Fractions explained for parents: what school teaches now and how to help at home

What is a fraction, in words a child understands?

A fraction is a number that tells you how many equal parts of one whole you have. Cut a sandwich into 4 equal pieces and eat 3, and you ate 3⁄4 of it. The word that does the work is equal: 4 pieces of different sizes are not fourths.

Each fraction has two numbers, and children mix them up at first:

• The denominator (bottom) says how many equal parts the whole is cut into. It names the size of the piece.
• The numerator (top) says how many of those parts you have. It counts the pieces.

A fraction with 1 on top is called a unit fraction: 1⁄2, 1⁄3, 1⁄4, 1⁄8. Schools build every other fraction out of these. The Common Core standards describe 3⁄4 as three parts of size 1⁄4 (3.NF.A.1), and the same idea is how your child will add fractions later: 3⁄4 is 1⁄4 + 1⁄4 + 1⁄4.

The way into fractions for most children is sharing. The What Works Clearinghouse practice guide Developing Effective Fractions Instruction for Kindergarten Through 8th Grade recommends that teachers build on children's informal understanding of sharing to introduce fractions. Two children and one cookie is a lesson on halves before anyone writes a numeral.

Why do schools put fractions on a number line?

Schools put fractions on a number line because a fraction is a number with a size, and the line shows that size. Many parents learned fractions only as pizza slices, and that picture hides something important: 3⁄4 is a number that sits between 0 and 1, the way 7 sits between 6 and 8.

The grade 3 standard asks children to understand a fraction as a number on the number line (3.NF.A.2). Here is what that looks like at home:

1. Draw a line and mark 0 at one end and 1 at the other.
2. Split the space between them into 4 equal jumps.
3. The first mark is 1⁄4. Two jumps from 0 is 2⁄4. Three jumps is 3⁄4.
4. Four jumps lands on 1, so 4⁄4 = 1.

The What Works Clearinghouse guide recommends using number lines as a central representational tool from the early grades onward. The US National Mathematics Advisory Panel reached a similar view in 2008: it called difficulty with fractions pervasive and a major obstacle to algebra, and named the ability to place fractions on a number line as one key link between understanding fractions and computing with them.

How do you explain equivalent fractions?

Equivalent fractions are different names for the same amount, such as 1⁄2 = 2⁄4 = 4⁄8. The fastest way to show it is a strip of paper.

1. Fold a strip in half and shade one half. That is 1⁄2.
2. Fold it in half again and open it. The shaded part now covers 2 of 4 equal parts: 2⁄4.
3. Fold it a third time. The shaded part covers 4 of 8 parts: 4⁄8.

The shaded amount never changed. The pieces got smaller and there were more of them. That is the whole rule, and the grade 4 standard says it in one line: multiply the top and the bottom by the same number and you get an equivalent fraction (4.NF.A.1).

• 1⁄2 = (1 × 2)⁄(2 × 2) = 2⁄4
• 2⁄3 = (2 × 2)⁄(3 × 2) = 4⁄6
• 3⁄4 = (3 × 3)⁄(4 × 3) = 9⁄12

Every step there is a times-table fact, which is why a child who is slow on the tables finds equivalent fractions hard. A few minutes of times tables practice pays off here more than extra fraction worksheets.

Play free in your browserTimes TablesGrade 3+

How do you compare fractions?

Children compare fractions by asking what the two fractions share: the same denominator, the same numerator, or neither. Each case has its own quick test.

The most common mistake is thinking 1⁄8 is bigger than 1⁄4 because 8 is bigger than 4. The What Works Clearinghouse guide names that exact pattern: students view fractions with larger denominators as larger. The fix is a question about people: would you rather share one pizza with 3 friends or with 7? Your child knows the answer at once, and the answer is the rule.

One more point from the standards is worth saying out loud: a comparison only works when both fractions refer to the same whole. Half of a small pizza is not the same amount as half of a large one. Comparing sizes is a skill children already practice with whole numbers, and a round of comparing numbers is a good warm-up for the greater-than and less-than signs.

How do you add fractions with different denominators?

Fractions with different denominators can only be added once both are cut into pieces of the same size. The grade 5 standard calls this replacing given fractions with equivalent fractions with like denominators (5.NF.A.1).

Worked example: 1⁄3 + 1⁄4.

1. Find a number both denominators go into. Count up the 3s and the 4s: 12 is in both tables.
2. Rename each fraction. 1⁄3 = 4⁄12 (times 4 on top and bottom). 1⁄4 = 3⁄12 (times 3 on top and bottom).
3. Add the numerators, keep the denominator. 4⁄12 + 3⁄12 = 7⁄12.
4. Check the size. Both fractions are a bit less than 1⁄2, so the answer should be a bit more than 1⁄2. 7⁄12 is a bit more than 6⁄12. It fits.

The classic wrong answer is 1⁄3 + 1⁄4 = 2⁄7, made by adding tops and bottoms separately. The WWC guide traces that error to not seeing fractions as numbers with sizes, and suggests an estimate to catch it. The estimate works here: 2⁄7 is less than 1⁄3, and adding 1⁄4 to 1⁄3 cannot give less than you started with. The grade 5 standard uses the same check, recognizing 2⁄5 + 1⁄2 = 3⁄7 as wrong because 3⁄7 is less than 1⁄2.

A chocolate bar with 12 squares makes the example real. One third of the bar is 4 squares, one quarter is 3 squares, and together they are 7 of the 12 squares.

What fractions does each grade learn?

Fraction work in the Common Core runs from grade 3 to grade 5, with a fixed set of denominators at each step. What math each grade should know covers the other strands for every grade.

Grade 5 also teaches that 3⁄4 means 3 ÷ 4. That link makes division facts useful here too: a quick round of division facts or a division worksheet for 3rd graders keeps them sharp. Practice pitched at a fourth grader's level is on the math game for 4th graders page.

Free printable worksheetDivision — 3rd GradersWorksheet · PDF

Kitchen-table activities that teach fractions

Short, hands-on activities at home teach fractions better than a stack of worksheets, because your child can see and move the equal parts.

• Measuring cups. Ask how many 1⁄4 cups fill the 1⁄2 cup. Two do, which is 2⁄4 = 1⁄2 poured out in front of you. Then try three 1⁄3 cups into the 1 cup.
• Folding paper. Fold a sheet into halves, fourths and eighths, and shade the same amount each time. Line the strips up and name the matching fractions.
• Sharing a chocolate bar. Use a bar with 12 squares. Ask for 1⁄2, 1⁄3, 1⁄4 and 1⁄6 of it in squares: 6, 4, 3 and 2.
• Cutting a pie or a pizza. Cut it into 8 equal slices. Ask whether 3⁄8 or 1⁄2 is more, and let your child count to check.
• A number line on the fridge. Tape a strip of paper from 0 to 1 and add a new fraction each evening.

Keep it to ten minutes and stop while it still feels easy. More ideas for math you can do together are on the math games for parents and kids page.