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GRADE 5 • UNIT 5
Table of Contents
Addition and Multiplication with Volume and Area
Overview of Unit Topics and Lesson Objectives
Lessons
Topic 1: Concepts of Volume
1-3
Lesson 1: Explore volume by building with and counting unit cubes.
Lesson 2: Find the volume of a right rectangular prism by packing with cubic units and counting.
Lesson 3: Compose and decompose right rectangular prisms using layers.
Topic 2: Volume and the Operations of Multiplication and Addition
4-9
Lesson 4: Use multiplication to calculate volume.
Lesson 5: Use multiplication to connect volume as packing with volume as filling.
Lesson 6: Find the total volume of solid figures composed of two non-overlapping rectangular
prisms.
Lesson 7: Solve word problems involving the volume of rectangular prisms with whole number
edge lengths.
Lesson 8: Apply concepts and formulas of volume to design a sculpture using rectangular prisms
within given parameters.
Lesson 9: Apply concepts and formulas of volume to design a sculpture using rectangular prisms
within given parameters.
Topic 3: Area of Rectangular Figures with Fractional Side Lengths
10 - 15
Lesson 10: Find the area of rectangles with whole-by-mixed and whole-by- fractional number side
lengths by tiling, record by drawing, and relate to fraction multiplication.
Lesson 11: Find the area of rectangles with mixed-by-mixed and fraction- by-fraction side lengths
by tiling, record by drawing, and relate to fraction multiplication.
Lesson 12: Measure to find the area of rectangles with fractional side lengths.
Lesson 13: Multiply mixed number factors, and relate to the distributive property and the area
model.
Lesson 14: Solve real world problems involving area of figures with fractional side lengths using
visual models and/or equations.
Lesson 15: Solve real world problems involving area of figures with fractional side lengths using
visual models and/or equations.
Topic 4: Drawing, Analysis, and Classification of Two-Dimensional Shapes
16-21
Lesson 16: Draw trapezoids to clarify their attributes, and define trapezoids based on those
attributes.
Lesson 17: Draw parallelograms to clarify their attributes, and define parallelograms based on
those attributes.
Lesson 18: Draw rectangles and rhombuses to clarify their attributes, and define rectangles and
rhombuses based on those attributes.
Lesson 19: Draw kites and squares to clarify their attributes, and define kites and squares based on
those attributes.
Lesson 20: Classify two-dimensional figures in a hierarchy based on properties.
Lesson 21: Draw and identify varied two-dimensional figures from given attributes.
Unit 5: Vocabulary
Review Familiar Terms and Symbols1
Angle - the joining of two different rays sharing a common vertex
Area – the number of square units that covers a two-dimensional shape (A = length x width)
Attribute/property – a quality or characteristic
Cube - three-dimensional figure with six square sides
Distributive Property – breakdown one or two factors of a multiplication problem into its addends,
multiply each by the other factor, and then add the products together to get the whole answer
Examples:
54 x 2 = (50 + 4) x 2
= (50 x 2) + (4 x 2)
= 100 + 8
= 108
38 x 12 = (30 + 8) x (10 + 2)
= (30 x 10) + (30 x 2) + (8 x 10) + (8 x 2)
= 300 + 60 + 80 + 16
= 456
Face – any flat surface of a three-dimensional figure
Kite - quadrilateral with two pairs of two equal sides that are also adjacent; a kite can be a
rhombus if all sides are equal
Line Segment – a straight path that connect two points
Parallel lines - two lines that do not intersect
Parallelogram - four-sided closed figure with opposite sides that are parallel and equal
Perpendicular – two lines are perpendicular if they intersect, and any of the angles formed
between the lines are 90° angles
Perpendicular bisector - line that cuts a line segment into two equal parts at 90°
Plane - flat surface that extends infinitely in all directions
Polygon – closed figure made up of line segments
Quadrilateral – closed figure with four sides
Rectangle – parallelogram with four 90° angles
Rectangular prism - three-dimensional figure with six rectangular sides
Rhombus – parallelogram with four equal sides
Right angle - angle formed by perpendicular lines; angle measuring 90°
Right rectangular prism – rectangular prism with only 90°angles
Solid figure - three-dimensional figure
Square units – squares of the same size—used for measuring
Three-dimensional figures – solid figures that have a length, a width, and a height
Trapezoid- quadrilateral with at least one pair of parallel sides
Two-dimensional figures – figures on a plane
1
These are terms and symbols students have used or seen previously.
New Vocabulary for 5th Grade Unit 5
Base – one face of a three-dimensional solid – often thought of as the surface on which the solid rests
Bisect – divide into two equal parts
Capacity - the amount of liquid that fills a container; filling
Cubic centimeter – all sides measure 1 centimeter; abbreviation cm3– centimeters cubed
Cubic units – cubes of the same size used for measuring volume
Diagonals – straight line joining two opposite corners (vertices) of a shape
Height - adjacent layers of the base that form a rectangular prism
Hierarchy - series of ordered groupings of shapes
Milliliter – unit of capacity equal to one-thousandth of a liter; abbreviation is mL
Space - the amount of cubes that will fit inside a solid; packing
Unit cube - cube whose sides all measure 1 unit; cubes of the same size used for measuring volume
Volume of a solid – measurement of space or capacity
Suggested Tools
Area model
- students will begin to use area models to multiply mixed
numbers.
Centimeter cubes
Centimeter grid paper
- graph paper with 1 square centimeter boxes.
Isometric dot paper
- similar to graph paper; no lines only
dots that when connected will create 1 cm cubes.
Protractor

Set square or right
angle template
Lesson by Lesson Suggestions
4th Grade Review (see recommended resources for IXL skills to help practice and review
these skills)
It is expected that students have mastered the concepts of area and perimeter before starting this unit.
Students should be able to calculate the area and the perimeter of quadrilaterals (4 sided figures). Area =
length x width. Perimeter = 2 x length + 2 x width or for a square 4 x side.
Example 1:
Rectangle
- different measurements for length and width
length = 2 in
width = 4 in
Example 2: Square
- length and width are equal
length = 5 cm
width = 5 cm
Perimeter = (2 x 2in) + (2 x 4in) = 12 in
Perimeter = (4 x 5 cm) = 20 cm
Area = 2 in x 4 in = 8 in2
Area = 5 cm x 5 cm = 25 cm2
The exponent of 2 is used to show “square units” or “units squared.
Lessons 1 - 3: Concepts of Volume
In these lessons students will explore volume. They will understand that cubic units are used to measure
volume. Using unit cubes, students build three-dimensional shapes, including right rectangular prisms, and
count to find the volume. They will also make connections between area and volume. Next, students pack
rectangular prisms made from nets with centimeter cubes. This helps them visualize the layers of cubic
units that make up volume. Finally, students compose and decompose a rectangular prism from and into
layers of unit cubes and reason that the number of unit cubes in a single layer matches the number of unit
squares on a face.
Example 1:
The following
solids are made
up of 1-cm cubes.
Find the volume
of each figure,
and write in the
chart.
Example 2:
Dwight says that the figure below, made of 1-cm cubes, has a volume of 5 cubic centimeters. Explain his
mistake.
He failed to count the cube
that is hidden. The cube on
the second layer has to be
sitting on a cube below it.
Steps for drawing 1 cubic centimeter using the isometric dot paper.
Step 1: Connect four dots to make a rhombus. This will represent one square face of the cube,
viewed at an angle.
Step 2: Draw three straight segments to the right from the two vertices on the top and the one on
the bottom right.
Step 3: Draw two segments to represent the missing edges.
Steps for 2 cubic centimeters.
Step 1: Connect four dots to make a rhombus.
Step 2: Add another rhombus that shares its right edge, just like your cubes.
Step 3: Draw four straight segments to the right from the three vertices on the top and the one on
the bottom right.
Step 4: Draw three segments to represent the missing edges.
Application Problem:
Jack and Jill both have 12 centimeter cubes. Jack builds a tower
that is 6 cubes high and 2 cubes wide. Jill builds one that is 6
cubes long and 2 cubes wide. Jack says his structure has the
greater volume because it is taller. Jill says that the structures
have the same volume. Who is correct? Draw a picture to
explain how you know.
Jack’s tower has 12 cubes.
Jill’s tower has 12 cubes.
Jill is correct because both have a volume of 12 cubic centimeters.
Jack’s is standing upright and Jill’s is lying down.
Filling a box or rectangular prism with cubic units
The model below represents a net (pattern) of a rectangular prism or box. If you think of taking a cereal box and
cutting it open to form a flat shape, this would create a net of the cereal box.
The shaded part is the base of the box.
We can tell that it would take 4 cubes to cover the base.
The flaps show that there are 2 layers.
4 x 2 =8
So the volume of this prism is 8 units3
Example #1: If this net were to be folded into a box or rectangular prism, how many cubes would fill it?
It would take 16 cubes to cover the shaded part
which is the base or bottom layer.
The flaps show that there are 3 layers.
16 x 3 = 48
So the volume of this rectangular prism is 48 units3
It would take 48 cubes to fill the box.
Example #2: How many centimeter cubes would fit inside the box? Explain your answer
using words.
The front of the box has 4 rows with 5 cubes in each row which equals 20 cubes.
The box is 2 layers deep. (20 x 2 = 40)
So the volume of this box is 40 cubic centimeters or 40 cm3
It would take 40 centimeter cubes to fill the box.
Decompose right rectangular prisms using layers
There are 3 different methods to finding the volume of a rectangular prism. Using the prism to the
right, look below at the three approaches. The prism is made of centimeter cubes.
Method 1: We could think of drawing
horizontal lines to show the 5 layers of
12 cubes each. This resembles layers of
cake.
Method 2: We could think of drawing
vertical lines to show 3 layers of 20 cubes
each. This resembles bread slices.
Method 3: We could think of drawing
both a horizontal and a vertical line to
show the front and back layers. There are
4 layers of 15 cubes each. This resembles
books standing up.
20 cm3 + 20 cm3 + 20 cm3 = 60 cm3
12
cm3
+ 12
cm3
cm3
+ 12
+ 12
60 cm3
cm3
+ 12
cm3
=
3 x 20 cubic centimeters = 60 cm3
5 x 12 cubic centimeters = 60 cm3
No matter which method is used, the volume is the same. Students use the layers
that are easier for them to visualize. A good practice is to use a second approach
to check the volume determined from the first approach.
15 cm3 + 15 cm3 + 15 cm3 + 15 cm3 = 60 cm3
4 x 15 cubic centimeters = 60 cm3
Lessons 4 - 9: Volume and the Operations of Multiplication and Addition
In these lessons students come to see that multiplying side lengths (length x width x height) or multiplying
the area by the number of layers will have equivalent volume.
Find the volume by multiplying side measures
Volume =(3 cm x 2 cm) x 4 cm
= 6 cm2 x 4 cm
= 24 cm3
Volume =(2 cm x 4 cm) x 3 cm
= 8 cm2 x 3 cm
= 24 cm3
Volume = (4 cm x 3 cm) x 2 cm
= 12 cm2 x 2 cm
= 24 cm3
All three generate the same volume. This shows that the order does not matter when multiplying
the measure of each side. The dimensions can be multiplied in any order due to the commutative
property of multiplication.
Calculate the volume by multiplying the area of one face by the number of layers
Area using layers from top to
bottom
(find the area of the top layer)
3 cm x 2 cm = 6 cm2
There are 4 layers of 6 cm2.
Volume = 6 cm2 x 4 cm
= 24 cm3
Area using layers from left to right
2 cm x 4 cm = 8 cm2
It is 3 layers wide of 8 cm2.
(left to right – resembles bread slices)
Volume = 8 cm2 x 3 cm
= 24 cm3
All three yield the same volume.
Example 1:
Eddie says more information is needed to find the volume of the rectangular prism. Explain why Eddie is
mistaken and calculate the volume.
Eddie can multiply the area of the face by the
width of 5 in.
Volume = 60 in2 x 5 in
= 300 in3
Example 2: What is the volume of a jewelry box with a length of 10 centimeters, a width of 4
centimeters, and a height of 3 centimeters?
Volume = (10 cm x 4 cm) x 3 cm
= 40 cm2 x 3 cm
= 120 cm3
The volume of the jewelry box is 120 cm3.
*Remember the order does not matter when multiplying the measure of each side.
Example 3: A rectangular prism has a volume of 30 cubic feet. Its height is 5 feet. Which are
possible dimensions for the base of the prism?
A. 1 foot x 6 feet
B. 3 feet x 10 feet
C. 3 feet x 3 feet
D. 12 feet x 12 feet
Correct Answer: A. (1 ft x 6 ft) x 5 ft = 30 cubic feet or 30 ft3 - The dimensions of the base must equal 6 when
multiplied because 30 ÷ 5 = 6. This is the volume divided by the height = the base.
Lesson 5: Liquid Volume
From an activity in Lesson 5, students will conclude that 1 cm3 is equivalent to 1 mL. Milliliters are units of
capacity which tell the amount of liquid a container will hold. There are 1,000 mL in a liter.
Example 1: Find the volume of the prism and then shade the beaker to show how much liquid would fill the box.
Volume = (8 cm x 5 cm) x 10 cm
= 40 cm2 x 10 cm
= 400 cm3
Since 1 cm3 equals 1 mL, 400 cm3 equals 400 mL.
Lesson 6: Total volume of a solid figure compose of two or more non-overlapping prisms
Prism A
Length – 3 inches
Width – 5 inches
Height – 2 inches
Volume = 3 in x (5 in x 2 in)
= 3 in x 10 in2
Prism B
Length – 6 inches
Width – 5 inches
Height – 4 inches
Volume = (6 in x 5 in) x 4 in
= 30 in2 x 4 in
= 30 in3
= 120 in3
Total volume = 30 in3 + 120 in3 = 150 in3
Application Problem:
A planting box pictured below is made of two sizes of rectangular prisms. One type of prism measures 2 inches
by 5 inches by 12 inches. The other type measures 12 inches by 4 inches by 10 inches. What is the total volume
of three such boxes?
Prism A
Volume = (2 in x 5 in) x 12 in
= 10 in2 x 12 in
= 120 in3
There are two prisms ‘A.’ 120 in3 x 2 = 240 in3
2 4 0 in3
+ 4 8 0 in3
7 2 0 in3
Prism B
Volume = (12 in x 4 in) x 10 in
= 48 in2 x 10 in
= 480 in3
The total volume of the planting box is 720 cubic inches.
Lessons 10 - 15: Area of Rectangular Figures with Fractional Side Lengths
In these lessons students will find the area of rectangles with fractional side lengths. Students will use an
area model to find partial products. Students using tiling to find the area of rectangles. Tiling is a strategy
used to find area of rectangle by covering the entire figure with square units and fractional parts of square
unit.
Example of tiling
Example 1: Randy made a mosaic using different color rectangular tiles. Each tile measured 3 ½ inches x 2
inches. If he used six tiles, what is the area of the whole mosaic in square inches?
The drawing below resembles an area model used in earlier units when students multiplied whole numbers
and decimal fractions. Now the area model has fractional parts.
The 3 ½ is thought of as 3 + ½.
Using tiling, each whole square
represents 1 square inch. To
represent ½ inch, the whole
square is cut in half and only
half is showing in the model.
There are 6 whole squares and
two ½s.
The total area for each tile is 7 in2.
Since there are 6 tiles, the area of the whole mosaic is 42 square inches or 42 in2 (6 x 7).
Method 1: Algorithm using the distributive property:
3x 2 = (3 + ) x 2
= (3 x 2) + (x 2)
=6+1
=7
In lesson 13 students will be given a choice of method to find the area of a rectangle with mixed
number side lengths. Students will be encouraged to choose the most efficient and effective method.
Method 2: Algorithm without using the distributive property; the mixed number is changed to an
improper fraction:
3x 2 =
7
2
x2=
7𝑥2
2
=
14
2
=7
Eventually students will just record partial products rather than draw individual tiles.
Example 2:
Francine cut a rectangle out of construction paper to complete her art project. The rectangle measured
4inches x 2inches. What is the area of the rectangle Francine cut out?
Add the partial products together to find the area.
8 in2 + 1 in2 + 1 in2 + in2 = 10in2
The area of the rectangle cut out is 10square inches.
Method 1: Algorithm using the distributive
property
4x 2= (4 + ) x (2 + )
=(4 x 2) + (4 x ) + (x 2) + ( x )
1
=8+1+1+
1
8
Method 2: Algorithm without using the distributive
property; mixed numbers are changed to
improper fractions:
1
1
42x24
9
9
=2x4=
81
8
= 10 
= 10 8
**The algorithm is provided so students are exposed to a more formal representation of the
distributive property. However, students are not required to be as formal in their
calculations. Using an area model to keep track of their thinking is sufficient.
Example 3: Find the area of a rectangle that measures km x 2km. Draw an area model if it helps.
Example 4: John decided to paint a wall with two windows. Both windows are 3ft by 4ft rectangles.
Find the area the paint needs to cover.
The paint needs to cover 71 ½ square feet.
Lessons 16 - 21: Drawing, Analysis, and Classification of Two-Dimensional Shapes
In these lessons students will analysis the properties and defining attributes of quadrilaterals.
Trapezoid
There are actually two definitions for a trapezoid:
1. A quadrilateral with only one pair of opposite sides parallel
2. A quadrilateral with at least one pair of opposite sides parallel
Most mathematicians define a trapezoid using the second description which is the
characteristics the student will use in this unit when talking about the attributes/properties of
a trapezoid.
Parallelogram
Attributes/Properties: a quadrilateral and opposite sides are parallel
The diagonals of parallelograms bisect each other. Bi – means two and sect
means cut, so bisect means to cut in two parts.
These two parts are equal in length.
**Since a parallelogram has two pairs of parallel sides then it has at least
one pair of parallel sides. Therefore, all parallelograms are also classified
as trapezoids.
Example Questions with Answers:
1. When can a quadrilateral be called a parallelogram?
A quadrilateral can be called a parallelogram when both pairs of opposite sides are parallel.
2. When can a trapezoid also be called a parallelogram?
A trapezoid can be called a parallelogram when it has more than one pair of parallel sides.
Rhombus
Attributes/Properties: a quadrilateral, all sides are equal in length, and opposite
sides are parallel
The attributes indicate that a rhombus can also be classified as a
parallelogram and all parallelograms are also classified as a trapezoid.
The diagonals of a rhombus bisect one another
but because they bisect each other at 90° angles,
we call these diagonals perpendicular bisectors.
Rectangle
Attributes/Properties: a quadrilateral, 4 right angles, and opposite sides are
parallel
Since opposite side are parallel, we can classify the rectangle as a parallelogram
and a trapezoid.
The diagonals of a rectangle do
bisect each other and the two parts
are equal in length.
Example Questions with Answers:
1. When can a trapezoid also be called a rhombus?
A trapezoid can be called a rhombus when all sides are equal in length.
2. When can a parallelogram also be called a rectangle?
A parallelogram can be called a rectangle when all angles measure 90°.
3. A rhombus has a perimeter of 100 cm. What is the length of each side?
Since all sides of a rhombus are equal in length, I divided 100 by 4 sides which gives me a length of
25 cm. So the length of each side of the rhombus is 25 centimeters.
Square
Attributes/Properties: a quadrilateral, 4 right angles, 4 sides of equal length, and
opposite sides are parallel.
- Since a square has 4 right angles, it can also be classified as a rectangle.
- Since a square has 4 sides of equal length, it can also be classified as a rhombus.
- The opposite sides are parallel so a square can also be classified as a
parallelogram. If it is classified as a parallelogram then it is also classified as a
trapezoid.
The diagonals of a square bisect each other at 90° angles just like a rhombus. These
diagonals are called perpendicular bisectors.
Kite
Example Problems:
Attributes/Properties: a
quadrilateral and adjacent
sides or sides next to each
other are equal.
The diagonals of a kite may
intersect outside, but they are
still perpendicular. The
diagonals are not the same
length. Only one diagonal
bisects the other.
Look at the two shapes.
Can these shapes be classified as a kite?
The specific name for each shape is a square and a rhombus. Both have 4
equal sides. Therefore the adjacent sides are equal. So they can be classified
as a kite.
Can a kite ever be a parallelogram? Yes, since a square and a rhombus can be classified as a kite
and these shapes do have opposite sides that are parallel, then a kite at times can be classified as a
parallelogram.
Recommended Resources
Videos
What is meant by 1-D, 2-D, and 3-D
https://www.khanacademy.org/math/pre-algebra/measurement/volume-introductionrectangular/v/how-we-measure-volume
Identify and label three-dimensional figures
https://learnzillion.com/lessons/1485
Or use quick code: LZ1485
Difference between a square unit and a cubic unit
https://learnzillion.com/resources/51151
Lessons 1-5:
Volume
http://www.learnalberta.ca/content/mesg/html/math6web/index.html?page=lessons&lesson=
m6lessonshell15.swf
https://learnzillion.com/lessons/1796
Or use quick code: LZ1796
Finding the volume by analyzing the layers
https://learnzillion.com/lessons/1264-find-volume-by-counting-cubes quick code: LZ1264
Understand the connection between cubic volume and liquid volume
https://learnzillion.com/lessons/1798
Or use quick code: LZ1798
Use multiplication (V = l x w x h) to find the volume of a solid figure
https://learnzillion.com/lessons/1803
Or use quick code: LZ1803
https://www.khanacademy.org/math/enem/conhecimentos-geometricos/comparimentosareas-volumes/v/measuring-volume-as-area-times-length
Find the volume of a solid figure using the area of the base x height formula (V = B x h)
https://learnzillion.com/lessons/1804
Or use quick code: LZ1804
Find missing edge lengths on composite 3D prisms
https://learnzillion.com/lessons/1591
Or use quick code: LZ1591
Find the volume of complex rectangular prisms
https://learnzillion.com/lessons/1809
Or use quick code: LZ1809
Area of Rectangular Figures with Fractional Side Lengths
https://www.khanacademy.org/math/pre-algebra/fractions-pre-alg/mixed-number-mult-div-prealg/v/multiplying-mixed-numbers
Lessons 16-21:
Quadrilateral Properties
https://www.khanacademy.org/math/cc-fifth-grade-math/cc-5th-geometry-topic/cc-5thquadrilaterals/v/quadrilateral-properties
Attributes of polygons
https://learnzillion.com/lessons/3448-identify-attributes-of-polygonsOr use quick code: LZ3448
Classify and compare quadrilaterals
https://learnzillion.com/lessons/3481
Or use quick code: LZ3481
Identify quadrilaterals based on attributes
https://learnzillion.com/lessons/1708
Or use quick code: LZ1708
Identify rectangles
https://learnzillion.com/lessons/2448
Or use quick code: LZ2448
Identify rhombuses
https://learnzillion.com/lessons/2640
Or use quick code: LZ2640
Identify squares
https://learnzillion.com/lessons/2695
Or use quick code: LZ2695
Kites as a mathematical shape
https://www.khanacademy.org/math/cc-fifth-grade-math/cc-5th-geometry-topic/cc-5thquadrilaterals/v/kites-as-a-mathematical-shape
**This site has a direct link to homework pages.**

http://www.oakdale.k12.ca.us/ENY_Hmwk_Intro_Math
(Click on 5th Grade – Select the Module 5– Select the lesson)
Direct Link for Unit 5
http://www.oakdale.k12.ca.us/cms/page_view?d=x&piid=&vpid=1428998199814
GAMES
How many cubes
http://www.interactivestuff.org/sums4fun/3dboxes.html
Compute volume
http://www.xpmath.com/forums/arcade.php?do=play&gameid=118#.UgGYD1PodJM
IXL skills covered in this unit:
Review Skills from 4th grade
B.1 Identify 2-dimensional and 3-dimensional shapes
B.2 Types of triangles
B.4 Regular and irregular polygons
B.5 Number of sides in polygons
B.10 Similar and congruent
B.11 Nets of 3-dimensional figures
B.12 Types of angles
B.13 Measure angles with a protractor
B.15 Perimeter
B.16 Area of squares and rectangles
B.17 Area of triangles
B.18 Area of parallelograms and trapezoids
B.19 Area of compound figures
B.20 Area between two rectangles
B.21 Area and perimeter of irregular figures
B.22 Area and perimeter: word problems
New 5th Grade Skills
B.6 Which figure is being described?
B.7 Classify quadrilaterals
B.23 Volume of rectangular prisms made of unit cubes
B.24 Volume of irregular figures made of unit cubes
B.25 Volume of cubes and rectangular prisms
B.27 Three-dimensional figures viewed from different perspectives
B.30 Lines, line segments, and rays
B.31 Parallel, perpendicular, intersecting
N.22 Estimate products of mixed numbers
N.23 Multiply a mixed number by a whole number
N.24 Multiply a mixed number by a fraction
N.25 Multiply two mixed numbers
N.27 Multiplication with mixed numbers: word problems
N.28 Multiply fractions and mixed numbers in recipes
N.29 Complete the mixed-number multiplication sentence