If Not Methods: Mixed Numbers

Continuing our series on teaching the tough topics of arithmetic…

Our own school math experiences led many of us to think that math is all about memorizing and following specific procedures to get right answers. But that kind of math is obsolete in our modern world.

The math that matters today is our ability to recognize and reason about numbers, shapes, and patterns, and to use the relationships we know to figure out something new.

But what if our children get stumped on a mixed-number calculation like 2 ⁵/₁₂ + 1 ³/₄?

Notice, Wonder, Create

If we want children to understand math, to build true fluency, then we need to get them thinking and making sense of the subject. We need to start where they are and help them reason about the numbers.

As we take turns noticing details and wondering about this problem, we might say things like…

• I notice addition, so the answer will be greater than either of the original numbers.
• I wonder, does addition always make numbers bigger?
• I notice the sum has to be more than 2 + 1 = 3.
• I notice that ⁵/₁₂ is almost ¹/₂, and ³/₄ is more than ¹/₂, so I predict the answer will be a bit more than four.
• I wonder, would drawing a picture help?

And this process of working through the idea will lead to growth, building those thinking mindsets that serve as a foundation for future learning.

Five meanings of a fraction

There are five different ways we can think about what a fraction means:

• part-whole: seeing fractions as parts of some whole thing, visualizing geometric shapes cut into equal portions
• measure: reasoning about fractions as marks on the number line, measuring out a given length
• operator: finding a fractional part of some number
• quotient: considering the numerator divided by the denominator
• ratio: thinking about a proportional relationship

The first three models — part-whole, measure, and operator — provide insight into the meaning of mixed numbers, helping us make sense of this math calculation.

We’ll talk about the final two (more abstract) fraction concepts next week.

Part-whole: A Mixed Number is a Sum

Every mixed number is the sum of a whole number plus a fraction.

2 ⁵/₁₂ = 2 + ⁵/₁₂
1 ³/₄ = 1 + ³/₄

We can choose to transform each sum into a single (improper) fraction or to work with the parts separately. In this case, the fractional part we are trying to subtract is greater than our starting fraction, so let’s choose improper fractions.

2 + ⁵/₁₂ = ²⁴/₁₂ + ⁵/₁₂ = ²⁹/₁₂
1 + ³/₄ = ⁴/₄ + ³/₄ = ⁷/₄

These are called improper fractions because the numerators are greater than the denominators. But there is nothing mathematically improper about them. In fact, by the time we get to algebra, it’s mixed numbers that become obsolete.

So now our calculation looks like this:
²⁹/₁₂ + ⁷/₄ = ?

We can transform fourths into twelfths by cutting each fourth into three equal pieces.

The number of pieces (numerator) goes up in the same proportion as the size of the pieces (denominator) goes down. Since we cut each fourth into three parts, we now have three times as many pieces.

²⁹/₁₂ + ²¹/₁₂ = ⁵⁰/₁₂

As we mentioned in last week’s post, ⁵⁰/₁₂ is a perfectly good answer. Simplifying fractions or mixed numbers is merely a cosmetic change that doesn’t affect the value (or correctness) of our answer.

⁵⁰/₁₂ = (50 ÷ 2) / (12 ÷ 2)
= ²⁵/₆ = ²⁴/₆ + ¹/₆ = 4 ¹/₆

Fraction as measure: Number line

Addition can be imagined in two ways: putting like things together, or growth (increase). The growth model is easy to picture as jumping forward on a number line.

2 ⁵/₁₂ + 1 ³/₄ = ?

We can start at 2 ⁵/₁₂ and jump forward one whole unit.

2 ⁵/₁₂ + 1 = 3 ⁵/₁₂

Then we still need to add the fractional part, and we know that ³/₄ is the same amount as ⁹/₁₂.

3 ⁵/₁₂ + ³/₄ = 3 ⁵/₁₂ + ⁹/₁₂
= (3 ⁵/₁₂ + ⁷/₁₂) + ²/₁₂

Final answer = 4 ²/₁₂ = 4 ¹/₆

Fraction as operator: Choosing a convenient unit

When we think of a fraction as operator, we think of finding that fractional part of a given number.

2 ⁵/₁₂ + 1 ³/₄ = ?
2 ⁵/₁₂ of something plus 1 ³/₄ of the same thing = how much in all?

Since this is an abstract number calculation, we can choose any convenient “something” we like. We have twelfths, so let’s count eggs in dozen-size cartons.

2 ⁵/₁₂ dozen eggs = 2 cartons + 5 extra eggs = 29 eggs

1 ³/₄ dozen eggs = 1 carton + ³/₄ of another
= 12 eggs + 9 eggs = 21 eggs

Total = 29 eggs + 21 eggs = 50 eggs

50 eggs = 48 + 2 = 4 dozen + 2 extra eggs = 4 ²/₁₂ dozen

Then we can switch back from dozens to abstract numbers, giving the final answer of:
2 ⁵/₁₂ + 1 ³/₄ = 4 ²/₁₂ = 4 ¹/₆

Mental math: Do the easy part first

When we do math in our heads, it doesn’t “look” like the traditional steps on paper. We pick whatever seems easiest and do that first.

2 + 1 = 3, plus the fraction bits…

So our calculation becomes:
3 + ⁵/₁₂ + ³/₄ = ?

Another thing about mental math is that we don’t have to do the calculation exactly as it is written. We can use a friendlier number first, then adjust our answer at the end.

For example, ⁵/₁₂ is almost ⁶/₁₂ = ¹/₂, which gives us:

3 + ¹/₂ + ³/₄ = 3 + ¹/₂ + (¹/₂ + ¹/₄)
= 3 + (¹/₂ + ¹/₂) + ¹/₄
= 3 + 1 + ¹/₄ = 4 ¹/₄

But remember, we rounded the ⁵/₁₂ up, so we added ¹/₁₂ too much. We need to take that extra part away.

4 ¹/₄ − ¹/₁₂ = 4 ³/₁₂ − ¹/₁₂
= 4 ²/₁₂ = 4 ¹/₆