Unit Plan- Maed

Catherine Shelfer

Ratio, Proportion, and Percent A Sixth Grade Mathematics Unit Plan

Created by Catherine Shelfer

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Catherine Shelfer

Table of Contents I.

Introduction and General Information

2

II.

Depth of Content Knowledge and Unit Content Content Standards Unit Summary Unit Plan Outline

5

III.

Subject Matter Content

7

IV.

Unit Goals

8

5 5 6

Enduring Understandings Essential Questions Students Will Know… Students Will Be Able To… V.

Content

8 8 8 9 10

Unit Summary Graphic Organizer Unit Planning Calendar

10 10 11

Acceptable Evidence Examples of Evidence Methods of Assessment Grading Outline

12

VII.

Lesson Plans Day 1—Ratio and Rates Day 2—Proportions Day 3—Proportions and Measurement Day 4—Similar Figures and Indirect Measures Day 5—Mid-Unit Quiz and Scale Drawings and Maps Days 6—Percents, Decimals, and Fractions Day 7—Using Percents Day 8—Unit Review Day 9—Unit Test

14 14 16 18 19 21 22 24 26 26

VIII.

Appendix

27

IX.

Summative Reflection

45

VI.

12 12 13

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Catherine Shelfer

Introduction The following unit plan is designed for a sixth grade mathematics class. The unit will take place over nine days and it will cover ratio, proportion, and percent. Specifically, we will look at: ratios and rates; proportions and customary measurements; similar figures; indirect measurements; scale drawings and maps; percent, decimals, and fractions; and percent applications. Students need to learn about ratio, proportion, and rate because these topics are useful in everyday life, for example, computing test scores or lottery odds, tipping in a restaurant, converting measurements, reading maps, and comparing statistics. Also, ratio, proportion, and percent are a basic concepts in understanding more advanced mathematics principles and applications. The unit is designed to reach all learners. The lessons are paced and "scaffolded" so that students of any level can master the skills, and then apply the skill with critical thinking. There are many places for students to ask questions and questions are encouraged. In addition, there are multiple opportunities to reach all learners through journals, examples, models, group activities, pair projects, graphic organizers, etc. Throughout the unit, a variety of instructional strategies are used to accommodate various learning styles or special needs. The curriculum to be covered in this unit is in-line with the North Carolina Standard Course of Study for 6th grade mathematics. The first competency goal the unit will cover outlines the understanding and computation of rational numbers. Students will develop meaning for percents which involves connecting the variety of representations as well as making estimates in appropriate situations. In addition, students will compare and order rational numbers and should develop fluency in addition, subtraction, multiplication and division of nonnegative rational numbers. Students will also estimate the results of computations and be able to judge the reasonableness of solutions. Finally, the students will develop flexibility in solving problems by selecting strategies and using mental computation, estimation, calculators or computer, and paper and pencil. The second goal the unit covers will teach students to demonstrate an understanding of simple algebraic expressions. Students will use graphs, tables and symbols to model and solve problems involving rate and ratios. The goal of the unit is to provide students with the lessons needed for a full understanding of percents. 3

Catherine Shelfer

The students will be able to write ratios and rates, find unit rates, write and solve proportions, use ratios to identify similar figures and use proportion and similar figures to find unknown measures. Students will also be able to read and use map scales and scale drawing. They will learn to write percents as decimals and fractions and write decimals and fractions as percents. Students will be able to apply their knowledge of percents, ratios and proportions to solve a variety of complex problems by selecting the appropriate strategy. They will also be able to relate ratio, proportion, and percent to previous topics and to future problems. Finally, the unit will teach students to compare, analyze, and interpret data using ratios, proportion and percent, create related charts and graphs, and use their knowledge to identify situations in the real world or classroom where ratio, proportion or percent is used to solve or simplify. After the implementation of this unit plan students will have clear knowledge of ratio, proportion, and percent and understand that they are a means of simplifying and organizing for comparison. They will know that percent is a ratio compared to 100, that a ratio is a comparison of two quantities using division, a proportion is an equation that shows two equivalent ratios, and that proportions make conversions in measurements and identify similar figures. Lastly, students will have knowledge that proportions and similar figures are used to find unknown measures of items we cannot measure directly and acknowledge that a scale drawing is of a real object that is proportionally smaller or larger than the real object.

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Catherine Shelfer

Step 1: General Information Unit Title: Proportional Reasoning Unit Topic: Ratio, Proportion, and Percent Course Content: Mathematics Grade Level: 6 Length of Class Time: 85 minutes Length of Unit: 9 Days Student Population, Characteristics, and Accommodations: Based on Clinical Observation, Community House Middle School, 6th grade population In the two classes that I observed, there are three LEP students who work with a specialist outside of class and have access to additional notes, if needed. However, they are expected to take their own notes and participate. There are three students with a 504 plan. Two students have ADD and one has Dyslexia. All of the plans allow for extra time on tests, preferential seating, and access to notes. Average Class Size: 28 students RACE % ENROLLED African American 11.5 White 66.7 Asian 9 Hispanic 10 American Indian less than 1 Multi-Racial 2.7 The unit will be taught to a sixth grade mathematics class of mixed academic levels. The unit contains seven different lessons regarding ratio, proportion, and percent, and will take nine days to complete.

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Catherine Shelfer

Step 2: Demonstrating Depth of Content Knowledge and Defining Unit Content Link to Content Standards: North Carolina Standard Course of Study Competency Goal 1: The learner will understand and compute with rational numbers. 1.02 Develop meaning for percents. a) Connect the model, number word, and number using a variety of representations. b) Make estimates in appropriate situations. 1.03 Compare and order rational numbers. 1.04 Develop fluency in addition, subtraction, multiplication, and division of nonnegative rational numbers. a) Estimate the results of computations. b) Judge the reasonableness of solutions. 1.07 Develop flexibility in solving problems by selecting strategies and using mental computation, estimation, calculators or computers, and paper and pencil. Competency Goal 5: The learner will demonstrate an understanding of simple algebraic expressions. 5.04 Use graphs, tables, and symbols to model and solve problems involving rates of change and ratios. Summary of Unit: The unit will cover ratio, proportion, and percent. Specifically, we will look at: ratios and rates; proportions and customary measurements; similar figures; indirect measurements; scale drawings and maps; percent, decimals, and fractions; and percent applications. Students will be able to… 1. Write ratios and rates and find unit rates 2. Write and solve proportions 3. Use ratios to identify similar figures 4. Use proportion and similar figures to find unknown measures 5. Read and use map scales and scale drawings 6. Write percents as decimals and fractions 7. Write decimals and fractions as percents 8. Find the missing value in a percent problem 9. Apply their knowledge of percents, ratios, and proportions to solve a variety complex problems by selecting appropriate strategies 10. Estimate, when appropriate, using ratio, proportion, and percent 11. Relate ratio, proportion, and percent to previous topics and to future problems 12. Compare, analyze, and interpret data using ratio, proportion, and percent 13. Identify situations in the real world or class room where ratio, proportion, or percent is used to solve or simplify 14. Read and create graphs that involve ratio, proportion, and percent

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Catherine Shelfer

Required Prior Knowledge: 1. Division/multiplication 2. Number line 3. Decimals, rounding decimals 4. Fractions, reducing fractions 5. Base 10 number system Important Subject Matter Elements 1. Percent means “out of 100” 2. Fractions, decimals, and percents can express the same number in different forms 3. A ratio is a comparison of two numbers; a percent is a number compared to 100 4. A proportion makes two ratios equal 5. Charts, graphs, and diagrams display data 6. Similar figures have the same shape but not necessarily the same size 7. A scale drawing is of a real object that is proportionally smaller or larger than the real object Unit Plan Outline Day 1—Ratio and Rates Students will write ratios and rates and find unit rates Day 2—Proportions Students write and solve proportions using ratios Day 3—Proportions and Measurement Students use ratios and proportions to find measurements Day 4—Similar Figures and Indirect Measures Students use ratios to identify similar figures Students use proportions and similar figures to find unknown measures Day 5—Short Mid-Unit Quiz and Scale Drawings and Maps 8 question quiz from days 1 to 4 Students read about and use map scales and scale drawings Days 6—Percents, Decimals, and Fractions Students use ratio and proportion to write percent as decimals and fractions Students use ratio and proportion to write decimals and fractions as percent Day 7—Using Percents Students find missing values in real-world percents problems Day 8—Unit Review Review study guide and answer questions about unit Day 9—Unit Test

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Step 3: Subject Matter Content • •

• • • •

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Textbook: Middle School Math: Course 1 Worksheets from Middle School Math: Course 1, Resource Book Graphic Organizers and Charts Activity from www.LearnNC.org Activity from National Council of Teachers of Mathematics website http://illuminations.nctm.org Teacher resources include worksheets, activities, models, word problems, and diagrams. PowerPoint and Excel

Bibliography Holt Middle School Math Course 1, Chapter 8, Resource Book. New York: Holt Rinehart and Winston, 2002. Illuminations: Welcome to Illuminations. The National Council of Teachers of Mathematics. 12 October 2009 . LEARN NC. 12 October 2009 . Middle School Math Course 1 Algebra Readiness with CD ROM (Middle School Math). New York: Holt, 2004.

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Step 4: Indentifying Unit Goals Enduring Understandings: Students will understand… 1. Math allows us to organize data for easier comparison 2. Basic mathematic principles can simplify situations, so we can often solve a more complex problem 3. Math concepts translate into real world scenarios; a variety of applications and adaptations must be employed to solve problems according to their constraints 4. Estimations are helpful in everyday life, especially as a consumer 5. Mathematics is a process; what we learn today builds upon previous lessons and serves a base for future lessons Essential Questions 1. How does percent relate to our previous lesson on ratio? How do we use ratios when solving percent problems? 2. How are ratios related to proportions? 3. How are equivalent ratios like equivalent fractions? 4. Why does 55% rest between 0 and 1 on the number line? 5. Tell of a time (other than assignments and tests) when you will use your knowledge of ratio, proportion, and percent. 6. How and when can we use our math knowledge in the world of sports? 7. As a consumer, when is our knowledge of estimation useful? 8. When would you use scale drawings? 9. When would you use indirect measure? 10. Pretend you are an English teacher, when would you use ratio, proportion, and percent in your daily activities? 11. Why do I need to know ratio, proportion, and percent? Students will know… 1. Ratio, proportion, and percent are a means of simplifying and organizing data for comparison 2. A variety of representations for numbers between 0 and 1 (decimal, fraction, percent) 3. Percent is a ratio compared to 100 4. A ratio is a comparison of two quantities with division 5. A proportion is an equation that shows two equivalent ratios 6. Proportions make conversions in measurements and identify similar figures 7. Proportions and similar figures are used to find unknown measures of items we cannot measure directly 8. A scale drawing is of a real object that is proportionally smaller or larger than the real object

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Catherine Shelfer

Students will be able to… 1. Write ratios and rates and find unit rates 2. Write and solve proportions 3. Use ratios to identify similar figures 4. Use proportion and similar figures to find unknown measures 5. Read and use map scales and scale drawings 6. Write percents as decimals and fractions 7. Write decimals and fractions as percents 8. Find the missing value in a percent problem 9. Apply their knowledge of percents, ratios, and proportions to solve a variety complex problems by selecting appropriate strategies 10. Estimate, when appropriate, using ratio, proportion, and percent 11. Relate ratio, proportion, and percent to previous topics and to future problems 12. Compare, analyze, and interpret data using ratio, proportion, and percent 13. Identify situations in the real world or class room where ratio, proportion, or percent is used to solve or simplify 14. Read and create graphs that involve ratio, proportion, and percent

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Catherine Shelfer

Step 5: Scope and Sequencing of Content Unit Summary The unit will cover ratio, proportion, and percent. First, students will learn about ratios and rates. Students will learn how to find a unit rate and write ratios. Then, students will extend their knowledge of ratios by writing and solving proportions. Students will learn that a proportion is made up of two equivalent ratios. Also, students will apply their knowledge of proportions to convert measurements, identify similar figures, calculate indirect measures, and explore scale drawings. Next, the unit moves into percents. Students learn the definition of percent and how fractions and decimals relate to percents. Then, the students use their knowledge of ratios and proportions to calculate percents. Finally, the students will investigate real-world applications of percents.

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Catherine Shelfer

Unit Planning Calendar Day 1 Ratio and Rates Skill and Concepts: Students will write ratios and rates and find unit rates Important Subject Matter: A ratio is a comparison of two quantities with division Essential Questions: How are equivalent ratios like equivalent fractions? Vocabulary: rate, ratio, unit rate

Day 4 Similar Figures and Indirect Measures Skill and Concepts: Students use ratios to identify similar figures Students use proportions and similar figures to find unknown measures Important Subject Matter: Similar figures have the same shape but not necessarily the same size Proportions and similar figures are used to find unknown measures of items we cannot measure directly Essential Questions: When would you use indirect measure? Vocabulary: congruent, similar figure

Day 2 Proportions Skill and Concepts: Students write and solve proportions Important Subject Matter: A proportion makes two ratios equal Essential Questions: How are ratios related to proportions? Vocabulary: proportion

Day 5 Mid-Unit Quiz Scale Drawings and Maps Skill and Concepts: Students read and use map scales and scale drawings Important Subject Matter: A scale drawing is of a real object that is proportionally smaller or larger than the real object Essential Questions: When would you use scale drawings? Vocabulary: scale

Day 3 Proportion & Measurements Skill and Concepts: Students use proportions to find measurements Important Subject Matter: Proportions make conversions in measurements Essential Questions: Tell of time when knowledge of proportions and measurements might be helpful.

Day 6 Percents, Decimals, & Fractions Skill and Concepts: Students write percent as decimals and fractions Students write decimals and fractions as percent Important Subject Matter: Percent means “out of 100” Fractions, decimals, and percents can express the same number in different forms A percent is a number compared to 100 Essential Questions: How does percent relate to our previous lesson on ratio? How do we use ratios when solving percent problems? Why does 55% rest between 0 and 1 on the number line? Vocabulary: percent

Day 7 Using Percents

Skill and Concepts: Students find missing values in percents problems Students solve percents problems that relate to real life Important Subject Matter: Math concepts translate into real world scenarios; a variety of applications and adaptations must be employed to solve problems according to their constraints Charts, graphs, and diagrams display and compare data Essential Questions: Tell of a time when you will use your knowledge of ratio, proportion, and percent. How and when can we use our math knowledge in the world of sports? As a consumer, when is our knowledge of estimation useful?

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Catherine Shelfer

Step 6: Determining Acceptable Evidence Examples of acceptable evidence 1. Explains how to represent percentages by either moving the decimal two spaces to the right or by multiplying by 100 2. Reveals understanding that a percent in less than 1 when asked to mark a percent on the number line 3. Estimates the tip at a restaurant 4. Constructs and/or interprets a pie chart that reflects the results of a class election 5. Calculates net income and creates a monthly budget 6. Analyzes and compares the same products at different stores, considering discounts and tax 7. Recognizes and records uses of percent in day-to-to activities 8. Solves percent problem using ratio and proportion 9. Converts a group of fractions, decimals, and percents to the same form and organizes from greatest to least 10. Uses proportion and similar figures to find the measure of something that could not be directly measured 11. Identifies instances when indirect measure is important 12. Determines the unit rate of items in a store and identifies the importance of unit rate as a consumer 13. Sets up and solves proportions 14. Applies proportion knowledge to various situations, such as identifying similar figures, finding indirect measures, converting measurements, determining percent, etc Methods of Assessment 1. Diagnostic “Are You Ready?” Worksheet will be assigned the week before we begin the unit on ratio, proportion, and percent. The worksheet will assess: vocabulary, simplifying fractions, writing equivalent fractions, writing fractions as decimals, writing decimals as fractions, and multiplying decimals. 2. Formative Formative assessments are included in the essential questions and checks for understanding. I will observe and interact with the students during their group activities in selected lessons; therefore, I will see their thought processes and possible misconceptions. Also, students will work examples (guided by me) throughout each lesson, either in their notes or on the board (if they volunteer). During the examples we will discuss each step and why we are taking that step. Students self-assess their learning and understanding during the math journals, reflections, KWL, and vocabulary chart. 3. Summative Daily homework, in class activities, warm-ups, a mid-unit quiz, lesson reflections, math journal, class participation, and a unit test will be used as summative assessments.

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Grading Outline Assessment 5 homework assignments Reflection Vocabulary Self-Awareness Chart 4 Warm-Ups Currency Activity Shadow Activity Basketball Challenge 2 Math Journal Entries Mid-Unit Quiz GIST Think-Pair-Share Sale, Sale, Sale Activity Unit Test Class participation Total

Points/Assignment 10 10 40 10 20 30 20 10 30 10 20 50 100 10

Total Points 50 10 40 40 20 30 20 20 30 10 20 50 100 10 450

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Step 7: Unit Strategies and Activities Day 1—Ratio and Rates • Objective: Students will be able to write ratios and rates and find unit rates •

•

Materials: KWL Chart, overhead projector/Smart Board, Vocabulary Chart Procedure ○ Attention Grabber with KWL: Have you ever heard the term rate or ratios? When? How was the word used in a sentence? Allow the entire class to brainstorm and share what they KNOW about rate and ratio; write information in the "K" column of the KWL chart on overhead projector or Smart Board. After brainstorming, ask students what they WANT TO KNOW about rate and ratio. Record questions in the "W" column of the KWL chart. After lesson, come back to the KWL chart. Ask students what they LEARNED about rate and ratio and fill in the "L" column of the chart. See if any student can answer the questions posed in the "W" column. ○ Ratio Comparison of two quantities using division Compare two groups using division Example: Number of Dogs at Kennel, by Breed ✔ Pugs 8 ✔ Mixed Breeds 10 ✔ Poodles 9 ✔ Boston Terriers 4 ✔ Labradors 12 ✔ Maltese 5 ✔ ✔ ✔ Compare the number of pugs with the number of poodles. ✔ Can be written three ways 8 to 9 8:9 89 ✔ Order is very important! Pug to Poodles is different than Poodles to Pugs Ratios can compare: ✔ A part to a part ✔ A part to the whole ✔ The whole to a part Practice writing ratios using dog breed chart. ✔ Maltese to Labradors 5 to 12 5:12 512 ✔ Mixed Breeds to total number of dogs at the kennel 10 to 48 10:48 1048 ✔ Total number of dogs at the kennel to Boston Terriers 48 to 4 48:4 484 ○ Equivalent Ratios Ratios that name the same comparison. Found by multiplying or dividing both terms of a ratio by the same number. Example: ♥♥♥♥  ✔ Write three equivalent ratios to compare the number of stars to the number of hearts. 15

Catherine Shelfer Number of starsNumber of hearts

○ Rate

= 46 There are 4 stars and 6 hearts

4÷26÷2

= 23 There are two stars for every 3 hearts

4×36×3

= 1218 If you triple the ratio, there will be 12 stars for every 18 hearts

Compares two quantities that have different units of measure Example: A 15 ounce can of soup costs $1.35. Rate = pricenumber of ounces = $1.3515 Unit Rate ✔ When the comparison is to one unit ✔ Divide both terms by the second term ✔ Unit Rate = $1.35÷1515÷15 = $0.091 $0.09 for 1 ounce of soup ✔ Compare unit rates of two or more items to find the better deal ✔ Which is a better deal? A 2 liter bottle of soda that costs $2.02 or a 3 liter bottle of soda that costs $2.79? $2.022 liters = $1.011 liter

$2.793 liters = $0.931 liter

○ Reflection: Define ratio and give examples of ratios. Explain how to find equivalent ratios. Why is the ratio 2 cats: 6 dogs different from the ratio 6 dogs: 2 cats? •

Add terms to vocabulary self-awareness chart: rate, ratio, unit rate

○ Homework: Exercises from text book •

Closure: Return to the KWL chart. Ask students what they LEARNED about rate and ratio and fill in the "L" column of the chart. See if any student can answer the questions posed in the "W" column

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Catherine Shelfer

Day 2—Proportions • Objective: Students will be able to write and solve proportions •

Materials: Vocabulary Chart, “Patriotic Proportion” worksheets

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Procedure ○ Warm up with ratios Brown Bears Giraffes Monkeys

3 Write each ratio: giraffes to monkeys = 2:17 2 polar bears to all bears = 4:7 17 monkeys to all animals = 17:26 4 all animals to all bears = 26:7

Polar Bears ○ Proportion

An equations that shows two equivalent ratios 21

= 42

42

= 126

21

= 63

Cross Multiplication to find missing values in proportions Example: 34

= n16  Cross Multiply

3×16 = 4×n  Products are equal 48 = 4n  4 is multiplied by n 484

= 4n4  Divide both sides by 4 to undo multiplication

n = 12 Write a proportion for the following:      Find the missing values in the proportions: 86

= n3 (n = 4)

t5

= 2820

p40 ○

(t = 7)

= 38 (p = 15)

Small group activity: Currency Exercise from Holt’s Middle School Math: Course 1

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Catherine Shelfer

“The value of the US Dollar as compared to the values of currencies from other countries changes every day. The graph shows the recent value of various currencies compared to the US Dollar. Use the graph to answer the following questions. 1. What is the value of 9.72 European Euros in US Dollars? 2. You have $100 US Dollars. Determine how much money this is in Euros, Canadian Dollars, and Mexican pesos. 3. A watch in Israel costs 82 shekels. In the US, the watch costs $25. In what country does it cost less? 4. Would you prefer to have five US Dollars or five Canadian Dollars? Why?” •

Add terms to vocabulary self-awareness chart: proportion

•

“Patriotic Proportion” challenge worksheet for homework

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Closure: Give an example of a proportion. Tell how you know it is proportion.

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Day 3—Proportions and Measurement • Objective: Students will be able to make conversions within the customary system. •

Materials: “Basketball Challenge” worksheets, Math Journals

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Procedure ○ Warm up with missing values in proportion 41 = 24x (x=6) x30 = 56 (x=25) 1416 = x40 (x=35) 8x = 1218 (x=12)

○ Attention Grabber: How many centimeters in 1 meter? (100) How many centimeters are in 4 meters? (400) Demonstrate proportion with this info. (This shows that proportions convert measurements.) 100 cm1 m = 400 cm4 m ○ Review common customary measurements Length 1 foot = 12 inches 1 yard = 3 feet 1 mile = 5280 feet

○

Weight 1 pound = 16 ounces 1 ton = 2000 pounds

Time 1 minute = 60 seconds 1 hour = 60 minutes 1 day = 24 hours 1 week = 7 days 1 year = 365 days 1 year = 12 months

Capacity 1 cup = 8 fluid ounces 1 pint = 2 cups 1 quart = 2 pints 1 gallon = 16 cups

Examples: The units must be in the same order in both ratios 1 yard3 feet = 185 yardsx feet (x = 555 feet) 1 Hour60 minutes = x Hour360 minutes (x = 6 hours) 1 ton2000 pounds = x tons55,000 pounds (x = 27.5 tons)

○ “Basketball Challenge” worksheet in pairs in class •

Students write about proportion and measurement in Math Journal for homework

•

Closure: Describe a situation in which you would need to convert measurements.

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Catherine Shelfer

Day 4—Similar Figures and Indirect Measures • Objective: Students will be able to use ratios to identify similar figures; Students will be able to use proportions and similar figures to find unknown measures •

Materials: Protractors, Overhead Projector or Smart Board, Shadow Activity sheets, Vocabulary Chart, Discussion Web, yard sticks, string, scissors, calculators

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Procedure ○ Warm up: Students draw two congruent angles with protractors. Ask how they know they are congruent. (Same measure) Then draw and label two pairs of line segments: 1 inch and 2 inch, 3 inch and 6 inch. Ask students if the measurements of these lines can form a proportion. Yes, 12 = 36 ○ Attention Grabber with Discussion Web: On the board, make a list of things that are too tall for us to measure. Pick one really tall item from the list, let's say The Empire State Building Display the Discussion Web on a projector or Smart Board. Write the question in the center of the discussion web: Can we find the height of The Empire State Building using our mathematical knowledge? Let students answer the question with yes or no, but they MUST give a reason. Write the reasons in the corresponding columns on the web. Return to the chart after the Shadow Activity to fill in and reflect on the conclusion. ○ Similar Figures Two or more figures are similar if they have exactly the same shape. They may be different sizes. Corresponding sides have proportional lengths Corresponding angles are congruent Example ✔ Draw and label two similar rectangles on the board.

✔ Locate and discuss the corresponding sides and corresponding angles. Sides: Angles: AB corresponds to WX A corresponds to W BC corresponds to XY B corresponds to X CD corresponds to YZ C corresponds to Y AD corresponds to WZ D corresponds Z ✔ Set up proportions with the data. ABWX = ADWZ or 26 = 39

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If you cannot use corresponding side lengths to write proportions, or if the corresponding angles are not congruent, then the figures are not similar. Example: Missing Values with Similar Figures ✔ The two triangles are similar. Find the missing length x and the measure of angle b.

812 = 6x (x = 9 cm)

Angle A is congruent to angle B, so angle B must equal 65° ○ Indirect measure One way to find height that you cannot measure directly is to use similar figures and proportions On a sunny day, a tree cast a shadow that was 228 feet long. A 6 foot tall man standing near the tree casts a 12 foot long shadow.

Both the person and the tree form right angles with the ground and their shadows are cast at the angle. What does this mean? We can form two similar right triangles and use proportions to find the missing height. 6h = 12228 h = 114 feet ○ Shadow Activity in pairs From National Council of Teachers of Mathematics http://illuminations.nctm.org •

Add terms to vocabulary self-awareness chart: congruent, similar figure

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Answer Shadow Activity questions for homework

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Closure: Return to the discussion web after the Shadow Activity to fill in and reflect on the conclusion

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Day 5—Short Mid-Unit Quiz and Scale Drawings and Maps • Materials: Quizzes, GIST sheets, Math Journals, Vocabulary Charts •

Procedure ○ Administer 8 question quiz from days 1 to 4 ○ Early finishers begin reading lesson on scale drawings with GIST strategy , then write about scale maps in Math Journal

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Add terms to vocabulary self-awareness chart: scale

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Math Journal for homework: Write about scale maps in Math Journal.

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Students only need to be familiar with the scale drawing and map section. They should know that it is related to proportion.

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Closure: How is the scale on a map useful? If the scale is 1 inch = 5 inches, what does this mean? How can this info be used? How is this related to our previous lessons?

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Catherine Shelfer

Days 6—Percents, Decimals, and Fractions • Objective: Students will be able to express percent as decimals and fractions and express decimals and fractions as percent •

Materials: Common Equivalents Charts sheets, Vocabulary Charts, “Problem Solving” worksheets

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Procedure ○ Warm-Up: Review fractions and decimals Write the following fraction as a decimal: ¾, 9/10 Write the following decimal as a fraction: 0.375, 0.05 Reduce the following fractions: 8/10, 21/63 ○ Attention Grabber: Where and when have you seen/heard percentages in everyday life? Does anyone know anything about percent? ○ Percent a percent is a ratio whose second term is always 100 means per 100 Using ratio terms of “part to whole”, the percent is the part and the whole is 100 100% means “the whole thing” Example: 8% means 8 out of 100 or 8100 10x10 grid has 100 squares 8 out of 100 squares are shaded to show 8%

Percent is also a fraction whose denominator is 100 Example: Changing percent to fraction and reducing Given 40%, express as a ratio 40100 40100 is also a fraction which can be reduced to 25 A percent is less than 1 Draw number line on board. Who can show me where 50% lies on the number line? Why does it lie there? Percent can be represented as a decimal Example: Moving decimals using base 10 knowledge to show less than 1 Water frozen in glaciers makes up 75% of the worlds fresh water supply. Write 75% as a decimal. 75% = 75100 75100 means 75.00 ÷ 100.00 We move decimal two places to the left  00.75 ÷ 1.00 = .75 .75 is less than 1 23

Catherine Shelfer

Fractions decimals and percents appear in real life. To understand the data, we should be able to change from one form to another. Examples: Write the decimal as percent using place value 0.3 0.3 = 310  3 × 1010×10 = 30100 = 30% Write the decimal as percent using multiplication by 100 0.7431 0.7431 x 100= 74.31% (move decimal two place to the left) Write the fraction as a percent—denominator is factor of 100 45 4× 205×20 = 80100= 80%

Write the fraction as a percent—denominator is not a factor of 100 38

3 ÷ 8 = 0.375 (use division or calculator) 0.375 = 37.5% (move decimal two places to the left) ○

Students complete common fractions, percents, and decimals chart in pairs Fraction

15

14

13

25

12

35

23

34

45

Decimal

0.2

0.25

0.333

0.4

0.5

0.6

0.666

0.75

0.8

Percent

20%

25%

33.3%

40%

50%

60%

66.6%

75%

80%

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Add terms to vocabulary self-awareness chart: percent

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"Problem Solving" worksheet for homework

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Closure: Which method do you prefer for converting decimals to percent? Why?

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Catherine Shelfer

Day 7—Using Percents • Objective: Students will be able to calculate missing values in real-world percents problems •

Materials: “Sale, Sale, Sale” instructions, calculators, “Unit Review” worksheets, construction paper, rulers

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Procedure ○ Review of percent and ratio and proportion What do you remember from yesterday’s lesson about percent? What is a percent? Can anyone tell or show me what a ratio is? Can anyone tell or show me how a proportion is related to ratio? ○ Attention Grabber: Does anyone know how to tip a waiter? ○ Percents of numbers other than 100 can be found with ratios and proportion and multiplication Examples: What is 80% of 40? Set up proportion with given info 80100 = x40 Cross multiply and solve for x. 100x = 3200 x = 32 Find 20% of 150. 20% = 0.20  write the percent as a decimal 0.20 x 150 = 30  multiply using the decimal A shirt is $20 and sales tax is 8%. How much will the sales tax be? Set up proportion with provided info 8100 = x20 Cross multiply and solve. 160=100x x = $1.60 Mary is downloading a file from the internet. So far 75% of the file has downloaded. If 30 minutes has passed since she began, how long will it take to download the rest of the file? 75100 = 30m  30 minutes is a part (75%) of the entire time needed to download 3,000= 75m 40 = m The time needed to download the entire file is 40 minutes. So far, the file has been downloading 30 minutes. Because 30-40 = 10, the remainder of the file will download in 10 minutes. ○ Percents can be represented in charts and graphs and percent can be calculated from info on charts and graphs. ○ Pie chart displays percent very effectively… the entire pie represents 100 and the pieces represent the percents. Example: Results of a vote for 6th graders t-shirt color 60% want blue 16% want red 20% want green 4% want purple (chart will be projected) 25

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If 10 kids voted for green shirts, how many kids are in the class? Since 10 kids represent “part” of the class, we are looking for the “whole” this time. Set up proportion 20100 = 10x Cross multiply and solve for x. 20x = 1000 x = 50 ○ Estimation of percent is appropriate in some instances, round numbers for easier calculation ○

Calculating mentally: 10% of a number move decimal one place to the left

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1% of a number move decimal two places to the left

○ 5% of a number, half of 10% ○ Estimation can serve as method to check our work Example: A store sign reads “10% off the regular price.” If Nicole wants to buy a CD whose regular price is $14.99, about how much will she pay for it after discount? Round $14.99 to $15 10% of $15 = $1.50 move decimal one place to the left Discount = $1.50 $15 - $1.50 = $13.50  Subtract discount from original price ○ "Sale, Sale, Sale!!" Activity in groups of 4 From LearnNC.org ○

If students finish early, then they can start on their homework.

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"Unit Review" worksheet for homework

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Closure: What do you think is important to know about percent? Why do you need to understand percent?

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Day 8—Unit Review • Materials: “Sale, Sale, Sale” instructions and handouts, calculators, Vocabulary Charts, completed “Unit Review” worksheets, Math Journals •

Procedure ○ Finish “Sale, Sale, Sale” activity, if needed ○ Review, complete, correct, revise vocabulary self-awareness charts ○ Review study guide and answer questions about unit ○ Share reflections and math journals

Day 9—Unit Test

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Vocabulary Self Awareness Chart Word

+

√

-

Example

Definition

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Procedure: 1. Examine the list of words you have written in the first column. 2. Put a “+” next to each word you know well, and give an accurate example and definition of the word. Your definition and example must relate to what we are studying. 3. Put a “√ “ next to any words for which you can write only a definition or example, but not both. 4. Put a “-“ next to words that are new to you. This chart will be used throughout our unit. By the end of the unit you should have the entire chart completed. Since you will be revising this chart, write in pencil.

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Length 1 foot = ____ inches

Weight 1 pound = ____ ounces

1 yard = ____ feet 1 ton = _____ pounds 1 yard = ____ inches 1 mile = _____ feet 1 mile = ____ yards Time 1 minute = ____ seconds

Capacity 1 cup = ____ fluid ounces

1 hour = ____ minutes 1 pint = ____ cups 1 day = ____ hours

1 quart = ____ pints

1 week = ____ days

1 gallon = ____ cups

1 year = ____ days 1 year = ____ months

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Fraction

14

12

Decimal

0.2

0.33…

0.4

Percent

20% 25% 33.3…% 40% 50%

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0.6

0.66 …

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0.75 75% 80%

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“Sale, Sale, Sale!” Activity (from LearnNC.org)

1. Students work in groups of 4. 2. The teacher models a scenario, buying pizza. As guided practice, the class works on the problem together to find the new sale price offered by each restaurant, and then applies the NC 6% sales tax for total cost. Pizza Hut: $13.99 at 20% off + tax = $11.86 Dominos: $14.00 at 1/4 off + tax = $11.13 Franks: Pay only 3/5 of $16.50 + tax = $10.49 3. Students are given the sale ads of retailers and their competitors. Students are directed to use the example on the board as a model for organization and layout of information. Each product to purchase has three retailers for which to compare and this could take a whole side of a piece of paper (8.5 X 11) divided into 3 sections using a ruler. 4. The goal is to find the best deal.

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Summative Reflection Before this mathematics education course, I had studied lesson planning and pedagogy, but not specifically for mathematics. Of course teachers across the curriculum use similar planning tactics but mathematics has special requirements. And, to be honest, until this course, I am not sure that I even considered mathematics’ unique needs. Throughout the course, we examined an array of factors that affect student learning in mathematics, factors such as classroom management, motivation, manipulatives/technology, standards, questioning/communication, homework, cooperative learning, assessments, etc. Prior to this course, I had not thought about ALL of these factors and the effects they have on each other. All of these factors create a tangled web that we, teachers, must consider when planning for our students. For instance, we must plan lessons to reach as many students as possible with thoughtful use of manipulatives, while managing student behavior, asking meaningful questions, and meeting standards! On top of that, we must keep track of grades, communicate with parents, and keep students motivated. In general, I learned that there are many aspects that enhance or hinder student learning in the mathematics’ classroom. And we must consider these aspects during our planning in order to provide an effective learning experience for our students. In addition to the aforementioned factors, we must also consider the combination of characteristics that our students possess. These combinations make each student unique and contribute to their learning needs or preferences. Thus, it is essential that we know our students, so we can plan appropriate learning experiences for them. I believe that it is important to know our students as individuals. We must challenge our students and encourage them; show them that we believe in them. We must acknowledge and embrace the students’ differences but treat all of them fairly and as an individual.

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Many instances in the course served as reflection-points for me and forced me to ponder how I would address the many factors effecting mathematics education. For example, until recently, I never thought about how I would use manipulatives in my classroom, or if my way was the most beneficial to my students. Additionally, I learned a lot from the ideas of my classmates. The discussion boards challenged my opinions and showed me other people’s perspectives. After reading about and discussing the variety of topics in this course, I feel like I am knowledgeable on many aspects that effect mathematical learning and know how to plan or respond properly. Now, I understand that an effective mathematics educator must consider a combination of factors and the effects of those factors on students. Furthermore, to effectively teach to a diverse group of students, we must know and embrace the characteristics that make our students unique. Our students’ needs and the world of education are ever-changing. It is our responsibility to stay abreast of the issues in mathematics education and constantly communicate with our peers and students in order to be the most effective teachers we can be.

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