My students have been learning how to compare fractions with like and unlike denominators. To assess students' knowledge of fractions so far, I write two fractions with the same denominator on the board (3/12 and 6/12) I ask students to tell me how the fractions are alike. (Students determine that both fractions have the same denominator.) Then, I write a 9/24 which is equivalent to 3/12 and 6/12. I ask students how are 9/24 like the other two fractions. (Students are able to say that the fractions are equivalent.)
I may probe a bit more to see if they are ready to order fractions. For instance, I ask the following questions: What did you notice about the fractions? Explain? Can someone think of a way we can organize fractions by their size? What other math concept can we use to arrange fractions?
Students see that we can use greater than and less than to arrange numbers in order from least to greatest.
We will be using the following Mathematical Practices in this lesson:
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. Then I say, we have learned how to find equivalent fractions, and today you will learn how to use equivalent fractions to compare and order fractions.
Teacher and Student Talk:
How can you determine if two fractions are equivalent? (They are different names for the same amount.) I take note of students who are unable to respond to the given question. After that, I model how to determine if two fractions are equivalent.
The best way to think about equivalent fractions is that they are fractions that have the same overall value. Equivalent fractions represent the same part of a whole. For example, if we cut a pie exactly down the middle, into two equally sized pieces, one piece is the same as one half of the pie.
And if another pie (the same size) is cut into 4 equal pieces, then two pieces of that pie represent the same amount of pie that 1/2 did. So we can say that 1/2 is equivalent (or equal) to 2/4. I draw an illustration on the board to point out that it is the same size, however, the pieces are different.
Pose a Question:
Joe skated 2/3 mile, Carl skated 3/5 mile, and Pete skated ½ miles. Write these distances in order from least to greatest.
To model, I use large fraction strips to create models of the given fractions. Then I place the fraction strips side by side to show students a visual picture of the given order.
Then I write the fractions in order from least to greatest. I ask students to explain how they would solve this given problem using other solution methods. It is my goal to only model what students are expected to do at this point.
I ask them to arrange the fraction strips in order by size. (small to large)
In a whole group discussion I ask students to explain the given question.
How would you write 3/10, 5/10, and 4/10 in order from least to greatest?
I want to see if students will illustrate fraction strips to solve their problem, or will they be able to solve it on their own. I also want to check their understanding of ordering fractions. This helps me to adjust the complexity of the lesson.
After that, I pair students with a partner. I want them to have time ordering fractions from least to greatest. I give them a set of fraction strips, and I write several fractions on the board. I ask students to arrange the listed fractions in order from least to greatest. I notice some students use fraction strips to determine the size, and then write the fraction on their paper in the correct order. Some students ask is there another way to order fractions. I encourage them to explore new ways, however, the concept I am to help them understand is that using visual models is beneficial when ordering fractions. As students continue to work, I circle the room to check for understanding.
What did you notice?
Can you explain?
Did your partner find another way to order fractions? Explain?
Did the fraction strips help you? How?
Without the fraction strips it was kind of hard to determine which fraction was larger/ smaller.
After that, students are asked to return to their assigned seats. I remind students to list the fractions in order from least to greatest, not greatest to least, since this was one of the misconceptions I noticed while students were working in pairs. Then, I give each student their Work Assignment, and remind them to draw strips to assist them if they need help with understanding.
As students are working, I circle the room to monitor students' thinking. I take notes of any student who seem to be having difficulty understanding how to order fractions from least to greatest. I will use the notes to help clear up misunderstandings in a smaller group setting.