Number Line Subtraction

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Objective

SWBAT subtract positive and negative fractions with like denominators by graphing on a number line.

Big Idea

Students show what they already know about number line positions and work in pairs to explore subtraction of rational numbers on number lines

Do Now

10 minutes

Students enter the room silently according to the Daily Entrance Routine. The SMARTboard displays the directions for today’s Do Now:

  1. Please take a seat quickly and silently.
  2. Take out your homework from last night.
  3. Enter your answers into clickers. We will review together as a class.

Three students are identified for urgency in following directions and neatness in work. They are handed a sticky note that reads “please go to the whiteboard (or black board) and copy your work and answer for #__”.

I plan on reviewing questions #2, 3, and 6 with the use of these students work. After 6 minutes have passed the assessment will be stopped and we will begin by reviewing the answers and work on the board for the indicated problems. I ask students to evaluate the work as correct or incorrect . Whenever I ask students to put work up on the board I try to include work that is correct as well as work that includes errors being made by a large percentage of the class. It is also equally important to select students with unique solutions to a problem. This allows students to communicate about math by building arguments and justifications for their opinions (MP3).

For question 2, I expect students to have a vertical number line model/visual. I walk around the room as students discuss the work on the board to ensure that students are copying this model on their paper or fixing it if needed. Question 3 is a new skill and strategy, so as I walk around I make sure students are drawing arrows in the correct direction. Question 6 is a spiraling question where I am checking in with individual students as I walk around if they are not writing expressions or are not understanding how to translate real world examples as integers.

Class Notes + Intro to Lesson

15 minutes

Students are asked to put their homework away and turn to their CW section to receive their Cornell notes for today. We begin by providing a definition for “rational number” as “any number that can be expressed as a ratio or a fraction”. I also provide an alternate definition, “any number that can be expressed as a quotient of two integers”. This part of the notes holds many great opportunities for vocabulary review. We also review that integers are a type of rational number, but fractions and decimals are not identified as integers. I also provide a Vocabulary Card Index at this point with the word rational number and ask students to include two non-examples of a rational number, pi and the square root of two. Later in the week we will continue our discussion of rational vs. irrational numbers and I will introduce the terms repeating and non-repeating.

We continue by reviewing the bullet points that follow next to the topic “subtracting rational numbers on the number line”. We begin by identifying the factions and decimals that are missing on the number line included in the notes. Students are expected to identify these points with their partners given 2 minutes. I work with a small group to help them identify. Then, we discuss the organization of fractions and decimals to the left of zero. I can describe it as “numbers moving in reverse, with negative signs in front of them”, but I prefer to hear student descriptions first. I have 2 students share how they remember the order and locations of fractions and decimals on the number line and then share my own description. The aim is to provide all students 3 strategies for conceptualizing a difficult concept. We also review directions indicated by + and – signs, as well as the additive inverse and its usefulness whenever we see double negatives.  We use the included example to draw a visual. When we discuss the additive inverse and its usefulness, we talk about problems like the example displayed on the board:

–3/2 – (–1/2) 

If we use the additive inverse we can break this problem into something simpler:

–3/2 + (+1/2) 

Which also equals:

–3/2 + 1/2 

Students who can successfully reproduce this strategy and explain it are making use of structure to help them solve a rigorous problem (MP7)

Task

20 minutes

Students who earned an 80% or above in their HW will be entered into a random name generator application on the SMARTboard to earn booth seats for the task. Students who earned less than a 60% will be working with me. All other students will be given the choice to work with a partner of their choice or independently. 

 

As with the previous day's lesson, look out for students struggling to:

  • convert fractions and decimals
  • double signs (like in problem #4)
    For problems including double negatives I teach different strategies:
    • Same good, different bad, ugh!
      This is a strategy passed down through many years by a math teacher who taught 7th and 8th grade math for about 15 years at our school before becoming our principal. He would dress up as a caveman, reciting the quote above, "same good different bad, ugh!" The "ugh" grunt is the students' favorite part of the repetition. It reinforces the idea that when the signs are the "same" the result is "good" or positive. When the signs are "different", the results are "bad" or negative.  
    • consider the number line...
       

Closing

10 minutes

Once there are 10 minutes left in class I will ask all students to return to their seats. I will target questions 2 and 4 as review with the whole class. Question two uses a mixed number subtracting from a smaller fraction. I want to be able to have students analyze the solution to the problem and attempt to make a rule for solving without a number line. Question 4 involves multiple subtractions signs and I foresee it being one of the most difficult questions in the task. My work for all problems reviewed with the smaller group will already be displayed on the board and will useful for all students to check. Clarification questions may be asked at the time. Homework will be distributed and students will pack up for their next class.