CHAPTER 1
Basic Multiplication Facts
Building conceptual understanding of the basic multiplication facts is essential in helping students gain fluency with complex multiplication problems.
Basic Multiplication Facts: Using Physical Models for Conceptual Understanding
Multiplying Counters
(CCSS: 2.0A.C.3; 3.0A.A.1)
Manipulatives Needed:
• Counters of any kind (e.g., buttons, beans, washers, beads, small blocks, etc ...)
• Small bowls or containers for counters
Preparation:
• Distribute an ample supply of counters and up to 9 small bowls for each student or group of students.
• Copy and cut apart the Multiplying Counters cards (1 set for whole instruction and/or 1 set per group of students (pages 4-6)).
Procedure:
• Model a set of multiplication facts to the class. For example, model the 2 multiplication facts by illustrating with 2 bowls containing 1 counter each (2x1=2), 2 bowls with 2 counters (2x2=4) and continue through 2x9. As you model each fact, verbalize the mathematical sentence, e.g.,, I have 2 groups of 2 which equals 4.
• Shuffle the Multiplying Counter cards and use them to call out problems for students to represent with their counters and bowls. Students should share their findings.
Variation:
• Each group places a set of shuffled Multiplying Counters cards in the center of their workspace. Each student picks a card and creates the multiplication fact with their counters and bowls.
• Students should communicate their multiplication counter sets, e.g.,, I have 3 groups of 6, which equals 18.
Show me 3 x 6 (3 groups of 6). How many do you have altogether? 18 [3x6]
Multiplying Counters
[Card set to copy and cut out – page 1 of 3]
1x1 1x2 1x3 1x4 1x5 1x6
1x7 1x8 1x9 2x1 2x2 2x3
2x4 2x5 2x6 2x7 2x8 2x9
3x1 3x2 3x3 3x4 3x5 3x6
3x7 3x8 3x9 4x1 4x2 4x3
[Card set to copy and cut out – page 2 of 3]
4x4 4x5 4x6 4x7 4x8 4x9
5x1 5x2 5x3 5x4 5x5 5x6
5x7 5x8 5x9 6x1 6x2 6x3
6x4 6x5 6x6 6x7 6x8 6x9
7x1 7x2 7x3 7x4 7x5 7x6
[Card set to copy and cut out – page 3 of 3]
7x7 7x8 7x9 8x1 8x2 8x3
8x4 8x5 8x6 8x7 8x8 8x9
9x1 9x2 9x3 9x4 9x5 9x6
9x7 9x8 9x9
Basic Multiplication Facts: Using Physical Models for Conceptual Understanding
Unlocking the Facts
(CCSS: 2.0A.C.3; 3.0A.A.1)
Manipulatives Needed:
• Unifix™ cubes (up to 9 colors) or Interlocking cubes
Preparation:
• Supply each group of 4 students plenty of Unifix ™ or Interlocking cubes. They will need up to 9 different colors to accommodate all the facts.
• Duplicate 9 copies of Unlocking the Facts (page 9) for each group to record their findings.
Procedure:
• Guide students through the process of constructing groups of colored cubes to represent the basic multiplication facts. It is essential to utilize the colors and the order of the fact, e.g.,, 3 groups of 5 in each group = 15. At this point do not introduce the communicative property. See the chart below to assist you in directing students to build various facts.
• Students record their findings on the Unlocking the Facts worksheets (page _____)
3x5=15
Special variations:
• You may also use this activity as a learning center in the classroom.
• If you don't have enough manipulatives, you can easily build the foundation through 5's and then have students use crayons to represent the rest of the facts using the Unlocking the Facts worksheets. You may find this an effective strategy once they have conceptual understanding.
Answers: Use the fact charts to correct their work.
Unlocking the Facts
1's 2's 3's 4's 5's 6's 7's 8's 9's
1. Circle the multiplication fact above that you are working on.
2. Color in squares that match your work and write the multiplication sentence for each. For example, if working on 2's, use 2 colors and color in 2 groups of 1 each and write 2x1=2.
Name ____________________________ Date ________
Basic Multiplication Facts: Using Physical Models for Conceptual Understanding
Linking Lego™ Array
(CCSS: 2.0A.C.3; 2.0A.C.4; 3.0A.A.1)
Manipulatives Needed:
• Lego™ blocks (up to 9 different colors)
• Lego™ building plates
Preparation:
• Supply each student or group of students an ample supply of legos and 1 10"x10" lego building plate per student/group. They will need up to 9 different colors to accommodate all the facts.
• Copy the Multiplying Counters cards (1 set for whole instruction and/or 1 set per group of students (pages 4-6).
Procedure:
• Model how to represent the basic multiplication facts with legos. For example 3x4=12 would be 3 different colors of legos with 4 in each row snapped into the lego building plate grid. Instruct students to leave a small space between the lego blocks so that each is distinguishable.
• Use the chart on page 9 to guide in the building of all the facts and to correct students' work.
Variations:
• This activity can easily be set up as a learning center. You may use graph paper to have them record their findings as they build various multiplication facts.
Lego Races
• Divide students up into teams to race against each other in building multiplication facts with their legos.
• Use the cards to draw and call out different multiplication fact problems. The team who displays the array the fastest stands up and says, for example, "3x4=12" and shows their board to the class.
Basic Multiplication Facts: Using physical models for Conceptual Understanding
Multiplication Book
(CCSS:2.0A.C.3; 2.0A.C.4; 3.0A.A.1)
Manipulatives Needed:
• 1 in graph paper (10 sheets per student) (page 13)
• 1 cm graph paper (10 sheets per student) (page 14)
• 9 different colors of crayons, markers, or colored pencils
• 9x12 in. light-colored construction paper (15 sheets per student)
• Scissors
• Glue or glue sticks
• 9 different colors of 1" colored squares (for teacher demonstration)
Preparation:
• Supply each student with all the above materials except for the colored squares.
Special Note:I use different sizes of graph paper according to the grade level. Use the larger squared graph paper for younger students and the smaller squared graph paper for older students. You can also mix and match sizes so that they fit on the pages better. For example, use larger squared graph paper for 1's – 5's and smaller squared graph paper for 6 - 9 multiplication facts.
Procedure:
• Using the colored squares on 1" graph paper, illustrate how they will create the pages of their multiplication books. For example, in creating the 3's page, students will use 3 colors to fill in horizontal rows of designated amounts, e.g.,, 3 rows of 1,2,3,4,5,6,7,8,9. Instruct students to cut out 3x1, 3x2, 3x3, etc ... and glue them onto on sheet of construction paper and then write the number sentence beside each representation of the multiplication fact.
3x1=3 3x2=6 3x3=9
• Note that for each new subsequent multiplication set of facts, another color will be added, thus 4's will have 4 colors and so forth until the 9's with 9 colors.
• Students will create a page/s for each multiplication set of facts, thus they will have a page for 1's, 2's, 3's, 4's, 5's, 6's, 7's, 8's, and 9's.
• Monitor students' work carefully. Make sure they are showing each fact correctly. They will begin to see patterns and the properties of multiplication.
• After they have created all their pages representing the multiplication tables through 9, have them create a cover page titled, "My Multiplication Book" with their name and any design they want to add.
• Staple/bind all the pages together.
Special note: This book can now serve as a wonderful manipulative tool for not only learning the basic facts but also conceptual understanding of number theory concepts (,. factors, multiples, prime and composite). You will notice that students will begin to visualize the facts and patterns. Some will comment that they can see the basic multiplication facts in their heads!
One-Inch Graph Paper
One-Centimeter Graph Paper
Basic Multiplication Facts: Using Physical models for Conceptual Understanding
Table it!
(CCSS:2.0A.C.3; 3.0A.A.1)
Manipulatives Needed:
• Two 0-9 Spinners per student
• Multiplication books created from pages 11-12.
• Table It worksheet
Preparation:
• Assemble the spinners (page 16) and distribute two to each student.
• Provide each student with a copy of the Table It! worksheet (page 17).
• Student multiplication books can be used to reinforce multiplication facts.
Procedure:
• Instruct students to write numbers 0-9 in random order along the top and left side of the Table it worksheet.
• Students will spin each spinner to create their multiplication fact. After they find that fact in their books, write the answer in the correct square on their worksheet.
• Continue activity until they have completed the table.
Variations:
• Set a time limit and see who can create the largest number of different combinations and fill up their worksheet the fastest.
• Have students work in groups of 2-4 with one worksheet, each taking turns to spin and solve.
Special note: This method of utilizing the multiplication book and working with the multiplication facts out of order helps with concrete visualization of the basic facts. Answers will vary according to how each student set up their chart. Even though assessing their work will take a little longer, it is worth the effort.
8x6=48
To assemble, duplicate the pattern onto cardstock. Use a brad to secure safety pin in center of spinner.
Table it!
Name ________________________________ Date ____________
Basic Multiplication Facts: Using Physical Models for Conceptual Understanding
Uncovering the Facts
(CCSS: 3.0A.A.1; 3.0A.A.3 )
Preparation:
• Print the Uncover the Fact Finder pattern onto cardstock for each student and cut along the dotted lines (page 19).
• Provide each student with a copy of the Uncover the Fact worksheet
Procedure:
• Students place their Uncover the Fact Finder on top of the Uncover the Fact Grid worksheet, revealing any size rectangle. This rectangle represents a multiplication fact with factors (vertical and horizontal numbers on grid). The product is found by counting the squares.
• Students use the Fact Finder moved to different positions to reveal all the different multiplication facts and fill in the fact grid with products by counting the squares in each set of factors. (e.g.,, 3x7=21 shown below)
Uncover the Fact Finder
Answers:
Uncover the Fact Finder
Uncover the Fact Grid
Name ________________________________ Date ____________
CHAPTER 2
Properties of Multiplication
Building conceptual understanding with the closure, identity, zero, commutative and associative properties of multiplication enables students to gain fluency with the application of basic multiplication facts.
Properties of Multiplication: Using Physical Models for Conceptual Understanding
Closure Property [axb=c]
(CCSS: 3.0A.B.5; 3.0A.D.9)
Manipulatives Needed:
• Counters of any kind (e.g., buttons, beans, washers, beads, small blocks, etc ...)
• Small bowls or containers for counters
• 1 in or 1 cm graph paper (see pages 13-14 for graph paper)
• Colored pencils, markers or crayons
• Multiplication books created from pages 11-12
Preparation:
• Distribute manipulatives to individuals or groups of students. You may also use the manipulatives for class demonstration.
Background concept – The closure property of multiplication states that axb=c (e.g., If you multiply any two whole numbers (excluding 1 or 0), you will get a new unique number).
Procedure:
Closure property with counters and containers:
• Guide students through the process of understanding the closure property by using counters and containers. For example, ask students to show you 3 containers (groups) with 2 counters in each. How many do we have? 6. 3x2=6. 5 containers with 3 counter in each. How many? 15. 5x3=15. Continue until students are able to deduct that any time 2 numbers (exclusive of 0 and 1) are multiplied together, a new unique number is created.
2x8=16 3x4=12
Closure property with graph paper and Multiplication books:
• Use graph paper to illustrate various rectangles representing multiplication facts exclusive of 0 and 1. (e.g.,, mark off 3 groups of 7 – 3x7. How many do we have? 21. 8 groups of 2 – 8x2. How many do we have? 16.
• An excellent way to illustrate the closure property is for students to use their multiplication books they created on page and find all the multiplication sentences that are an example of the closure property.
Properties of Multiplication: Using Physical Models for Conceptual Understanding
Identity Property
[ax1=a/1xa=a] Teaching Tip for students: You never lose your identity when you multiply by 1.
CCSS: 3.0A.B.5; 3.0A.D.9)
Manipulatives Needed:
• Counters of any kind (e.g., buttons, beans, washers, beads, small blocks, etc ...)
• Small bowls or containers for counters
• 1 in or 1 cm graph paper (see pages 13-14 for graph paper)
• pencils, markers or crayons
Preparation:
• Distribute manipulatives to individuals or groups of students. You may also use the manipulatives for class demonstration.
Mathematical concept – The identity property of multiplication states that ax1=a or 1xa=a. (i.e., if you multiply any number by 1 you will always end up with that number).
Procedure:
Identity property with counters and containers:
• Guide students through the process of understanding the identity property with counters and containers. For example, ask students to show you 3 containers (groups) with 1 counter in each. How many do we have? 3. 3x1=3. 5 containers with 1 counter in each. How many? 5. 5x1=5. Show 1 container with 7 counters. How many counters? 7. 1x7=7. Continue until students are able to deduct that any time 1 is a part of our groups in multiplication, the value remains the number (of counters) we have.
1x8=8 3x1=3
Identity property with graph paper:
• Use the graph paper to illustrate various rectangles representing groups of 1. (e.g., mark off 1 group of 7 – 1x7. How many do we have? 7. 8 groups of 1 – 8x1. How many do we have? 8. Continue outlining various multiplication problems all using a factor of 1. Continue until students are able to understand the principle of the identity property.
Once students are able to apply this concept you can have lots of fun with multiplying large numbers by 1. 1x5,439 = ? 5,439! This is great boost for their attitudes towards math.
1x7=7
6x1=6
Properties of Multiplication: Using Physical Models for Conceptual Understanding
Zero Property
[Ax0=0 or 0xa=0] Teaching tip for students: If you multiply any number by (zero) nothing, you will end up with nothing! (c)
(CCSS: 3.0A.B.5; 3.0A.D.9)
Manipulatives Needed:
• Counters of any kind (e.g., buttons, beans, washers, beads, small blocks, etc ...)
• Small bowls or containers for counters
• 1" or 1 cm graph paper (see pages 13-14 for graph paper)
• Colored pencils, markers or crayons
Preparation:
• Distribute manipulatives to individuals or groups of students. You may also use the manipulatives for class demonstration.
• Mathematical concept – The zero property of multiplication states that nx0=0 and 0xn=0 (i.e. if you multiply any number by 0 you will always end up with 0)
Procedure:
Zero property with counters and containers:
• Guide students through the process of understanding the zero property with counters and containers. For example, ask students to show you 3 containers (groups) with 0 counters in each. How many do we have? 0. 3x0=0. 5 containers with 0 counters in each. How many? 0. 5x0=0. Show 1 container with 0 counters. How many counters? 0. 1x0=0. Continue until students are able to deduct that any time 0 is a part of our groups in multiplication, the answer (product) will always be zero.
1x0=0 3x0=0
Zero property with graph paper:
• Use the graph paper to illustrate what happens when you multiply any number by zero in trying to show the number of squares multiplied by 0. (e.g., mark off 3 groups of 0 – 3x0. How many do we have? 0. 8 groups of 0 – 8x0. How many do we have? 0. Continue outlining various multiplication problems all using a factor of 0. Continue until students are able to understand that it is impossible since the answer is always zero.
Once students are able to apply this concept you can have lots of fun with multiplying large numbers by 0 just like with 1. 0 x 45,439 = ? 0. This is another great boost for their attitudes towards math.
