## Exploring Equal Sets

3-5
1

This lesson builds on the previous lesson and encourages students to explore another model for multiplication, the familiar set model. Students find products using equal sets and present results in the form of a table. The students apply their knowledge about multiplication in the creation of pictographs.

To set the stage for this lesson, you may wish to read one of the books listed in the Bibliography of Books About Multiplication, What Comes in 2's, 3's, and 4's.

You may also wish to discuss some examples from science such as:

• If a starfish has 5 arms, how many arms will 2 starfish have?
• If a spider has 8 legs, how many legs will 4 spiders have?

This is also a good time to relate multiplication to addition, in that multiplication is repeated addition. In the example, "If a starfish has 5 arms, how many arms will 2 starfish have?", students may recognize the multiplication fact 5 × 2 = 10, and they may recognize the addition fact 5 + 5 = 10.

Provide students with counters, pieces of string or yarn, and a workmat (large construction paper). Tell them that they will be making equal sets and finding out how many counters there are in all. Ask them to make 5 sets of 4 counters, with each set inside its own yarn circle. Then tell them to determine in any way they wish how many counters they have used. Next display an empty table with several rows in which the three columns are labeled "Number of Sets, Number in Each Set, and Number in All". Have them suggest what will go in each column (5, 4, 20). Then have them work in pairs to create new equal set models for addition. When they have identified the product, help them enter their findings on their Equal Sets Activity Sheet.

You may wish to review the terms factor, multiple, and product.

Allow the children time to make several entries, then ask them if they see any similarities among the entries. If examples of the order property are not mentioned, prompt them to notice such entries. Encourage students to also notice rows in which the last column shows the same number.

To provide an application for this model, have the children create pictographs of favorite fruit. If the children are not familiar with pictographs in which an icon stands for multiple data points, you might demonstrate one or find an example in their Social Studies textbook. Ask the students to vote for their favorite fruit, limiting the choices to a set number of possible selections. Then ask them to tally the collected data. Now assign them to groups and have each group construct a pictograph. Before they begin, you may wish to help the groups find an appropriate number for each icon to represent and also to decide what to do if the number choosing that fruit is not a multiple of the number they chose.

When the children are ready, call them together to share the pictographs and describe how each of the entries on it was constructed.

Assessment Options

1. At this stage of the unit, it is important for students to know how to:
• Model multiplication using the set model
• Construct a pictograph
• Recognize and use the order principle.
2. These guiding questions may assist you in understanding the students’ level of knowledge, but others may seem appropriate as your dialogue with the students progresses. You may find it helpful add to your recordings on the Class Notes Recording Sheet you began earlier in this unit. This data may be helpful as you plan strategies for regrouping students and for remediation or extension activities.

Questions for Students

1. Suppose you had 5 groups of 0. What would be the product? How about 0 groups of 5?

[0; 0.]

2. Would you get the same product if you had 4 groups of 3 instead of 3 groups of 4?

[Yes, the product would be 12 in either case.]

Teacher Reflection

• Which students met all the objectives of this lesson? What extension activities are appropriate for these students?
• Which students did not meet the objectives of this lesson? What mathematical ideas need clarification? What misconceptions did they demonstrate?
• What adjustments would I make the next time I teach this lesson?

### Hopping on the Number Line

3-5
In this lesson, students generate products using the number line model. This model highlights the measurement aspect of multiplication and is a distinctly different representation of the operation. The order (commutative) property of multiplication is also introduced. Students are encouraged to predict products and to answer puzzles involving multiplication.

### Modeling Multiplication With Streets and Avenues

3-5
This lesson encourages students to explore the array model of multiplication, a model that lays an important foundation for the later study of area. The lesson focuses on the factors 4 and 6. Students will also explore products with 0 or 1 as a factors. First students make arrays with counters, and then they create a second concrete example of the array model using toothpicks. They also write problems which involve multiplication.

### Balance Beam Discoveries

3-5
This lesson encourages students to explore another model of multiplication, the balance beam, and another relationship, the inverse of multiplication. This exploration leads naturally into representing multiplication facts in equation form. In addition to extending their understandings of the concept of multiplication, students begin to practice the multiplication facts by playing the Product Game.

### Learning Objectives

Students will:

• Explore the results of adding equal sets.
• Construct a pictograph with icons representing multiple data points.

### NCTM Standards and Expectations

• Understand various meanings of multiplication and division.
• Understand the effects of multiplying and dividing whole numbers.
• Develop fluency in adding, subtracting, multiplying, and dividing whole numbers.

### Common Core State Standards – Mathematics

• CCSS.Math.Content.3.OA.A.1
Interpret products of whole numbers, e.g., interpret 5 x 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 x 7.

• CCSS.Math.Content.3.OA.A.3
Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

• CCSS.Math.Content.3.OA.C.7
Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 x 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.

• CCSS.Math.Content.3.OA.D.8
Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

• CCSS.Math.Content.3.MD.B.3
Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step ''how many more'' and ''how many less'' problems using information presented in scaled bar graphs. For example, draw a bar graph in which each square in the bar graph might represent 5 pets.

• CCSS.Math.Content.4.OA.A.2
Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.

• CCSS.Math.Content.4.OA.A.3
Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

Grade 4, Num & Ops Base Ten

• CCSS.Math.Content.4.NBT.B.5
Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.