The goal for today is to help students explore why fractions are equivalent.
I write the fraction 4/12 on the board. I ask students to give me a math vocabulary word for the number on top and a math vocabulary word for the number on bottom. I want to math sure my students understand the importance of saying the math vocabulary words. I also want see if they still have basic knowledge of how a fraction is set up. This will help me to adjust the complexity of this lesson.
I ask students would they like to have a slice of my chocolate supreme cake. Students love cake, so it is really a good way to engage them in this activity, and to get them thinking about what size their slice of cake will be.
Then I ask students to choose a size 2/8 or 4/16 (of the cake.) I invite students to surround the table. I want them to get a closer look at the different sizes. I carefully place a unit fraction by the size of cake it represents. I call out the fraction equivalent to each slice of cake, and encourage students to raise their hands when I call out the fraction size they would like to have. I ask, are all the sizes equal. Students are quick to say no! It is ok if they don’t understand that they are because at this point I am just determining what they already know. Meeting students where they are academically is a key asset for implementing a successful lesson.
In this lesson we will be using the following Mathematical Practices:
MP.2. Reason abstractly and quantitatively.
MP.4. Model with mathematics.
MP.7. Look for and make use of structure.
MP.8. Look for and express regularity in repeated reasoning.
I invite students to the carpet to model how to reduce fractions to their simplest form. This is important if I want students to see that a fraction reduced to it's lowest can in fact be equivalent to a given fraction.
Students need to be able to see a visual when referring to size. So, I represent 2/8 and 4/16 with an area model. However, I draw the model horizontal and draw a line through the middle. I am careful to draw each model equally, so that students can compare the size. The number of equal parts doubles and the size of parts is halved.
Students begin to notice connections between the model and fractions in the way both parts and wholes are counted. This concept helps when they begin to generate a rule for writing equivalent fractions. I continue to model this concept until students are responding mathematically when they generate their problem solving ideas.
Some students may need to hold the compared pieces in their hands to see the similarities. However, allowing students to practice will help them grasp the concept of how to reduce fractions into their simplest form.
Before moving into smaller groups I take a moment and ask students if can they think of another way to reduce fractions other than using illustrations? I also, ask students to think of a way to create a mathematical model that explains how to reduce fractions into simplest form, and can be used as a reference guide to assist them in learning about reducing fractions.
I give students about ten minutes or so to compare and explore the relationship among fractions. As students are working I ask them to compare two fractions with different denominators by creating common denominators or numerators. I explain that the models should be the same size, however, the sectioning should be equal and different.
Key questions to ask are: how are they alike and how are they different? Students notice they are the same size, however, the parts are different. Some are larger, and some are small. I ask student volunteers to write the fraction that represent their pictorial representation.
After students' time is up, I ask student volunteers to share what they have learned so far.
I ask students to move into their groups so that we can work together on reducing fractions to simplest form. I explain to them that they are going to be simplifying fractions. In order to simplify a fraction, the numerator and denominator has to be divisible by the same number. I ask students to identify the numerator and the denominator to make sure that they understand what they need to do.
I write the fraction 12/24 on the board. I ask students to think of an easy way to figure out what number goes into both numbers equally by using factors of 12 and 24. As students are responding I write down the numbers in large print on the board.
12 = 1, 2, 3, 4, 6 and 12
24 = 1, 2, 3, 4, 6, 8, 12 and 24
After I finish writing, I will have students compare the factors and find all the numbers that each number has in common. Some students listed 1, 2, 3, 4, 6, and 12. I explain that these are all common factors, but to find the fraction in simplest form, use the biggest common factor. I ask what the biggest common factors of 12 and 24 are. Students should clearly see that twelve is the largest common factor. I say, now that we have determined what the biggest common factor is, we will use that number and divide it by each part of the fraction. (Numerator and denominator).
I ask students to think of a way that we can divide 12 into the numerator and denominator using division. I want them to make the connection between fraction and division to help them better solve reducing fractions to simplest form.
Then I say, if we were to divide 12 and 24 by 12 what will the new numbers be. (So if 12/12 = 1 and 24/12 = 2) What will the reduced fraction be? (½)
To check for understanding I give them another example, but this time I pick a fraction that is already reduced. (1/3) I ask them to go through the same process and simplify the fraction. I want to see if students will realize that if the fraction cannot be reduced; there will be no common factors (besides 1) between the two numbers.
While students are working, I ask if they could explain why some fractions have the same common factors, and some fractions do not have the same common factors. Checking for connections and student understanding is critical throughout this lesson to see if students are mastering reducing fractions to simplest form.
I ask students to return to their seats. I give them problems to solve writing the fraction in simplest form. I also ask them to explain whether or not the fractions are equivalent. I ask students to illustrate and explain how they solve at least two of their given problems. I have them to write two of the problems and explanations in their math journals to use whenever they need help in reducing fractions and finding equivalent fractions.