Estimating With Fractions--Let's Draw!

Who: The following lesson/activity is geared towards 3rd through 5th grade students. This lesson is also great for students who like to draw.

Prior Knowledge: Students should have a good idea of what a fraction is. Students should also know how to divide a whole and a collection into fractional parts.

Lesson Objective: By the end of this lesson you will be able to draw pictures that represent fractions that are close to 0, 1/2 and 1.

Materials: You will need paper, crayons and a stencil if you have one!

Ready to Draw!

Step 1: Draw three identical polygons and number them 1, 2 and 3. For example, I could draw three triangles that are the same shape and size (use a stencil if you need to).

Step 2: In polygon #1, shade in an amount that would represent a fraction that is very close to 0. In polygon #2, shade in an amount that would represent the fraction 1/2. In polygon #3, shade in an amount that would represent a fraction that is very close to 1. These are your benchmark pictures that we will use for the rest of the activity.

Step 3: Look at the following fractions. Draw any polygons you want and shade in the amount that you think would represent the fraction. DO NOT separate your polygons into fractional pieces. For example, for the fraction 6/7, do not draw a rectangle and divide it up into 7 pieces and then shade in 6. Simply shade in the rectangle to the amount that you think would equal that particular fraction. Finally, place each picture under the benchmark picture you think it goes with, is it closest to 0, 1/2 or 1?

Here is your list of fractions:

1/10, 3/18, 19/20, 10/50, 5/10

3/12, 98/100, 4/12, 8/10, 8/15

2/7, 3/4, 12/20, 9/18, 21/50

Thinking Behind the Lesson:

Estimating fractions is a concept that is over looked far too often in textbooks, curriculums and even standards. However, this concept is so important for helping students develop a fractional number sense. Van de Walle states in his book Elementary and Middle School Mathematics: Teaching Developmentally, that students need to be able to know "about" how small or big a specific fraction is. The most important benchmarks for helping students acheive this goal are 0, 1/2 and 1 (Van de Walle p. 251).

This activity pushes students to think "about" how small or big a particular fraction is. It also reinforces these benchmarks which will help them in their development of a solid fractional number sense. While other activities may use manipulatives, such as Cuisinaire Rods, I designed an activity that would provide students with the opportunity to draw and create. It is important to include student choice as well as art into our lessons. My hope was that this activity would fulfill both of these important aspects.

Reflection:

*The above activity is helpful in pushing students to think critically about how small or big specific fractions are and where they land on the spectrum between 0 and 1. While this may be a great place for your child to start thinking about this concept, to make this activity more accessible for more struggling learners, you may want to help them draw their benchmark pictures. It may also be helpful for them to use fraction strips

or another maninpulative so that they can see the fraction (drawing the fraction may be too difficult at first, but may be more appropriate later on). This may help them place the fraction with the appropriate benchmark number.

*For students who are catching on quickly and mastering this concept, push them with more challenging fractions. Another way to challenge your student is to ask them to use mental math instead of drawing a picture. Make sure they justify their answer with a detailed explanation of how they placed their fraction on the spectrum between 0 and 1. You may also want to ask them follow-up questions after they state that their fraction is close to 1/2, for example. Ask them, is it more than 1/2 or less than 1/2. Ask them is it close to 1/4 or 3/4. These questions will push your students to think more critically.

References:

Van de Walle, J.A. (2004). Elementary and middle school mathematics: Teaching developmentally. Boston: Pearson.

Beginning to Add Fractions with Unlike Denominators!

Who: The following lesson/activity is geared towards 3rd through 5th grade students. This is also a great lesson for auditory and visual learners.

Prior Knowledge: Students should be able to add and subtract fractions with like denominators. Students should be able to change improper fractions to mixed numbers. Students should have be able to estimate fractions and state whether they are more or less than a half or more or less than a whole.

Lesson Objective: By the end of this lesson you will have a better understanding of how to add fractions with unlike denominators.

Materials: Get out paper and a pencil as well as your fraction strips!

Ready to Add!

For the following lesson I am going to direct you to a video that will introduce you to adding fractions with unlike denominators. The lesson will give you helpful tips and strategies for adding fractions in a real world scenario. Get ready to use your fraction number sense! After the video I will give you some more problems to try solving.

http://www.youtube.com/watch?v=UnMOM-_kMbQ&feature=relmfu

What a great lesson huh?!? I would recommend trying the 30-day trial! If not, we can use the same strategies that you learned during the lesson using your fraction strips.

Let's Practice!

To practice what you just learned, I want you to follow the steps below with each set of fractions.

Sets of Fractions:

A. 1/2 + 3/4
B. 4/5 + 1/10
C. 1/3 + 5/6
D. 3/8 + 1/2
E. 5/6 + 1/12
F. 8/10 + 1/2

1. Write a word problem to go along with your two fractions. Think about hte problem that the lesson used.

2. Ready to solve?!? Estimate first! Remember the estimation lesson we did and be sure to ask your self, is the fraction more than half or less than half. Is the fraction more than a whole or less than a whole? Once you ask yourself those questions for each fraction, go ahead and find a reasonable estimate for your problem.

3. Use your fraction strips to solve the problem like she did in the lesson OR find a common denominator! Watch the video again if you need to review what she did. Make sure you make your improper fractions proper!

4. Once you have your answer, compare it to your estimate, were you close?

5. Finally, write a complete sentence stating your answer! For example, T'mya and Sarah have 1 and 1/2 pizzas.

6. Post your word problems and answers on the blog so I can check them out!

Thinking Behind the Lesson:

Adding fractions with unlike denominators can be a tough concept to teach and a tough concept to learn. Before exploring this concept, students should be well on their way to developing a solid fractional number sense. In the past, I have jumped into adding and subtracting fractions too soon, before my students really understand what a fraction is and how it compares to a whole. This has been a mistake on my part and has been detrimental to my students' abililty to understand this concept. Yes, we can teach our students the algorithm for adding fractions with unlike denominators, but it is much more important for our students to truly understand what is happening.

In John A. Van de Walle's book, Elementary and Middle School Mathematics: Teaching Developmentally , he states that students should use a variety of models and strategies when teaching fraction computation. He discusses the importance of allowing students to explore different ways of solving the problems and to defend their solutions as these ideas will help children build a foundation and familiarity with fractions (Van de Walle p. 265). This short video exposes students to two different ways of solving an addition problem with unlike denominators. It also stresses the importance of estimation and fractional number sense.

Reflection:

*This can be a helpful introduction or reinforcement for introducing your student to adding fractions with unlike denominators. However, as Van de Walle states in his book (see above), it is important to expose our students to many different strategies and models when teaching fraction computation. Students will also need a lot of time to explore this concept in order to truly develop an understanding.

*Another great model or strategy for teaching this concept is by using Cuisinaire Rods. Check out this video:

http://www.youtube.com/watch?v=0BSvSGIWvMs&feature=relmfu

*In the practice part of this lesson, I have given fraction sets where students will only need to change 1 denominator. Once your student becomes familiar with the concept, challenge them by giving them two fractions where you would need to come up with a DIFFERENT common denominator. For example, 1/2 + 2/3.

*Continue to reinforce this concept through real world application. Cooking is a great place to do this!

References:

Van de Walle, J.A. (2004). Elementary and middle school mathematics: Teaching developmentally. Boston: Pearson.

Equivalent Fractions with a Hershey's Milk Chocolate Bar

Who: The following lesson/activity is geared towards 3rd through 5th grade students. This is also a great lesson for tactile and visual learners.

Prior Knowledge: Students should already understand the concept of what a fraction is. Students should know how to divide up a whole into halves, quarters, thirds, etc.

Lesson Objective: By the end of this lesson you will be able to define what an equivalent fraction is. You will also be able to find common equivalent fractions (such as fractions equal to 1/2, 1/3, 1/4).

Materials: You will need a BIG Hershey’s Milk Chocolate bar (one that has 12 pieces). It also might be helpful to have or make fraction strips, but this is not essential.

Ready to Find Equivalent Fractions

For this activity I am going to direct you to a GLOG I made that allows you to explore equivalent fractions with a Hershey's Chocolate Bar! Simply click on the link below. This link will take you to a Glog that I have made to help students discover and understand equivalent fractions. While this is a great activity, students who are new to this concept will need many other opportunities to engage with this topic.

http://kande9un.edu.glogster.com/fractions-are-fun/

Thinking Behind the Lesson:

While I do not like to motivate or reward children with food, relating mathematical concepts, especially fractions to familiar things can help students understand and make meaning of the learning more readily. Fortunately or unfortunately, food lends itself very well to teaching students about fractions. Whether it is cutting the pizza up or the Hershey Bar, students are immediately engaged and seem to grasp the concept more easily because they see how this relates to their everyday lives. Students are also able to practice and model this lesson with family and friends because the materials are readily available. Gardner emphasizes that making learning meaningful and accessible to all learners by teaching in a variety of ways, is essential to their learning (Slavin p. 118). Similar to other lessons, this activity is line with Gardner’s thoughts and theories. It not only engages the learner, but it is a great activity for visual and tactile learners as well! When learning about fractions, especially equivalent fractions, it is essential for students to be able to manipulate materials in order to discover how two different fractions can be equal to one another. This lesson is a great way for students to see and model this concept.

Reflection:

*In reflecting on this lesson/activity, I am wondering if this lesson/activity could have been improved by adding more visuals. I went back and forth on this, as I didn’t want to interfere with the self-discovery aspect of this lesson. The real objective of this lesson was to allow students to discover what an equivalent fraction was and what that looked like. I am hopeful that my guiding questions are not too prescriptive and that they still allow students the opportunity to explore and figure the concept out for themselves. If students are struggling with following the steps without visuals, please open the attachment at the end of the Glog, as these pictures should help students see how two different fractions can be equal to one another.

*If you want to make this lesson more discovery and exploratory based, do not read the book first. Have students do the activity first and then read the book after, as a way to reinforce the concept. The book is a way of getting students thinking about equivalent fractions and then the activity helps them solidify the concept a bit more. Think about the type of student you have and make your decision based on your learner.

*To allow students to practice and discover more equivalent fractions, either buy or cut up the fraction strips that are attached to the Glog. Simply ask students to come up with as many pairs of equivalent fractions as possible. Do not give them any other guidance besides that.

*Finally, ask your student to find equivalent fractions around the house. If they find some good examples, have them take pictures of them and attach them to the Blog. For example, I could take a picture of a carton of eggs that only has 6 eggs inside. This would be an example of how 6/12 = ½ .

References:

Slavin, Robert E. (2009). Educational psychology: Theory and practice. Upper Saddle River, New Jersey. Pearson.

Baking With Improper Fractions!

Who: The following lesson/activity is geared towards 3rd through 5th grade students. This is also a great lesson for tactile learners and those who like to bake!

Prior Knowledge: Students should already understand the concept of what a fraction is. Students should also be proficient at finding a fraction of a whole. Students should be familiar with what improper fractions and mixed numbers are, but they are not able to convert between the two.

Lesson Objective: By the end of this lesson you will be able to convert an improper fraction to a mixed number. I also hope that you will be able to describe this process to someone at home and explain its importance!

Ready to Bake!

To get started, I would like you to look at the recipe below and get out the following ingredients and baking tools that you will need, such as measuring cups, spoons, bowls, cooking sheets, etc.

Problem 1: The head baker for Fire Hook Bakery has a big problem! Her recipe for chocolate chip cookies got all messed up and she needs YOUR help! All of the measurements for the ingredients look really funny to her! She doesn't know what to do?!? I told her not to worry, that I had some really smart math students who could help her out!

So, let's check out this recipe and see if we can change it back to "normal" for her so she can bake her cookies!

11/4 cups all-purpose flour
4/3 teaspoon baking soda
3/2 teaspoon salt
1 cup (2 sticks) butter, softened
5/4 cup granulated sugar
5/4 teaspoon vanilla extract
2 large eggs
2 cups chocolate chips
7/4 cup chopped nuts

As you can see, this recipe is a little different looking than most recipes. If you are not sure what I mean, ask someone at home to show you a recipe for cookies. Compare these recipes, what do you notice is different? Do you notice that this recipe has NO mixed numbers and that they are all improper fractions? Well, in order for the baker to make these cookies, she needs these improper fractions to be converted to mixed numbers because right now she doesn't know how much of each ingredient to add and she can't make her cookies!

Please get out your flour and your measuring cups. How much flour do we need? That's right, 11/4. Now, why is this fraction an improper fraction? (Because the numerator is bigger than the denominator, right? And the fraction is then bigger than a whole!). So we need to change this improper fraction to a mixed number, so let's do some experimenting! What we need to do is figure out how many wholes we have and how much we have left over! So if we are talking about fourths, what measuring cup do we need? 1/4, good! I want you to see how many whole cups you can make with 11/4 and how many quarter cups you have left over. Hint: Scoop out 1/4 cups of flour and put this in your 1 cup measuring cup. Keep doing this until you have scooped out 11 quarter cups. Every time you fill up your 1 cup, record this on your paper, pour your flour into the bowl and then keep going until you have used 11/4 cups of flour.

How many whole cups did you make with your 11/4? 2 whole cups, right?
How many quarter cups were left over? 3, right? So that makes 3/4 of a cup, right?
So all together you have 2 wholes and 3/4 cups of flour or 2 3/4.

Great job! You just converted an improper fraction to a mixed number! Make sure you write this new amount down on your recipe!

The baker is going to be so happy! Now, I want you to do the same thing and change all of the improper fractions to mixed numbers! When you are finished, please submit the recipe to me and I will check your work! Have fun, be careful and enjoy your cookies!


Challenge Yourself!

If that was really fun and you want to challenge yourself, see if you can convert a mixed number to an improper fraction. Google any recipe that you would like to make and see if you can experiment with your ingredients to convert your mixed numbers to improper fractions!
Submit both recipes to me and I will check your work!


Thinking Behind the Lesson:

When I was in my first few years of teaching I always taught my students how to convert between improper and mixed numbers by using what I thought was a simple algorithm (because that is the way I was taught). Although some of my students caught on to this method, it was very frustrating for others and most of them did not understand the meaning behind it or what they were doing. They simply memorized a formula. I learned that I needed to make this concept much more visual and hands on for my students. I also learned that the more they figured things out for themselves, the better they would understand and retain the information. After reading about fraction concepts in John A. Van de Walle's book, I was even more convinced that I did not need to teach my students any conventional methods or rules for converting between the two. In his book, Van de Walle states that students should figure out how to convert between improper and mixed numbers by using any materials they wish or by drawing models. They should also be pushed to give an explanation and be able to justify how they arrived at their answer. He also goes on to state that there is no reason to provide a rule or algorithm, as students will develop their own rule AND it will be in their own words and with complete understanding (Van de Walle p. 260). Based on my experience and Van de Walle's methodology, I thought allowing students to figure out how to convert improper fractions to mixed numbers by experimenting with food would be a great way for them to gain an understanding of this concept and make learning more meaningful.

This activity is also very engaging for tactile students who like to bake or cook!

Reflections:

*While I recommend students have a good understanding of what improper fractions and mixed numbers are before taking part in this activity, as I reflect on this lesson, I think that this activity could also be a great way to introduce both of these concepts. For example, if students are given the fraction 11/4, they can use the flour and their measuring cups to figure out that once they use 4 quarter cups, they have reached a whole! This can then help them understand that 11/4 is more than a whole, and then they can be introduced to the vocabulary of improper fractions and mixed numbers.

*If your student is not a big fan of cooking, or if you do not want your student in the kitchen by themselves, you could always do this activity with non-edible ingredients such as dried beans, sand or other manipulatives.


References:

Van de Walle, J.A. (2006). Teaching student-centered mathematics: Grades k-3. Boston: Pearson.

Fractions of a Collection--Get Out Your Favorite Toys!

Who: The following lesson/activity is geared towards 3rd or 4th grade students who are not fluent with their multiplication facts. This is also a great lesson for tactile learners!

Prior Knowledge: Students should already understand the concept of what a fraction is. Students should also be proficient at finding a fraction of a whole.

Lesson Objective: By the end of this lesson you will be able to find a fraction of a collection. I also hope that you will be able to explain this process to someone at home and also apply this concept to a real life problem or situation!

Ready to Have Fun!

To get started, I would like you to get out some toys that you have a lot of. For example, you could get out Lego people or toy cars. You will need 12 of those toys.

Problem 1: Your best friend is coming over and you want to share some of your legos with him/her. Please find 1/2 of your collection. How many will you share with your friend? How many will you keep for yourself? Now I want you to think of what you just did to find 1/2 of your collection. Think it over, write down or discuss your strategy with someone at home. I want you to think about the following questions:

*Why did you make two seperate groups?
*Why did you give your friend 1 of those groups?

Now if you think you have a good understanding of what you did, try the next problem.

Problem 2: Your best friend is being kind of greedy and wants to play with more of the toys than you. He asks you for 2/3 of your toys. How many toys should you give him now? How many toys will you be left over with?

Think of the following questions:

*Think of what you did in the problem above.
*Think of what a fraction means or represents--(part of a whole, remember?)
*How many groups should you make?
*How many groups will you give your friend?
*How many toys are in those groups? (Can you skip count?)
*How many groups will you keep?
*How many toys will you play with? (Can you skip count?)

So, can we agree that 1/2 really means 1 out of 2 groups and 2/3 really means 2 out of 3 groups? If this is the case, can you figure out the following problem?

Problem 3: You really want to buy a new Lego set, however you don't have enough money! The lego set costs $30, but you only have 4/5 of that amount! How much money do you have? Hmmm, how can we solve this problem?

Think of the following questions:

*Can we use toys to represent the money?
*How many toys do you need to use?
*How many groups should you make with your toys?
*How many toys are in each group?
*How many groups represents the amount of money you have?
*How much money is that? Can you skip count by a certain number?
*How much MORE money do you need in order to buy the new Lego set?

Wow, you are doing awesome! Now here is your challenge! I need YOU to create a word problem and post in on the blog for me! I don't just want you to create it, but I want you to solve it and explain how you solved it too! Remember your complete sentences!

Thinking Behind the Lesson:

While there are many ways to teach a student how to find a fraction of a collection, including using multiplication, I like to make this lesson as engaging, applicable and hands on as I can! I find that teaching through Gardner's 9 intelligences

is a great way to teach any concept, but especially something tough like fractions. The more your student can touch and manipulate objects to understand the concept and make sense of the learning, the better! In this activity, your student is able to play with toys, understand why you would want to find a fraction of a collection, and physically figure it out for himself/herself. All of these components make the learning more meaningful for your student and allows fractions to come to life!

Reflections:

*While these problems above might be engaging and applicable for some students, they may be boring and unintersting to others. Pick a topic that is motivating for your student, and then rewrite the problems to suit his or her interests.

*While I did not reinforce this in the lesson, it is important to reinforce with your student when making groups they always need to be equal! For example, remind them that 1/2 of a collection means 1 collection divided into 2 EQUAL groups. This concept of equal groups will help them when finding fractions of a collection!

*These problems might move too fast for some students. You may want to stick with unit fractions for a while, until your student gets the concept down, such as 1/3, 1/4, 1/5 etc.

*When making groups, I also find that using mats are very useful. For example, you can fold a piece of paper into 4ths. If students are finding 2/3 of 12. They make 3 equal groups first with their chosen toy/manipulative. They can place the groups in 3 of the seperate boxes, this way they have a place for all of their manipulatives and groups, thus helping them organize better.

Make Fractions More Fun At Home!

I am creating this blog as a resource for home-schooled students, students who are home due to an illness or injury and these students' parents or caretakers. The following blog entries will be focused around lessons and activities that have to do with fractions. While some fraction concepts can be tough, boring and filled with silly algorithms that just don't make sense, I hope these lessons and activities will change your view of fractions, your comfort level with fractions and simply make fractions come to life for you! These lessons will be targeted at students in grades 3 - 5.

While I have a specific audience in mind, I hope this blog can come in handy for teachers, students and parents alike, who are looking for a different way to teach, reinforce or review fraction concepts. Enjoy, have fun and please don't hesitate to leave me feedback on how these lessons and activities went for you!