Teaching Fractions in Middle School: Using Models

This week we’ll take a look at the progression of fraction skills through the grades before our students reach us in upper elementary and middle school math class. And then we’ll look at models:-)

So, for a ‘quick’ overview (based on the Common Core Standards), in the earlier grades (1st – 4th ) students should be learning/mastering the following:
1st Grade Math:

• students partition circles and rectangles into two and four ‘shares’ and use the terms: halves, fourths, quarters, half of, fourth of, quarter of.

2nd Grade Math:

• students partition rectangles into rows and columns of ‘same-size square’ and count to find the total number
• students again partition circles and rectangles into 2 and 4 parts, but add on partitioning into 3 parts. They continue to use halves, half of, etc and add on thirds and a third of.
• students should be looking at wholes and identifying those as two halves, three thirds and four fourths.

In 3rd grade math, there’s a lot more going on!:

• students partition shapes into parts with equal areas, and identify each part as a unit fraction (1/3,1/4,1/2, etc)
• students begin to develop an understanding of fractions as numbers, represent fractions on a number line., and “understand two fractions as equivalent if they’re the same size, or the same point on a number line.”
• students recognize and generate simple equivalent fractions and should also be explaining why the fractions are equivalent
• students express whole numbers as fractions, like 5/1 = 5
• students compare two fractions with the same numerator or the same denominator ‘by reasoning about their size’

By 4th grade math, students should be doing SO many of the fraction things. I’m not going to list out each skill, but the general goals of fractions in 4th grade common core are:

• extend understanding of fraction equivalence and ordering
• comparing fractions with different numerators and denominators by using common denominators or benchmarks
• build fractions from unit fractions (includes understanding addition and subtraction as joining and separating parts, as well as decomposing fractions into sums of fractions)
• add and subtract mixed #s with like denominators (replacing each mixed number with an equivalent fraction)
• solve word problems w/addition and subtraction
• multiply fraction by a whole number and understand that a/b (like 5/4) is a multiple of 1/b (¼)
• solve word problems involving multiplication of a fraction by a whole number
• understand decimal notation for fractions, and compare decimal fractions​​

I think it’s important to  note that ‘using visual models’ is included in numerous places in these standards.

• Some students aren’t developmentally ready for all the fraction concepts
• Some classes may not have gotten as far in the curriculum as they needed to and may never have discussed certain fraction topics
• OR they did discuss the fraction topics, but very quickly or maybe with shortcuts, because there just wasn’t enough time

So that’s all well and good….students are supposed to have done all of this, should be proficient at these fraction skills, and should be ready to apply them and go deeper at the next grade level.

But what are math teachers at 5th, 6th, or higher grade levels supposed to do when this isn’t the case, and their math curriculum builds on that prior fraction knowledge? The ‘time’ issue continues to be an issue. Where does reteaching fit into the curriculum? (A question for many math concepts, but we’ll stick to fractions here!)

• On their models, students can add partitions to show both fractions as sixths, encouraging the continued development of equivalent fractions.

• The addition and subtraction of fractional parts is visual for middle school students when they use models.

This combination of process and modeling helps students connect the process with the meaning of the operations, or of simply finding equivalent fractions.

While you probably don’t want to redraw the lengths and the number line for every new problem, you can create and post some length/number line models around the classroom and provide models students can refer to on a regular basis.

This can again reinforce the concept of equivalent fractions, help students with comparing fractions, and transfer to fraction, decimal equivalences.