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Multiplication: Just the Facts, Ma’am
Mary Ebejer
EDG 630 -‐ 01
Teaching Mathematics K-‐8
November 30, 2010
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Table of Contents
Introduction ................................................................................................................................ 3
Unit Standards (GLCEs) ................................................................................................................ 3
“Big Ideas” .................................................................................................................................... 4
Assessments ................................................................................................................................. 5
Lessons: 1: Multiplication Illustration ............................................................................ 6
2: Fish Bowl ....................................................................................................... 8
3: Groupings All Around Us ........................................................................... 11
4: Game: Circle and Stars ............................................................................... 17
5: Creating Multiplication Tables ................................................................. 20
6: Billy Wins a Shopping Spree! ..................................................................... 29
References .................................................................................................................................. 31
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Introduction
This third and fourth grade math unit explores the various meanings and representations of multiplication as “repeated addition. The lessons in this unit rely on extensive use of manipulatives, as well as math songs and games and art activities to help students identify situations when multiplication would be useful, to reinforce their learning and to improve recall speed for multiplication facts. Many of the activities in the unit involve cooperative learning in pairs, small groups and as a class as a whole. Students should see their classroom as a place where cooperation and collaboration are valued and expected. It respects the principle that interaction fosters learning and that cooperative group work is basic the classroom culture.
Unit Standards
3 Multiply and divide whole numbers
3.N.MR.03.09 Use multiplication and division fact families to understand the inverse relationship of these two operations, e.g., because 3 x 8 = 24, we know that 24 ÷ 8 = 3 or 24 ÷ 3 = 8; express a multiplication statement as an equivalent division statement. 3.N.MR.03.10 Recognize situations that can be solved using multiplication and division including finding "How many groups?" and "How many in a group?" and write mathematical statements to represent those situations. 3.N.FL.03.11 Find products fluently up to 10 x 10; find related quotients using multiplication and division relationships. 3.N.MR.03.12 Find solutions to open sentences, such as 7 x __ = 42 or 12 ÷ __ = 4, using the inverse relationship between multiplication and division.
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“Big Ideas”
Lesson 1: Multiplication Illustration Kick-‐off lesson using M&M’s.
Lesson 2: Fish Bowl Multiplication is repeated addition.
Lesson 3: Groupings All Around Us Multiplication is a quick way to figure out how many you have altogether of something when things come in groups.
Lesson 4: Circle and Stars Game Students see multiplication as the combining of equal-‐size groups that
can be represented with a multiplication equation.
Lesson 5: Creating Multiplication Tables Students create personal laminated multiplication tables as they
learn to recognize both the geometry and patterns inherent in multiplication.
Lesson 6: Billy Wins a Shopping Spree! We use multiplication everyday to solve real-‐world problems.
Unit Songs: Multiplication songs
Introduce in morning playing a recording. During transition periods play songs. Teach song at end of math lesson of the day. Each student has book of songs. Sing song at closing.
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Unit Assessments
Teacher observation of class work, combined with evaluation of Student Portfolio and Math Journal Entries to serve as assessments of student understanding of multiplication, both its meaning and real-‐world uses.
Formative: Periodic Math Journal Entries
Group Work and Class Discussion Observations
Summative: Student Portfolios
Final Math Journal Entry “What I now know about multiplication.”
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Grade Level
Third and Fourth Time Needed
50 minutes
Materials
Large piece of butcher paper
Marker to record ideas
A math journal for each student
M&M's, jelly beans, and small candy
Lesson 1: Introduction to Multiplication
Introduction: Students will discuss and write about their current understanding of multiplication before we begin the unit of study. Recognize situations that can be solved using multiplication and division including finding "How many groups?" and "How many in a group?" and write mathematical statements to represent those situations. Preparation: The instructor will pass out a journal to each student. The journal will contain copies of everything that will be used in this unit including handouts, templates, and multiplication charts. The student journals will also contain blank paper for students to recorded their observations and thoughts as well as to use to generate any computations that may be needed. In my classroom this journal is comprised of a two pocket folder that contains brads for binding papers. Prior to beginning the unit on multiplication, ask the students to respond to this prompt in their Math Journals:
Write what you know about multiplication.
Their response will serve as a benchmark for their formative assessments for the unit. For this particular lesson, you will need bags of M&M's, jelly beans, or some other small candy. GLCE: .N.MR.03.10 Recognize situations that can be solved using multiplication and division including finding "How many groups?" and "How many in a group?" and write mathematical statements to represent those situations.
3. N.MR.03.12 Find solutions to open sentences, such as 7 x __ = 42 or 12 ÷ __ = 4, using the inverse relationship between multiplication and division.
N.MR.04.14 Solve contextual problems involving whole number multiplication and division.
Engagement (15 minutes): Teacher led class discussion: “Students, open your math journals to an empty page. As we discuss our ideas about multiplication you may write down your thoughts, ideas, and observations in the section titled ‘What I Know’. Write whatever you want to about the multiplication, spelling does not matter in this part. Please don’t erase anything you write. Who’s ready to begin?”
The instructor is to ask a series of questions that follows. Record the answers on the classroom KWL chart.
1. Has everyone heard about multiplication?
2. Who thinks they know what multiplication is?
3. Who thinks they could explain multiplication?
4. Who knows any multiplication facts?
5. Does anyone know how to solve a problem using more than one multiplication fact?
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Exploration (20 minutes): M&M multiplication Using real world story problems to solve multiplication facts This is a lesson to help students understand the uses of multiplication and practice problem solving while having fun. You will need bags of M&M's, jelly beans, or some other small candy.
Procedure
1) Students are divided into groups.
2) Give each group a bag of candy.
3) Explain that each group must share their candy with the other groups.
4) Now give each group a different problem to solve. For instance, if you have 5 groups with 4 students in each group tell your first group they must give every group 12 pieces of candy. What is the multiplication problem that would tell them how many pieces of candy they need? (12 X 5 = 60). Have them write the problem on the board and explain to the class how they solved their problem.
When each group receives their candy from another group they should write down the problem needed to show how many pieces of candy each student in the group will receive. (4 Students X ? = 60). At the end of the lesson let the students eat their candy
Teacher lead class discussion
“Now we will discuss the section titled ‘What We Want to Know’. As we discuss the things we want to learn about multiplication, you may write down your thoughts and ideas in the section titled ‘What I Want to Know’. Who’s ready to begin?”
The Instructor will ask for volunteers to tell the class what they hope to find out by studying this unit. Record the answers on the classroom KWL chart.
Journal Time
Students may record what they hope to learn in their journals.
Explanation (15 minutes): Setting the agenda
The instructor will explain that we are going to be studying multiplication for the next unit: the agenda for the unit:
• Students will bring home their journals daily and record their observations and discoveries about multiplication in their journal
• Students will locate arrays in real life, and either photograph them, draw them, or bring in examples of them.
• Students will create their own examples of multiplication through literature, music, and art. • As we study certain aspects of multiplication, you will record your data and observations in
your journals. • You will occasionally have other assignments that are to be recorded in your journals as well. I
will give you that information when we get to the appropriate lesson.
An instructor led Exploration of the journals:
• The instructor will show the students an example of the chart to record multiplication facts. • The instructor will show the students an example of the charts and templates they will use
during this unit. The instructor will remind them that we will not begin the individual lessons or activities until we have done them as a class.
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Grade Level
Third and Fourth Time Needed
50 minutes Materials
Clear container for “fishbowl”
Unifix cube “fish” (One color for each work group, three cubes for each student)
11” x 14” paper
Writing pencils
Colored pencils
Grid paper (For extension)
Lesson 2: Fish Bowl
Introduction: This lesson introduces students to the concept of “multiplication as repeated addition of equal sets.” First they will work either independently or in pairs to write and illustrate their solution to a “How many are there altogether?” problem, taking time to explore their thinking and clarify their understanding. Next, students will share their ideas with the class so others can try-‐on alternate ways of visualizing solutions to the same problem. Preparation: Prepare ahead of time small “packets” of Unifix cubes, one color for each work group, three cubes for each student. GLCE: 3.N.MR.03.10 Recognize situations that can be solved using multiplication and division including finding "How many groups?" and "How many in a group?" and write mathematical statements to represent those situations. Engagement (5 minutes): First divide the class up into even groups of 3-‐5 students each. Then pass out small bins of Unifix cubes to each group, giving each a single, unique color. Next, hold up a clear container (bowl, plastic bin, etc.). Tell the students that it’s a “fishbowl” and you want each of them to put three “fish” from their group’s bin into the bowl. Exploration (15 minutes): After discussing how many students put fish into the bowl, tell the class that you want to see if they can figure out how many are in the bowl altogether. On the board write:
There are ____ fish in the bowl. I think this because __________.
Tell them they can work in pairs or independently, but they need to explain their thinking with numbers and words. They can use pictures too if that would help. Explanation (30 minutes): Reconvene as a group and ask the students to share their thinking with the class. Acknowledge the different responses by asking thoughtful questions that extend their thinking and illuminate fuzzy logic. Students might show some of the following examples (24 students, 6 groups of 4):
a) 1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1 (etc) = 72 fish
b) 3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3 = 72 fish
c) 3+3+3+3 = 12 red fish 3+3+3+3 = 12 blue fish 3+3+3+3 = 12 green fish
3+3+3+3 = 12 yellow fish 3+3+3+3 = 12 brown 3+3+3+3 = 12 orange fis
Then add 12+12+12+12+12+12 and you get 72 fish!
d) 12 red fish + 12 yellow + 12 blue + 12 brown + 12 green + 12 orange = 72 fish
e) 6 groups of kids x 4 kids in each group x 3 fish for each kid = 72 fish
f) 24 kids x 3 fish each = 72 fish
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This is the time to explicitly make the connection that:
1) You can make groups of like things and add them together.
“What kinds of groups do we see here? 72 fish 1 time. 3 fish 24 times. 6 groups of three fish added together. 12 groups of fish 6 times.”
2) Adding up groups of things is quicker and easier than adding up singletons (a and b).
“For those of you who added each fish by itself up to 72 and those who added 3 fish 24 times, did you have any problems with your strategy? Do any other strategies look easier or faster?”
3) Multiplication is repeated addition of similar sized groups of things (c and d).
“Who can explain what I mean by, ‘multiplication is repeated addition’?”
4) Multiplication is commutative like addition, that is 2x4 = 4x2 = 8.
On the board write: 12 = 3 x 4 = 4 x 3 = 12
“Who can explain what I mean by, ‘multiplication is commutative’?”
5) You can group like numbers of things and add them to groups with larger or smaller numbers.
“Can you figure out how many fish there would be if there were 5 students in the green and orange fish groups?”
Sample answer:
3+3+3+3 = 12 red fish 3+3+3+3 = 12 blue fish 3+3+3+3+3 = 15 green fish 3+3+3+3 = 12 yellow fish 3+3+3+3 = 12 brown fish 3+3+3+3+3 = 15 orange fish
12 x 4 = 48 fish and 15 x 2 = 30 fish 48 fish + 30 fish = 78 fish!
“What if there were 8 students had red and 8 had yellow fish, 5 had blue and 5 had brown fish and only 3 students had green and 3 had orange?”
Sample answer:
3+3+3+3 = 12 red fish 3+3+3+3+3 = 15 blue fish 3+3+3 = 9 green fish 3+3+3+3 = 12 yellow fish 3+3+3+3+3 = 15 brown fish 3+3+3 = 9 orange fish
12 x 2 = 24 fish and 15 x 2 = 30 fish and 9 x 2 = 18 fish 24 fish + 30 fish + 18 fish = 72 fish!
“Hey, that’s interesting. That’s the same amount as we had the first time! Who knows why?” 6) Multiplying groups of things is even quicker and easier when you learn your math facts!
3 red x 4 kids = 12 red fish 3 blue x 4 kids = 12 blue fish 3 green x 4 kids = 12 green fish 3 yellow x 4 kids = 12 yellow fish 3 brown x 4 kids = 12 brown fish 3 orange x 4 kids = 12 orange fish
And 12 fish x 6 groups = 72 fish!
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“There’s one way that’s even faster. Can anyone see it? … 3 fish x 24 kids = 72 fish!”
“Do you think you could make similar groups for the other examples? Sure you could. Who wants to show us how?”
Invite at least two students come to the board and show their thinking. Ask the rest of the class if they agree. Be sure to ask them to explain their thinking if they head down the wrong path. Others in the class who may have gone there too will benefit.
Extension: Pass out grid paper and ask the students to represent their thinking in colorful arrays. Ask them to write a number sentence that means the same thing as their array. Ask for volunteers to explain their work. Ask thoughtful questions that extend their thinking and illuminate fuzzy logic. Evaluation: Monitor student’s oral and written responses to assess understanding of multiplication as repeated addition. Collect written responses as formative assessment.
Reference: Burns, M. (1995). Writing in Math Class: A Resource for Grades 2-‐8. Sausalito, CA: Math Solutions.
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Grade Level
Third and Fourth Time Needed
50 minutes Materials
Newsprint (One piece for each work group)
Drawing paper (At least one for each student)
11” x 14” paper
Writing pencils
Colored pencils
Lesson 3: Groupings All Around Us
Introduction: This lesson introduces students to the concept of “Multiplication is a quick way to figure out how many you have altogether of something when things come in groups.” First the class will work collaboratively brainstorming a list of objects in the world that always occur in groups of 2, 3, 4 … 12 and solving made up problems to find “how many?”. Next, students will work in small groups creating and solving their own made up problems. Finally, each group will share their ideas with the class so others can try-‐on alternate ways of visualizing solutions to multiplication problems. Preparation: Prior to beginning the lesson, determine how the class will be divided up into groups of 3-‐5 students each. Have sufficient newsprint for each group to have one piece. Also, be prepared to record lists generated by the class as a whole on newsprint posted on the wall, chart paper on an easel or on the white board. For the extension activity, each student will also need a piece of paper on which to write and illustrate a sample multiplication problem. GLCE: 3.N.MR.03.10 Recognize situations that can be solved using multiplication and division including finding "How many groups?" and "How many in a group?" and write mathematical statements to represent those situations.
3.N.FL.03.11 Find products fluently up to 10 x 10; find related quotients using multiplication and division relationships. Engagement (10 minutes): “Today we are going to brainstorm what sorts of things that come in groups of 2s, 3s, 4s, 5s all the way up to 12s. First we’re going to list as a class together examples of things that come in groups of two. Then, we’re going to break out into small groups. Each group will continue to brainstorm lists of things that come in groups of 3s through 12s and record their ideas on a large sheet of paper.” Now, together as a class, brainstorm a list of things that always come in twos, excluding things that sometimes come in twos. If students are unsure about an item, list it off to the side to research later. Once you have a good list of items, break up into the smaller work groups for the students to continue on their own. Be sure to remind them that since they are not listing groups of 1s and you have already listed groups of 2s together, each group will be exploring 10 lists total. Exploration (20 minutes): The first challenge of this activity will arise as the students figure out how they will work cooperatively to brainstorm and record their groups’ lists. Resist the urge to step in, confidently assuring them that they can figure it out for themselves. The next puzzle will be to figure out how to arrange their thinking on the large sheet of paper. Again, resist the urge to step in. Use this time to assess the creativity and uniqueness of each student’s thinking, as well as the students’ ability to cooperatively problem solve in a group setting. Explanation (20 minutes): Once all the groups have completed their lists, it’s time to discuss them together as a class.
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“Now we’ll go around the room, group by group. Each group will report just one thing from any one list, without telling us which list it’s on. Then the others in the class will have the chance to decide where it belongs. Once we agree, I’ll write it on the board under the correct number. Since you’ll want to report something from your list that has not already been suggested, take a few minutes now to have an alternative in case the one you chose has already been mentioned.” This part of the activity will involve group thinking and discernment. Some items will be obvious, legs on a dog and cans in a six-‐pack, for example; others may not be, such as legs on a stool or points on a star. You will need to talk this through problem together. Someone may suggest something that makes no sense. Others may be very creative, so be sure to ask students to explain their thinking. For example, a group my say “four holes in a shirt,” then offer they were thinking of the one for the neck, at the bottom and for each sleeve! Extension: These lists are a rich resource for generating problems that students can solve. Start by creating problems and linking them to their proper multiplication sentences.
1) For example, ask: “How many cans of Coke are in three six packs?”
If the students are able, have them tell you what sentence to write. If not, you write 3 x 6 = 18 on the board. Then ask: “What does the 6 tell us? What does the 3 tell us? What does the 18 tell us? How do you know that 18 is correct?”
2) Another activity would be to have students to write and illustrate multiplication problems for others to solve. They can write the problem out in words with an illustration on one side of the paper, then turn it over and write the complete multiplication sentence on the other side. That way, children can read each other’s problem, solve them, and check their solutions. Challenge students to see how many ways they can figure out the answer. Then ask volunteers to share their multiplication problems and their thinking for how they solved them.
It’s important that the solution is more than the answer that results from the multiplication; it is the entire multiplication sentence. The emphasis is on relating the multiplication sentence to the problem situation to develop children’s understanding.
3) Another extension activity would be to generate charts from the lists of 12 multiples. For example:
People Eyes Multiplication Sentence
1
2
3
(etc.)
2
4
6
(etc.)
1 x 2 = 2
2 x 2 = 4
3 x 2 = 6 (etc.)
Evaluation: Monitor student’s oral and written responses to assess understanding of multiplication as a quick way to figure out how many you have altogether of something when things come in groups, as well as their ability to work in groups effectively together. Collect written responses as formative assessment. Use extensions to challenge students who already have a basic understanding of multiplication or to provide additional practice to students who need help clarifying their understanding.
Reference: Burns, M. (1987). A Collection of Math Lessons: From Grades 3 through 6. Sausalito, CA: Math Solutions.
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Grade Level
Third and Fourth Time Needed
50 minutes Materials
One six-‐sided die (One die per group of 2-‐4 students)
Three 8 ½” x 11” sheet of paper for each student
Writing pencils
12-‐sided dice (For extension)
Lesson 4: Circles and Stars Game
Introduction: Through this game, students learn to see multiplication as the combining of equal-‐size groups that can be represented with a multiplication equation. Preparation: Divide the class up into groups of two to four students and distribute materials accordingly. GLCE: 3.N.FL.03.11 Find products fluently up to 10 x 10; find related quotients using multiplication and division relationships. Engagement (10 minutes): Invite the children to fold their pieces of paper in half, then in half again, creating four quadrants on each side. Explain the rules of the game.
1. The first player starts the first round by rolling the die. This number is the amount of circles he/she will draw in the first square on his/her paper. It is also the first number in his/her multiplication problem.
2. The player rolls the die again. This number is the amount of stars he/she will draw in each circle in that first square. It is also the second number in his/her multiplication problem.
3. Now the player writes the two numbers and the answer in a multiplication sentence right below the circles and stars.
4. Each player takes a turn until the group has repeated filled in all eight squares on their score sheets (front and back).
5. Add up all of your answers. Whoever has the most wins the game!
Model how to play the game then invite the class to play one round with guided practice.
Exploration (15 minutes): The students play several rounds of Circles and Stars. Explanation (15 minutes): Pose the following questions for students to discuss in small groups or as a class.
-‐ What is the fewest number of stars you can get in one round? Explain.
-‐ What is the greater number of stars you can get in one round? Explain.
-‐ What other observations did you make as you were playing this game? Explain.
-‐ What numbers did you represent in different ways? Compare with your partner. Explain.
-‐ I have a die that has a 0. What would you do if your first roll was a zero? Explain.
-‐ What would you do if your first roll was a 5 and your second roll was a zero? Explain.
Create Class Data Chart. (Prepare before the lesson.) List all numbers 1-‐36 on a chart
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using column format. (Thirty-‐six is the largest product possible using a six-‐sided die.)
Select one student to bring up one of his/her recording sheet. Together model how to use tally marks to record the student’s scores for each round on the Class Data Chart. Then invite the groups to come up and record their scores from all of their games on the Class Data Chart. Suggest that if one partner reads each score, the other partner can record tally marks.
Discuss the data. After all students have played several games and recorded their products for each round on the class chart, engage students in conversation about the data chart, asking questions like:
-‐ Why did I write the numbers 1-‐36 on the chart?
-‐ Are there numbers that are impossible using a 1-‐6 die? Explain.
-‐ Why do some numbers have more tally marks than other numbers? Explain.
-‐ What are the ways to get 2 as an answer? Ways for 6? Ways for 12? (Students might think about this with a partner or in small groups. Record equations.)
-‐ Which number(s) 1-‐36 has the most combinations using two 1-‐6 dice? What numbers can I skip count by to say this number? (Relate numbers on dice to factors in multiplication equations.
-‐ You can skip count by both factors to figure out the product. Is this always true? (Ask students to test this idea. Some may want to test larger numbers.)
-‐ Is there a product that can only be represented one way? Why? Explain.
-‐ What other observations do you notice about the data?
-‐ How might this data be useful for thinking about multiplication combinations (facts)? Extension: Invite those looking for a challenge to:
1. Change the die to a higher number sided die (e.g. 12 sided) to make the multiplication problems more difficult.
2. Use two dice at the same time and choose which order to put them in for your circles and stars. Commutative property of multiplication rule says you get the same answer no matter what order.
3. Write the fact family for each problem you roll to practice multiplication and division sentences.
Example: 3 x 4 = 12 4 x 3 = 12 12 ÷ 3 = 4 12 ÷ 4 = 3 Evaluation: Monitor student’s oral and written responses to assess understanding of multiplication as repeated addition. Collect score sheets for formative assessment.
Reference: Cleveland County Schools. http://tinyurl.com/circlesandstarsdirections. Accessed November 26, 2010. (Adapted from Math by All Means; Multiplication Grade 3 by Marilyn Burns.)
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Circles and Stars Multiplication Game Mary Ebejer and Becki West
Objective
Students will be able to form simple multiplication problems using 1 die by grouping them in circles and using stars to represent the numbers then multiplying them to find the product. Materials Needed
1 Die (6 sided) Paper Pencil Directions
1. Fold the paper into separate sections, usually four squares on front and four on back.
2. The first player starts the first round by rolling the die. This number is the amount of circles he/she will draw in the first square on his/her score sheet. It is also the first number in his/her multiplication problem.
3. The first player rolls the die again. This number is the amount of stars he/she will draw in each circle in that first square. It is also the second number in his/her multiplication problem.
4. Now the player writes the two numbers and the answer in a multiplication sentence right below the circles and stars.
5. Each player takes a turn until the group has filled in all eight squares on their score sheets (front and back).
6. Add up all of your answers. Whoever has the most wins the game! Challenges
1. Change the die to a higher number sided die (e.g. 12 sided) to make the multiplication problems more difficult.
2. Use two dice at the same time and choose which order to put them in for your circles and stars. Commutative property of multiplication rule says you get the same answer no matter what order.
3. Write the fact family for each problem you roll to practice multiplication and division sentences.
Example: 3 x 4 = 12 4 x 3 = 12 12 ÷ 3 = 4 12 ÷ 4 = 3
From: http://tinyurl.com/circlesandstarsgame. Accessed November 26, 2010. (Variation on Marilyn Burn: “Circles and Stars.” Math By All Means. ©1991 The Math Solution Publications.)
**
2 x 4 = 8
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Grade Level
Third and Fourth Time Needed
5 days
50 minutes/day Materials
For each group:
24 1” square tiles
For each student:
8 ½” x 11” paper ruled with ½” squares (stack of extras on hand)
Writing pencils
Colored pencils
Scissors
“Rectangles” Worksheet
Lesson 5: Creating Multiplication Tables
Introduction: In this 5-‐day lesson, students will create arrays for multiplication fact families 0-‐12 and cleverly transfer them to create a multiplication table to laminate for their own personal use. Preparation: Prior to beginning the lesson, ask students to respond to this prompt in their math journals:
Write what you know about the 0-‐12 multiplication table.
Their response will serve as a benchmark for their formative assessments. GLCE: 3.N.FL.03.11 Find products fluently up to 10 x 10; find related quotients using multiplication and division relationships. DAY ONE: MAKING RECTANGLES Engagement (10 minutes): Divide the class up into groups of four students. Invite one person from each group to come up and count out 25 tiles and bring them back to their group. Exploration (40 minutes): “Each group will have 25 tiles. I would like you to work with a partner in your group for this first task. (A group of three will work if there is an odd number.) I want you and your partner to take 12 tiles and arrange them into a solid rectangle. Your rectangle should be all filled in completely. Don’t use the tiles just to outline a rectangle.”
Students create their rectangles.
“Look at your group’s rectangles. Raise your hand if both the rectangles are the same.”
“Now raise your hand if your rectangles are different.”
Some may not raise their hands at all because they have the same shape but a different orientation, e.g. 6x2 and 2x6 or 4x3 and 3x4. Ask the students to describe their rectangles so you can draw them on the board. Show that the rectangles are the same dimension, just in a different position, so they are the same.
Rectangles that are the same shape and orientation but used different colors are also the same.
Have a member from each group come up and draw their rectangle on the board until all factors of 12 are represented (1x12; 2x6; 3x4). Ask them to write “12” on each rectangle.
“Let’s try another number. This time, work as a group instead of with a partner. See if you can find all the ways to build rectangles with sixteen tiles. Draw each rectangle you find on the grid paper, write 16 inside, and cut it out. If you finish that and others are still working, do the same for the number 7.” (Write 16 and 7 on the board.)
If anyone asks, a 4x4 square counts because a square is a rectangle.
Once you’re sure everyone understands the directions, they can continue making rectangles for numbers 1-‐25. Suggest that they continue using the tiles if that helps.
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“Draw each rectangle you find on the grid paper, write the number on it and cut it out. You will be cutting out a lot of rectangles so draw them close together to conserve paper. Also, don’t forget the number 12. We already did it on the board, but you will need to draw and cut out rectangles for that one too. Also, you will want to figure out a way to keep track of which ones you have finished. So take a minute to get organized before you begin. Any questions?”
(If the paper isn’t long enough to cut out the longest rectangles, it’s okay to tape two pieces together.)
As the time for the activity runs out, give each group a legal-‐size envelope. Ask them to put their names on it and put all of their rectangles inside, as well as any extra paper and scraps of paper still big enough for more rectangles. Put their envelopes and tiles on the supply table. Tomorrow, when it’s time for math, they can get their envelopes, some tiles and paper and continue working.
DAY TWO: FINISH RECTANGLES; BEGIN SUMMARIZING Engagement (10 minutes): On the board write the numbers 1-‐12 across the top, with about 6-‐8” between each. As the groups finish, ask them to organize their rectangles by number. Then ask one group at a time to come tape their rectangles to the board under the corresponding number. Be sure to ask if any other group has any other rectangles after each set of rectangles is posted. If a group is missing a set or two of rectangles, this would be a good time to make them. Explanation (40 minutes): Distribute “Exploring Our Rectangles” worksheet to each student and invite groups to investigate the patterns together.
You can leave the rectangles posted on the board for the next day’s lesson.
DAY THREE: MAKING OUR MULTIPLICATION TABLES Engagement (10 minutes): Invite the students to come up to the board to take a good look at al of the rectangles they have posted. After a few minutes, invite them to sit down on the floor near the rectangle display and ask them how it went working in groups on their rectangles. (“What worked well?” “What could have gone better?”) Exploration (40 minutes): Work through each of the questions on the “Exploring Our Rectangles” worksheet, listing the answers on the board, discussing the patterns, and giving new vocabulary when appropriate. For example, for rectangles that have a side with two squares on them, write 2, 4, 6, 8 10. 12, 14, 16, 18, 20, 22, 24.
“What do you notice about these numbers?” (They skip every other one.)
“Who could continue the numbers in this pattern?”
“What is another name for these numbers?” (Even)
“These numbers are also multiples of 2 because each can be written as two times something … 2 times 2 is 4 (write 2 x 2 = 4).”
Other patterns to make note of include multiples of 3, 4 and 5, as well as squares, like 1, 4, 9, 16 and 25. Ones with only one rectangle like 1, 2, 3, 5, 7, 11, 13, 17, 19, 24 are prime.
Next, introduce the idea of transferring their rectangles to a chart.
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“Here’s what I want you to do next. I’ll demonstrate on the board; then you’ll each do this individually. You’ll use your own sheet of squared paper, but you’ll share your group’s rectangles.”
Tape a piece of the squared paper to the board. Take the 3-‐by-‐4 rectangle and place it on the squared paper in the upper left-‐hand corner. Then lift the lower right-‐hand corner and write the number 12 in the square. Explain:
“If I drew a rectangle around the 12, I would outline the 3-‐by-‐4 rectangle I used to locate the 12. Now I’ll use the same rectangle, but in another position.”
Rotate the rectangle and again place it in the upper left-‐hand corner. Again, lift the lower right-‐hand corner and write 12 in the square. Do the same for the 2-‐by-‐6 and the 1-‐by-‐12 rectangles, writing 12 in the four additional squares.
Demonstrate the process again using the rectangles for the number 9, showing that rotating the 3-‐by3 rectangle doesn’t matter since the lower right-‐hand corner will be the same square either way.
Invite the students to return to their seats and follow this process for each of their rectangles that would fit on the squared paper. They can use the rest of class to finish.
DAY FOUR: INVESTIGATING PATTERNS ON OUR MULTIPLICATION TABLES
Engagement (10 minutes): Ask students to take a look at their squared paper and the chart they are creating. Does anyone recognize it? Exploration (40 minutes): Discuss the patterns in what they have done. Look at rows with patterns they are familiar with … 2s, 5s and 10s. Model how you continue to fill in the rest of each row and column. Some students may also know the 3s. You can show them how to continue skip counting using a calculator, pressing 3 then +, then = repeatedly until that row and column are filled in. Invite the class to go back to their desks and fill in the rest of the numbers themselves. Also tell them that as they fill in their tables you want them to make note of any special patterns on special 3” x 11” strips of paper. Explanation (15 minutes): When everyone has finished, post and compare what the students have found. Some of the patterns will include:
In even numbered rows and columns, all of the products are even numbers.
In the odd numbered rows and columns, the products are odd, even, odd, even, odd, even.
In the 5 row and column, the products end in 5, 0, 5, 0, 5, 0.
For the 10x column, you just have to add a 0.
Everything in the 11 row and column has a double digit.
In the nines row and column, all of the products add up two nine.
Plus many more!
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DAY FIVE: INVESTIGATING MORE PATTERNS ON OUR MULTIPLICATION TABLES Engagement (10 minutes): Pass out several sheets of multiplication tables to each student and ask them to get out their colored pencils or crayons. Tell them that today they are going to investigate even more patterns on the multiplication table.
Begin by modeling the multiples of 6.
“First I need to make a list of the multiples of 6. Read them to me from the 6 row or column of your multiplication table.” (The list will go up to 72.) Now demonstrate how you will cross off the number 6 wherever it occurs on the chart, then the number 12 wherever it occurs, and so on.
“What is the largest number on the 12-‐by-‐12 table?” (144) “So we need to continue the list of multiples to get as close to 144 as we can. Let’s add 6 to 72 to get the next number (and so on).”
“We could continue adding 6s or we could use a calculator. Do you think we will land exactly on 144? Is 144 a multiple of 6?” Invite students to explore their thinking out loud.
Exploration (40 minutes): Now invite the students complete what you’ve started on the multiples of 6 chart in their small groups, then the multiples of the ten remaining numbers (2-‐5 and 7-‐12) – making sure to use separate charts for each number.
“As we did here, you’ll want to first list the multiples of the number, then color in all of the multiples of that number on a fresh multiplication table. Be sure to color in every square for that multiple. For example, for multiples of 6, we crossed off all four 6s that occurred on the chart and all six 12s. Continue until you have colored in all of the multiple squares and see what patterns emerge.”
As the children work, write the numbers 2-‐12 on the board leaving room underneath each so group representatives can post sample charts for discussion when everyone is done. Explanation (15 minutes): Discuss the students’ findings during the last 15 minutes of class. Example questions for their consideration include: What did you notice? Which of the numbers have just stripes? We colored in the multiples of only two square numbers, 4 and 9. What did you notice about them? Evaluation: Monitor student’s oral and written responses to assess understanding of factor patterns that emerge on the multiplication table. Also, ask the students to respond to this prompt in their math journals: What do you know about 7 x 6?
References: Burns, M. (1987). A Collection of Math Lessons: From Grades 3 Through 6. Sausalito, CA: Math Solutions. Burns, M. (1991). Math By All Means: Multiplication Grade 3. Sausalito, CA: The Math Solutions Publications.
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Name __________________________________
EXPLORING OUR RECTANGLES
1. Which numbers have only one rectangle? List them from smallest to largest.
2. Which rectangles have a side with two squares on them? Write the numbers from smallest to largest.
3. Which rectangles have a side with three squares on them? Write the numbers from smallest to largest.
4. Do the same for rectangles with four squares on a side.
5. Do the same for rectangles with five squares on a side.
6. Which numbers have rectangles that are squares? List them from smallest to largest. How many squares would there be in the net larger square you could make?
7. What is the smallest number that has two different rectangles? Three different rectangles? Four?
From A Collection of Math Lessons: From Grades 3 through 6. (c)1987 Math Solutions.
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Grade Level
Third and Fourth Time Needed
50 minutes Materials
Copies of “Billy Wins a Shopping Spree” worksheet
Writing pencils
Lesson 6: Billy Wins a Shopping Spree!
Introduction: In this lesson, students will solve a real-‐world problem – Billy Wins a Shopping Spree – using their growing knowledge of multiplication, demonstrating that they understand both the meaning of and practical use for multiplication. Preparation: Divide the class up into groups of two to four students and distribute materials accordingly. GLCE: 3.N.MR.03.09 Use multiplication and division fact families to understand the inverse relationship of these two operations, e.g., because 3 x 8 = 24, we know that 24 ÷ 8 = 3 or 24 ÷ 3 = 8; express a multiplication statement as an equivalent division statement.
3.N.MR.03.10 Recognize situations that can be solved using multiplication and division including finding "How many groups?" and "How many in a group?" and write mathematical statements to represent those situations. Engagement (10 minutes): Tell the class that Billy is a fortunate boy who won a $25 shopping spree at the Science Museum Store. They will find a list of the items that he can purchase and the price for each item on their worksheet. Explain that Billy can spend up to $25 on any selection of the listed items. If he doesn’t spend the entire amount, he can’t keep the change, instead he will have a store credit that he can use later. He can’t spend more than $25 and cannot use any other money that he might have … or ask his parents for some. They do not need to calculate any sales tax.
Draw a model of the receipt on the board:
Science Museum Store Receipt
___ items @ $3.00
___ items @ $3.00
___ items @ $3.00
Total
Store Credit
$__________
$__________
$__________
$__________
$__________
Instead of duplicating blank receipts for the students to fill in, have them prepare their own. This experience will help them learn how to organize their work on paper.
They need to record Billy’s transaction two different ways:
1) In words, describing what he bought, how much each item cost, the total amount he spent and the amount of any store credit he can use later; and
2) On the receipt that they prepare.
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Exploration (25 minutes): Invite the students to “shop” for Billy, writing their transactions both ways. Explanation (15 minutes): Use class discussion to have some of the children present different ways they found to spend exactly $25. This will reinforce the idea that problems can have more than one solution. Extension: Find the different combinations of $3, $4 and $5 items that equal exactly $25. When students search for solutions by trial and error, they get great deal of number practice. Make sure, however, that they understand the focus on the number of items at a particular price, not the section of particular items. For example, buying five Koosh balls is the same solution as buying three Koosh balls, an inflatable world globe, and a dinosaur model kit. In each case, Billy spends $25 buy buying five items @ $5. Evaluation: The students’ written and oral responses will serve as a component of the summative assessment of their understanding of multiplication, both its meaning and real-‐world uses.
For their final Math Journal entry for the unit, invite them to respond to the prompt: “What I now know about multiplication.”
Reference: Burns, M. (1991). Math By All Means: Multiplication Grade 3. Sausalito, CA: The Math Solutions Publications.
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References
Burns, M. (1987). A Collection of Math Lessons: From Grades 3 through 6. Sausalito, CA: Math Solutions.
Burns, M. (1991). Math By All Means: Multiplication Grade 3. Sausalito, CA: The Math Solutions
Publications. Burns, M. (1995). Writing in Math class: A Resource for Grades 2-‐8. Sausalito, CA: Math Solutions. Cleveland County Schools. http://tinyurl.com/circlesandstarsdirections. Accessed November 26,
2010. (Adapted from Marilyn Burn: Math by All Means: Multiplication Grade 3. ©1991 The Math Solution Publications.)
San Marco Unified School District. http://tinyurl.com/circlesandstarsgame. Accessed November
26, 2010. (Adapted from Marilyn Burn: “Circles and Stars.” Math by All Means: Multiplication Grade 3. ©1991 The Math Solution Publications.)
