Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.
Official wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
Two fractions can name exactly the same amount even when their numbers look different. In fourth grade, students learn why: if you cut every piece of a fraction model into the same number of smaller pieces, you get more pieces, each one smaller, but the shaded amount does not change. Cutting each third of a bar into 4 equal parts turns 2/3 into 8/12, because there are now 4 times as many parts in the whole and 4 times as many shaded parts.
The key idea is that multiplying the numerator and the denominator by the same number is not a trick. It describes what happens when you partition a model more finely. Students should be able to explain this with area models, fraction strips and number lines, and then use it to generate equivalent fractions (3/5 = 6/10 = 9/15) and to check whether two fractions are equivalent. The standard limits the denominators to 2, 3, 4, 5, 6, 8, 10, 12 and 100, which keeps the models drawable.
Students who think 2/3 = 3/4 because they added 1 to each part need to see the two fractions drawn on the same-size bar. The pieces are different sizes, so the amounts are different.
Seeing 8/12 next to 2/3, some students say 8/12 is larger because the numbers are bigger. Fold or cut a strip in front of them so they see the shaded part stay exactly the same.
Equivalence 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, so models must be drawn the same size.
Writing 3/4 = 6/4 shows the student has scaled the shaded parts but not the total parts. Ask how many parts the whole is now cut into.
Use a fraction model to decide whether 2/3 and 8/12 are equivalent, then write the multiplication that shows it.
Answer: Yes. 2/3 = 8/12, because cutting each third into 4 pieces gives 12 pieces with 8 shaded, and the shaded amount stays the same.
Start with paper strips that students fold themselves: a strip folded into thirds and then folded in half again makes the jump from 1/3 to 2/6 visible and physical. Move to drawn area models, then to number lines, where equivalent fractions land on the same point. Only after students can explain the model should the rule n × a / n × b become the quick method.
On state tests this standard usually appears as find the missing number (3/4 = ?/12), pick every fraction equivalent to a given one, or explain with a model. It is the foundation for 4.NF.A.2 (comparing fractions) and for adding fractions with unlike denominators in fifth grade.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: 9
The denominator went from 4 to 12, which is 4 × 3. Multiply the numerator by the same number: 3 × 3 = 9. So 3/4 = 9/12.
Answer: D) 4/10
Multiply the numerator and denominator of 2/5 by 2: (2 × 2)/(5 × 2) = 4/10. Adding 1 to both (3/6) does not keep the amount the same.
Answer: 12
The numerator doubled from 5 to 10, so the denominator must double too: 6 × 2 = 12. So 5/6 = 10/12.
Answer: B) 4/12
There are 3 × 4 = 12 pieces in the bar, and the shaded third holds 4 of them. The shaded part is 4/12, the same amount as 1/3.
Answer: A) No, 2/3 is less than 3/4 when both are drawn on the same whole
On equal-size strips 2/3 = 8/12 and 3/4 = 9/12, so 3/4 is larger. Equivalent fractions come from multiplying (or dividing) both parts by the same number, not adding.
Answer: 70/100
Multiply the numerator and denominator by 10: (7 × 10)/(10 × 10) = 70/100.
Answer: C) 4/6
3/6, 5/10 and 6/12 all have a numerator that is half the denominator. 4/6 simplifies to 2/3, which is more than one half.
A full lesson with slides, activities and an exit ticket on equivalent fractions, pitched to grade 4 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 4.NF.A.1 with an answer key, ready in about a minute.
Make a worksheet →Turn equivalent fractions into a quiz students answer online that marks itself, with a class summary for you.
Build a test →It means fourth graders can explain why fractions such as 1/2, 2/4 and 4/8 are the same size, using a drawing or model, and can use that idea to make new equivalent fractions.
Grade 4 fraction work in Common Core is limited to denominators of 2, 3, 4, 5, 6, 8, 10, 12 and 100, so every fraction can be shown with a manageable model.
In third grade (3.NF.A.3) students recognize simple equivalent fractions with a model. In fourth grade they explain the general reason, (n × a)/(n × b), and use it with a wider set of denominators.