Math Background

Lesson: Adding and Subtracting Fractions With Like Denominators
Developing the Concept

When we teach students to add and subtract, we encourage them to work from right to left. For whole numbers, this means starting in the ones place and then working with the tens, then the hundreds, and so forth. To reinforce this habit, encourage students to compute with the fraction parts of mixed numbers before the whole-number parts. While this is not essential to producing a correct answer in addition, it is important in some subtraction problems, so developing the habit of always working right to left may help students to avoid errors.

Materials: rulers marked in half-, fourth-, and eighth-inches; pencils; erasers

Preparation: none

  • Say: Outside the classroom, you find fractions most often when you're measuring things. Today, we'll be measuring objects and adding measures.
  • Ask: How can you measure an object that is longer than your ruler?
    One efficient way is to work with a friend and move your rulers one over the other until you've counted how many whole feet and how many twelfths of a foot it takes to go from one end to the other of the object. Another way is to measure one foot, put a dot at the end of the ruler, and then move the 0 mark to the dot to measure the next foot (or part of a foot). Of course, if you do that, you need to erase the dots when you've finished.
  • Say: I'm going to use this ruler to measure the length of the board. Help me keep track of the whole feet.
    Measure and count each foot, and then have a student come up to read, to the nearest inch, the part of the foot that completes the measurement. Write the measure on the board. Let's say the board is 3 feet 8 inches long. You can complete this lesson with any measure, but 8 inches, or eight-twelfths of a foot, is a useful fraction to work with at some time in the lesson because the greatest common factor of 8 and 12 is neither 8 nor 12.
  • Ask: Can you write 3 feet 8 inches as a mixed number of feet?
    Talk about how 8 inches is eight-twelfths of 12 inches and discuss the fact that this fraction is not in simplest form. Look for the greatest common factor, 4, and simplify the fraction to two-thirds. The mixed number that represents 3 feet 8 inches is 3two-thirds feet.
  • Ask: The material for the chalk tray comes in long strips. I need to make chalk trays for two boards just like ours. How long is the strip of material that I need to buy?
    Help students decide that they need to double 3two-thirds feet. Since they don't yet know how to multiply fractions, adding 3two-thirds and 3two-thirds seems like a good way to go about it. Encourage them to add the fractions first. You can simplify the sum of the fractions before or after adding the whole numbers. Since students will eventually encounter problems in which they don't have that choice, make sure they have practice simplifying after they've got a sum. The reasoning should go something like this:
    3two-thirds + 3two-thirds = 6 four-thirds. Since four-thirds > three-thirds, I've got another whole foot. four-thirds = three-thirds + one-third and 6 four-thirds = 6 + three-thirds + one-third, so 6 four-thirds = 6 + 1 + one-third = 7one-third. The answer to the question is You need to buy 7one-third feet of material.
  • Ask: A piece of chalk-tray material is 8 2/3 feet long. How much material would be left after I bought enough for our two trays?
    Have the students reason through the subtraction. Have them subtract fractions first, then whole numbers. Then write the problem on the board.
    8 two-thirds
    − 7 two-thirds
    1 one-third  feet

    The answer to the question is There will be 1one-third feet left.

    Pose a number of problems that require students to measure things in the classroom, and then add and subtract measures. Be careful to keep denominators within one problem the same. Measuring in feet and inches will provide you with denominators of 12, 6, 4, 3, and 2. For demonstration, you might measure the windowsills, the molding around the base of the wall, a number line, or anything else that has a distinctly linear quality to it.

    Measuring in inches and fractions of inches will provide you with denominators of 8, 4, and 2. Students at their desks might measure pencils, crayons, erasers, ribbons or strings of beads, or anything made in long strips and cut to length so you can ask sensible questions about the length of material required for more than one. While measuring with fractions of inches, students may wish to use the ruler as a way to check their work: Instead of adding 2one-half inches to 2one-half inches, they might just start at the 2one-half-inch mark, then count off 2one-half more inches to find the sum.

Wrap-Up and Assessment Hints
When students are comfortable using their rulers to generate the numbers for addition and subtraction problems, pose some problems that involve the denominators 2, 4, and 8. Allow them to refer to their rulers if they need to. When they find these denominators easy, pose problems with denominators of 3, 6, and 10.

Houghton Mifflin Math Grade 4