Equivalent fractions

By the end of the lesson, students can produce at least two fractions equivalent to a given fraction and explain, using a model or by multiplying the numerator and denominator by the same nonzero number, why the fractions name the same value.

Before this lesson: Students can identify the numerator and denominator of a fraction, multiply whole numbers, and explain that a fraction names part of a whole.

• 6 strips of paper of equal length per group of four
• Scissors and pencil
• One printed activity sheet per student
• Manila paper and marker for each group's answer board
• Whiteboard or projector for the class example

50 minutes · one classroom screen

Start with a bar divided into 2 equal parts with 1 part shaded. Ask the class how much is shaded (1/2). Draw the same length bar divided into 4 equal parts with 2 shaded. Ask again how much is shaded. Students will say 2/4. Ask: 'Is the same amount of the bar shaded in both pictures?' Let them argue, then place the two bars one above the other so they can see the shaded lengths line up exactly. Both bars name half of the same whole, so 1/2 = 2/4. Write that sentence on the board and keep it there for the rest of the lesson. Now show the arithmetic: 1/2 = (1 x 2)/(2 x 2) = 2/4. Say clearly: you multiply the numerator and the denominator by the same nonzero number. Cutting every piece into two smaller pieces gives you twice as many pieces, and each piece is half as big, so the amount stays exactly the same. Show that dividing works too: 4/8 divided by 4 in both parts becomes 1/2, which is why we can simplify. Then show what does not work: 1/2 is not 2/3. Adding 1 to the numerator and 1 to the denominator changes the size of the pieces you are counting, so the value changes. Close the explanation by writing two rules on the board: multiply or divide both parts by the same nonzero number and the value does not change; adding the same number to both parts does not generally preserve the value.

1. 1Make an equivalent fraction for 1/3 by multiplying by 3: (1 x 3)/(3 x 3) = 3/9. Check with a bar: three equal parts with one shaded covers the same length as nine equal parts with three shaded.
2. 2Make an equivalent fraction for 3/4 by multiplying by 2: (3 x 2)/(4 x 2) = 6/8. Multiplying the denominator by 2 cuts each fourth into two eighths; multiplying the numerator by 2 counts twice as many of those smaller pieces.
3. 3Simplify 6/9 by dividing both parts by 3: (6 ÷ 3)/(9 ÷ 3) = 2/3. Ask the class to check with a bar before moving on, because simplification is the same idea read backwards.
4. 4Non-example to work through together: is 2/3 equivalent to 3/4? Compare the cross products: 2 x 4 = 8 and 3 x 3 = 9. They are not equal, so 2/3 and 3/4 are not equivalent.

1. 1Together, fill in the blank: 1/2 = ?/6. Ask what was multiplied in the denominator, then what must be multiplied in the numerator.
2. 2Together, fill in the blank: 3/5 = ?/15.
3. 3Together, decide whether 4/6 and 2/3 are equivalent, and justify the answer in one sentence.
4. 4Together, write two fractions equivalent to 2/5, and check one of them with a bar drawing on the board.

1. 1Write two fractions equivalent to 2/3.
2. 2Write two fractions equivalent to 5/8.
3. 3Complete: 1/4 = ?/12
4. 4Complete: 3/7 = ?/21
5. 5Simplify 10/15.
6. 6Simplify 8/12.
7. 7Are 3/5 and 9/15 equivalent? Show how you know.
8. 8Are 4/9 and 8/16 equivalent? Show how you know.

• Exit slip: write one fraction equivalent to 4/5 and explain in one sentence why it is equivalent.
• Show a bar with 3 of 6 parts shaded. Ask: 'Without counting again, how do you know this is also one half?'
• Ask one student to explain why 1/2 is not 2/3, using the words numerator and denominator.

• Adding the same number to the numerator and denominator makes an equivalent fraction, e.g. 1/2 = 2/3.

  What to do: Ask the student to shade 1/2 and 2/3 on two equal bars. The shaded lengths do not line up. Name the reason out loud: each piece of a third is not the same size as each piece of a half, so counting one more of them does not give the same amount.

• A fraction with larger numbers is always larger, so 2/4 must be bigger than 1/2.

  What to do: Use the folded strips from the activity. The two shaded parts of the strip divided into four end at exactly the same place as the one shaded part of the strip divided into two. More pieces, smaller pieces, same amount.

• Equivalent fractions are different fractions because the numbers are different.

  What to do: Separate the two ideas: the symbols are different, the value is the same. Ask the student to write the amount as a decimal or as a fraction of one whole and compare.