# Fractions Part 5: Equivalent Fractions

This is part 5 of a series of fractions posts. Thanks for sticking around! Through explorations with fraction manipulatives, pictures, and drawings, we hope students begin to notice there may be different ways to express the same area using fractional terms. To cut a sandwich into halves and eat one of the halves is the same as cutting the same sandwich into fourths and eating two of the fourths. Read on for several freebies about equivalent fractions.

Students can gain experience finding equivalent fractions using models in several ways: Fraction strips, area models, set models, bar / length models, and number lines. Then with a strong understanding using concrete and pictorial models, the student is ready to apply paper-pencil methods to name equivalent fractions. But remember to use the same size whole: As in this picture using pattern blocks, 2/3 of a trapezoid is NOT the same as 2/3 of a hexagon.

Fraction Strips: If you don’t have sets of fraction strips, here is a free resource Fraction, decimal and percent strips charts via Kim Tran (TPT).  Commercially available strips are also nice. But, with some 1″ strips of construction paper, students can create their own and probably learn a lot about the relationship between halves, fourths, and eighths as well as thirds, sixths, and twelfths in the process of partitioning and cutting them.

TIP:  Make sure students know how to read a fraction strip chart. A couple of years ago I was working with a third grade class and assumed they could readily see that 5/10 was equivalent to 1/2. But after confusing looks, I realized that I needed to physically show them how to follow a line vertically down the chart to find other fractions that were in line (by placing a ruler or long pencil along the vertical line). Another example.  “To find another fraction equivalent to 1/3, find the line at the end of the 1/3 section and trace it vertically down the page to see if there are other fractions that stop along that same line. You should see in the sixth’s line that 2/6 lines up, and in the ninth’s line that 3/9 lines up and in the twelfths line that 4/12 lines up.”

After cutting and labeling strips, then explore equivalent fractions (those with the same size length). Students should be able to generalize that different fractions can used to represent the same area.

Area models:  By covering or partitioning shapes, students should notice that even though the same area is covered or shaded,  the number and size of the parts can change. Continue reading

# Fractions Part 4: Compare Fractions

Starting in 3rd grade, students start using words and symbols to read and write fractions (Oklahoma Academic Standards OAS 3.N.3.1), construct fractions (3.N.3.2), compose and decompose them (3.N.3.3), and order and compare them using models and number lines (3.N.3.4).  Fourth and Fifth graders continue to refine these skills. In this post, I will address different ways to compare fractions (keeping in mind the concrete-pictorial-abstract progression) by comparing numerators, comparing denominators, comparing to half, and utilizing knowledge of unit fractions. Students should have extensive experience utilizing models such as fraction strips, fraction circles, pattern blocks, number lines, pictures, and drawings to help build the concepts of fractional parts before being asked to put a <, >, or = sign between two fractions. See the end for a FREE comparing fractions guide.

In my opinion, determining if (or how) two fractions are equivalent is also a very important step when comparing fractions. However, regarding the OAS, students are not asked to represent or rename equivalent fractions until 4th grade (4.N.2.1). I will address equivalent fractions in the next post – just know that sometimes this skill goes hand in hand with comparing fractions. AND keep in mind that most of the standards for fractions through 4th grade stipulate “using concrete and pictorial models, fraction strips, number lines.” Students in 4th grade should not be expected to do abstract paper-pencil steps to simplify or “reduce” fractions to simplest terms, nor cross multiply to compare, etc. They need hands-on experience to more fully understand the concepts about fractions that are so difficult to grasp abstractly. Then in 5th grade students should have enough visual pictures in their head to solve operational problems with fractions. OK, that’s my soapbox. Don’t make it harder than it should be.

Materials to use: pattern blocks, fraction strips, fraction circles, cubes, tiles, two-color counters, Cuisenaire rods, number lines, paper plates, graham crackers, candy (m and m’s, skittles, etc.)

Ways to Compare (when using same size wholes – you can’t compare 3/4 of a donut with 1/2 of a birthday cake):

• Using unit fractions:  If the fraction is a unit fraction, it has a 1 as a numerator. This should form the first type of comparison:  1/2 > 1/3 and 1/5 < 1/4 and 1/6 > 1/10, etc. This type of comparison is critical to fractional understanding.
• Same denominator: When the denominators are the same, then compare the numerators. 2/4 > 1/4.
• Same numerator:  When the numerators are the same, compare the denominators. For example: When comparing 2/5 with 2/10, since fifths are larger parts than tenths, 2/5 will be larger than 2/10. This is hard for some students to think about, because the smaller the number designated for the denominator, the larger the part (when comparing the same size whole).
• Unit fractions one away from the whole:  These are fractions in which there is one unit to be added to make it a whole (1). The numerators of these fractions will be one less than the denominator.  11/12 is 1/12 away from the whole (1). 7/8 is 1/8 away from the whole (1). Example: To compare 3/4 with 5/6, use manipulatives or a number line to see that 3/4 is 1/4 away from 1, while 5/6 is 1/6 away from 1.  Since 1/4 is a bigger part than 1/6, then 3/4 < 5/6.
• Less than half? More than half?  Learn all of the fractions that equal half. While this might sound simple, students often have misconceptions that 1/2 is the only way to describe half, or that a 5 must be in the fraction to be half (because 5 is the midpoint when used on a number line for rounding). I ask students to recall their addition facts dealing with doubles from 2nd grade. Since 2 + 2 = 4, two is half of 4, and 2/4 = 1/2. Repeat that with other forms of 1/2. Students should learn that finding half of an even-numbered denominator should be figured quickly (7/14, 9/18, 25/50, 50/100, etc.). Then use knowledge of half to determine if a fraction is less than half or more than half. Since 7/14 = 1/2, then I know that 6/14 < 1/2 and 9/14 > 1/2.

# Fractions Part 3: Misconceptions

The fractions focus today will be on some basic concepts that students should understand before they work to compare them, determine equivalent fractions, simplify them, use mixed fractions, or add / subtract them.  I am including a FREE copy of my Fraction Basics reference guide (click here), along with a photo of an anchor chart I made for a fourth grade class.

I have been rereading a book I love about fractions called “Beyond Pizzas and Pies, 1st Edition.” It has great examples of children’s misconceptions about fractions and lessons on how to try to remediate them. A recurring theme in the book is that while kids can learn “tricks” to help them solve fraction problems, they often do little to help students conceptualize what fractions are. Here’s a link to Math Solutions regarding this book: Beyond Pizzas and Pies (2nd Edition) Following are five  examples from the book that made an impact on me and my teaching (which I will go into more detail about on future posts). Continue reading