Finding the Distance Between Signed Fractions on a Number Line
Lesson 21 of 27
Objective: SWBAT determine the distance between two signed fractions on a number line and express the distance as the absolute value of their difference
In the introduction we review the meaning of absolute value - the distance from a number to 0 on a number line. Distances are always positive. All rational numbers have an absolute value . I will display the blank number lines on the SmartBoard. I will cold call students to determine the absolute value of various fractional points that I will mark on the number lines. For the first few problems, I will show the line segment from 0 to the number over the number line. I will also make sure to label the absolute value notation | | for each number. Students often confuse this symbol with parentheses so I will make sure they note the differences. As students see these values represented with the absolute value symbol they are using MP2 as they see a distance represented using symbols such as | -3 1/4 | = the distance 3 1/4. With a nod to MP3 and MP6, I will ask some students to explain how they know their answers are valid. This simple explanation helps students develop the habit of using precise language to explain and justify their reasoning.
Distances on Number Lines
I have presented students with 16 distance problems to solve using a given number line. Students will be asked to make their own number line (MP5) in order to solve the last 4 problems. About half of the problems involve common denominators. If students have difficulty finding the distances between the fractions with unlike denominators, I may ask how can we easily compare 2/3 to 1/6? Or what interval is represented by each tick mark on the number line? How can we represent the values on the number line? In addition to assessing their ability to work with unlike denominators, I will be checking their ability to count spaces between intervals. If I see a problem with a distance I may ask a question like: Where do the two points belong on the number line? What is the distance (how many intervals) between the two points? What is the value of each interval? Students often make mistakes when counting distances. A common error is for students to count the starting point as the first interval instead of appropriately calling it 0. When I see this error I may ask: "What is your starting point? When you're at the starting point, what distance are you from the starting point?". Hopefully a student will now see that they are 0 distance from the starting point. To mitigate the chance of this happening, I may tell students to count the spaces between the tick marks of the number line. I often like to make click sounds that accompany each move - it helps me and my students keep track of the number of moves.
Each distance is represented using the absolute value symbols so students should see that there are two ways to write the distance as the absolute value of their difference. Again this presentation is here so students see that the symbols represent the distance (MP2).I am teaching this lesson before we learn to add or subtract integers, so I may allow my students to use calculators to verify that their counting on the number line matches the calculations for each problem. (MP5) For each group of problems, I have included a couple problems with numbers that do not explicitly appear on the number line or numbers that are inbetween tick marks. I expect to hear students say "The number is not there." For example problem e uses 2 1/2 on a number line with 1/5 intervals. Some of the questions above, about dealing with unlike denominators would be approporiate for students struggling to place the value.
At the end of this section, we will quickly review answers. I will ask students if they see a shortcut or easier way to find the distance between a positive and negative number without counting each tick mark. This is to see if students see the structure (MP7) that the sum of a negative numbers' abs value and the positive numbers' abs value is also their distance. By this point, students may see this shortcut more quickly as they have already explored distances with integers and decimals in previous lessons.
This goes with pages 1-2 of the resource:
The problem solving section is on pages 3-4 of the resource:
Now students apply what they know in a "real-world" problem solving context. In problem 1, students will have to figure out the appropriate scale for their number line (MP6), find the distance between a positive and negative point, and represent the distance using two different equivalent expressions. Students should resolve issues with their partners, but if stuck I may ask questions like: "How far is the Station A or Z from station Ml? Is station a/z to the east or west of station M?" In part C, it is okay if students choose to write a sum of each values' absolute value or as the absolute value of the difference of the two values. Either way, they will have to have at least 1 absolute value of a difference represented.
Problem 2 has similar tasks as problem 1 except that there are more points to find and a vertical number line is a more appropriate (though not necessary) model. I have included a blank number line. In problem B, students are asked to find the distances between every house. Here is a small opportunity for students to make sense of a problem (MP1) as they must find a method to make sure they find all 6 distances.
The exit ticket is a similar problem to the two problems in the problem solving section. In fact, it is most similar to the first problem. Students work on this independently, but it is okay for them to refer to any of their class work as an aid.