Mini Lesson: Comparing Fractions

A quadrat of ways to compare fractions made easy:

 

I.  Number Lines are a great visual and serve as an introduction to measuring tape for use in projects.  My method of teaching this skill is as follows . . .

  1. Draw a line and then find/eyeball where half way is.  Mark this as 1/2 regardless of what fractions are to be compared.
  2. Next label the ends with whole numbers.  This can be 0 and 1 (as the anchor chart above is displayed) or 12 to 13.  It really does not matter.  *Side note – this is for introduction and therefore the reason we’re starting with whole numbers as the book ends.
  3. If comparing fourths, then under the whole numbers write the matching fractions.  0 = 0/4, while 1 = 4/4.  Then match 2/4 under 1/2.  Guide your learners by asking them to help.  What is half of 4? (2) What is half of 2? (1).  Point out that the numerator doubles under this spot, ie 1/2 =2/4.
  4. Now time to teeter totter back and forth on each side of the 1/2 (or 2/4) marker.  Starting on the left side next to 0, “What comes in between 0 and 2?” (1)   “So let’s think then, if we have 0/4 and 2/4, what could come in between these two fractions?” (1/4)  While marking the 1/4 make it a spatial activity by looking for halfway between 0/4 and 2/4.  This helps not to cram the fractions too close together.  The same is then repeated on the right side of the 1/2 way mark using the same questioning, “What number is between 2 and 4 when counting?” (3)  “If this is true, then what should come between 2/4 and 4/4?” (3/4)  Again marking it midway between the 2/4 and 4/4 markers.
  5. Now select a fourth and compare it to 1/2 (using 1/2 as a benchmark).  Ex – is 1/4 <,>, or = to 1/2?  Responses should be – Well according to our number line 1/4 is less than 1/2 by one jump, or one-fourth of a jump, or I know 1/4 is less because it comes before the 1/2.  1/2 is closer to the whole number (4/4).

 

II.   Creating Common Denominators, a vantaztic way to bring in multiplication skills.  Looking at the example in the anchor chart given   1/3, 4/6, and 2/12.  All of these denominators have 12 in common.  The most important piece (opinion) to this skill is for the learner to continually question, “Can I multiply this number to reach the other? “

  • Can I count by 3 to reach 6? (yes)
  • What if I count by 3 to reach 12? (yes)
  • Is 6 a multiple of 12? (yes)

If the answer be yes, count by those multiples to reach the common goal.  If no, then combine those denominators to create a common goal.  Take these fractions as example:  3/5,  1/4,  and  7/10.

  • Can I count by multiples of 5 to reach 4? (no)
  • Can I count by multiples of 4 to reach 5? (no)
  • Can I count by multiples of 5 to reach 10? (yes)
  • Can I count by multiples of 4 to reach 10? (no)
  • If 5 and 10 are a match then how can I use 4?

Take 4 and 5 and combine these numbers with the might of multiplication to create 20.  Now . . .

  • Can I count by multiples of 10 to reach 20? (yes)  The common denominator will be 20.
  • 3/5 = ?/20 . . . well if the denominator 5 x 4 =20, then multiply the numerator 3 x 4 =12 (12/20)
  • 1/4 = ?/20 . . . well if the denominator 4 x 5 = 20, then multiply the numerator 1 x 5 = 5 (5/20)
  • 7/10 = ?/20 . . .  well if the denominator 10 x 2 = 20, then multiply the numerator 7 x 2 = 14 (14/20)
  • Finally put them in order from least to greatest. 5/20 (1/4), 12/20 (3/5), 14/20 (7/10)

Number lines and Creating Common Denominators not only allow learners to compare fractions to see which is <,>, or=, but these two methods also allow learners to see exactly how much larger or smaller fractions can be from one another.

  • 3/5  > 1/4 by 7/20
    • The reasoning:  {3/5 = 12/20 and 1/4 = 5/20, so 12/20 – 5/20 =7/20}
  • 1/4 < 7/10 by 9/20
    • The reasoning: {1/4 = 5/20 and 7/10 = 14/20, so 14/20 – 5/20 = 9/20}

III.  Pictures:  Visuals are the tops, but only if mastered correctly.  Too many times learners create visuals ending up in two different lengths because they have added fractional pieces at the ends.  Pictures will work if they are congruent (same shape, same size, and same size pieces).  Muy importante to create the two or three shapes first. Lined paper serves as a great guide, keeping the figures spaced according to the number of lines down, and lining shapes under on another to keep as much consistency in length and form as possible.  This method best suits the meticulous and/or the artistic.  Another great way to compare fractions by how much though.

IV. Cross Multiplication, a.k.a The Butterfly Method.  This is my least favorite mostly because it is not specific enough to compare “by how much” one fraction is than the other.  Granted in the examples from the anchor chart we can see 1/3 is greater than 2/12, and that when multiplying 12 x 1 = 12 and 3 x 2 = 6, so learners can see that 1/3 is 6 more than 2/12, but 6 what?  Out of how many pieces?  This method is most successful for struggling learners unable to complete the needed steps for number lines or common denominators.  No shade being thrown at butterflies, they are a necessary being to ecosystems.

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