Mixed Fraction Arrays
5th Grade
Multiplying Mixed Fractions
Unit Five – PBIT Five
Launch
Student Goals: Use an area model to multiply mixed fractions
Grouping: pairs
Student materials: pencil, student sheet, four colored pencils, inch grid paper for final problem
Teacher materials: inch grid paper for demonstration, four colors of markers or pencils
A brief introduction to
a problem led by the
teacher.
The Launch phase must be kept brief. It is critical that, while giving students a clear picture of what is expected, the teacher leaves the potential of the
task intact. He or she must be careful to not tell too much and consequently lower the challenge of the task to something routine, or to cut off the rich
array of strategies that may evolve from a more open launch of the problem.
Adapted from The Connected Mathematics Project 2, Michigan State University, 2006
Students have been using an area model to multiply fractions. Whether the problems were fraction x whole number or fraction x fraction, the square has represented one whole. This PBIT
will continue to investigate multiplying mixed fractions. Students were reminded of arrays as a model for multiplication in the last PBIT. We will use arrays again to help make sense of a
reasonable estimate for a problem and also to see how the array is extended when multiplying mixed fractions.
Begin by discussing the models with a progression of problems from whole number arrays to mixed fraction arrays.
1
1
½
1
½
1
1x1
1x½
1½ x 1½
2
2
½
2
½
1
1x2
1x2½
Suggested questions:
What multiplication problem is modeled with a single square?
How can we build on the model for 1 x 1 to show 1 x 1 ½ ?
How can we build on the model for 1 x 1 ½ to show 1 ½ x 1 ½ ?
What multiplication problem is modeled with these two squares?
How can we build on the model for 1 x 2 to show 1 x 2 ½ ?
How can we build on the model for 1 x 2 ½ to show 2 ½ x 2 ½ ?
2½ x 2½
Explain to students that they will be working with a partner to draw models for other fraction multiplication problems. After they have drawn models we will come back together as a whole
class and discuss how to solve the problems.
Page 1
Mixed Fraction Arrays
Explore
5th Grade
Students work on the
problem while
teacher observes and
supports their work.
Multiplying Mixed Fractions
Unit Five – PBIT Five
Students will vary in their progress in completing the problem. The teacher should move around the classroom observing individual performance and
encouraging on-task behavior. The teacher helps students persevere in their work by asking appropriate questions and providing confirmation and
redirection when needed. The Explore phase is the preparation work for the Summary phase, think about how to use students’ work and strategies during
the Summary phase.
Adapted from The Connected Mathematics Project 2, Michigan State University, 2006
As you circulate among the students check to see if students are adding on to the arrays correctly.
Suggested question for struggling students:
How long is the original array?
How long do you want it to be?
What will you have to do to the length to make it that long?
How wide is the original array?
How wide do you want it to be?
What will you have to do to the length to make it that wide?
As students extend the arrays watch to see that they are creating a new array shaped like a rectangle. Sometimes students lose the idea that they are forming a new array
when extending the length and width.
Select students that have modeled the arrays correctly to share during the Summary discussion.
Page 2
Mixed Fraction Arrays
Summary
5th Grade
Multiplying Mixed Fractions
Whole-class
discussion led by the
teacher.
Unit Five – PBIT Five
The Summary phase should begin when most students have made enough progress in working through the problem. Students present and discuss their
solutions and strategies they used in the problem. During the discussion, the teacher helps students deepen their conceptual understanding of the
mathematics in the problem and guides them to refine their strategies into efficient and effective techniques or algorithms. The teacher has planned how
the discussion will unfold by observing student work during the Explore phase and planning questions to ask as students share their work.
Adapted from The Connected Mathematics Project 2, Michigan State University, 2006
When the majority of students have completed their arrays, begin the Summary discussion.
The intent of the Summary discussion is to help students make the connection of solving the multiplication problem to the parts of the array. It is critical to students’ understanding that they
realize multiplying mixed numbers requires finding the total of all the parts or sections of the array. A very common misunderstanding of multiplying mixed fractions is to multiply the whole
numbers and multiply the fractions ( 2 ½ x 3 ¼ = 6 1/8 ). By focusing on each section of the array model, students will develop understanding of multiplying mixed fractions correctly. This
also helps students understand the Distributive Property.
Begin the Summary discussion by having a student share his/her arrays for Part A. Focus the discussion on having the student share how they figured out how to add on to the array for each
part. Make sure each new section is outlined in a different color. There should be four sections and four colors. Have students make adjustments to their models if necessary.
Next, discuss the first problem and finding the total number of squares in the array. The discussion should focus on one section of the array at a time.
2
½
1
½
Suggested discussion:
-To solve the problem 1 ½ x 2 ½, we are going to focus on the different sections of the array.
-The original array was 1 x 2. Where is that in this array? Do you have this outlined in a color? How many squares are in the 1 x 2 array? Where is 1 x 2 in the
expression? Write 1 x 2 = 2 on your paper.
-Now let’s look at the next section of the array. It is smaller than the 1 x 2. The sides are ½ x 1. Can someone explain this? Do you have this outlined in a
different color? How many squares are in the ½ x 1 array? Where is ½ x 1 in the expression? Write 1 x ½ = ½ on your paper.
-Next let’s looks at the next section of the array. It is also smaller than the 1 x 2. The sides are ½ x 2. Can someone explain this? Do you have this outlined in
a different color? How many squares are in the ½ x 2 array? Where is the ½ x 2 in the expression? Write ½ x 2 = 1 on your paper.
-The last section of the array is the small square in the corner. It is the smallest section. The sides are ½ x ½. Can someone explain this? Do you have this
outlined in a different color? How many squares are in the ½ x ½ array? Where is ½ x ½ in the expression? Write ½ x ½ = ¼ on your paper.
-To solve the problem 1 ½ x 2 ½ we have drawn an array. We have recorded all the problems for the different sections of the array. If we add them all
together, we can find the total of this array and the answer to 1 ½ x 2 ½ .
1x2=2
1x½=½
½x2=1
½x½=¼
2+½+1+¼ = 3¾
So, 1 ½ x 2 ½ = 3 ¼
Continued on next page…
Page 3
Mixed Fraction Arrays
5th Grade
Multiplying Mixed Fractions
Unit Five – PBIT Five
Finally, discuss the second problem and finding the total number of squares in the array. The discussion should focus on one section of the array at a time.
3
2
¼
½
Suggested discussion:
-To solve the problem 2 ¼ x 3 ½, we are going to focus on the different sections of the array.
-The original array was 2 x 3. Where is that in this array? Do you have this outlined in a color? How many squares are in the 2 x 3 array? Where is 2 x 3 in
the expression? Write 2 x 3 = 6 on your paper.
-Now let’s look at the next section of the array. It is smaller than the 2 x 3. The sides are 2 x 1/2 . Can someone explain this? Do you have this
In a different color? How many squares are in the 2 x ½ array? Where is the 2 x ½ in the expression? Write 2 x ½ = 1 on your paper.
-Next let’s looks at the next section of the array. It is also smaller than the 2 x 3. The sides are ¼ x 3. Can someone explain this? Do you have this outlined in
a different color? How many squares are in the ¼ x 3 array? Where is the ¼ x 3 in the expression? Write ¼ x 3 = ¾ on your paper.
-The last section of the array is the small section in the corner. It is the smallest section. The sides are ¼ x ½. Can someone explain this? Do you have this
outlined in a different color? How many squares are in the ¼ x ½ array? Where is ¼ x ½ in the expression? Write ¼ x ½ = 1/8 on your paper.
-To solve the problem 2 ¼ x 3 ½ we drew an array. We recorded all the problems for the different sections of the array. If we add them all together, we can
find the total of this array and the answer to 2 ¼ x 3 ½ .
2x3=6
2x½=1
¼ x3=¾
¼ x ½ = 1/8
6 + 1 + ¾ + 1/8 = 7 7/8
So, 2 ¼ x 3 ½ = 7 7/8
Give students a sheet of inch grid paper. Pose the problem 2 ¾ x 3 ½. Have students work with a partner to draw an array and write the four problems from the four sections of the array.
Partners can work together on one paper. If you feel students may need help getting started, discuss with the class the whole number array this problem will come from. Have students draw
the 2 x 3 array on their paper and then let them work in partners to finish the problem. After most students have successfully drawn the array, bring everyone back for a discussion on solving
the problem. Lead the discussion in the same way as the two previous problems.
Page 4
Mixed Fraction Arrays
Part A
Use the grid below to draw arrays.
1)
2)
3)
4)
An array for 1 x 2 is drawn on the grid below. Trace the array with a color.
Add on to the array so that it shows 1 x 2 ½. Trace the new section with a new color.
Add on to the array so that it shows 1 ½ x 2 ½. Trace the new section with a new color.
Trace the small section in the corner with a new color.
5th Grade: Unit Five
PBIT Five
Mixed Fraction Arrays
Part B
Use the grid below to draw arrays.
1)
2)
3)
4)
An array for 2 x 3 is drawn on the grid below. Trace the array with a color.
Add on to the array so that it shows 2 x 3 ½. Trace the new section with a new color.
Add on to the array so that it shows 2 ¼ x 3 ½. Trace the new section with a new color.
Trace the small section in the corner with a new color.
5th Grade: Unit Five
PBIT Five