Illuminations: All About Multiplication

All About Multiplication

Modeling Multiplication With Streets and Avenues

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.

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

Materials
 
One Hundred Hungry Ants (see Bibliography)
Unifix cubes
Toothpicks
Index cards
Glue
Chart paper
Number of Streets Activtiy Sheet

Instructional Plan
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 four streets crossed by three 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 activtiy sheet.

Number of Streets Activity Sheet 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.

Questions for Students
 

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.]

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

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

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

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

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.

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?

NCTM Standards and Expectations
 
Number & Operations 3-5
  1. Develop fluency in adding, subtracting, multiplying, and dividing whole numbers;
  2. Understand the place-value structure of the base-ten number system and be able to represent and compare whole numbers and decimals.
  3. Understand various meanings of multiplication and division.
This lesson prepared by Grace M. Burton.
  
1 period   

NCTM Resources

Principles and Standards for School Mathematics
National Council of Teachers of Mathematics Thinkfinity Verizon Foundation
More and Better Mathematics for All Students
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