Let’s Learn!
How do you compare fractions that have the same numerator?
Goal:
Goal:
Daniel is excited to read about the super celebration. Superheroes Donte and Ian cut the cookie cakes into different numbers of pieces. Each cake is the same size and shape. Donte and Ian want to know which cake has the biggest pieces and which cake has the smallest pieces.
Each part of each cookie cake is a fraction with a numerator and a denominator. Donte will use his fraction powers to show the parts of the first two cakes!
Click on the box to see the fractions of the first two cookie cakes!
Two cookies. The oatmeal raisin cookie is partitioned into 2 equal parts. The double chocolate cookie is partitioned into 3 equal parts.
The oatmeal cookie's parts are labeled \({ \frac{1}{2} }\).
The double chocolate cookie's parts are labeled \({ \frac{1}{3} }\).
The oatmeal raisin cookie cake has two equal parts. Each of these parts is \({ \frac{1}{2} }\). The double chocolate cookie cake has three equal parts. Each of these parts is \({ \frac{1}{3} }\).
Each part of the cookie cakes has the same numerator, 1. That is because each part is 1 part out of the whole of each cookie. Ian used his fraction superpower to show one piece of each cookie cake and its fraction.
Which piece is bigger, the oatmeal raisin half or the double chocolate third? Donte says that the half looks bigger than the third, but he wants to make sure. Ian flew in two number lines to compare the two fractions: \({ \frac{1}{2} }\) and \({ \frac{1}{3} }\) . Because the fractions have different denominators, the number lines have different numbers of parts too!
First, Ian shows \({ \frac{1}{2} }\) on the first number line by drawing a purple line to \({ \frac{1}{2} }\). Click on the empty number line to show \({ \frac{1}{2} }\).
Number line from 0 to 1 with 2 spaces between. The number line is labeled from \({ \frac{0}{2} }\) to \({ \frac{2}{2} }\)
Number line from 0 to 1 with 2 spaces between. The number line is labeled from \({ \frac{0}{2} }\) to \({ \frac{2}{2} }\). One part is filled in to \({ \frac{1}{2} }\).
Then Ian shows \({ \frac{1}{3} }\) on the second number line by drawing a brown line to \({ \frac{1}{3} }\). Click on the empty number line to see how he shows \({ \frac{1}{3} }\).
Number line from 0 to 1 with 3 spaces between. The number line is labeled from \({ \frac{0}{3} }\) to \({ \frac{3}{3} }\)
Number line from 0 to 1 with 3 spaces between. The number line is labeled from \({ \frac{0}{3} }\) to \({ \frac{3}{3} }\). One part is filled in to \({ \frac{1}{3} }\).
Ian flew in red lines to show where the fractions end on the number lines. Even though the number lines are divided into different fractions, the lines themselves are the same size. Donte used his fraction powers to see that \({ \frac{1}{2} }\) is greater than \({ \frac{1}{3} }\). \({ \frac{1}{2} }\) is greater than \({ \frac{1}{3} }\) because it is closer to 1 on the number line!
Two number lines. Top number line from 0 to 1 with 2 spaces between. The number line is labeled from \({ \frac{0}{2} }\) to \({ \frac{2}{2} }\). One part is filled in to \({ \frac{1}{2} }\). Bottom number line from 0 to 1 with 3 spaces between. The number line is labeled from \({ \frac{0}{3} }\) to \({ \frac{3}{3} }\). One part is filled in to \({ \frac{1}{3} }\).
Each number line has a dotted line extending from their filled in part. The top number line is longer.
The 2 in the denominator of \({ \frac{1}{2} }\) is less than the 3 in the denominator of \({ \frac{1}{3} }\). This is because the smaller the denominator, the bigger the parts! When you cut the same size space into more parts, the parts get smaller.
\({ \frac{1}{2} \gt \frac{1}{3} }\). The denominators are circled.
2 is less than 3
Ian says that you can also use lines with dots or just dots to show each fraction.
Two number lines. Top number line from 0 to 1 with 2 spaces between. The number line is labeled from \({ \frac{0}{2} }\) to \({ \frac{2}{2} }\). One part is filled in to \({ \frac{1}{2} }\). Bottom number line from 0 to 1 with 3 spaces between. The number line is labeled from \({ \frac{0}{3} }\) to \({ \frac{3}{3} }\). One part is filled in to \({ \frac{1}{3} }\).
Ian’s number lines with lines and dots
Two number lines. Top number line from 0 to 1 with 2 spaces between. The number line is labeled from \({ \frac{0}{2} }\) to \({ \frac{2}{2} }\). Dot on \({ \frac{1}{2} }\). Bottom number line from 0 to 1 with 3 spaces between. The number line is labeled from \({ \frac{0}{3} }\) to \({ \frac{3}{3} }\). Dot on \({ \frac{1}{3} }\).
Ian’s number lines with dots
Ian also says that you can use \({ \lt }\) (less than), \({ \gt }\) (greater than), or \({ = }\) (equal to) to compare fractions. He came up with a super rule for comparing fractions:
If the numerator is the same, the fraction with the smaller denominator is the greater fraction!
Click each slide to compare the fractions with the superheroes!
Alani wants a piece of the double chocolate cookie cake. It is cut into thirds. Joe wants a piece of the strawberry frosted cookie cake. It is cut into fourths. Which superhero will get more cookie cake?
Left: \({ \frac{1}{3} }\) piece of a double chocolate cookie cake.
Right: \({ \frac{1}{4} }\) piece of a frosted cookie cake.
Alani and Joe put a fraction on each piece of cookie cake. They noticed that the fractions have the same numerator and different denominators. They think that the \({ \frac{1}{3} }\) piece looks bigger than the \({ \frac{1}{4} }\) piece. They need to check their work by comparing the fractions. Click the Show Me button to compare the fractions by plotting them on number lines and by looking at their denominators!
Alani’s fraction is the brown dot on the first number line. Joe’s fraction is the pink dot on the second number line.
Two number lines. Top number line from 0 to 1 with 3 spaces between. The number line is labeled from \({ \frac{0}{3} }\) to \({ \frac{3}{3} }\). Dot on \({ \frac{1}{3} }\). Bottom number line from 0 to 1 with 4 spaces between. The number line is labeled from \({ \frac{0}{4} }\) to \({ \frac{4}{4} }\). Dot on \({ \frac{1}{4} }\).
3 is less than 4!
\({ \frac{1}{3} \gt \frac{1}{4} }\). The denominators are circled.
So \({ \frac{1}{3} \gt \frac{1}{4} }\)!
Alani’s piece is bigger. She will get more cookie cake!
Donte wants two pieces of the chocolate chip cookie cake. It is cut into sixths. Ian wants two pieces of the banana cookie cake. It is cut into fifths. Which superhero will get more cookie cake?
Left: \({ \frac{2}{6} }\) of a chocolate chip cookie cake.
Right: \({ \frac{2}{5} }\) piece of a banana cookie cake.
Donte and Ian circled their pieces but did not put fractions on them. They noticed that the fractions have the same numerator and different denominators. They think that \({ \frac{2}{6} }\) of the chocolate chip cookie cake looks smaller than \({ \frac{2}{5} }\) of the banana cookie cake. They need to check their work by comparing the fractions. Click the Show Me button to compare the fractions by plotting them on number lines and by looking at their denominators!
Donte’s fraction is the black dot on the first number line. Ian’s fraction is the red dot on the second number line.
Two number lines. Top number line from 0 to 1 with 6 spaces between. The number line is labeled from \({ \frac{0}{6} }\) to \({ \frac{6}{6} }\). Dot on \({ \frac{2}{6} }\). Bottom number line from 0 to 1 with 5 spaces between. The number line is labeled from \({ \frac{0}{5} }\) to \({ \frac{5}{5} }\). Dot on \({ \frac{2}{5} }\).
6 is greater than 5!
\({ \frac{2}{6} \lt \frac{2}{5} }\). The denominators are circled.
So \({ \frac{2}{6} \lt \frac{2}{5} }\)!
Ian’s pieces are bigger. He will get more cookie cake!
Joe and Donte want more cookie cake, but the cakes are not done yet! They know that the first cake will be cut into sevenths. The second cake will also be cut into sevenths. Joe and Donte both want three pieces. Which superhero will get more cookie cake?
Left: \({ \frac{3}{7} }\).
Right: \({ \frac{3}{7} }\).
Can the superheroes still compare without seeing the cookie cakes? Yes! The fractions have the same numerator and the same denominator. They can use number lines to show their fractions! Click the Show Me button to compare the fractions by plotting them on number lines and by looking at their denominators!
Joe’s fraction is the blue dot on the first number line. Donte’s fraction is the orange dot on the second number line.
Two number lines. Top number line from 0 to 1 with 7 spaces between. The number line is labeled from \({ \frac{0}{7} }\) to \({ \frac{7}{7} }\). Dot on \({ \frac{3}{7} }\). Bottom number line from 0 to 1 with 7 spaces between. The number line is labeled from \({ \frac{0}{7} }\) to \({ \frac{7}{7} }\). Dot on \({ \frac{3}{7} }\).
7 is equal to 7!
So \({ \frac{3}{7} = \frac{3}{7} }\)!
Both superheroes will get the same amount of cake!
The superheroes want to share the cakes with their teachers. There is a piece of the banana cookie cake (\({ \frac{1}{5} }\)), a piece of the chocolate chip cookie cake (\({ \frac{1}{6} }\)), and a piece of the double chocolate cookie cake (\({ \frac{1}{3} }\)) left over. Which cake has the most left over?
Left: \({ \frac{1}{5} }\).
Middle: \({ \frac{1}{6} }\).
Right: \({ \frac{1}{3} }\).
How can we compare these three fractions? By putting them in order from greatest to least or least to greatest! The superheroes want to show the fractions from greatest to least. That means the greatest fraction goes first and the least fraction goes last. Click the Show Me button to compare the fractions by plotting them on number lines and by looking at their denominators!
The banana cookie cake’s fraction is the red dot on the second number line. The chocolate chip cookie cake’s fraction is the black dot on the last number line. The double chocolate cookie cake’s fraction is the brown dot on the first number line.
Three number lines. First number line from 0 to 1 with 3 spaces between. The number line is labeled from \({ \frac{0}{3} }\) to \({ \frac{3}{3} }\). Dot on \({ \frac{1}{3} }\). Second number line from 0 to 1 with 5 spaces between. The number line is labeled from \({ \frac{0}{5} }\) to \({ \frac{5}{5} }\). Dot on \({ \frac{1}{5} }\). Third number line from 0 to 1 with 6 spaces between. The number line is labeled from \({ \frac{0}{6} }\) to \({ \frac{6}{6} }\). Dot on \({ \frac{1}{6} }\).
The dot that is closest to 1 is the greatest fraction. The dot that is closest to 0 is the least fraction. That is because lesser numbers are on the left and greater numbers are on the right. To put the dots in order from greatest to least, go from right to left!
\({ \frac{1}{3} }\), \({ \frac{1}{5} }\), \({ \frac{1}{6} }\)
\({ \frac{1}{3} }\) is the greatest. That is because it has the denominator with the least value. \({ \frac{1}{6} }\) is the least. That is because it has the denominator with the greatest value. The double chocolate cookie cake has the most left over!
Question:
How can you compare or order fractions if you do not have a number line or fraction picture?
When fractions have the same numerator, look at their denominators to compare or order them!