Finding the Distance Between Signed Decimals on a Number Line
Lesson 16 of 27
Objective: SWBAT determine the distance between two signed decimals 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 as the distance from a number to 0 on a number line. Distances are always positive. We will see that even decimal values 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 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 as using symbols such as | -5.2 | = the distance 5.2. To include a bit of 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 17 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. As students are working with their partners, I will be walking around assessing 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? 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." Problem e, has a 3.1 which is greater than the last given tick mark. I will ask them to tell me where 3.1 is in relation to 3. Problem l uses a number line in fourths yet -1.05 is given. I will ask them to point where they think -1.05 is based on the values given. If they struggle, I may ask them to represent the value in terms of money (-$1.05). Then ask, "What tick marks is this closest to?".
At the end of this section, we will quickly review answers. I will then 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 they see the structure (MP7) that the sum of a negative numbers abs value and the positive numbers abs value is also their distance. Several lessons earlier, we had a very similar lesson, so hopefully this shortcut becomes more apparent to the students.
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). Students should resolve issues with their partners, but if stuck I may ask questions like: "How far is the scuba diver/hang glider from sea level? Is the scuba diver/hang glider above or below sea level?" Part C asks students to write two different expressions to represent the distance. It is okay if they 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 distance represented.
Problem 2 is simlar, except that there are more points to find and a horizontal number line is more appropriate (though not necessary). 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 extension provides an opportunity for students to work more with the coordinate plane. Students may be surprised to see that decimal values can be placed on the coordinate plane.
The exit ticket is a similar problem to the first of the two problems in the problem solving section. Students work on this independently, but it is okay for them to refer to any of their class work while completing the exit ticket.