• At the beginning of the lesson, use the Admit Ticket to determine student understanding of fraction division.
• Observe students during the station rotation activity to evaluate student progress.

Say, “Today we are going to explore the division of fractions. There are times when we have to divide a fraction by a whole number, a whole number by a fraction, or a fraction by a fraction. If you have half a pie and you want to share it equally among three friends, how much of the original pie will each person get? If there are 15 feet of ribbon and you need to create sashes that are feet long, how many sashes can you make? Looking at these examples, you can see that knowing how to divide fractions can help you determine the correct amount.”

“We found a recipe to make trail mix.”

 cups peanuts

 cup chocolate chips

 cup raisins

 cup coconut

 cup sunflower seeds

“We realize the recipe would make too much trail mix, so we want to make only half of the recipe. How would we go about determining how much of each ingredient we need? Remember that we want this trail mix recipe. We do not want it to taste different, so we cannot divide one ingredient by two and forget to divide the other ingredients by two as well.”

Allow students time to dialogue with each other. Share responses. Look for similarities in thinking. Guide discussion toward the understanding that we would have to divide each of the ingredients by two.

“Today’s lesson will help us to understand how to divide in situations that involve fractions.”

Use graham crackers that are scored into fourths. If actual graham crackers are not available, model using linking cubes in sets of four. Ask three students to take a whole graham cracker (or set of linking cubes). Have these three students carefully break the graham crackers (or set of linking cubes) into four equal pieces. Ask students if they can think of a division equation or expression to represent this model and explain what it represents. Record the equation for students to see. (; there were three wholes and students broke each whole into four equal pieces, fourths.)

Review the terms dividend, divisor, and quotient. The dividend is the total amount we have, the divisor is the number we are dividing by, and the quotient is the answer. Ask students, “What will the quotient be?” (12)

“What do you notice about the quotient compared to the original number of pieces?” (The amount increased. When you divide a whole number by a fraction, the number gets bigger. Usually with division, the quotient is a smaller amount.)

Model another example with a picture. “Let’s say there are five packages of peanut butter cups (dividend). We know that each pack has two peanut butter cups.

“If we separate the peanut butter packages, how many peanut butter cups will we have all together? In other words, how many halves fit into 5?” (10)

“What is the division number sentence? How do you know?” (5 ÷ 1/2 =__, because we want to know how many groups of ½ go into 5.)

“Are you starting to see a relationship between the quotient and the dividend/divisor?” (The quotient gets larger. It seems that when you multiply the whole number (dividend) by the denominator in the divisor, you get the quotient.)

“Another way we can look at this is by asking the question: ‘How many ----- fit in ____?’ If you have the problem , you can ask the question: ‘How many s fit into 3?’ Another way to say this is, ‘How many one-half pieces did you get?’” Ask students how they can figure this out using a manipulative, picture, or repeated subtraction. If you use manipulatives like fraction circles, you can take three wholes and place half pieces on top of the three wholes. After modeling, ask students, “Now count the number of halves. How many halves are there?” (6 halves)

If drawing a picture, draw three wholes. Then divide each whole in half and count up how many halves: six halves.

If using repeated subtraction, remind students that just as repeated addition is another way to do multiplication (3 + 3 + 3 + 3 = 12; 3 × 4 = 12), repeated subtraction is another way to do division (12 – 3 = 9; 9 – 3 = 6; 6 – 3 = 3; 3 – 3 = 0; 12 ÷ 3 = 4). Using this strategy, you can find by doing . Be sure to point out to students that  was subtracted 6 times, so the answer to the division problem using the strategy of repeated subtraction is 6. Do additional problems as necessary.

“What do you think will happen if we start with a fraction and divide by a whole number?” Allow students to dialogue and discuss with each other. Share predictions and discuss reasoning. (Possible responses: The quotient will be larger than both the divisor and the dividend because we were just practicing similar problems. The quotient will be smaller because when you start with a small number and divide by a larger number, you will get an even smaller number.)

“Let’s see if any of our predictions were on target. If we have  , what would be the quotient? We can look at it by asking ourselves the question, ‘how can we put    into 3 equal groups?’ Any ideas?” (I know the denominator tells me what size piece I have: fifths. I know I have three of these pieces. So, each group can get one of these pieces. The quotient would be .)

Do additional problems similar to these until students start to see a pattern: , reduced to . Try another: .

“Is anyone starting to see a pattern?” (The denominator stays the same because that tells us what size piece we are referring to. We break up the numerator into the whole number we are dividing by.)

“Can anyone come up with an equation that would fit this pattern?” Allow students time to dialogue with each other. Ask students to share their equations and record the equations on chart paper/white board. Once the equations are recorded, discuss them as a class if indeed these equations fit the pattern. Address any misconceptions.

“Now what happens if we cannot break up the numerator into equal groups as in the equation: ?” Ask students to share their thoughts. (Possible answers include: You will have to somehow break up the leftover pieces. You might have to use equivalent fractions. The denominator will no longer be the same.)

“Let’s see if any of our ideas will help us to solve the following problem:

? I have a half. If I draw a whole and divide it in half, how can I break a half into two equal parts?”

“I know by the drawing that  is equivalent to . Now I can break   into two groups as we did earlier. Each group would then get . So .”

“If we look at the answer, we see that the quotient is smaller than the dividend. Is that what we originally thought?”

Station Rotation

Create four stations for additional practice. Divide students into four groups, and allow them to work at each station for approximately seven to ten minutes. Any student who needs assistance can go to Station 4, receive help, and then return to the station rotation when ready.

• Station 1: Four Square - Division by a Fraction (M-5-2-2_Four Square - Division by a Fraction and KEY.docx). Provide an answer key for students to check their answers.
• Station 2: Four Square - Division by a Whole Number (M-5-2-2_Four Square - Division by a Whole Number and KEY.docx). Provide an answer key for students to check their answers.
• Station 3: Recipe Conversion (M-5-2-2_Recipe Conversion and KEY.docx). Take the recipe introduced earlier in the lesson and divide each ingredient by 4. Rewrite the recipe with the adjusted amounts for each ingredient.
• Station 4:At this station, students will meet with you to review any challenges they may be experiencing and to get answers to any questions they may have. Discussion of the recipe conversion can occur here or answers can be shared. You can also informally assess student understanding by asking students the following questions:
  • “What are the three parts of a division equation?” (dividend, divisor, quotient)
  • “When you divide a whole number by a fraction, what happens to the quotient?”(gets larger than the dividend)
  • “When you divide a fraction by a whole number, what happens to the quotient?” (gets smaller than the dividend)
  • “When might you have to divide fractions in the real world?” (baking, measuring, sharing items)

Use the data from task rotation to assess student understanding. Look over responses of the four squares that required written explanations and highlight any areas that may need to be readdressed. Allowing students the opportunity to check their work at a station rotation gives them the immediate feedback necessary to monitor their own understanding. Teacher observation and student interaction at Station 4 will also help promote understanding.

Extension:

• Routine: Emphasize proper use of vocabulary in lessons and classroom discussions. Allow students to work with partners or in small groups during some activities. Use an Admit Ticket, as discussed below, or review other activities to reinforce mathematical concepts and check for understanding.

An Admit Ticket (M-5-2-2_Admit Ticket and KEY.docx) is a strategy used to reinforce or assess skills previously taught. At the beginning of class, hand an Admit Ticket (task card/note card) to students as they enter the classroom. Students can be given one of the word problems created by students on the four-square station rotation sheets. Direct them to quickly solve the problem and check their accuracy and logic with a classmate. This gives you another opportunity to make note of which students show mastery and which students may need some additional practice.

• Small Group: Based on formative assessments, small-group instruction can be used to help strengthen understanding. Wholes made from construction paper can be used to represent items like candy bars or pizza. Then students can be asked to “break up” the whole into equal groups after being given a scenario. For example, you have three pizzas, and you want to divide each pizza into sixths. How many pieces will there be in all? Students can cut apart or fold the whole to represent the process of division.
• Expansion: Ask students who quickly grasp division of fractions by a whole number to locate a recipe of choice and halve or quarter the ingredients.

Students can also explore the division algorithm of “invert and multiply.” Remind students that when you divide something by a whole number, you are dividing it into equal parts. This is the same as taking the fractional part, which is multiplication. Look at the problem done earlier in the lesson, __. Using the invert-and-multiply algorithm, the problem becomes . Remind students that dividing by 2 is the same as multiplying by .

Use another example: ___. Using the invert-and-multiply algorithm, the problem becomes , which is equivalent to . Dividing a number into 3 parts is the same as taking  part, or multiplying by . Once students see a few examples like these, ask them how they can apply what they just learned to an equation like ÷= __. Have students try other examples and explain in words why the invert-and-multiply method works when dividing fractions.