Higher Order Thinking Activities

Assignment #1 – Estimating the height of a tree

Topic – Geometry (Similar Figures)
Grade Level – 7^th
Depth of Knowledge Level – 3

Description – Students will apply their knowledge of the attributes of similar triangles to estimate the height of a tree on the school campus. Using the length of the tree’s shadow, along with their own heights and the lengths of their shadows, they will draw diagrams of the similar triangles formed using these measurements. By using the relationship between the lengths of the two shadows, students will apply that scale factor to determine the height of the tree. This project requires students to apply concepts they’ve studied in class in a real-world setting. Some students will likely benefit from the hands-on nature of this activity.

Assignment #2 – Bargain shopping for a birthday gift

Topic – Numbers & Operations (Decimals & Percents)
Grade Level – 6^th
Depth of Knowledge Level – 3

Description – Students will select an item to research that they would like to receive for their upcoming birthday. Using online resources, they will be required to find the best deal for their product, and then determine the final cost including tax and shipping. Then, students will be challenged to compare the affects of various coupons provided by the teacher (such as 5% off, $10 off, but 1 get one for half price, etc.) on the total cost of the item. This real-world problem will engage the students since they will get to research the gift of their choice. Also, throughout the activity they will have to determine which operations to use in order to calculate totals, tax, and discounts.

Assignment #3 – Graphing mix-up

Topic – Algebraic Relationships (Graphing & Equations)
Grade Level – 7^th
Depth of Knowledge – 2

Description – Students will be told that a personal trainer recently created several graphs from the workout data of her clients. Unfortunately, all of the data and graphs fell off her desk and got all mixed up. Students will have to match the graphs with the various forms of data such as tables, raw data, written descriptions, and algebraic equations. For example, students will have to analyze a written log of a client’s workout (“he started working out too hard and then had to take a 5 minute break before continuing his workout”) and match it to the graph that fits the situation. Also, there will be a few graphs missing that will have to be constructed by the students based on the data. This project requires interpretation of data, identifying patterns, comparing data and graphs, and creating graphs.

Assignment #4 – All about Skittles

Topic – Data Analysis (Measures of Center, Organizing Data, Graphing)
Grade Level – 6^th
Depth of Knowledge – 2

Description – Students will receive a bag of Skittles and will count/record how many of each color Skittle they have. Once the data is compiled for the whole class, students will analyze the data using a variety of measures including mean, median, mode, range, stem-and-leaf plots, bar graphs, circle graphs and line plots. Also, they will need to calculate what fraction (in simplest form) and percent of the class’s Skittles was each color. Technology can be integrated by using Excel to input and organize the data in the form of a circle graph or bar graph. This project helps students build a deeper understanding of various forms of data analysis using hands-on data collection.

Assignment #5 – Cooking a family recipe

Topic – Numbers & Operations (Fractions, Decimals, Unit Comparisons)
Grade Level – 6^th
Depth of Knowledge Level – 3

Description – Students will select a favorite recipe for this project. In addition to having to scale the recipe up (or down) for various numbers of servings, they will have to determine the cost per serving. Unit rate comparisons will be made for various ingredients within their recipes. For example, students will have to decide whether they should purchase 2-15 ounce cans of beans or 1-30 ounce can. This project will require students to analyze different ingredients, determine which mathematical operations are needed for their calculations, and connect their in-class learning to other situations. An extension of this project could be having students plan a food booth for a local festival and determine the price and expected profit based on different amounts of sales.

Assignment #6 – Creating an advertisement

Topic – Numbers & Operations (Ratios, Fractions, Percents)
Grade Level – 7^th
Depth of Knowledge Level – 3

Description – Students will begin by selecting two (or more) competing products (such as Coke & Pepsi). They will then create a survey that will be given to their classmates asking about their preferences and impressions of the products. The data they collect will be analyzed using fractions, percents, and ratios and students will use some of this data to create advertisements for each of the products. For example, if the survey results showed that 75% of the students preferred Pepsi over Coke, then this could be included in the advertisement. Students will create their advertisements on the computer using PowerPoint, Word, or a graphic-design application. At the end, students would present their ads to the class and then evaluate the persuasiveness of each advertisement. This activity requires significant strategic thinking and drawing conclusions from statistical analysis.

Assignment #7 – Designing the “perfect” box

Topic – Geometry (Surface Area, Volume)
Grade Level – 7^th
Depth of Knowledge Level – 3

Description – Students will be challenged to design to the ideal box to hold 24 building blocks to be sold in toy stores. They will have to consider the possible dimensions (1 x 24, 2 x 12, 3 x 8, and 4 x 6) and evaluate the strengths and weaknesses of each possibility. Comparison of each box’s surface area and volume will be required. Once they’ve selected a design, students will have to calculate the cost of constructing the box based on the cost of the materials being used. For example, students will be told that cardboard costs $0.17 per square yard and they’ll have to convert the surface area of their box from square inches (or square centimeters) to square yards. Students will get a better understanding of the considerations that businesses need to analyze when designing packaging for their products. They will also get a better grasp of how to create the most efficient packaging (a cube).

Backward Design Lesson
Mike Royal

CED 505 – Fall 2010

Unit Name: Bits & Pieces III	Students will be able to use decimals and percents to solve real-world problems involving addition, subtraction, multiplication, and division.
Student Expectations	Missouri GLEs Numbers & Operations: 1a - apply and understand whole numbers to millions, fractions and decimals to the thousandths (including location on the number line) 1b - recognize and generate equivalent forms of fractions, decimals and benchmark percents 2b - describe the effects of multiplication and division on fractions and decimals 3c - multiply and divide positive rational numbers 3d - estimate and justify the results of multiplication and division of positive rational numbers
Essential (Guiding) Questions	How can decimals and percents be used to better understand the world around us? What are some similarities and differences of fractions and decimals? How will multiplying and dividing decimals change their values?
Assessment	Students will choose a catalog and fill out an order form that will require them to calculate the total cost of the order including tax and shipping. All work will have to be shown for the decimal calculations. Also, students will have to determine the cost of an item if it were on sale for 25% off and another item that was 1/3 off. They will then have to compare two deals to see which one would be better (ex. $5 off or 20% off).
Learning and Teaching Activities
Introductory Activity	Begin by having a discussion with the students about examples of how decimals and percents are used in the real world. Use this discussion as a means of assessing students’ general level of understanding of decimals and percents. If students are struggling coming up with examples, you may want to give them general topics to think about such as shopping or measuring.
Instructional Activities	Learning activities include the following: · Estimating with Decimals: Estimating the total cost of several items by rounding decimals to the nearest half of a dollar. · Adding & Subtracting Decimals: This process is first demonstrated by linking adding fractions, which students have already mastered, to adding decimals. Then, students will solve problems involving the total amount of highway cleaned up by a combination of student volunteers. · Developing Algorithms for Adding & Subtracting Decimals: Students will use what they know about place values and fractions to create an algorithm that will help them solve any problem involving adding and subtracting decimals. · Multiplying Decimals: Students will solve a variety of problems involving shopping for apples (ex. 3.2 pounds at $1.70 per pound). · Factor-Product Relationships: Students will explore the effects of multiplying numbers by powers of ten (ex. 21x100, 21x10, 21x1, 21x0.1, 21x0.01). · Deciphering Decimal Situations: Students will explore various problems involving decimals while determining which operation should be used to solve each problem. · Using Common Denominators to Divide Decimals: Students will begin to learn how to divide decimals by changing them to fractions with common denominators and then dividing the numerators. · Dividing Decimals: Students will use the knowledge they’ve attained in the previous lessons to create algorithms for dividing decimals. · Determining Tax: Students will explore ways to calculate sales tax. · Computing Tips: Students will select items from a local restaurant’s menu and will then determine how much of a tip they would have to leave for the waiter. · Finding Bargains: Students will calculate the final cost of CDs after taking percent discounts off the original prices. · Clipping Coupons: Students will calculate percent discounts based on coupons in local papers. · Making Circle Graphs: Students will create circle graphs based on percent data.
Assessment Activities	Rubric for Summative Assessment 4+ EXEMPLARY RESPONSE • Complete, with clear, coherent explanations • Shows understanding of the mathematical concepts and procedures • Satisfies all essential conditions of the problem and goes beyond what is asked for in some unique way 4 COMPLETE RESPONSE • Complete, with clear, coherent explanations • Shows understanding of the mathematical concepts and procedures • Satisfies all essential conditions of the problem 3 REASONABLY COMPLETE RESPONSE • Reasonably complete; may lack detail in explanations • Shows understanding of most of the mathematical concepts and procedures • Satisfies most of the essential conditions of the problem 2 PARTIAL RESPONSE • Gives response; explanation may be unclear or lack detail • Shows some understanding of some of the mathematical concepts and procedures • Satisfies some essential conditions of the problem 1 INADEQUATE RESPONSE • Incomplete; explanation is insufficient or not understandable • Shows little understanding of the mathematical concepts and procedures • Fails to address essential conditions of problem 0 NO ATTEMPT • Irrelevant response • Does not attempt a solution • Does not address conditions of the problem
Student Product
Extension	Throughout the unit problems can be made easier or harder by changing the percents and tax rates. For example, for a special-needs student, they could be asked to solve a problem using a tax rate of 5% and discount of 25%, while more advanced students could be given a tax rate of 7.25% and discount of 12%. Also, special needs students could use calculators for some/all of this unit, while other students would not be allowed to use calculators. Also, for making circle graphs, data could be differentiated to meet the needs of various students. For example, special needs students might make a circle graph using the following data: 50%, 25%, 15%, 10%. An advanced student would be given more difficult data such as the following: 43%, 32%, 13%, 12%.

References:

Lappan, Fey, Fitzgerald, Friel, and Phillips. Bits and Pieces III - Computing with Decimals and Percents. Boston, MA: Pearson / Prentice Hall, 2009. Print. Connected Mathematics 2.

Backward Design

Teachers are planners. From the academic sequence for the entire year to the specific activities used in a daily lesson, teachers are always planning to best meet the needs of their students. It is important, however, that teachers take a purposeful approach when they plan their activities, lessons, units, and sequence. One way to improve the chances of creating meaningful, effective lessons involves using the planning process called “Backward Design.” As opposed to using existing activities to achieve specific learning goals, the Backward Design process begins by determining the desired end result and designing all of the learning activities around a few big ideas. This method of curriculum design, developed by Grant Wiggins & Jay McTighe, involves three main stages of planning. First, the teacher determines the primary understandings students should achieve by the end of the unit. Next, the teacher must design assessments that will measure students’ mastery of the big ideas. Finally, the teacher creates learning activities that will “promote understanding, interest, and excellence” (Tasmanian DOE, p. 1). By following the steps of Backward Design, teachers will be better equipped to plan meaningful lessons that efficiently achieve specific, measurable learner outcomes.

Before designing activities, lessons, or assessments, the teacher must begin by defining the learner objectives he aims for students to achieve. This big idea is often more broad than a specific grade-level expectation or content standard. This step in the curriculum-design process needs to focus on the enduring understandings students will build throughout the unit. As opposed to simply “knowing” content, “understanding” requires a more thorough mastery of the concepts. Students have a true understanding of a concept if they are able to explain, interpret, apply, have perspective, empathize and have self-knowledge (“Steps for Backward Design”). Typically, this step in the planning process also involves creating a few essential questions that will “guide student inquiry and focus instruction for uncovering the important ideas of the content” (Tasmanian DOE, p. 3). This initial stage of the planning process is of utmost importance since it will shape the direction of the unit, including the assessments and learning activities.

The second step in the Backward Design process is designing the summative assessment. Again, since this is being created before any learning activities, the teacher must have the essential questions and big ideas in mind when creating the assessment. While formative assessments throughout the unit will drive instruction, the summative must effectively gauge students’ mastery of the big ideas of the unit. In addition, summative assessments should “require the application of skills, concepts, and understandings, rather than a mere reporting of information” (“Steps for Backward Design”). While the summative assessment is a critical component of the unit, significant emphasis must also be placed on creating other meaningful formal and informal formative assessments such as quizzes and observations. When assessments are created and sequenced in a planned and purposeful manner, teachers can more effectively monitor student progress and design lessons that will guide students toward mastery of the intended content.

After determining the main themes and designing assessments within a unit, teachers must then plan lessons that will help students develop the skills they need in order to be able to complete the summative assessment. In addition to identifying specific skills that must be developed, teachers must also “design the sequence of learning experiences that students will undertake to develop understanding” (Tasmanian DOE, p. 6). A benefit of creating the summative assessment before the learning activities is that each activity can be designed in a way that will get students one step closer to the deep levels of understanding required to answer the essential questions. It is important that the learning activities are varied and require students to do authentic tasks that challenge them to explore the topic at hand. Finally, the fact that the end goal is determined before the lessons are constructed allows the teacher to scaffold in a way that will help students achieve success.

By using Backward Design and starting with the end goal in mind, teachers can create more cohesive and focused lessons. Students will be better prepared for the summative assessment, and will develop a deeper understanding of the concepts. In my school district, teachers are responsible for writing curriculum every six years. The Backward Design model is the starting point for all curriculum writing in our schools. One of the frustrating aspects of it is that it is difficult to integrate all of the Missouri grade-level expectations into the units. In order to make sure that our students are prepared for standardized tests, it is important to address each GLE at some point throughout the year. When starting out with a central theme or big idea, it is sometimes challenging to find enough GLEs that can be addressed to warrant the amount of instructional time the unit will take. Despite the steep learning curve to master Backward Design, the end result can be powerful, effective, and well-conceived lessons that will change the ways students think and teachers teach.

Bibliography

Australia. Tasmanian Department of Education. Principles of Backward Design. Web. 20 Nov. 2010. .

"Steps for Backward Design." Greece Central School District. Web. 20 Nov. 2010. .

Literature Review: How New Technologies Have (and Have Not) Changed Teaching and Learning in Schools

Despite the explosion of technological breakthroughs in the world around us, Halverson and Smith deliver sobering thoughts on the impact these tools are having in today’s classrooms. The school system has fought significant structural and pedagogical change even though the technology that is available today could have a much more positive impact on student achievement if such changes became the norm. While the authors note that the advances in technology have, in some cases, “fundamentally transformed schools,” many of these transformations are not considered ideal by classroom technology experts (Halverson & Smith, p. 49). In addition to exploring the ways our classrooms have changed with the influx of educational technology, the authors also examine how technology will impact the classrooms of tomorrow.

One important area of emphasis in this article is the authors’ attempt to differentiate between two types of educational technology. The first, technologies for learning, “support the interests of the technology designers” (Halverson & Smith, p. 49). This type of technology is prevalent in today’s classroom, and is usually designed around achieving specific learning goals. An educational math game that allows the user to practice fraction skills would be an example of a technology for learning. The other type of technology, technology for learners, allows for more creativity and input from the learner as to how it will be used to achieve the desired goals. As opposed to being specialized for achieving one particular goal, technology for learners is more flexible and better prepares students to be real-world problem solvers. Software that helps students create graphic organizers and idea maps would be an example of these multi-use, technology for learner tools.

The recent history of classroom technology has been full of both predictable and unexpected breakthroughs and failures. During the 1990s, schools pumped money into educational technology. Although this created an impressive infrastructure of technology tools, it did little to change the classroom practices in most schools. The highlight of the late 1990s was the spike in Internet availability in the classroom. While only 35% of public schools had Internet access in 1994, that number jumped to 97% by 2000 (Halverson & Smith, p. 50). Another positive outcome of the increased focus on classroom technology “was the development of a robust technology infrastructure to meet the demands of the high-stakes accountability policies of the 2000s” (Halverson & Smith, p. 50). Through the early 2000s, the vast majority of classroom technology was technology for learning as opposed to technology for learners. Classroom teaching techniques largely remained the same as they had been before technology entered the schools. Some of the greater technology changes occurred with schools’ administrative practices due to the need to address high-stakes accountability by doing more data analysis using student information systems. While many of the tools have changed in the last couple decades, life in the classroom remains remarkably similar.

Looking ahead, the authors suggest that technology may continue to be used for learning as opposed to for learners. Perhaps the tendency to purchase and utilize this type of technology is because it is easier to implement and has a specific, often measureable impact on particular learning outcomes. Another potential roadblock for using technology for learners is that it takes more training for teachers to successfully implement this type of technology. A tendency of technology for learners is that unintended, but often positive, goals may be achieved since it leads to more input from the learner as to the direction the learning progresses. In the current environment of data-driven instruction, when the results of a new technology initiative are uncertain, administrators and teachers will be less likely to recommend implementing the technology since it may not result in the intended outcomes. Despite the significant positive impact technologies for learners can have in the classroom, all signs point to technology for learning continuing to dominate the educational landscape unless philosophical changes affect how educational technology purchasing decisions are made.

In order to make the case for using technology for learners in the classroom, the authors focus on two impactful examples that have been successful. The first technology for learners that has had a positive impact on students is that of virtual charter schools. Most of these online schools have the following three main components: “structured content and assessment, online mentoring, and a learning management system” (Halverson & Smith, p. 51). These components facilitate the crucial characteristics of traditional schools that have, until recently, been impossible via an online course. The various communication tools, such as discussion boards, chat rooms, and video conferencing, allow for students to interact with the other students and teacher. These interactive sessions provide meaningful, collaborative learning experiences that closely mirror those of a face-to-face discussion. The second technology for learners discussed by the authors is participating in fantasy sports leagues. Despite the obvious entertainment components of fantasy sports, the online communities created in these leagues sometimes “blur the lines between learning and entertainment” (Halverson & Smith, p. 52). Success in a fantasy sports league requires participants to use various online resources to make informed decisions. Also, the virtual environment created for the league allows players to communicate with each other and share in the experience of the season. These two examples of technologies for learners show the potential for similar educational experiences when technology is used effectively in our classrooms.

The future of technology in education continues to be one of uncertainty. While some teachers and schools are already implementing technology for learners in ways that greatly change the teaching and learning experience, this is certainly not the norm. Technology continues to be underutilized in most classrooms, and oftentimes purchasing decisions are made to address specific goals as opposed to improving the all-around learning experience of the students. Continued focus on technology for learning will result in classrooms remaining disturbingly similar to those of many decades past. While technology could be used to reinvent classroom learning experiences, this would require teachers and administrators to step out of their comfort zones and take a chance. If implemented with proper professional development, technology for learners can engage our most distractible and excite our least interested students.

Bibliography:

Halverson, Richard, and Annette Smith. "How New Technologies Have (and Have Not) Changed Teaching and Learning in Schools." Journal of Computing in Teacher Education Winter 26.2 (2009): 49-54. Print.

Essential Questions Lesson Plan

Essential Questions Lesson

Michael Royal

CED 505

Fall 2010

Lesson Title:	Similarity Around Us
Grade:	Subject: 7th Grade Mathematics
Overview:	Students will explore how geometric similarity can be used to make sense of the world around us. They will apply the concepts learned in class to create and solve problems involving similar figures.
Essential Question:	How can the characteristics of similar figures help us better understand the world around us?
Subsidiary Questions:	What are some basic truths about sides, perimeters, and areas of similar figures? How can scale factors be used to find missing information about similar figures? What are some characteristics that are the same and different for similar figures? What is the difference between similarity and congruence?
Connection to Standards:	Missouri Grade Level Expectations Addressed: Geometric and Spatial Relationships (7th Grade) 1B - Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships: describe relationships between corresponding sides, corresponding angles and corresponding perimeters of similar polygons 3B - Apply transformations and use symmetry to analyze mathematical situations: describe the relationship between the scale factor and the perimeter of the image using a dilation (contractions- magnifications; stretching/shrinking) 4B - Use visualization, spatial reasoning and geometric modeling to solve problems: draw or use visual models to represent and solve problems
Activities, Tasks & Procedures:	After exploring the characteristics of similar geometric figures, the teacher will ask students the following questions: “What are some examples of similar figures in the real world?” and “How can the concepts we’ve practiced in class be applied to solve problems about similar figures in the real world?” Students will already have significant experience with finding missing lengths and angles of similar figures by calculating and applying the scale factor from one figure to another. If they are unable to see the connection between similar figures and real-world applications, sample problems such as these could be provided: - Cropping photos - Using photos to determine the height of another object - Determining the height of a tree/building/flagpole - Finding the height of a wall using a mirror - Creating a model of a larger object (ex. Model car) - Drawing a scale floor plan for a room/building After discussing examples of similar figures in the real world, students will be asked to select one of the examples of similar figures above to further explore. Their task will be to examine how geometric similarity is used in the real-world example they selected. They will write a mathematical problem that could be solved using similarity that pertains to their real-world example. They will be required to draw a scale diagram showing how their real-world example links to the similarity concepts developed in class. Throughout this process, the teacher will guide students as they make connections between what has been learned in class and their real-world example of similarity. They will provide students with help in the mathematics involved in each example as needed. Also assistance might be needed in drawing scale diagrams to depict each of the real-world examples. The technology that would be used during this lesson would be using a drawing, presentation, or word-processing program to make a diagram depicting the similar figures from the real-world example. For example, if a student decided to investigate how you can determine the height of the school’s flagpole using similar figures, their diagram would include the triangle created by the flagpole, its shadow and the line that connects the end of the shadow to the top of the flagpole. They would also need to draw the triangle of the other object they used to determine the height of the flagpole. Students could also use technology to investigate ways of applying the mathematics learned in class to the real-world similar figures they are researching. Once students have selected and explored a real-world situation that involves similar figures, they will be required to write a detailed explanation of their findings including the mathematical computations they used in their research.
Assessment:	Math - Problem Solving : Similarity Around Us Teacher Name: Mr. Royal Student Name: ________________________________________ CATEGORY Advanced - 4 Proficient - 3 Basic - 2 Below Basic - 1 Mathematical Reasoning Uses complex and refined mathematical reasoning. Uses effective mathematical reasoning Some evidence of mathematical reasoning. Little evidence of mathematical reasoning. Mathematical Errors 90-100% of the steps and solutions have no mathematical errors. Almost all (85-89%) of the steps and solutions have no mathematical errors. Most (75-84%) of the steps and solutions have no mathematical errors. More than 75% of the steps and solutions have mathematical errors. Diagrams and Sketches Diagrams and/or sketches are clear and greatly add to the reader's understanding of the procedure(s). Diagrams and/or sketches are clear and easy to understand. Diagrams and/or sketches are somewhat difficult to understand. Diagrams and/or sketches are difficult to understand or are not used. Explanation Explanation is detailed and clear. Explanation is clear. Explanation is a little difficult to understand, but includes critical components. Explanation is difficult to understand and is missing several components OR was not included.
Samples of Student Work:
Teacher Commentary Reflection:	I’m in the middle of a unit on similarity, ratios, and proportions with my 7th graders right now, and many of them struggle with the concept of scale factor. Also, I’m not sure how many of the students will be able to come up with examples of similar figures in the real world without significant prompting. I anticipate using this lesson as a final project at the end of the similarity unit. This lesson, and the required knowledge to complete it, addresses several GLEs and would require students to demonstrate their mastery of these important concepts.