.7 Developing training programs and training sessions

Chapter 6: Skill drills
1.2A Planning, creating and presenting a Project &ndash folio [ONLINE ONLY]
1.2B Creating and presenting an Investigation &ndash report [ONLINE ONLY]
1.3A Strategies for improving your results on the Examination &ndash combination response [ONLINE ONLY]
1.3B Conducting a survey and presenting the results [ONLINE ONLY]
1.3C Using the internet to find relevant, credible and reliable sources [ONLINE ONLY]
2.3 Devising a personal tactical strategy
2.4 Evaluate the effectiveness of decision making in authentic game settings
2.6 Evaluate the effectiveness of your personal tactical strategy
3.4 Implement your values in physical activities
3.7 Ethical dilemmas exercise
3.8 Apply the decision-making framework to an ethical dilemma
4.5A Use heart rate recovery as a measure of fitness
5.1 Evaluate the effectiveness of a training session on a particular energy system
5.2 Determine personal performance capacities for a physical activity using customised fitness testing
5.3 Using heart rate to determine if you are in the correct training zone
5.4 Determine the importance of applying the training principles when developing a training program
5.5 Determine the impact of volume, intensity and skill work for development of a specialised movement sequence
5.6 Determine the importance of periodisation
5.7 Evaluating the importance of a correctly structured warm up

### Authors

Crystal Hede
Crystal Hede has been the Head of Health and Physical Education at The Glennie School for over 10 years. She has lead curriculum change and held the position of IT mentor, supporting the broader staff body in the implementation of technology to enhance teaching and learning. She has also been a district panellist for senior Physical Education.

Kate Russell
Kate Russell has taught Health and Physical Education in Queensland for 14 years, including as Head of Department at St Saviour&rsquos College. As a district review panellist, Kate has contributed to the development of the Physical Education syllabus over many years. Kate now specialises in the field of child behaviour, psychology and development, working to help parents and educators build positive relationships with the children in their care.

Ron Weatherby
Ron Weatherby has been a Health and Physical Education teacher for over 30 years and has been Head of Department at Lockyer District State High since 1997. Ron has been involved with senior Physical Education curriculum development at all levels since 1995, acting as a panellist, district review panel chair and state member over this time. Ron was also a member of the review panel for the new senior Physical Education syllabus and is currently presenting at workshops and developing resources for its implementation in 2019.

Monique Brennan
Monique Brennan is an experienced Health and Physical Education teacher and Head of Department who has taught across a range state and Catholic schools. Monique led the implementation of the Australian Curriculum: HPE for Brisbane Catholic Education, providing guidance and support for teachers. Monique is passionate about the lifelong positive effects of quality health and physical education and currently leads Middle Years Curriculum at Carmel College in Brisbane.

Wayne Gore
Wayne Gore is the Head of Physical Education at Anglican Church Grammar School (Churchie) in Brisbane. He is a committed educator with over 20 years&rsquo teaching experience. Wayne is a management committee member with ACHPER Australia Queensland Branch and has also worked as a Panel Member, QCAA Endorsement Assessor (trial).

Ben Williams
Ben Williams is a Lecturer in Health and Physical Education in the School of Education and Professional Studies at Griffith University. He is a member of the QCAA's state review panel and has been a member of many health and physical education industry advisory panels. He is also President of the Australian Council for Health, Physical Education and Recreation's Queensland Branch (ACHPER QLD). Before completing his PhD and joining Griffith University, Ben was a Health and Physical Education teacher at The Gap State High School.

### Student Resources

This resource includes a physical copy of the Student book and access to obook assess which is a cloud-based obook that students can use anywhere, anytime, on any device.

• a complete digital version of the Student book with added note-taking and bookmarking functionality
• free Oxford Concise Dictionary look-up feature
• targeted instructional videos by some of Queensland&rsquos most experienced Physical Education teachers, designed to help students prepare for assessment tasks and exams
• a range of engaging worksheets for every chapter, designed to consolidate and extend understanding of key content from the syllabus
• additional case studies and opportunities for extension
• a range of interactive, auto-correcting, multiple-choice assess quiz questions.

### Teacher Resources

This resource is supported by the Physical Education for Queensland Units 3 & 4 2E Teacher obook assess (ISBN: 9780190313289).

Teacher obook assess is available FREE to booklisting schools or schools that purchase a class set of 25 or more copies. Contact your Oxford Education Consultant via www.oup.com.au/contact to discuss your requirements and request a demonstration.

obook assess is a cloud-based obook that teachers and students can use anywhere, anytime on any device.

• detailed course planners, teaching programs and lesson plans
• answers to all questions and assessment tasks in the Student book
• chapter summary PowerPoint presentations and revision notes ideal for individual or whole-class revision
• printable (and editable) practice exam with answers
• all student resources listed below.

With obook assess, teachers can:

• set in-platform assessment tasks with the ability to create groups and tailor instructions to meet the different needs of abilities of different students
• monitor student progress and graph results
• view all available content and resources in one place.

Teachers will have access to the following student resources via obook assess:

## 4.2E: Exercises - Mathematics

Meeting the health needs of the staff and community….Helping families….extending lives…alerting the minds of our youth….having positive goals….a healthy form of competition (literally)…..

Standard 1: Personal Health and Fitness

Students will have the necessary knowledge and skills to establish and maintain physical fitness,

participate in physical activity, and maintain personal health.

apply prevention and risk reduction strategies to

• demonstrate the necessary knowledge and skills to

• analyze the multiple influences which affect health

1. Students will use an understanding of the elements

of good nutrition to plan appropriate diets for

themselves and others. They will know and use the

appropriate tools and technologies for safe and

• understand the relationships among diet, health, and

physical activities evaluate their own eating patterns

and use appropriate technology and resources to make

food selections and prepare simple, nutritious meals

• apply principles of food safety and sanitation

• recognize the mental, social, and emotional aspects of

• apply decision making process to dilemmas related to

This is evident, for example, when students:

_ plan a personal diet that accommodates nutritional needs,

activity level, and optimal weight

_ prepare a meal with foods from the food groups described in the

I Culture – A people’s way of life, language, customs, arts, belief systems, traditions, and how they evolve over time.

V Individuals, Groups, and Institutions – The impact of educational, religious, social, and political groups and institutions and the integral roles they play in people’s lives.

VII Production, Distribution, and Consumption – The role of resources, their production and use, technology, and trade on economic systems.

VIII Science, Technology, and Society – The significance of scientific discovery and technological change on people, the environment, and other systems.

IX Global Connections – The critical importance of knowledge and awareness of politics, economics, geography, and culture on a global scale.

1.2c: understand the relationship between the relative importance of United States domestic and foreign policies over time

1.3d: classify major developments into categories such as social, political, economic, geographic, technological, scientific, cultural, or religious

** 2.1a: know the social and economic characteristics, such as customs, traditions, child-rearing practices, ways of making a living, education and socialization practices,

gender roles, foods, and religious and spiritual beliefs that distinguish different cultures and civilizations

2.4c: view history through the eyes of those who witnessed key events and developments in world history by analyzing their literature, diary accounts, letters,

artifacts, art, music, architectural drawings, and other documents

Students will use a variety of intellectual skills to demonstrate their understanding of how the United States and other societies develop

economic systems and associated institutions to allocate scarce resources, how major decision-making units function in the U.S. and other

national economies, and how an economy solves the scarcity problem through market and nonmarket mechanisms.

1. The study of economics requires an understanding of major economic concepts and systems, the principles of economic decision making, and the interdependence

of economies and economic systems throughout the world.

Student Performance Indicators:

4.1a: explain how societies and nations attempt to satisfy their basic needs and wants by utilizing scarce capital, natural, and human resources

4.1b: define basic economic concepts such as scarcity, supply and demand, markets, opportunity cost, resources, productivity, economic growth, and systems

4.1c: understand how scarcity requires people and nations to make choices which involve costs and future considerations

4.1d: understand how people in the United States and throughout the world are both producers and consumers of goods and services

4.1e: investigate how people in the United States and throughout the world answer the three fundamental economic questions and solve basic economic problems

4.1f: describe how traditional, command, market, and mixed economies answer the three fundamental economic questions

4.1g: explain how nations throughout the world have joined with one another to promote economic development and growth

2. Economics requires the development and application of the skills needed to make informed and well-reasoned economic decisions in daily and national life.

Student Performance Indicators:

4.2a: identify and collect economic information from standard reference works, newspapers, periodicals, computer databases, textbooks, and other primary and

4.2b: organize and classify economic information by distinguishing relevant from irrelevant information, placing ideas in chronological order, and selecting

appropriate labels for data

4.2c: evaluate economic data by differentiating fact from opinion and identifying frames of reference

4.2d: develop conclusions about economic issues and problems by creating broad statements which summarize findings and solutions

4.2e: present economic information by using media and other appropriate visuals such as tables, charts, and graphs to communicate ideas and conclusions

5.1a: analyze how the values of a nation affect the guarantee of human rights and make provisions for human needs

Students will communicate their mathematical thinking coherently and clearly to peers, teachers, and others.

Share organized mathematical ideas through the manipulation of objects, numerical tables, drawings, pictures, charts, graphs, tables, diagrams, models, and symbols in written and verbal form

Students will recognize and use connections among mathematical ideas.

Understand and make connections and conjectures in their everyday experiences to mathematical ideas

Students will recognize and apply mathematics in contexts outside of mathematics.

Recognize and provide examples of the presence of mathematics in their daily lives

Apply mathematics to problem situations that develop outside of mathematics

Investigate the presence of mathematics in careers and areas of interest

Recognize and apply mathematics to other disciplines and areas of interest

Students will create and use representations to organize, record, and communicate mathematical ideas.

Use physical objects, drawings, charts, tables, graphs, symbols, equations, or objects created using technology as representations

Students will use representations to model and interpret physical, social, and mathematical phenomena.

Use mathematics to show and understand social phenomena (e.g., construct tables to organize data showing book sales

7.R.10 Use math to show and understand social phenomena (e.g., determine profit from sale of yearbooks)

Students will determine what can be measured and how, using appropriate methods and formulas.

Measure capacity and calculate volume of a rectangular prism

Identify customary units of capacity (cups, pints, quarts, and gallons)

Identify equivalent customary units of capacity (cups to pints, pints to quarts, and quarts to gallons)

Students will collect, organize, display, and analyze data.

Develop the concept of sampling when collecting data from a population and decide the best method to collect data for a particular question

7.M.4 Calculate unit price using proportions

Students will collect, organize, display, and analyze data.

Identify and collect data using a variety of methods

7.S.6 Read and interpret data represented graphically (pictograph, bar graph, histogram, line graph, double line/bar graphs or circle graph)

Students will determine what can be measured and how, using appropriate methods and formulas.

Solve equations/proportions to convert to equivalent measurements within metric and customary measurement systems Note: Also allow Fahrenheit to Celsius and vice versa.

Opportunities for parents and guardians to meet with schools to discuss nature, purposes, and educational values of different program types and services, including the option to transfer their pupils to such programs at other schools or the option to withdraw their pupils from participation in an instructional bilingual education program only .

Procedures for the distribution of school-related information to parents of LEP students in the language they understand (maximum 1 page) :

The description must include:

The methods employed by the school district to ensure that the parents and guardians of LEP pupils are informed fully, in a language they understand, and in a timely manner of school related-activities and any information that are pertinent to the education of their pupils.

Middle School Students and the Grade 8 Benchmark

Early adolescence embodies an exhilarating range of characteristics and

contradictions. Physical, mental, and emotional fluctuations render

middle school youngsters amenable to an environment that affirms their

fledgling self-identity and developmental capacities. Sequential music

study develops the following skills and understandings:

n Physical/Social: Students acquire vocal and instrumental dexterity

discover leadership skills and engage in increased peer interaction and

n Cognitive: Students analyze, differentiate, create, and compare performances,

repertoire, and experiences.

n Aesthetic: Students develop self-expression as music makers integrate

music learning with personal observations and choices.

n Metacognitive: Students consider and assimilate a range of musical

experiences to make appropriate responses.

n compose and perform a piece of music in response to a powerful personal or musical experience.

dramatize a scene from a musical play such as West Side Story using voice and instruments and

drawing attention to the relationship between movement/gesture and music.

n write a poem and musical underscoring to express their emotional reaction to pivotal events in

n demonstrate awareness of the ways in which music information, resources, and tools may be accessed.

n lead a research project that draws parallels between a culture’s geography, natural resources, climate, ancestry, and its music—

n Apply learning from other arts and disciplines, such as math, science, language arts, social studies, technology,

music, visual arts, dance, and film/video, to extend their understanding of theater.

n Reflect on and discuss theater’s connection to their own lives through examining the themes and lessons of the

n Identify and articulate the cultural and historical components of the work and how these components create a

particular world of behaviors.

n Recognize that behaviors and themes particular to the world of the play also connect to our understanding of the

Create a computer-generated

• application of the principles

of design (balance, contrast,

English language arts skills

Examine a work of art as a

primary document based on

visual evidence, write hypotheses

Examine a work of art over an

extended period of time. Keep

a record of observations as

evidence of the way a viewer’s

perceptions deepen over time.

Use notes as a basis of discussion.

Questions

Are we able to do more exercises now that we are eliminating junk food and eating more raw healthy food?

Do we have more or less energy?

Teachers can compare the amount of exercises that children can do compared to before the change in diet.

Students can write a paper on the benefits to the body in reference to how they feel physically and mentally answering the questions above.

Vocabulary words on healthy foods

Use pictures to promote vocabulary in reference to foods.

Students will create campaign ideas for the stores in the neighborhood to provide better produce. They will discuss and write about how their culture influences their diet and the type of healthy food they eat at home. They will also discuss and debate whose responsibility it is to keep the community healthy.

Answer the question. Do we, America eat healthier now than we did in the past?

Keep a record of the costs to each student and family to change their diet. How much money do they save on junk food by eliminating it from their diet. What is the potential amount of money that can be saved over an extended period of time? Do a survey of the family and class and school on the topic of their choice in reference to this campaign. Perhaps they could ask what is the favorite thing about this change in diet.

The students will continue in the Omnivore’s Dilemna and the teacher will decide how to implement a writing assignment connected to the campaign. They will also assist in the writing involved in the other subjects.

The student will make a list of healthy foods on a color chart. They will discuss how a colorful plate of food is a healthier plate of food. Students will keep a record of their daily diet. Teacher and students will discuss which vitamins are being taken into the body. See attachment.

Singing and playing music that promotes health. Perhaps creating jingles or singing/playing existing jingles that promote health.

Students will compare and contrast healthy and toxic food. They will research articles on the topic and present them.

Students will do a report on the following question notating the times commercials come on television. Do commercials promote healthy eating during the time shows come on for little kids? Teens? Why do you think that is? Is there a responsibility from the creators of children’s programing to provide commercials that are not harmful to children?

They will find and summarize articles in the newspaper in reference to healthy eating.

They will look at and discuss advertisement in magazines and reference commercials.

They will work in groups to create a commercial, acting out a scene to promote health.

Draw campaign slogans to promote our goal. Commit to the same diet that we are promoting…..no junk food, more fruits and vegetables. Taking a look at old advertisements.

Write a play on eating healthy and perform it

Dean will announce “It’s crunch time!” Students will be served a snack they can crunch on.

There will be a daily homework assignment of no junk food, more fruits and vegetables.

Commit to eliminating junk from the diet for one week or longer. Commit to the same diet that we are promoting…..no junk food, more fruits and vegetables.

Commit to the same diet that we are promoting…..no junk food, more fruits and vegetables. Encourage their homeroom to last the longest on the health campaign. Bring in a healthy snack. Use star bucks to encourage students.

Commit to the same diet that we are promoting…..no junk food, more fruits and vegetables. Promote the idea that no junk food packages/containers are allowed on their floor. Encourage academy to extend time and keep the diet longer than the other academies.

Commit to the same diet that we are promoting…..no junk food, more fruits and vegetables. Try to get the lunch room to serve food with a minimum of 3 choices of fruits and/or vegetables that are a different color each day. Promote a letter home to parents having them commit to being involved with dinner at home and snacks at home. The letter should available in all languages that represent the student population. Promote healthy competition between the academies to last the longest following the campaign. Develop a prize for the class and staff member that lasts the longest. I recommend a free field trip for a class or top classes.

Commit to the same diet that we are promoting…..no junk food, more fruits and vegetables. Encourage their colleagues to last the longest on the health campaign. Bring in a healthy snack.

## Scientific Notation and Significant Figures

In the previous example you should have noticed that the answer is presented in what is called scientific notation.

&hellipis a way to express very small or very large numbers
&hellipis most often used in "scientific" calculations where the analysis must be very precise
&hellipconsists of two parts: A Number and a Power of 10. Ex: 1.22 x 10 3

For a number to be in correct scientific notation only one digit may be to the left of the decimal. So,

egin 1.22 & imes 10^3 ext < is correct> 12.2 & imes 10^2 ext < is not>end

How to convert non-exponential numbers to exponential numbers:

This is a large number and the implied decimal point is at the end of the number.

To convert this to an exponential number we need to move the decimal to the left until only one digit resides in front of the decimal point. In this number we move the decimal point 5 times.

&hellipand thus the exponent we place on the power of 10 is 5. The resulting exponential number is then:

egin 21 & o 2.1 imes 10^1 16600.01 & o 1.660001 imes 10^4 455 & o 4.55 imes 10^2 end

Small numbers can be converted to exponential notation in much the same way. You simply move the decimal to the right until only one non-zero digit is in front of the decimal point. The exponent then equals the number of digits you had to pass along the way.

The first non-zero digit is 5 so the number becomes 5.56 and we had to pass the decimal point by 4 digits to get it to the point where there was only one non-zero digit at the front of the number so the exponent will be -4. The resulting exponential number is then:

egin 0.0104 & o 1.04 imes 10^ <-2> 0.0000099800 & o 9.9800 imes 10^ <-6> 0.1234 & o 1.234 imes 10^ <-1>end

So to summarize, moving the decimal point to the left yields a positive exponent. Moving the decimal point to the right yields a negative exponent.

Another reason we often use scientific notation is to accommodate the need to maintain the appropriate number of significant figures in our calculations.

### Significant Figures

There are three rules on determining how many significant figures are in a number:

1. Non-zero digits are always significant.
2. Any zeros between two significant digits are significant.
3. A final zero or trailing zeros in the decimal portion ONLY are significant.
• 2003 has 4 significant figures
• 00.00300 has 3 significant figures
• 00067000 has 2 significant figures
• 00067000.0 has 6 significant figures

### Exact Numbers

Exact numbers, such as the number of people in a room, have an infinite number of significant figures. Exact numbers are counting up how many of something are present, they are not measurements made with instruments. Another example of this are defined numbers, such as

There are exactly 12 inches in one foot. Therefore, if a number is exact, it DOES NOT affect the accuracy of a calculation nor the precision of the expression. Some more examples:

• There are 100 years in a century.
• Interestingly, the speed of light is now a defined quantity. By definition, the value is 299,792,458 meters per second.

In order to present a value in the correct number of significant digits you will often have to round the value off to that number of digits. Below are the rules to follow when doing this:

The application of significant figures rules while completing calculations is important and there are different ways to apply the rules based on the type of calculation being performed.

### Significant Figures and Addition or Subtraction

In addition and subtraction the number of significant figures that can be reported are based on the number of digits in the least precise number given. Specifically this means the number of digits after the decimal determine the number of digits that can be expressed in the answer.

### Significant Figures and Multiplication or Division

In multiplication and division the number of significant figures is simply determined by the value of lowest digits. This means that if you multiplied or divided three numbers: 2.1, 4.005 and 4.5654, the value 2.1 which has the fewest number of digits would mandate that the answer be given only to two significant figures.