Review 2: Zooming In on a Number Line - Working with Rational Numbers
Lesson 2 of 7
Objective: Students will be able to find and position integers and rational numbers on a number line, find numbers that come between two other numbers.
See the Do Now video in my Strategy Folder for more details about how I begin class. Often, I create do nows that have problems that connect to the task that students will be working on that day. For this do now, I picked to multiple choice problems from my released state test that are important: using rates and graphs to answer a question and simplifying numerical expressions that include exponents.
For #1 I ask students to share out how they found the weight of 100 bags. Some students found the weight of one bag and then multiplied by 100. Other students picked one coordinate and used it strategically to find the weight of 100 bags. For example, a student saw that 10 bags weighed 600 pounds, so he created equivalent rates and multiplied 600 pounds by 10. I had one student saw that 30 bags would weigh 1,800 pounds. He multiplied 1,800 pounds by 3 to find the weight of 90 bags. He then looked at the graph and saw the weight of 10 pounds was 600 pounds. So he added 600 + 5,400. I think it is important to take some time to have students share their different strategies – it not only helps students to see other approaches but it helps me to see how my students think about and attack a problem.
For #2, I tell students that I think A is the correct answer. This exponent mistake is the one I see most often with my students. I have students share their thinking and help to identify the correct answer.
After the Do Now, I have a student read the objectives for the day. I ask a student what it means to “zoom in” on something. I share with students that we will practice zooming in on a number line and brainstorming numbers that will fall between two other numbers.
On page 2 I have students work independently for 3 minutes. I ask students to make sure their fractions are in simplest form. We go through the answers and review the procedure of changing one version of a number into another. We then take questions and share answers. Students will most likely struggle with the fractions that turn into repeating decimals (5/6, 2/3). I am looking for students to recognize a pattern and that after seeing a digit is repeating in the quotient 3-4 times that it is a repeating decimal. Here students are engaging with MP8: Look for and express regularity in repeated reasoning. I ask students:
- What does it mean to have a repeating decimal?
- How can you be sure that the same digit will keep repeating?
Number Line Review
For the number line review, I emphasize that on a horizontal number line the numbers get larger as you move from left to right. The numbers get smaller as you move right to left. I have students label the first two examples with a fraction and a decimal. For the number line at the bottom of the page I will place dots on the number line and ask students to label them. For example, I may place dots at 1 ½ , ½ , - ½, - 1 ½, -4 ½ and ask students to label them. A common mistake is for students to get confused when labeling a negative fraction like -1 ½ . A student may say that the dot is on -2 ½. I start with having the student identify the two integers on either side of the dot. Then I have the student use the two integers on either side to make a comparison the fraction is ____________(less than) -1 because it is to the left of -1. The fraction is ___________ (greater than) -2 because it is to the right of -2. So it must be -1 ½ .
After completing page 4 together, students complete the problems on page 4 independently or with a partner. Based on your students needs and levels create a list of number pairs that students can choose from to “zoom in”. Simple pairs would include whole numbers, while challenging pairs would include one decimal and one fraction. I stress to students that it is important that they are precise with their answers and are able to explain their thinking (MP6: Attend to precision).
During this time I circulate and check students work and look at the strategies they are using. I look for students who are splitting the number line in ½ and then in ½ again to find numbers; students who are extending out one decimal place, then using consecutive values; and using a larger denominator, then using consecutive numerators. I also look to see if students are more comfortable generating fractions or decimals, or if they are comfortable generating both. This will help me decide which examples to go over in the closure section of the lesson.
Once students complete their work on the zooming in they work independently on the multiple choice problems.
Closure and Ticket to Go
I ask students to stop their work and return to their zooming in work on page 5. I ask a student(s) who I saw using one of these strategies (splitting the number line in ½ and then in ½ again to find numbers; students who are extending out one decimal place, then using consecutive values; using a larger denominator, then using consecutive numerators) to show their work with the document camera and explain their mathematical thinking. If one of these strategies did not show up with my students it I will model it.
I pick one example and I ask students, “Do you think you could find five more fractions between these numbers? How about 5 more decimals?” I have students brainstorm in partners and then share out to the entire class. I want students to recognize that there is an infinite amount of numbers between the two given numbers. We could always create a larger common denominator or extend out one more decimal place. See the Closure video in my Strategy Folder for more details about how I end class.
With the last few minutes, I have students complete the Ticket to Go independently. Based on how my students have done with the practice, I select values for the boxes in the example. See the Ticket to Go video in my Strategy Folder for more details.