The Pantry as a Math Lab: Teaching Fractions Without a Worksheet

When I was homeschooling my children, fractions made considerably more sense when somebody was waiting for the biscuits.

On a worksheet, ½ + ¼ = ¾ is an exercise. There is a correct answer, and the child is supposed to learn the procedure that produces it. But in our kitchen, that same child might be standing beside me with a recipe calling for three-quarters of a cup of milk and no three-quarter-cup measure.

Now we needed to know.

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I wasn’t trying to make math more entertaining. I was trying to

make it make sense.

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That little shift—from you need to learn this to we need to figure this out—was a big part of the way I homeschooled, and there was a very personal reason for it.

I have dyscalculia, sometimes described as "math dyslexia." Numbers on a page have never behaved for me the way they seem to behave for people who are naturally good at math. As a child, I could memorize a procedure and still not really understand what I was doing. Give those same numbers a physical meaning, though, and I had something I could hold onto.

Geometry was always the easiest kind of math for me because I could see it. A triangle was a triangle. I could look at its sides and angles, draw it, turn it around, and understand the relationships in front of me. The math had a shape.

Fractions began to make sense in much the same way when they stopped being marks on paper. A half cup made sense. I could see it. I could pour two of them into a one-cup measure and prove to myself that two halves really did make a whole. Three-quarters wasn't just a 3 sitting over a 4; it was a half cup and a quarter cup sitting on the counter in front of me.

In many ways, this was the only way I ever truly learned math myself.

So when I homeschooled my children, I taught math the way I had learned to understand it. Of course they learned the notation and the procedures, but whenever I could, I gave the numbers something real to represent. We measured them, weighed them, divided them, doubled them, spent them, baked them, and put them into jars.

I wasn't trying to make math more entertaining.

I was trying to make it make sense.

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A math problem changes when the answer is something

you actually need.

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When Numbers Meant Something

Of course my children learned what numerators and denominators were. They learned how fractions were written and how to work with them on paper. But I didn't want the symbols to be the only thing they understood.

Our measuring cups were already fraction manipulatives.

I could put a one-cup measure beside the half cup and quarter cup and let a child pour two half cups into the one-cup measure. Then we could try four quarter cups. A half cup and a quarter cup together showed us exactly what three-quarters looked like.

My child wasn't merely being told that two halves made a whole. She could see it happen.

Better yet, she could use it.

If our recipe needed ¾ cup of something and we didn't have a ¾-cup measure, how could we get it?

Now ½ + ¼ = ¾ wasn't a problem somebody had invented to see whether she remembered a rule. It was useful information standing between us and whatever we were making.

That was the kind of math lesson I loved.

We Started With What We Were Making

A recipe could become an entire math lesson without my announcing that we were about to "do math."

If a recipe made twelve muffins and we only wanted six, we had to halve it.

Two cups of flour became one. One cup became half a cup. Half a cup became a quarter cup.

If we needed twenty-four muffins instead, we doubled it.

I didn't have to invent a reason for multiplication or fractions. The recipe had already given us one.

"We need twice as much milk. The recipe says ¾ cup. How much do we need?"

Sometimes a child could solve that problem with the measuring cups before being able to explain the arithmetic neatly on paper. That never bothered me. In fact, I considered it useful information. It meant the understanding was developing before the notation caught up.

Once the child had figured it out with the cups, we could talk about how to write down what had just happened.

The symbols described something the child already understood.

Measuring Cups Were Math Manipulatives

Even when we weren't cooking, the measuring cups could teach quite a lot.

A younger child could arrange the cup, half cup, third cup, and quarter cup from largest to smallest. We could see how many quarter cups it took to fill the half cup, how many halves filled the whole cup, or whether three one-third cups really did equal one cup.

Water was particularly good for this, although I learned to put the whole operation on a tray unless I wanted to include an impromptu lesson on mopping.

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Three-quarters wasn’t just a 3 sitting over a 4. It was a half cup and a quarter cup sitting on the counter in front of me

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As the children got older, the questions could grow with them. If two quarter cups equal one half cup, how many eighths would make a half? If we needed 1½ cups, how many half-cup measures would we use?

Instead of beginning with a page full of fraction problems, we could discover the relationships first.

Tablespoons Were Fractions Too

Then there were the measuring spoons.

Sixteen tablespoons equal one cup.

That one fact could keep us busy for quite a while.

If a cup contained 16 tablespoons, how many tablespoons were in half a cup? A quarter cup? Three-quarters of a cup?

Then I could turn the question around. If a recipe called for 12 tablespoons, how many cups was that?

With a younger child, we could simply count tablespoons into a cup measure and watch what happened. An older child could calculate the conversion first and then measure it to see whether the answer was right.

The kitchen had a lovely habit of checking our math for us.

Then We Brought Out the Scale

A kitchen scale opened up another whole layer of mathematics.

We could put a bowl on the scale, tare it, and weigh a cup of flour. Then we'd scoop another cup and weigh that one.

Were they exactly the same?

Usually not.

That gave us something else to think about. A "cup" was a unit of volume, but grams measured weight. Those weren't simply two different ways of writing the same thing.

As my children became ready for more advanced work, a recipe could lead us naturally into ratios, percentages, and conversions.

If a bread recipe used 500 grams of flour and we wanted half a batch, how much flour would we need? What about one and a half batches? If the water weighed 70 percent as much as the flour, how much water would we need?

The same activity could be happening around the same kitchen table while different children were doing very different levels of mathematics.

One child might be finding the half-cup measure for me. Another might be doubling the recipe. An older child could be working with weights, ratios, or percentages.

That was one of the things I loved about homeschooling several children together. They didn't all have to be doing identical work in order to be learning from the same experience.

Preserving Season Was Full of Math

Canning and preserving gave us another wonderfully practical reason to calculate.

If a batch of jam was supposed to yield six half-pint jars and we doubled it, how many jars would we need to prepare?

Easy enough.

But what if we only had pint jars?

What if we had five pint jars and four half-pint jars? Would that be enough?

Now fractions, multiplication, addition, equivalence, volume, and estimation mattered because I wanted to know whether we had enough jars ready before I had a pot of hot preserves waiting for somewhere to go.

With an older child, I could take the problem farther.

If twenty pounds of peaches cost $32 and gave us a certain number of jars, how much did the fruit in each jar cost? What happened when we added the cost of sugar and lids? What did the whole batch cost us?

And if we were going to sell those jars, what would we have to charge?

Without changing subjects, we had wandered from arithmetic into ratios, unit pricing, percentages, cost, and profit.

Except we hadn't really wandered at all.

That was where the math was.

One Kitchen Table, Several Ages

This was one of the reasons I liked teaching through ordinary life. Real work doesn't come labeled "third grade" or "seventh grade."

I could give a little one a genuine job: "Find me the half-cup measure."

Another child could measure two halves and discover that they made a whole.

Someone else could halve the recipe.

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Real work doesn’t come labeled “third grade” or “seventh grade.”

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An older child could double it.

A teenager could convert the recipe from volume to weight, compare the unit price of two different bags of flour, work with ratios, or calculate the cost per finished loaf.

We could all be involved in the same work while each child was doing mathematics appropriate to his or her ability.

I didn't need four separate lessons scattered across four separate workbooks. I needed one real task and questions appropriate to the children sitting around my table.

I Wanted the Problem to Come First

Looking back, I think this was one of the most important parts of the way I taught.

Instead of teaching a procedure first and promising my children that someday they might need it, I tried whenever possible to let them encounter the need first.

We have ⅓ cup of honey, but we're doubling the recipe. How much do we need?

We have eight jars. Will the whole batch fit?

This bag of flour costs less, but that bag is bigger. Which one is actually the better buy?

The bread recipe is written for two loaves, and we only want one. What do we do with ¾ teaspoon of yeast?

Those weren't questions I invented because it was Tuesday and the curriculum said we were studying fractions.

We actually needed the answers.

And that changed the question in the child's mind. Instead of wondering, What answer does Mom want me to put in the blank? the question became, How are we going to figure this out?

That was the question I wanted my children learning to ask.

The Pantry Was Already Full of Math

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Don’t just teach the answer. Give them a reason to need the answer.

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I didn't need special manipulatives to teach fractions. I had them nested inside one another in a kitchen drawer.

I didn't always need a worksheet about measurement conversions. I had tablespoons, cups, flour, water, recipes, and a scale.

And I certainly didn't need to manufacture word problems about imaginary people buying imaginary bags of flour. We could go to the store and find out which bag really was the better buy.

Our pantry was full of fractions, ratios, weights, volumes, percentages, estimation, multiplication, division, and money.

More importantly, it was full of reasons to use them.

That was the kind of education I wanted for my children—not simply knowing how to produce the right answer, but knowing how to think when a real problem was sitting in front of them.

Don't just teach the answer. Give them a reason to need the answer.

That's when math stops being something that happens on a worksheet and becomes a tool a child knows how to use.

Seed Math Activities: Growing Math Skills at the Kitchen Table

Kitchen Table School

Some of the best math manipulatives don't come in a brightly colored box from an educational supply company.

Sometimes they come in a seed packet.

Beans, peas, sunflower seeds, corn kernels, and pumpkin seeds are just about perfect for hands-on math. They're small enough to count and sort, different enough to compare, and plentiful enough that nobody needs to panic when one rolls under the refrigerator.

And unlike plastic counting bears, seeds have a story.

A bean isn't merely something that represents the number seven. It's something a child can hold in his hand today, push into the soil tomorrow, and—with a little water, sunshine, and patience—pick from a plant months from now.

That's the kind of learning I love.

Start With a Bowl of Seeds

You don't need anything elaborate for seed math. Gather several kinds of dried beans or seeds from the pantry, garden, or inexpensive seed packets.

Choose larger seeds for younger children, especially if you have little ones who are still inclined to investigate things with their mouths. Seed activities with small children should always be supervised.

Then put the seeds on the kitchen table.

Before you give any instructions, see what your children do with them.

They may begin sorting them without being asked. They may line them up from smallest to largest. Someone may make a picture. Someone else will almost certainly announce that one bean looks like a potato.

They're already observing, comparing, classifying, and organizing.

In other words, they're doing math.

Count the Seeds

For your youngest mathematicians, begin with simple counting.

Put a small pile of seeds in front of your child and ask her to count them. Have her move each seed from one side of the table to the other as she counts.

That little movement matters. It helps children understand one-to-one correspondence—that each object counted represents one number.

Try making number cards from 1 through 10. Let your child choose a card and place the correct number of seeds on it.

Older children can work with larger numbers. Give them a handful of beans, have them estimate the quantity first, and then count to see how close they came.

Don't skip the estimating. Being able to make a reasonable guess about quantity is an important mathematical skill.

Sort and Classify

Mix several kinds of seeds together and ask your child to sort them.

Don't tell him how.

That's the interesting part.

He might sort them by color, size, shape, or type. Once he's finished, ask:

Why did you put these together?

Now rearrange them using a different rule.

A child who sorted beans by color the first time might sort them by size the second time. Older children can create increasingly complicated classification rules.

Try:

• light and dark

• round and long

• large, medium, and small

• edible seeds and seeds we don't normally eat

• seeds from fruits and seeds from vegetables

• seeds that grow underground crops versus above-ground crops

Classification is foundational mathematical thinking—and it crosses beautifully into science.

Make Patterns

Start a pattern with your seeds:

Bean, bean, corn.
Bean, bean, corn.
Bean, bean...

What comes next?

Once your child understands the idea, let her make patterns for you to complete.

Begin with simple AB patterns and gradually make them more complicated:

ABAB
AABAAB
ABCABC
ABBABB
AABCAABC

For older children, turn the activity around. Make a complicated seed pattern and ask them to describe the repeating unit.

Suddenly you've moved from preschool pattern play toward the kind of thinking they'll eventually use in algebra.

Practice Addition and Subtraction

Seeds make wonderful counters.

Try a tiny garden story:

"I planted five bean seeds in one row and four in another. How many seeds did I plant altogether?"

Let the child physically build the problem with seeds.

For subtraction:

"We planted ten sunflower seeds, but three didn't sprout. How many plants grew?"

Move three seeds away.

The symbols 10 − 3 = 7 are abstract. Ten sunflower seeds sitting on the table are not.

Once children understand what the numbers mean, the symbols have something to attach themselves to.

Build Multiplication Arrays

Now plant an imaginary garden.

Make three rows with four bean seeds in each row.

Ask:

How many rows are there?
How many seeds are in each row?
How many seeds altogether?

Then write:

3 × 4 = 12

You've just created a multiplication array.

And unlike an arbitrary worksheet array, this one reflects something gardeners actually do.

Try different garden beds:

2 rows of 6 peas
4 rows of 5 beans
5 rows of 8 corn seeds

Children who are beginning multiplication can build each problem. Children who already know their multiplication facts can predict the answer first and then use the seeds to check themselves.

Discover Division

Take 24 beans and announce that you have four garden rows.

How many seeds should go into each row if every row gets the same number?

Let your child distribute the beans one at a time.

Then write:

24 ÷ 4 = 6

Try changing the question:

What if we want six seeds in each row? How many rows can we plant?

Same seeds. Same numbers. Different way of thinking.

Explore Fractions

Seeds make fractions wonderfully visible.

Put 12 beans on the table.

Ask your child to divide them into two equal groups.

Each group is one-half.

Put them back together and divide them into four equal groups.

Each group is one-fourth.

Then begin asking questions:

What is half of 12?
What is one-fourth of 12?
What is three-fourths of 12?

Older children can write the corresponding equations after building them.

You can also mix seed varieties.

Put out 10 seeds—5 beans, 3 corn kernels, and 2 pumpkin seeds.

What fraction are beans?

5/10, which can also be written 1/2.

Now fractions aren't mysterious pieces of pizza that nobody actually gets to eat. They're right there on the table.

Measure and Compare

Seeds are terrific for informal measurement.

How many bean seeds long is your pencil?

How many sunflower seeds wide is a notebook?

Which seed makes the best measuring unit?

Children will quickly discover an important principle: measurement only works well when the units are reasonably consistent.

That's why measuring a pencil with a mixture of giant lima beans and tiny lentils gives rather questionable results.

It's also a delightful way to introduce the reason we use standard units.

Graph Your Seeds

Sort a mixture of seeds by type and count each group.

Then make a simple bar graph.

Younger children can make a concrete graph by lining up the actual seeds in columns.

Older children can transfer their results to graph paper.

Ask questions about the graph:

Which group has the most?

Which has the least?

How many more beans are there than corn kernels?

How many seeds are there altogether?

What fraction of the seeds are sunflower seeds?

One bowl of seeds has now taken you from counting all the way to data analysis.

Estimate Germination

Here's where seed math gets even better.

Plant some.

Before planting, count your seeds. Suppose you plant 20 radish seeds.

Ask your child to predict how many will germinate.

A week later, count the seedlings.

If 16 of your 20 seeds sprouted, younger children can simply compare 16 sprouts with the 20 seeds planted.

Older children can calculate the germination rate:

16 ÷ 20 = .80

or

80% germination

Now percentages have a reason to exist.

Keep a record and compare different kinds of seeds. Which had the best germination rate? Does the age of the seed matter? Does soaking make a difference?

At this point, your math lesson has wandered cheerfully into science, gardening, record keeping, and experimental design.

I see no reason to chase it back into its proper subject.

Plan a Tiny Garden

For older elementary children, take the math outside.

Give your child the dimensions of a small garden bed—or let him measure one.

Then look at the planting directions on the seed packet.

If beans should be planted four inches apart, how many will fit in a four-foot row?

How many rows will fit in the bed?

How many seeds will you need?

Do you have enough seeds in the packet?

How many plants could the entire bed hold?

Now you're working with measurement, multiplication, division, area, estimation, and practical problem solving.

And there is an actual garden at the end of the equation.

Let Several Ages Learn Together

This is one of my favorite things about hands-on learning.

You don't necessarily need a separate lesson for every child.

Put the same bowl of seeds in the middle of the table.

Your preschooler can sort and count.

Your early elementary child can practice addition and subtraction.

Another child can make multiplication arrays.

An older student can calculate fractions, percentages, germination rates, or garden spacing.

They're working with the same materials and sharing the same experience, but each child is doing mathematics at an appropriate level.

That makes life considerably easier when you're teaching several children at once.

Math Is Already Everywhere

Children sometimes get the impression that math lives in textbooks.

It doesn't.

Math is in recipes and fence posts. It's in egg cartons and feed buckets. It's in measuring garden beds, doubling bread recipes, figuring out whether six bales of hay will fit in the truck, and deciding how many bean plants will fit along a trellis.

It's even hiding in a handful of seeds.

So pour some beans onto the kitchen table and see where the lesson goes.

You may start with counting.

You may end up planting a garden.

Around here, I'd call that a very successful school day.