Back to The Line
Lesson 4 of 19
Objective: SWBAT combine positive and negative fractions with like denominators by graphing on a number line.
Students will complete a worksheet that will assess their knowledge of the locations of positive fractions on the number line (Webb’s DOK1). Students will input their answers into clickers and we will review together.
Because the SMART Response program can instantly generate this data, we will be able to target any misunderstandings in this topic. For example, students often get confused when they are asked to divide a section of the number line between two integers into given fractional pieces. For example, if students must identify and draw fifths between 0 and 1, they may draw five different tick marks in the given space, creating six fractional pieces instead of 5. Some students may not yet understand the concept of a fraction and thus have trouble knowing where to start when identifying positions. Completing this activity ensures that students are ready for negative and positive fractions on the number line, or it ensures that I will be ready to help students who are struggling with the basic identification and understanding of fraction positions on the number line. I will be focusing on asking students to review the meaning of a denominator and the numerator (i.e. the denominator represents the number of pieces between each consecutive pair of integers on the number line while the numerator indicates the number of pieces in question). Thus, 3/5 is three pieces out of five, or 3 pieces to the right on a number line between two integers.
Intro to Lesson + Task
There are 4 different parts of the task. We will review the first problem in each part and then students will work with neighbors in groups of 4 to complete the rest of the problems for that section. Part 1 addresses the identification level of understanding. Students will need to simply label each of the tick marks shown on their number lines. I will model #1 for students, indicating that negative fractions are graphed to the left on the number line. The points labeled in red indicate the positive numbers and in blue are negative numbers. This is consistent with our chip models and should be helpful when conceptualizing operations with positive and negative fractions.
By correctly identifying the positions of positive and negative fractions students are using MP8 to understand the organization of rational numbers along the number line. By identifying different types of fractions (i.e. halves, thirds, fourthds, etc), students will begin to understand the idea that there are an infinite amount of point between two consecutive integers. While students are only expected to identify the un-labeled position on each number line, I may ask students who struggled with the Do Now to label the whole numbers as well.
Part 2 of the task will ask students to combine positive fractions and decimals and show their work on the number line. This is a great opportunity to practice counting by sixths for #1 and by 0.5s for #2. Question 7 presents a great opportunity for students to connect fraction conversions to decimals as this questions asks students to add decimals and the number line given is split into fifths. Look out for students who may not have full mastery of these conversions; they may require a reminder of the process for converting.
Part 3 of the task will introduce negative fractions. All fractions will have the same denominator. Students will combine a positive and a negative fraction, alternating the type of fraction at the beginning of the expression. It will be important for me to target a group of struggling students to work with them to complete this section and to draw it correctly. We will be displaying answers on the board for the rest of the class to see. Look out for students who are only completing the algorithm without using the number line. Hold on to this expectation! Understanding the number line can help with more challenging, abstract problems involving variables on the number line (7NS1c).
Part 4 of the task will have students combining two negative fractions. All arrows will point to the left. This visual will help students begin to understand that fractions and decimals follow the same rules as integers when combining two negatives. Look out for confusion with double signs as exhibited by #12. Students are able to use different strategies: simplify double signs to make 1 sign (same signs, add, different signs subtract) OR remember that (+) says "right" and (-) says "reverse" when there are multiple signs next to each other.
All answers will be displayed and drawn on the SMARTboard, white board, and chalkboard during the Task. Students will have 5 minutes to discuss answers with neighbors if they are finished or to work independently to copy down the work and ask clarifying questions.
There are a couple of problems I want to make sure I review with a small group of students. Thus, after 5 minutes of review with neighbors, I will be pulling a small group of 6 - 8 students outside of my room or to a corner of the room to review the answers to problems #7, #8, and #12. These are the problems I noticed most kids struggling to depict on the number line. The students who remain in the larger group will be led by a student helpers (or a pair) through the same review. The following questions will be on an index card to help guide my student helpers through the review, they are the same questions I use with my small group:
- Which problem was the most difficult (ask no more than 5 students for that opinion)
- how many of you agree that this was the toughest?
(in my group this might require more scaffolding questions):
- where should we start? what is the first number in the problem?
- in what direction should we move? how do we determine in what direction we move?
- how to we decide the direction when there are double signs (+ (-) )?
At the end of this section homework will be distributed and students will pack up for their next class.