Unit 2 Table of Contents Unit 2: Fractions and Mixed Numbers Learning Objectives 2.2 Instructor Notes The Mathematics of Fractions and Mixed Numbers Teaching Tips: Challenges and Approaches Additional Resources Instructor Overview Tutor Simulation: Increasing the Size of a Recipe Instructor Overview Puzzle: Fraction Matchin' Instructor Overview Project: Let's Get Cooking! 2.4 2.11 2.12 2.14 Common Core Standards 2.21 Some rights reserved. See our complete Terms of Use. Monterey Institute for Technology and Education (MITE) 2012 To see these and all other available Instructor Resources, visit the NROC Network. 2.1
Unit 2 Learning Objectives Unit 2: Fractions and Mixed Numbers Lesson 1: Introduction to Fractions and Mixed Numbers Topic 1: Introduction to Fractions and Mixed Numbers Learning Objectives Identify the numerator and denominator of a fraction. Represent a fraction as part of a whole or part of a set. Topic 2: Proper and Improper Fractions Learning Objectives Identify proper and improper fractions. Change improper fractions to mixed numbers. Change mixed numbers to improper fractions. Topic 3: Factors and Primes Learning Objectives Recognize (by using the divisibility rule) if a number is divisible by 2, 3, 4, 5, 6, 9, or 10. Find the factors of a number. Determine whether a number is prime, composite, or neither. Find the prime factorization of a number. Topic 4: Simplifying Fractions Learning Objectives Find an equivalent fraction with a given denominator. Simplify a fraction to lowest terms. Topic 5: Comparing Fractions Learning Objectives Determine whether two fractions are equivalent. Use > or < to compare fractions. Lesson 2: Multiplying and Dividing Fractions and Mixed Numbers Topic 1: Multiplying Fractions and Mixed Numbers Learning Objectives Multiply two or more fractions. Multiply a fraction by a whole number. Multiply two or more mixed numbers. Solve application problems that require multiplication of fractions or mixed numbers. 2.2
Topic 2: Dividing Fractions and Mixed Numbers Learning Objectives Find the reciprocal of a number. Divide two fractions. Divide two mixed numbers. Divide fractions, mixed numbers, and whole numbers. Solve application problems that require division of fractions or mixed numbers. Lesson 3: Adding and Subtracting Fractions and Mixed Numbers Topic 1: Adding Fractions and Mixed Numbers Learning Objectives Add fractions with like denominators. Find the least common multiple (LCM) of two or more numbers. Find the common denominator of fractions with unlike denominators. Add fractions with unlike denominators. Add mixed numbers with like and unlike denominators. Solve application problems that require the addition of fractions or mixed numbers. Topic 2: Subtracting Fractions and Mixed Numbers Learning Objectives Subtract fractions with like and unlike denominators. Subtract mixed numbers without regrouping. Subtract mixed numbers with regrouping. Solve application problems that require the subtraction of fractions or mixed numbers. 2.3
Unit 2 Instructor Notes Unit 2: Fractions and Mixed Numbers Instructor Notes The Mathematics of Fractions and Mixed Numbers This unit tackles one of the most disliked topics in basic mathematics fractions. As much as students will want to avoid them, fractions will crop up in every part of this and every other math course and are surprisingly common in the outside world as well. By the time they've finished Unit 2, students will be well on their way to accepting, perhaps even appreciating, fractions. They'll know the terminology of fractions and how to simplify, compare, add, subtract, multiply, and divide these kinds of numbers. They'll also have seen numerous examples of the application of fractions to real-life situations. Teaching Tips: Challenges and Approaches Fear of Fractions The main challenge in teaching fractions is simply overcoming your students' almost universal fear of them. Fractions look complicated and can't easily be punched into a standard calculator, so to most developmental math students, they'll seem to be far more trouble than they're worth. The best way to overcome this reluctance to deal with fractions is to tie them into everyday experiences familiarity breeds confidence. If students understand where fractions are going to be encountered and how they are relevant, they'll be more eager to understand them. For example, one of the most common uses of fractions is in cooking and recipes all your students have eaten, and many of them have cooked. So in the topic texts for this unit we often use food-related examples to define terms and illustrate fraction problems. First, a measuring cup reminds students what fractions are: 2.4
[From Lesson 1, Topic 1, Topic Text] Later, pizzas are used to help define improper fractions and mixed numbers: [From Lesson 1, Topic 2, Topic Text] 2.5
Relating fractions to common and comfortable situations like these reassures students that learning about fractions is neither frightening nor foolish. Divisibility Rules Even after students get over their reluctance, some will have trouble carrying out calculations with fractions. Finding common denominators and adding and subtracting fractions are the most challenging issues. We suggest reviewing the divisibility rules before tackling these procedures. The divisibility rules for 2, 3, 4, 5, 6, 9 and 10 make it much easier to simplify fractions and will also help students find common denominators quickly. The rules are explained and illustrated this way in the course: [From Lesson 1, Topic 3, Topic Text] Common Denominators Common denominators must be found to compare, add, and subtract fractions. Most students can do this fairly easily when the denominators are small, such as 3 and 4 or 4 and 6. But, what 2.6
if the denominators are larger, like 12 and 15? Many students cannot sift through this problem in their heads. We suggest providing students with several alternate methods for finding common denominators: using the divisibility rules, listing multiples of each denominator, and finding the prime factorizations. Operations The hardest challenge is likely to be adding and subtracting fractions. In the past, students learned how to first add, then subtract, multiply and finally divide fractions, in the same order they learned to carry out these operations with whole numbers. Currently, the practice is to teach multiplication and division of fractions before addition and subtraction. Multiplication and division are easier because there's no need to find a common denominator. In this course, we follow this new sequence. Once students are adept at doing a few operations on fractions, learning the last two procedures won't be quite so intimidating. Multiplication of fractions is definitely the easiest of the four basic operations. Be sure to explain that students should simplify fractions and convert mixed numbers to improper fractions before multiplying. Remind them that a whole number can be turned into a fraction by putting it over one without changing its value. Once students understand how to multiply with fractions, they'll have little trouble with division, since these problems can also be solved by multiplying. Just teach them a phrase like keep change flip to help them remember how (keep the first fraction (or mixed or whole number), change the division sign to a multiplication sign, and flip the second fraction). It can be helpful to remind students that they will need to put a whole number over one before finding the reciprocal (i.e. flipping). Also, a mixed number will need to be converted to an improper fraction before finding the reciprocal. Some students will say that the reciprocal of can t just flip the fractional part of the mixed number. 3 2 8 is 8 2 3 you Students generally have the most difficulty with adding and subtracting fractions. In a problem with like denominators, some will want to add both the numerators and denominators. You can point out that two apples added to five apples is seven apples. No matter how many apples you add, they're still apples. Similarly, two-ninths added to five-ninths is seven-ninths, not seveneighteenths, because no matter how many ninths you add, they're still ninths. When unlike denominators appear, some students will again try to solve these problems by adding both the numerators and the denominators. Others will struggle to find common denominators. Encourage them first to work with the divisibility rules or factorization. Some students have learned to find a common denominator by multiplying all the denominators in the problem together. This is okay to do, but then the answer will likely need to be simplified, a step that students often forget. 2.7
The same overall techniques for addition of fractions apply to subtraction find a common denominator, convert the fractions to this common denominator form, and then subtract the numerators. But there is a good chance some regrouping might have to be done. This regrouping process, for example renaming 8 as 8 7 8 or 2 6 3 as 5 5 3, needs to be explained in fact, it is beneficial to give a series of examples of regrouping before even getting to a subtraction problem. An unfortunate result of teaching students that common denominators are needed to add or subtract fractions is that they then try to use them when the next multiplication or division problem comes along. A reminder that common denominators are not needed when multiplying or dividing fractions might be necessary. Keep In Mind All students will have studied fractions before. One thing that can be counted on with developmental students is that if they haven t used fractions for any length of time, their skills will be rusty. Typically they will also recall not liking fractions and they will avoid using them whenever possible. When the study of algebra is started and the answer to an algebraic 1 equation is x, you can count on getting the answer x.14. Why? Students are just 7 more comfortable with decimals as compared to fractions. At this point, it must be explained to 1 students that x and x.14 are not equivalent answers. 7 Students sometimes confuse the numerator and denominator of a fraction. Pointing out that denominator and down both begin with d may help. In this course, improper fractions should be simplified to either whole or mixed numbers. It should be noted that because proper fractions have a numerator that's less than the denominator, this means a fraction equivalent to 1, like 7, is an improper fraction. 7 Remind students to check that all their fractions are behaving "properly" by making sure that an improper fraction is not left as a final answer. In every problem presented in the course materials, students will see that answers are simplified: 2.8
[From Lesson 2, Topic 1, Presentation] It may help students to understand how to convert improper fractions to mixed numbers by pointing out that the process is just a division problem: [From Lesson 2, Topic 1, Topic Text] 2.9
Additional Resources In all mathematics, the best way to really learn new skills and ideas is repetition. Problem solving is woven into every aspect of this course each topic includes warm-up, practice, and review problems for students to solve on their own. The presentations, worked examples, and topic texts demonstrate how to tackle even more problems. But practice makes perfect, and some students will benefit from additional work. A good website to review all the skills learned in this unit can be found at http://www.aaastudy.com/fra.htm. After a brief review, an applet tests knowledge on many different fraction topics. Practice with simplifying and comparing fractions can be found at http://www.learningplanet.com/sam/ff/index.asp and more help can be found at http://www.visualfractions.com/comparel/comparel.html for comparing fractions. Knowing factors will help students find common denominators. A good practice for factor trees (which are obtained from the prime factors of a number) can be found at http://www.cut-theknot.org/curriculum/arithmetic/btreetesting.shtml. Summary The goal of this unit is to help students get over their fear of fractions and learn how to work with these numbers. After completing these lessons, they'll know how to simplify, add, subtract, multiply, and divide fractions. Students will understand and appreciate fractions more easily if they're shown how frequently their knowledge of fractions can be used in everyday life. 2.10
Unit 2: Fractions and Mixed Numbers Purpose Instructor Overview Tutor Simulation: Increasing the Size of a Recipe Unit 2 Tutor Simulation This simulation allows students to demonstrate their understanding of fractions and mixed numbers. Students will be asked to apply what they have learned to solve a problem involving: Adding Fractions Multiplying Fractions Dividing Fractions Applying Fractions to Real-World Situations Problem Students are presented with the following problem: You are working part-time in a bakery. You have to bake a lot of loaves of bread, but the recipes only give the ingredient amounts for one loaf. Your challenge will be to work with the fractions in the bread recipes and figure out the ingredients you'll need to bake multiple loaves. Recommendations Tutor simulations are designed to give students a chance to assess their understanding of unit material in a personal, risk-free situation. Before directing students to the simulation, make sure they have completed all other unit material explain the mechanics of tutor simulations o Students will be given a problem and then guided through its solution by a video tutor; o After each answer is chosen, students should wait for tutor feedback before continuing; o After the simulation is completed, students will be given an assessment of their efforts. If areas of concern are found, the students should review unit materials or seek help from their instructor. emphasize that this is an exploration, not an exam 2.11
Unit 2: Fractions and Mixed Numbers Instructor Overview Puzzle: Fraction Matchin' Unit 2 Puzzle Objectives Fraction Matchin' is a puzzle that mixes numerical and visual representations of fractions. The game tests a player's ability to recognize various forms of fractions and to convert mixed numbers into improper fractions. Figure 1. Players pick the pair of pie and shaded rectangle charts that match the central fraction. 2.12
Description Developmental Math An Open Program This puzzle has three levels, each with 10 games. Every game begins with a fraction in numeric form, surrounded by pie charts and shaded rectangle graphs. Players must pick the pair of graphs that represent the same value as the center fraction. In the 1st level, all given fractions are simple. In the 2nd level, all fractions are improper. In the 3rd level, mixed numbers are used. Players earn points by picking the correct pairs. Play does not proceed until the pairs are matched. Fraction Matchin' is suitable for both individual, group, and classroom play. 2.13
Unit 2: Fractions and Mixed Numbers Student Instructions Instructor Overview Project: Let's Get Cooking! Unit 2 Project Introduction Fractions are a large part of baking. Ingredient measurements are often given in quarter cup increments. In fact, small measurements are sometimes given in eighths of a teaspoon. In order to successfully bake, it is important to be able to work with fractions. Recipes will need to be doubled or halved depending on the quantity that needs to be made. Baking, therefore, requires an understanding of fractions. When you can work effortlessly with fractions you have the skills to bake up whatever you are in the mood for. Who doesn t like a warm chocolate chip cookie now and then? Task You are having a get together and are expecting 30 guests. You plan on serving Banana Bread, Chocolate Chip Cookies, and Sugar Cookies. Using the three recipes given, work with your group to create recipe cards to feed 30 people. Next, total up the ingredients needed. Then, check to see how much of each product needs to be purchased based on what is already on hand. Finally, create a display of your project that showcases your group s creativity. Instructions Complete each problem in order. Be sure to keep careful notes and save your work as you progress. You will work together to create a display at the end of the project. 1. Use your knowledge of fractions to re-write the recipe for Banana Bread. The card states that it serves 10 people. What will need to be done in order to make enough bread for 30 people? Show your work neatly and then re-write the recipe card. 2.14
2. Use your knowledge of fractions to re-write the recipe for Chocolate Chip Cookies. The card states that it makes 60 cookies. What will need to be done in order to make 30 cookies? Show your work neatly and then re-write the recipe card. 2.15
3. Use your knowledge of fractions to re-write the recipe for Sugar Cookies. The card states that it serves 20 people. What will need to be done in order to make enough to serve 30 people? Show your work neatly and then re-write the recipe card. 4. Use your new recipe cards to find the total amount of each ingredient needed. Use the table below to help you. Ingredient Recipe 1 + 2 + 3 (Don t forget to find common denominators before adding.) Total needed (Be sure to simplify any fractions.) Flour Sugar Butter Vanilla Baking Soda Eggs Salt Chocolate Chips Bananas 2.16
5. When taking inventory in the pantry, you found that you already have some of the ingredients. Use the following table to organize your work. Don t forget common denominators. HINT: If you need 5 eggs and you already have 2, how many do you need to buy? Write a number sentence to describe this situation. Which operation did you use? Use the same method to solve for the other ingredients. Don t forget common denominators. Ingredient Total needed from above Already in Pantry Needs to be bought Flour Sugar 1 3 cups 2 2 cups Butter Vanilla Baking Soda 3 cup 4 2 teaspoons 1 1 teaspoons 2 Eggs 2 Salt 1 teaspoon Chocolate Chips 1 pound 4 Bananas 4 Collaboration Compare your three recipe cards with another group. The cards should look identical. If a measured ingredient varies, look at the work to determine where the error was made. Then compare the table with total ingredients. Again, the table should look identical. Compare your work to find any errors. Finally, compare the last table. Do the answers agree? Are all fractions simplified? 2.17
Conclusion Developmental Math An Open Program Create a poster to display your work. Include the three new recipe cards, as well as the math work done to figure each new amount. Then include the table with the ingredient totals and the table with the amount to be purchased. Again, include the math on the poster. Finally, incorporate pictures and color to the poster to create a professional looking product. Instructor Notes Assignment Procedures Problem 2 1 This is a prime opportunity to discuss that dividing by two and multiplying by are equivalent 2 operations. Which operation will be the easiest and most reliable to perform? It depends on the 1 3 situation and the student. 4 divided by 2 seems easier than multiplying by ; however, 2 2 4 3 1 can be more difficult for some students than. Understanding that the operations are 4 2 equivalent can allow students to be more successful with fractions. Problem 3 3 Students may choose multiple methods to re-write this recipe card. They could multiply by or 2 they could multiply by 2 1 and then add the original amount. If time allows, discuss the multiple methods. Which method did the group choose? Why? Recommendations Have students work in teams to encourage brainstorming and cooperative learning. Assign a specific timeline for completion of the project that includes milestone dates. Provide students feedback as they complete each milestone. Ensure that each member of student groups has a specific job. Technology Integration This project provides abundant opportunities for technology integration, and gives students the chance to research and collaborate using online technology. The students instructions list several websites that provide information on numbering systems, game design, and graphics. 2.18
The following are other examples of free Internet resources that can be used to support this project: http://www.moodle.org An Open Source Course Management System (CMS), also known as a Learning Management System (LMS) or a Virtual Learning Environment (VLE). Moodle has become very popular among educators around the world as a tool for creating online dynamic websites for their students. http://www.wikispaces.com/site/for/teachers or http://pbworks.com/content/edu+overview Allows you to create a secure online Wiki workspace in about 60 seconds. Encourage classroom participation with interactive Wiki pages that students can view and edit from any computer. Share class resources and completed student work with parents. http://www.docs.google.com Allows students to collaborate in real-time from any computer. Google Docs provides free access and storage for word processing, spreadsheets, presentations, and surveys. This is ideal for group projects. http://why.openoffice.org/ The leading open-source office software suite for word processing, spreadsheets, presentations, graphics, databases and more. It can read and write files from other common office software packages like Microsoft Word or Excel and MacWorks. It can be downloaded and used completely free of charge for any purpose. Rubric Score Content Presentation/Communication 4 The solution shows a deep understanding of the problem including the ability to identify the appropriate mathematical concepts and the information necessary for its solution. The solution completely addresses all mathematical components presented in the task. The solution puts to use the underlying mathematical concepts upon which the task is designed and applies procedures accurately to correctly solve the problem and verify the results. Mathematically relevant observations and/or connections are made. There is a clear, effective explanation detailing how the problem is solved. All of the steps are included so that the reader does not need to infer how and why decisions were made. Mathematical representation is actively used as a means of communicating ideas related to the solution of the problem. There is precise and appropriate use of mathematical terminology and notation. Your project is professional looking with graphics and effective use of color. 2.19
3 2 1 Developmental Math An Open Program The solution shows that the student has a broad understanding of the problem and the major concepts necessary for its solution. The solution addresses all of the mathematical components presented in the task. The student uses a strategy that includes mathematical procedures and some mathematical reasoning that leads to a solution of the problem. Most parts of the project are correct with only minor mathematical errors. The solution is not complete indicating that parts of the problem are not understood. The solution addresses some, but not all of the mathematical components presented in the task. The student uses a strategy that is partially useful, and demonstrates some evidence of mathematical reasoning. Some parts of the project may be correct, but major errors are noted and the student could not completely carry out mathematical procedures. There is no solution, or the solution has no relationship to the task. No evidence of a strategy, procedure, or mathematical reasoning and/or uses a strategy that does not help solve the problem. The solution addresses none of the mathematical components presented in the task. There were so many errors in mathematical procedures that the problem could not be solved. There is a clear explanation. There is appropriate use of accurate mathematical representation. There is effective use of mathematical terminology and notation. Your project is neat with graphics and effective use of color. Your project is hard to follow because the material is presented in a manner that jumps around between unconnected topics. There is some use of appropriate mathematical representation. There is some use of mathematical terminology and notation appropriate to the problem. Your project contains low quality graphics and colors that do not add interest to the project. There is no explanation of the solution, the explanation cannot be understood or it is unrelated to the problem. There is no use or inappropriate use of mathematical representations (e.g. figures, diagrams, graphs, tables, etc.). There is no use, or mostly inappropriate use, of mathematical terminology and notation. Your project is missing graphics and uses little to no color. 2.20
Unit 2 Correlation to Common Core Standards Unit 2: Fractions and Mixed Numbers Common Core Standards Unit 2, Lesson 1, Topic 1: Introduction to Fractions and Mixed Numbers Grade: 8 - Adopted 2010 STRAND / DOMAIN CC.MP.8. Mathematical Practices CATEGORY / CLUSTER MP.8.3. Construct viable arguments and CATEGORY / CLUSTER MP.8.5. Use appropriate tools strategically. Grade: 9-12 - Adopted 2010 STRAND / DOMAIN CC.MP. Mathematical Practices CATEGORY / CLUSTER MP-3. Construct viable arguments and CATEGORY / CLUSTER MP-5. Use appropriate tools strategically. Unit 2, Lesson 1, Topic 2: Proper and Improper Fractions Grade: 8 - Adopted 2010 STRAND / DOMAIN CC.MP.8. Mathematical Practices CATEGORY / CLUSTER MP.8.3. Construct viable arguments and Grade: 9-12 - Adopted 2010 STRAND / DOMAIN CC.MP. Mathematical Practices CATEGORY / CLUSTER MP-3. Construct viable arguments and Unit 2, Lesson 1, Topic 3: Factors and Primes Grade: 8 - Adopted 2010 STRAND / DOMAIN CC.MP.8. Mathematical Practices 2.21
CATEGORY / CLUSTER MP.8.3. Construct viable arguments and Grade: 9-12 - Adopted 2010 STRAND / DOMAIN CC.MP. Mathematical Practices CATEGORY / CLUSTER MP-3. Construct viable arguments and Unit 2, Lesson 1, Topic 4: Simplifying Fractions Grade: 8 - Adopted 2010 STRAND / DOMAIN CC.MP.8. Mathematical Practices CATEGORY / CLUSTER MP.8.3. Construct viable arguments and Grade: 9-12 - Adopted 2010 STRAND / DOMAIN CC.MP. Mathematical Practices CATEGORY / CLUSTER MP-3. Construct viable arguments and Unit 2, Lesson 1, Topic 5: Comparing Fractions Grade: 8 - Adopted 2010 STRAND / DOMAIN CC.MP.8. Mathematical Practices CATEGORY / CLUSTER MP.8.3. Construct viable arguments and Grade: 9-12 - Adopted 2010 STRAND / DOMAIN CC.MP. Mathematical Practices CATEGORY / CLUSTER MP-3. Construct viable arguments and Unit 2, Lesson 2, Topic 1: Multiplying Fractions and Mixed Numbers Grade: 8 - Adopted 2010 STRAND / DOMAIN CC.MP.8. Mathematical Practices CATEGORY / CLUSTER MP.8.3. Construct viable arguments and Grade: 9-12 - Adopted 2010 STRAND / DOMAIN CC.MP. Mathematical Practices 2.22
CATEGORY / CLUSTER MP-3. Construct viable arguments and Unit 2, Lesson 2, Topic 2: Dividing Fractions and Mixed Numbers Grade: 8 - Adopted 2010 STRAND / DOMAIN CC.MP.8. Mathematical Practices CATEGORY / CLUSTER MP.8.3. Construct viable arguments and Grade: 9-12 - Adopted 2010 STRAND / DOMAIN CC.MP. Mathematical Practices CATEGORY / CLUSTER MP-3. Construct viable arguments and Unit 2, Lesson 3, Topic 1: Adding Fractions and Mixed Numbers Grade: 8 - Adopted 2010 STRAND / DOMAIN CC.MP.8. Mathematical Practices CATEGORY / CLUSTER MP.8.3. Construct viable arguments and Grade: 9-12 - Adopted 2010 STRAND / DOMAIN CC.MP. Mathematical Practices CATEGORY / CLUSTER MP-3. Construct viable arguments and Unit 2, Lesson 3, Topic 2: Subtracting Fractions and Mixed Numbers Grade: 8 - Adopted 2010 STRAND / DOMAIN CC.MP.8. Mathematical Practices CATEGORY / CLUSTER MP.8.3. Construct viable arguments and Grade: 9-12 - Adopted 2010 STRAND / DOMAIN CC.MP. Mathematical Practices CATEGORY / CLUSTER MP-3. Construct viable arguments and 2.23