Up to this point fractions have simply been discrete portions of tangible wholes; real parts that can be felt, seen, and tested. This third grade lesson plan, though, marks a water shed: we get to transfer our knowledge of ‘half a cookie’ to the more abstract concept of half a unit on a number line.
That students would be able to represent a fraction 1/b on a number line by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts, and that they’d understand that each part has size 1/b . Also, that they would understand that the endpoint of the part based at 0 locates the number 1/b on the number line.
- Number line experiment worksheet for each student (download here)
- Precut (colored) strips of paper, ½-1 inch thick and the length of zero to one on the student’s number lines, and a few longer strips of paper for you to use with the number line you will draw on the blackboard
- Apple or other similar easily dividable item
Tell your students that they’ve become so good at identifying and dealing with fractions—portions of pie, pizza, or apples and oranges—that today they can take what they’ve learned to a whole new sphere.
Pick up an apple. Tell them that it is an apple; you can feel it, you can measure it with a ruler, and you can divide it into two equal parts with a sharp knife. Remind them you can do the same with pizza, and with almost any physical object, provided your knife is sharp enough.
Then ask them if there are other things they could divide in half, things they can’t touch, feel, or cut with a sharp knife.
Write a list of ideas on the blackboard. Some ideas might be groups of things (or people), air, water, time, or space.
Validate each addition to the list as you write it down, and then tell them that today you’re going to look at fractions of three special things: portions of time, space, and mathematical units.
Talk about time first. Ask what it means when we say ‘half an hour’, and get as many versions of the answer as possible. If no-one suggests it, tell them that one way of thinking about it is as half the distance from one hour to the other.
Draw a diagram of your day on the blackboard; essentially, a number line that describes your day. At this point, though, don’t describe it as a ‘number line’ to your students. Put sitting up in bed, the first thing you do in the morning, away on the far left side of your diagram. Put going to sleep as the last thing, and in the middle put lunch.
Tell your students that the area between waking up and lunch is your morning; and then shade the first half, and tell them it is half your morning. If the morning was four hours long, from eight to twelve, and you were feeling miserable half the morning, ask them how long you were feeling miserable. (2 hours) How long were you feeling okay? (Also 2)
Observe that if you look at time in that way, time is very similar to distance. Talk about the distance between the bed and the lunch table in your drawing, and what half of it means. Talk about half of the way from the place you are standing to the window. Walk four steps away from your desk, counting as you go, and ask how many steps it is to your desk. Ask how many steps you would need to walk if you wanted to walk only half the way back to the desk (2). What if you wanted to walk just a quarter of the way back? (1)
Now erase everything from the blackboard and draw a simple number line going from zero to three. Ask them what this is called (a number line). Remind them that since a number line is math, we can use it to mean anything. We get to use the same number line, in exactly the same way, whether we’re talking about cookies, pizza, time, or distance.
Tell them for now you’ll pretend it’s talking about pizza. Put your chalk at zero, write a small dot, and test the students on basic number line usage: Here, you see, I have no pizza. If I buy two pizzas—one peperoni and one sausage—where would I show that on the number line?
Your students should guide you to move your hand to the two. Do so, make a dot there, and then go back to the zero.
That was the day before yesterday, explain. Yesterday, I also started with no pizza. I also bought pizza. But I wasn’t feeling very hungry and didn’t have much spare money, so I only bought half. How can I show that on the number line? There isn’t any place that says ‘half’.
Listen to any ideas they come up to. If someone suggests dividing the portion of the number line between zero and one in two parts and making a dot on the middle line, tell him you really like that idea.
Take a strip of paper exactly as long as the distance between zero and one; fold it in half, lay it on the number line from 0 to ½, and draw your ‘half pizza’ dot. Shade the area on the line between 0 and ½. Ask your students how long that segment is; compared to the length between 0 and 1 (1/2 the length). Ask them where the segment starts (0).
Pick up the folded paper strip again, and ask how long it is (1/2 of what it used to be). Since it is ½, ask them if it means ½ wherever you place it on the number line—do you have to begin measuring off at zero, or can you start somewhere else instead? Get feedback as to why or why not before you explain that since ½ is just half of one, and you have no ‘wholes’ to add it to, you always have to start on zero when you measure its placement.
Ask them how you’d find out where to place a do for 1/4th. Fold your strip into fourths, and use the folded strip as a measuring stick to place a dot at exactly 1/4th.
Ask about 1/3, unfold your strip and refold into thirds, and prepare to make a 1/3 mark. Ask where you should start the strip when you measure off the 1/3 (at 0).
Ask which mark is closer to the zero (1/4); which is furthest away (1/2).
Then pass out the student worksheets and strips of paper. Your students will be marking ½, 1/3, ¼ and 1/6 on their own number lines. If you have not introduced 1/6th previously, you may need to walk your student through that fraction by marking your own 1/6 on the blackboard.
Common Core Standards
Under 3.NF.2, the Common Core State Standards for Mathematics reads:
3.NF.2 Understand a fraction as a number on the number line; represent fractions on a number line diagram.
- Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.
Web Resources/Further Exploration
Here are some links to helpful web resources that might help your students learn to enjoy fractions. When they’ve decided that fractions are definitely fun, it’ll be easy to gain the familiarity and intuitive understanding they need to make a success of classroom work.