## Modeling Multiplication With Streets and Avenues

3-5
1

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.

To introduce the lesson, begin by reading One Hundred Hungry Ants. Ask children to arrange a set of 24 unifix cubes into arrays in as many ways as possible. As necessary, remind them that columns go up and down, and rows go across. Ask them to record the number of rows and columns each array contains.

Next, present the children with toothpicks, index cards and glue. Ask them to glue four toothpicks in parallel columns on the index card. Then ask them to place 3 toothpicks across the four columns at right angles. Explain that this represents a map of a town — the horizontal toothpicks represent streets, and the vertical toothpicks represent avenues, so this model shows three streets crossed by four avenues. Have them predict how many stoplights would be needed if one were placed at each intersection. Then, have them check their predictions.

Encourage students to try other numbers of streets and avenues and to again predict how many stoplights that would be needed.

As students explore various combinations, you may wish to have them record their information on a chart with columns headed Number of Streets, Number of Avenues, and Number of Stoplights, like the one shown on the Number of Streets Activity Sheet.

Then ask students to suggest the number of avenues for the next prediction, place that many toothpicks across the four glued toothpicks and predict, then count, the total number of stoplights needed.

Then have them glue on two more toothpicks, so that there are 6 streets and repeat the activity. Then challenge them to predict the number of stoplights needed when there is one avenue [6.], and the number needed when there are no crossing avenues [0.]. Encourage them to add these answers to their table.

As necessary, repeat with other numbers of streets, with one avenue, and no crossing avenues.

You may wish to conclude this lesson by having them glue a chosen number of toothpick "avenues" on top of the toothpick "streets" and marking the stoplights needed. Then ask them to record in equation form the multiplication fact displayed on their file card and write a problem using this fact on the back of the card. Encourage students to share their word problems with the class.

Assessment Options

1. At this stage of the unit, it is important for students to:
• Distinguish between factors and products
• Identify the effects of 0 and of 1 as a factor
• Write an equation using the multiplication sign

2. The guiding questions listed above may help you assess the students’ current level of knowledge in this area. You may find it helpful add to your entries on the Class Notes Recording Sheet you began earlier in this unit. It may provide useful information as you plan strategies for regrouping students and remediation or extension activities.
3. Samples of other "maps" might be constructed and children encouraged to sort them into sets needing an equal number of stoplights. You may wish to encourage children to describe orally or in writing the equations modeled.

Extension

Move on to the last lesson, Balance Beam Discoveries.

Questions for Students

1. What factor did you use for the streets? For the avenues? How many stoplights are needed? Can you write that in a number sentence?

[Student responses will depend upon which example they use.]

2. How many stoplights are needed if there are no avenues? Will that always be true?

[If there are no avenues, no stoplights are needed.]

3. How many stoplights are needed when there is only one avenue?

[The number of stoplights will be equal to the number of streets.]

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 instructional experiences do they need next? What mathematical ideas need clarification?
• What adjustments would you make the next time you 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.

### Exploring Equal Sets

3-5
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.

### 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:
• Construct array models.
• Explore the results of multiplying by 0 and by 1.
• Write a word problem using a selected multiplication fact.

### NCTM Standards and Expectations

• Understand the place-value structure of the base-ten number system and be able to represent and compare whole numbers and decimals.
• Understand various meanings of multiplication and division.
• Develop fluency in adding, subtracting, multiplying, and dividing whole numbers.

### Common Core State Standards – Mathematics

Grade 3, Algebraic Thinking

• 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.

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.

### Common Core State Standards – Practice

• CCSS.Math.Practice.MP1
Make sense of problems and persevere in solving them.
• CCSS.Math.Practice.MP4
Model with mathematics.
• CCSS.Math.Practice.MP6
Attend to precision.