Objective: Recognize equal groups, repeated addition ...



Andrea Roos

Sara Brownlie

Megan Robinson

Introduction

For our unit we chose to concentrate on multiplication and division at upper elementary grades. The topics of multiplication and division are important because they build off students’ prior knowledge of addition and subtraction. Both concepts have real life applications that students need to learn in order to apply to their own lives. One can hook the student’s interest by posing an extremely hard addition problem but, it could be easily solved by using multiplication. The teacher could stump the students in order to foster their interest in the subject. Multiplication and division supports the principles and standards for school mathematics as established by the National Council of Mathematics.

This unit relates to the following:

NCTM Standards

Numbers and Operations Standard

• Understand meanings of operations and how they relate to one another.

o Students will explore the meaning of multiplication and division through several lessons and different thought processes. Students will then find the relationship between these two operations.

o Students will see the relationship between repeated addition and multiplication and between repeated subtraction and division.

Problem Solving

• Build new mathematical knowledge through problem solving.

o Using prior knowledge of repeated addition and subtraction students will use problem solving to develop an understanding of multiplication and division.

• Apply and adapt a variety of appropriate strategies to solve problems.

o Students will be provided opportunities to use a variety of resources including tables, arrays, manipulatives, and other various supplies to help students solidified their knowledge and vary their use of strategies.

Illinois State Goals

State Goal 6-Number Sense

Benchmark 6.B.2

• Solve one-and two-step problems involving whole numbers, fractions and decimals using addition, subtraction, multiplication and division.

The scope of our unit is gradual progression of multiplication into division. We begin relating repeated addition to the new concept of multiplication. Through hands-on experiences and manipulatives, the students will explore and solidify their understanding of the concepts of multiplication. Following this, students will discover that division is closely related to multiplication. Similarly, students will investigate division through hands-on activities and manipulatives. Finally, we will conclude the unit by utilizing both multiplication and division skills in a final lesson. The objective for this unit is for students to perform basic multiplication and division facts. Further more, the students will learn how to relate the topics to real life situations. They will explore how multiplication is related to repeated addition and skip counting. Throughout this unit students will see how multiplication and division are interrelated through multiple problem solving strategies as it is modeled by our lessons. By the end of our unit students will be able to solve problems using common sense and an understanding of multiplication and division to identify information necessary in order to solve problems.

Multiplication and division are taught together to help children see the relation between the two operations. Multiplication helps students learn how many times a quantity is repeated and how to skip-count. Division is developed with two meanings. One is sharing, and the other is grouping. This is important for students at the upper elementary level because it helps them learn how divide something equally among friends or divide into groups. Students can work in cooperative learning groups to explore the properties of multiplication. Working in groups allows children to exchange ideas and provides an opportunity for them to verbalize their thoughts.

Concept Map

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Timeline

Week One

Activity 1-Introduction to Multiplication

• Objective = For the students to recognize multiplication as a form of repeated addition

• General Description = The teacher will have the students equal numbers of M&M’s into a specified number of groups. For example, the teacher may ask the students to put the candies into three groups of four. The students will have to do this a number of times with different size groups and different number of groups.

• Assessment = The students must complete the worksheet that accompanies the activity. In addition, a brief quiz will be given at the end of the activity to evaluate the students’ understanding.

Activity 2-Colorful Arrays

• Objective = Recognize equal groups, repeated addition, arrays, and multiplicative comparisons as multiplication.

• General Description = In this activity the students will draw a 3-by-6 array, and color each row a different color. The students will have to identify the number of rows, the number of squares as the second factor, and the total number of squares as the description. The students will have to write a number sentence that describes the product of the array.

• Assessment = The students will reflect on what they learned in their journals.

Activity 3 All Hands

• Objective = Use patterns to find products with factors of 0,1,2,5, and 9.

• General Description = The students will use their fingers to develop multiplication skills.

• Assessment = The teacher can assess the students by monitoring their response with their hands.

Activity 4-Two Tone Rectangles

• Objective = Students will use the distributive property to find products by breaking unknown facts into known facts.

• General Description = The students will have two outlined arrays. The students will have to shade the first grid one color. Then the students will have to color the second array so that it makes two arrays in one. The teacher should ask what multiplication facts the arrays show.

• Assessment = After the lesson the students should record in their journals what they learned that day. Also, the teacher should ask the students to make up two of their own arrays.

Week Two

Activity 5-Patterns with Multiplication

• Objective= Students should find product patterns in sets of consecutive numbers.

• General Description= Students will find patterns using four consecutive numbers. The students should come to the conclusion after numerous testing that with four consecutive numbers, if you multiply the two outer and two inner numbers, the product of the two inner numbers is always greater than the product of the two outer numbers.

• Assessment= Students will be assessed through self evaluations that they will complete after the completion of the activity. They will also reflect in their daily math journals.

Activity 6-Theme Park Website

• Objective= Student should be able to multiply facts through 12. They should also use strategies to solve problems using the following website:

• General Description= Students should follow the directions on the worksheet and go to a website where they explore the uses of multiplication through a theme park.

• Assessment= Student will be assessed based upon their responses on the worksheet.

Activity 7-Sorting by the product

• Objective = Students will find products of factors 10, 11, 12.

• General Description = For this activity students will make a set of multiplication flashcards for multiples of 10, 11, 12.

• Assessment = The students will write in their daily journal about the activity. The teacher could test the students to see if they know the flashcards without looking.

Activity 8 – School Store

• Objective = Make tables and use them to solve word problems.

• General Description = Students will create a paper “work mat” that will be used as a table to organize word problems. This will help them visually see patterns in multiplication as they use manipulatives to see how the cost increases by the same amount each time the pencils increase.

• Assessment = The students are assessed by the multiplication sentences they create based off the information on their tables. Students will then write in their journals reflecting on this strategy as an ongoing assessment for teachers to have an idea about the students’ understanding of problem solving strategies.

Week Three

Activity 9 – Division Towers

• Objective = Using sharing and repeated addition to solve word problems with division.

• General Description = Students use a specific number of manipulative blocks to make towers of equal size. They repeat this until they have all possible combination of tower sizes recorded into table form. They also will write the division sentences that correspond with their findings.

• Assessment = Students will write in their journal about what they learned from the activity. They will also be assessed on the information recorded in their tables.

Activity 10 – Dividing by Measuring

• Objective = To use measurement to demonstrate division by repeated subtraction.

• General Description = Students are given a string of 104 cm. With a partner, one measures out 8 cm while the other cuts each time. Eventually, they will have 13 strings each 8 cm long. They will repeat this with other lengths of string and record all their data in a table.

• Assessment = Students are evaluated on their data in the table and their division sentences that correspond. They will also be given a worksheet to review division sentences and strategies to help them solve these problems.

Activity 11– Investigating Arrays

• Objective = Discover relationships between multiplication and division through arrays and fact families.

• General Description = Each student is given a number card with a number somewhere between 10 and 48. They are to make arrays with this number on grid paper and create the multiplication and division sentences that relate to each array.

• Assessment = This is an ongoing assessment as students are discovering the relationship between multiplication and division. Teacher observation will be a helpful tool when assessing the arrays and sentences each student created.

Activity 12 – Replacing the Signs

• Objective = Relate multiplication and division by discovering the inverse property.

• General Description = The teacher will draw an array on the board or overhead. Using this, the class will create its division sentences and write it beneath the array. The teacher will then tape an “=” over the division sign and an “x” over the equal sign. This will create a multiplication sentence that is the inverse of the previous division sentence.

• Assessment = Students will be given worksheets as practice with this concept, which will be used as assessment. They will also reflect in their journals about how this helps them understand multiplication and division.

Week Four

Activity 13-Division Facts

• Objective= Students should be able to divide using a related multiplication fact.

• General Description= Students will be given the scenario: “Ellen has 35 cherries to place in bowls. She wants to put 7 cherries into each bowl. How many bowls does she need?” Then have each group find 35 divided by 7 using the bowls or cups to form equal parts. The students should also identify the dividend, divisor, and quotient in the division sentence. Students will then continue to do problems of this sort in a worksheet format.

• Assessment= The students will be assessed through their completed worksheets. They will also reflect in their journals.

Activity 14-Special Quotients

• Objective= Students should be able to see that they should give quotients of zero when the number divided is zero, give a quotient of 1 when a number is divided by itself, and give the number divided as the quotient when dividing by 1.

• General Description= Students will be placed in groups of five. Each group will be given five pencils, and they have to figure out how to divide the pencils among only one of them. Then take away one of the pencils so each group only has four pencils and redo the groups so there are only four in a group. Then do the same activity. Students should use these activities to figure out a rule when you divide a number by one.

• Assessment= Students will reflect in their journals. The teacher will also assess students through a worksheet that will be completed during the activity.

Activity 15-Multiple-Step Problems

• Objective = Students should be able to solve multiple-step word problems.

• General Description = In this activity each group in the class will divide into two teams labeled as A and B. Each student will then take a handful of counters. Each group should then proceed to find the difference between the number of counters selected by the two teams. This activity is to get students to practice their general problem solving abilities.

• Assessment = Students will reflect on this activity in their math journals. The teacher will also base assessment on the notes that the students take during the activity.

Activity 16 – Math Stories

• Objective = Write stories using provided multiplication and division facts.

• General Description = For this activity, students create both multiplication and division word problems about a specific theme.

• Assessment = Students are assessed on the quality of the word problem they created, and if the problem truly creates a situation that requires multiplication or division.

Activity 17 – 24 Game

• Objective: Students will use all basic mathematical operations (addition, subtraction, multiplication, and division).

• General Description: Students work in groups using 24 game cards. Students must use various operations to create an answer of 24 using the provided numbers. This game can be used frequently throughout the year whenever extra math time is available.

• Assessment: Students will be assessed based on teacher observation of group work and individual behavior.

Activities

An Introduction to multiplication

Educational Goal: For the student to recognize multiplication as a form of repeated addition.

Illinois Learning Standard:

• Goal 6 Demonstrate and apply a knowledge and sense of numbers, including numeration and operations (addition, subtraction, multiplication, division), patterns, ratios and proportions.

• 6.B Investigate, represent and solve problems using number facts, operations (addition, subtraction, multiplication, division) and their properties, algorithms and relationships.

• 6.B.1 Solve one-and two-step problems with whole numbers using addition, subtraction, multiplication and division.

Materials: Worksheet, pencil, candy provided by teacher (optional if choose to make the activity visual).

Activity/instructions: First the teacher should give an introduction to multiplication will be. Next, the teacher will allot time for the students to do a sample of multiplication problems with partners using the candy. The teacher will announce for the students different numbers of groups with the candies, and the first group done raising their hand with the correct answer gets to eat one of the piles. Next the teacher will distribute worksheets on grouping and multiplication problems and the students will have the rest of the class period to work on it with a partner, the teacher should pick the partners, trying to divide the students who understand better with students who are having trouble. Each student should have their own sheet, and if they do not finish it becomes homework. The following day an evaluation will be given in class, along with a review to the activity.

Introduction to multiplication

• Ask students where they might use multiplication in life.

• Start with the multiplication of zero and one. Anything times zero is zero and anything times one is itself. If the students understand move on to the multiplication of twos.

• Teach students to say 2 x 3 = 3 two times.

• Have the students count by twos and put the candies into groups, then tell the student to pull out four different groups of two and ask how many they get.

Name_________________

Multiplication

Directions: Use the candies to represent the numbers asked for; then draw what the candies look liked under the problem to solve.

1. 4+4+4 = 2. 7+7 = 3. 2+2+2+2+2 =

3fours = 2sevens = 5twos =

4. 3+3+3+3 = 5. 5+5+5+5+5+5 = 6. 6+6+6+6+6+6+6 =

4threes = 6fives = 7sixes =

Directions: Make the addition sentence and the multiplication sentence that goes with the picture.

7. 8. 9.

____+____ + ______ = _____ ______+ ______ = _____ __ + __ + __ + __ + __ =

______ x ______ = ______ ______x ______ = _____ _____ x _____ = _____

10.Draw a line to the connecting answers.

5+5+5+5 4 x 2

2+2+2+2 3 x 6

6+6+6 2 x 3

3+3 4 x 5

EVALUTAION

1. Draw three groups of five.

2. What multiplication sentence does this picture represent ______ x ______ =

3. What is the multiplication sentence of 4+4+4=

______ x ______ =

4. True or False is 6 sevens is the same 6 x 7 =

5. Write + or x in each blank

3 __4 =12

3__4 = 7

Rubric

Competence

Level Knowledge Articulation Work Habit

| |Student clearly |Student can apply the |Student is active and comprehends |

|1 |Understands all the |Information to get all the answers|enough to discuss with others in the |

| |information and problems | |class |

| |Students understands most |Student is able to figure out the |Student understands most but is timid|

|2 |of the problems and knows |answers to certain problems but |to talk with others in a group |

| |most of the information |cannot get them all | |

| | | | |

| |Student understands few of|Student can apply very little |Student does not talk in class and |

|3 |the problems and knows |information to get the answers |shows no understanding of the |

| |little information on the | |material being studied |

| |subject | | |

Score:

7-9 ( Meets Goal

4-6( Review And Re-Assess

0-3( Mediation Needed

Colorful Arrays

Objective: Recognize equal groups, repeated addition, arrays, and multiplicative comparisons as multiplication.

Materials: Centimeter grid paper

Crayons or markers

Procedure: In this activity the teacher will have the students draw a 3-by-6 array, and color each row a different color. Then the teacher could say, “How many squares are in the array (18 squares)?” Next, have the students identify the number of rows as the first factor, the number of squares in each row as the second factor, and the total number of squares in each as the product. Have the students label the factors in their drawings and write the number sentence describing the product in the array. For additional practice, students will practice the use of arrays on an online applet, . On this applet students will visually see multiplication problems through arrays.

Assessment: The students should write in their journals at the end of the lesson and reflect on what they learned that day. The teacher should also go around and do an ongoing assessment of all the students’ work. The teacher should see if the students understand the concepts of the lesson by looking over the students’ worksheets.

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All Hands

Objective: Use patterns to find products with factors of 0,1,2,5, and 9. [pic]

Materials: There are not any physical materials that are needed for this lesson.

Procedure: First, ask the students “how can you show multiples of 5 using your hands (Hold up all 5 fingers)?” Have student volunteers, work together to model multiples of 5 and record the results on the board. Next, ask the students, “What do you notice about the ones digits of the products (they are all either 5 or 0)?” Repeat this procedure for multiples of 0, 1, 2, and 9. To show 3 x 2, for example, have 3 students each hold up 2 fingers. To show 0 x 3, have the students hold up 3 fingers. To show 3 x 0 have 3 students hold up 0 fingers.

Assessment: The teacher can assess the student by monitoring their response with their hands. The assessment is done throughout the lesson. Also, at the end of the lesson the students should write in their journals about what they learned.

Two-Tone Rectangles

Objective: Use the distributive property to find products by breaking unknown facts into known facts.

Materials: Centimeter grid paper and markers.

Procedure: First have the students outline two 4-by-6 rectangular arrays on the grid paper. Have the students shade the first grid in one color. Then ask the students, “What multiplication fact does this array show” (4x6=24)? Next, have the students color the second array so that it makes two arrays: one 4-by-5 and the other 4-by-1. Ask the students, “What multiplication facts do these arrays show” (4x5=20 and 4x1=4, so 20 +4 = 24)? Lastly, have the students draw two arrays representing a multiplication fact and divide each into two arrays labeled with their multiplication sentences.

Assessment: After the lesson is finished the students should record in their journals what they learned that day. In addition, the teacher could assess the students by asking them to turn in two of the arrays they made up and see if they were done correctly.

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Patterns With Multiplication

Objective: Students should find product patterns in sets of consecutive numbers.

Materials: 1 calculator per pair (optional), pencil and paper

Procedure:

• Write the numbers 3, 4, 5, and 6 on the chalkboard in horizontal format. Use one color for 3 and 6, and another color for 4 and 5. Explain that sequential numbers like these are called consecutive numbers.

• Ask the students to figure out “What is the product of the inner two numbers? The two outer numbers?

• Encourage the students to try other sets of four consecutive numbers. Then ask them to describe any pattern that they find.

• If students choose numbers that are too difficult to multiply mentally, suggest that they begin with one-digit numbers.

• If students give a pattern after testing only one set of numbers, ask, “Can you be sure this is the pattern after using only one group of numbers? Why or why not?”

• If students are having difficulty discovering a pattern, suggest that they record their work in some type of organized table or list.

• Suggest that students use a calculator if they want to continue to test grater sets of four consecutive numbers.

• As a class, record three sets of consecutive numbers that the students suggest and the products of the inner and outer numbers.

• Ask the students “What is the product pattern in these three sets of consecutive numbers?”

• They should respond “ The product of the two inner numbers is 2 greater than the product of the two outer numbers.”

• Then ask the class “If the pattern holds for these three sets of consecutive numbers, will it hold for all the other sets of consecutive numbers that you have tested?”

• Wrap up the lesson by pointing out that many patterns exist within multiplication and that this pattern provides students with another strategy that they can use to find a product of an unknown fact.

Assessment: Assess students through having them reflect in their daily journal. Also the teacher can assess the students based on self-evaluations that the students can complete themselves with regards to how they participated in the activity.

Theme Park Activity

Objective: Student should be able to multiply facts through 12. They should also use strategies to solve problems.

Materials: Access to computer lab with internet, student handout printed from website, pencil (teacher refer to )

Procedure: Students should follow the directions on the worksheet and go to a website where they explore the uses of multiplication through a theme park.

Assessment: Student will be assessed based upon their responses on the worksheet.

Sorting by the Product

Objective: Find products of factors 10,11,12.

Materials: Calculator, index cards.

Procedure: First divide the students into groups. Have each group of students make a set of multiplication flashcards for multiples of 10, 11, and 12. On each card, they should put the factors on the front and the product on the back. They can use the calculator to ensure the products are correct. After, the students have made the flashcards, discuss with them any patterns they noticed. Ask, “What do you notice about the multiples of 10” (the last digit is always zero)? Also, ask the students, “What do you notice about multiples of 11 (up through 11x 9 the ones and tens digits are the same)? Then have each group mix their multiplication flashcards and place them in a stack with the products face-up. Have them sort the cards into multiples of 10, 11, and 12 with out looking at the factors.

Assessment: To assess the students have them write in their daily journals about the activity. The teacher could also, assess the students by testing to see if they know the answers to the flashcards without looking.

Front Back

School Store

Objective: Make tables and use them to solve word problems.

Material: Each group will need 10 pencils and a sheet of paper per student

Procedure: Students will work in small groups of 2-3 and each student will create a table work mat. This is made by folding a piece of paper in half lengthwise and then twice horizontally, into eighths.

In this work mat, students will organize their information. They can be given a variety of problem situations to do this. One example is for students to work with pencils. In the top row of boxes, place 1 pencil, then 2 pencils in the next box, etc. Each pencil costs 12 ¢.

|1 pencil |2 pencils |3 pencils |4 pencils |

|12 ¢ |24 ¢ |36¢ |48¢ |

In the second row, students will use money to find the total amounts as the number of pencils increase. After completing this table, students must write the multiplication sentences that represent each column.

Assessment: Students will be assessed based on their tables and related multiplication sentences. They will also be assessed by a journal reflection in which they are to explain how this activity was helpful. This is an ongoing assessment as students continue to develop problem solving strategies to help understand multiplication.

Division Towers

Objective: Using sharing and repeated subtraction to solve word problems with division.

Materials: Each group uses a set of Place-Value Blocks

Procedure: To begin this activity, each group is instructed to make towers of equal height using only 20 blocks. Students use their table making strategy to organize the data in this activity. They do this by recording the results of their tower heights and blocks in the first three columns. In order to complete the last column, the teacher must write on the board, “total number of blocks ÷ number of towers = number of blocks in each tower.” Above the appropriate section, the teacher will also add to the board the vocabulary for this activity, dividend, divisor, and quotient.

|Total Number of Blocks |Number of Towers |Number of Blocks in Each Tower |Division Fact |

|20 |4 |5 |20 ÷ 4 = 5 |

|20 |5 |4 |20 ÷ 5 = 4 |

|20 |10 |2 |20 ÷ 10 = 2 |

|20 |2 |10 |20 ÷ 2= 10 |

|20 |1 |20 |20 ÷ 1 = 20 |

|20 |20 |1 |20 ÷ 20 = 1 |

Assessment: Students will be assessed based on their completion of the table used in the activity. They will also reflect in their math journals about the activity and what they learned while doing it.

Dividing by Measuring

Objective: To use measurement to demonstrate division by repeated subtraction.

Materials: a meter stick, a ball of cord, a pair of scissors

Procedure: Seat the class in pairs. Have one student measure and the other cut. Students will complete the following directions:

1. Measure and cut a piece of cord 104 cm long.

2. Start at one end of the cord and cut pieces that are 8 cm long. Cut as many pieces as possible. Measure carefully.

3. Count the number of 8 cm pieces you have cut.

They can then repeat this process for other lengths of string. Like the previous activity, students will record their data in table form each time they use a different length string. The tables will look like this:

|Length of starting string|Length of singles pieces cut |Number of pieces cut |Division sentence |

| |from string | | |

|104 cm |8 cm |13 pieces |104 ÷ 8 = 13 |

|96 cm |6 cm |16 pieces |96 ÷ 6 = 16 |

Assessment: Students will be evaluated on their completion of the chart above, a related worksheet, and their ongoing understanding of the division concept. They will also be assessed through their description and reflection of the day’s activity in their math journal.

Investigating Arrays

Objective: Discover relationship between multiplication and division through arrays and fact families.

Materials: (whole class uses these) Number cards for 10, 12, 15, 16, 18, 19, 20, 24, 27, 30, 31, 32, 36, 40, 42, 45, and 48. For each pair, two sheets of centimeter grid paper, poster paper, scissors, and tape. Tile manipulatives

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Procedure: To prepare for this lesson, the teacher should draw a 2-by-4 rectangular array on the chalkboard, and then distribute a number card and grid paper to each pair of students.

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Using this table, the students will be reviewing their previously learned concept of rectangular arrays. They will be asked, “What is the multiplication sentence of this array?” [Students will respond with 2 x 4 = 8 or 4 x 2 = 8] The teacher should write these sentences below the array. Some students may not see how these are different arrays. The teacher can help them by rotating the array to give a new perspective.

The teacher should also help by modeling other possible arrays so students know what is expected:

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|2 |8 |

Using these numbers, students may come up with the following operations:

8 x 6 = 48 ( 48 ÷ 2 = 24 ( 24 x 1 = 24

This game can be used frequently throughout the year whenever extra math time is available.

Assessment: Students will be assessed based on teacher observation of group work and individual behavior. This will be used as the final activity as students combine their knowledge of operations. A test is unnecessary in this unit as students have given adequate feedback throughout the various activities. The teacher will be able to assess how they have progressed through the previous assessment tools.

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10 x 9

90

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