# Seeing Improper Fractions

Today I want to put in a good word for improper fractions. You know, those fractions where the numerator is larger than the denominator, like 5/3 or 289/18.

We don’t usually tell kids about improper fractions until kind of late in their fraction learning trajectory, after they’re comfortable with “normal” fractions like 1/2 and 3/4 and 2/3 and 5/8. So children naturally get used to thinking that a fraction is like a partially filled pie. 3/4, for example, means we have a pie with 4 pieces but there are only 3 left. And then once they’re really good at this idea, we spring 5/4 on them, and the kids think, “Huh? If there are only 4 pieces in the pie to begin with, how is it possible to have 5?”

Even adults can have a hard time with improper fractions.

The conceptual root of this problem (if you’ll allow me to go into math educator mode for a moment) is that when kids see only fractions less than one, they start to think that 3/4 means “3 out of 4 things” which is not quite right, because 5/4 is a totally legitimate fraction but “5 out of 4 things” doesn’t make any sense. A better way to think about 3/4 is as three 1/4’s, where it takes four 1/4’s to make a whole. 5/4 then means five 1/4’s, where it takes four 1/4’s to make a whole. When we think of a 1/4 as a unit, we can have as many of them as we want.

So moving back to parent mode, when we give our child chances to see improper fractions, like 5/4, in real life, and when we do this early in their fraction learning trajectory, we’re not only making improper fractions themselves easier, we’re helping them develop a strong and solid understanding of what a fraction is in the first place.

Here are a few ideas for seeing improper fractions in everyday situations:

Graham Crackers

Graham crackers are great for introducing fractions to young learners because they break naturally into halves and into fourths, and because those halves and fourths are an identifiable unit. A graham cracker square can be called a half. A small graham cracker rectangle can be called a fourth. Four small rectangles make a whole cracker – that’s why they’re fourths. And it’s not at all inconceivable that you could have 5 or 6 or 7 or more small rectangles: 5/4, 6/4, 7/4, and so on.

Measuring Cups

Measuring cups are also great for thinking of fractions as units. It takes three 1/3-cup measuring cups to fill up a 1-cup measuring cup; that’s why it’s called 1/3. 1/2-cup measures and 1/4-cup measures are similar. You could experiment and see how many 1/3-cup measures would fill a glass measuring cup up to the 2 cup line. That’s six thirds (6/3)! How many thirds would it take to fill it all the way to the top, above the line? 7 thirds? 8?*

Pizza

Or quesadillas, pies, mini bread loaves… The important thing is that a) you can cut the food into equal-sized servings, and b) you have more than one whole (whole pizza, whole quesadilla, whole pie, whole loaf). If a child can identify a piece of pizza as 1/8 of a pizza, and can count pieces as eights (one pieces is 1/8, three pieces are 3/8, etc.), they can also tell how many eights 10 pieces would be, or 15, or how many eights there are in 2 pizzas.

Ruler Measurements

It’s common to use fractions in measurements – a quarter inch, a half centimeter. We have to have a way of naming measurements that are in between whole number measurements. If you’re working on measurement with your child, you’re probably using mixed numbers (e.g., 2 1/2 inches) rather than improper fractions. But go ahead and try to make the leap. If something measures 2 1/2 inches, ask: “How many half inches is that?” or “How many quarter inches is it?” This is more for older children – this is more challenging than pizza where you can see and hold and count an eighth. But it never hurts to ask something that’s beyond the child, and come back to it later if you find it’s beyond the child’s current capacity.

* As an aside, a colorful set of plastic measuring cups and spoons (maybe even one with a 1/8-cup measure) is a great Christmas or birthday gift for a child. It’s not terribly expensive, and opens up opens up all sorts of opportunities for experiences, creativity, and one-on-one time with parents or older siblings and (bonus!) they have their very own tool for thinking about and reasoning with fractions in a completely natural context.

# Measure! Early Measurement in Everyday Interactions

A child’s very early math experience is made up in large part of learning how to quantify, or to use numbers to describe characteristics of their world. Usually we think of this skill in terms of counting. But another form of quantifying is measurement. Whereas counting answers the question “How many?”, measuring assigns a number to attributes such as height, weight, length, and temperature.

Measurement allows kids access to all sorts of interesting questions about their environment. It’s also useful—we use measurement in chores, cooking, art, sports, sewing, building, etc. And measurement provides fertile ground for starting to think about fractions: what, for instance, do we call a measurement that’s somewhere between 4 inches and 5 inches?

Formal measurement skills (such as using a ruler) develop out of informal measurement experiences. Here ere are some easy, natural ways to introduce early measurement ideas in conversations and play with your toddler or preschooler.

Comparison Questions

Ask your child, “Which is bigger, ___ or ___?” This is a simple question with infinite variations, and easily tailored to your child’s ability and interests.

• Use different measurement words for different attributes. “Which is bigger?” is appropriate for some comparisons, but you might also use, “Which is taller?” or “Which is longer?” or “Which is more full?” Use comparisons questions to compare length, height, weight, volume, width, area, or even coldness or loudness.
• Don’t shy away from silly questions! “Which is bigger, the real car or the toy car?” “Which is heavier, the piano or the pencil?” These silly questions can be very fun for a young child, and still get them to think about, notice, and talk about measurable attributes.
• When your child is ready, make comparisons of objects that have very similar measurements. Your child might just guess, and that’s okay. When they begin to show interest in actually knowing, you have a great gateway to introducing formal measurement. If it’s not clear whether the pink cup or the glass tumbler has more water in it, pull out the measuring cups and find out!

Find Something Bigger/Smaller

This is a game that can be played in the home, outdoors, at the grocery store, in the car. It is better for preschoolers than toddlers, but older children can join in. Start with an object, and then ask your child to find something bigger. Proceed from there, taking turns finding bigger and bigger objects.

• You can also, of course, find smaller and smaller objects – just be sure to start with something very big.
• You can also use different measurable attributes – find something longer, or heavier. The term “bigger” is actually vague, because big could mean lots of things. Is a bookshelf bigger than a couch or smaller? It depends on what you’re measuring! But don’t avoid the term “bigger” – using the word can give you insight into how your child is thinking about “big”, and can open up conversations that lead to noticing other size attributes.

Fill It Up

We usually measure things with tools, like rulers, measuring cups, or scales. But informal measuring techniques can help children understand what formal tools are actually doing. Most informal techniques involve “filling up”:

• Fill different-size containers (cups, bowls, tupperware, etc.) with “scoops” of water or Cheerios or flour (whatever you’re willing to clean up!). Use a single measuring cup (it doesn’t matter what size) to scoop with. Notice that some containers take lots of scoops and some take very few.
• Draw outlines of shapes on paper and fill up the outline with objects of the same size, like small blocks, cotton balls, dry pasta. Again, notice that some shapes take more and some take fewer. Or you can fill the same shape with different-sized objects – goldfish crackers first, and then wheat squares. Notice that the size of the objects relates to how many fit.
• Line silverware up end-to-end and see how many pieces of silverware it takes to go from one end to the other, or all the way around. Then use silverware to measure other lengths as well. You can also use pens, crayons, pretzel sticks – anything long and thin and approximately the same length.

As with all activities on this blog, the most important thing is to follow your child’s interest, and to have fun! Your child naturally wants to learn about his/her world, and you as a parent are there to give them more tools to do so.