Using
Measuring Cups to Learn About Fractions
Subjects: Adding fractions
Grades: 3-5
Pre-requisites: The students don't have to know much about fractions to do well in this lesson. Students do need to be able to add small numbers. Whether students have prior exposure to fractions or not, it is a good idea to do the following warm-up.
Good lead-up activity: (2) Fraction
Strips To remind students of fraction concepts simply have them fold a loose
leaf sheet of paper in half long ways. As they look at _, ask them to add
the other _ to tell what _ + _ equals. Next, ask them to fold the paper
back to one half. Then, they should be instructed to fold the folded paper
one more time, again long ways. This should make '. Ask the students how
many ''s it will take to make the 1 whole piece of paper again. By opening
the paper up again, the students can count the rectangles. They should come
up with the answer four ' 's =1 whole.
Lesson overview: (10)In this lesson, the students will be able to think more concretely about the operation of adding fractions using measuring cups and water.
(9)Materials: 11 ' measuring cups
6 _ measuring cups
3 1 cups
1/2 cups
1/4 cups ( every measurement should be marked on the side of this)
5 short pitchers of water (use food coloring for easy measurements)
Procedure: (6)
1. First, ask the students why we use fractions.
(7) When I tried this lesson out with 4th graders, one student said, "So we don't have to use real numbers." Another student said, "Teachers make us use these fractions to confuse us."
I asked, "So, fractions are fake numbers?"
Another student replied, "No, they have real numbers in them like _ has a '1' and a '2' in it, but you can't say 'one two' . Remember these misconceptions for end of the activity assessment.
2. Next, ask them to come up with a situation where you might need to add fractions. If it doesn't come up, ask them if anyone has ever followed a recipe. Explain that you have to know about fractions to cook.
3. Separate the students into groups of 5 or less. Introduce the materials. Allow the students time to manipulate the cups and water. Tell them they have five minutes to get used to pouring the water and to explore the stuff.
4. Now, ask the students to do exactly as they did with their papers at the beginning of class. (Take 1cup, half it, and then half the half) to get four ' cups full of water.
(1) Once they have completed that task, ask them to find four ways to make 2 " cups of a substance. Make sure they can communicate what they did to make it 2 " cups. *******They can draw pictures or write it in words or symbols to indicate what they did. As each group finds a new way to do it, allow them to put their solution on the board or overhead. (I let my students name their groups and put their names next to their own discoveries)**************************************************
(7) (14) When I did this, many students immediately
filled the 2cups glass, and then three ' cups. Then, some further broke
down the 2 cups into 1 cup+ 1cup, and kept the three ' cups. Other possibilities
emerged as students began to internalize exactly what was 'in' each measuring
cup. One student said, "Hey, remember, 1cup has two _'s in it. We can
make one with four _'s and three ''s." One of the most difficult concepts
for the students to grasp was the fact that " = ' + _. Yet, once they
figured that out, they easily broke _ down into ' + ' . When I asked one
group how they figured that out so fast, they said, " _ always gets
broken down since we have these ' cups."
6. Now that the students have come up with many different ways to make 2 " cups of a substance, allow them to pick the most interesting ones to present to the class. Take this opportunity to use their pictures, words, and symbols to properly write the expression. For example: One student might say, "You see, we did it by taking this cup (indicating ' cup) and this cup (indicating _) to make this part (points at the " part of 2 " )." You can help the student along by writing on the board what he is saying. So, in this case, you would write ' + _ = ". You could ask, "Is this what you're saying?"
(7)When I re-wrote what they were expressing in their words and drawings, they were surprised when they looked at the written form of what they had just said! One student argued, "I don't think I said that because you can't add fractions that have different bottoms (denominators). You can only add them when they have the same bottoms." I replied, "Did it work when you made it with the water and measuring cups?" He slowly replied, "I think so." I said, "Well then, I guess your rule doesn't always work." The students were incredulous.
What they should learn: (8)The
big idea here is for the students to see that adding fractions is as easy
as mixing waters in measuring cups. These students were stunned that you
could add ' and _ ! The idea is to help the students see that fractions
are numbers that represent how much- just like what they consider real numbers.
Once they start treating them like numbers that represent something - it
is a lot easier for them to understand why we use them.
Extensions: (13)From here, students are ready to subtract and multiply. Students could even learn to divide fractions using this same idea of measuring cups and water. For example, when faced with the problem: (2 " divided by _ ) My students quickly figured out that there are eleven ''s in 2 ", and there are two ' 's in _ . Soon, they were measuring it out: and figured out that there are five _'s and one ' in 2 " . Not once did I hear, "Hey, you just flip and multiply." This is a reference to the traditional way we were all taught. The rule was hammered into our brains: "Invert the denominator, and multiply it by the numerator."