Chapter 2

Recognizing Rectangular Lesson1

Arrays NCTM Standards 1, 2, 3, 6, 8, 10

STUDENT OBJECTIVES

Lesson Planner To identify the number of dots in rectangular arrays

= To find pairs of arrays that model the same multiplication fact

1 Daily Activities (TG P. 87) A . 8

Skills Practice and Review Open- -Ended Problem Solving/Headline Story

Adding 9 in Chains

221 Teach and 88-91 0

e TR: Activity Masters, AM13- -AM17 @ Counting Rows, Columns, and Dots

blank sheet of paper (TG pp. 88- -89)

@ Counting Hidden Dots (TG p. 90)

SH p. 17

Recognizing Rectangular Arrays (TG p.91)

Differentiated Instruction (IG P. 92)

Practice Book P9 Leveled Problem Solving (TG p. 92)

Extension Book E9 Intervention Activity (TG p. 92)

Spiral Review Book SR9 Extension Activity (TG p. 92)

Literature Connection (TG p. 84)

Lesson Notes

About the Lesson lessons prior to Lesson 2.6, without using this

This lesson explores the concept of multiplication using vocabulary and notation

rectangular arrays of dots. Students look at pairs of

related arrays and begin to see that in multiplication, the About the Mathematics order of the factors does not affect the product. Students

Students who can look at a rectangular array and

can be successful with this lesson by counting and may quickly respond with the number of elements in it are

be unaware they are learning about multiplication on the way toward mastering the basic multiplication

Think Math! presents multiplication as a shortcut, or and division facts. It is a small step from remembering

rather than a difficult topic convenience, to counting

that a 4- -by-5 array has 20 dots to producing the

students must learn. In this lesson, students are given sentence 4x 5 = 20. This process is gradual, so do

a counting task that could be tedious if they do not not expect mastery right away

such as skip-counting develop a shortaut of some kind

Exploring the ways arrays can be combined or split In Lesson 2.6, students are introduced to the

apart can help students as they work to acqurre vocabulary and notation of multiplication. Students

multiplication facts can complete the activities in this lesson and other

Use with Lesson Activity Book pp. 21-22

86 Chapter 2 - Lesson 1

1 paily Activities n

Developing Mathematical

Open-Ended Problem Solving Language Share the Headline Story with the class. Encourage students to create

Vocabulary: column, row problems that can be solved using information from the story

When you ask students to draw a

3-by-2 array of dots, either array shown Headline Story below is correct

0

Jan made stories and number

sentences for this picture

By not requring students to follow any

special rule for naming the arrays- --for

example, that rows (or columns) always

be named first you help your students VO

see that multiplication is commutative

(the order of the factors does not affect

the product) Possible responses: 4+4= 4: There are the same

number of green dots and blue dots: I had 8. I bought a sticker for Students should know the words column

have 4d and row, but it is not important for them

to associate a particular name or unique

multiplication fact with either of these T

two arrays

Skills Practice and Review MA

3

Adding 9 in Chains

Familiarize students with the terms

Say a number, and ask your students to column and row.

respond by adding 9 to the number. For example,

Point to a row of dots in if you say 0

8," students would respond with "'17. Beginning

a rectangular array, and tell students Choose a student to add 9 to that sum. Continue

26 selecting students and asking them to add 9 to the dots form a row. Then point to a

column, and tell students the dots form the previous sum. If students have dificulty

a column. Have students point out the adding 9, suggest that they add 10 and then 44

adjust their answer by subtracting 1 other rows and columns in the array 53

You might make a vertical list of the answers Intermediate Present a rectangular

that students supply. This list allows students array of dots. Ask students to copy the

to notice patterns, such as the ones digits 1

array and draw an additional row for

decreasing by 1 and the tens digits increasing the array. Then ask them to draw an

by 1. Repeat this activity with other starting additional column for the array

numbers as time allows

Advanced Challenge students to draw

an array that has 3 rows with 5 dots in

each row. Then ask students how many

columns the array has

87 Chapter 2 e Lesson 1

2I Teach and Practice

whole class

A Counting Rows, Columns, and Dots 10 0

MIN

Materials Purpose To introduce

For the teacher

multiplication situations and to Chapter AM13-AM17

count the number of dots in a Multiplication

2

NCTM Standards 1, 3, 8,10 rectangular array

Situations

Introduce The Student Dear Student

Letter introduces students to Do you know what situations use multiplication

instead of addition? Here are some examples situations that use multiplication

magine a rectangle made of dots. There are four

rows of dots and three dots in each row. How Have students read the letter many dots in all?

individually or together as a class Imagine a tiny town. It has four streets that

run north-south and three streets that

run east-west. Every north-south street

/OngoingAssessment crosses every east-west street. How many

Task Show one of the Dot intersections are there?

magine that you work in a sandwich Note students that Array Cards to your class shop. There are four kinds of

sandwich fillings. There are three

recognize the numbers of and ask how many dots are kinds of bread. How many "one

filling-one kind of bread

dots without counting one sandwiches

on the card can you make?

In this chapter, you will see by one. These students may

different ways to picture

Ask students how they counted and think about multiplication already be developing the as you start to learn

the dots multiplication facts. idea of multiplication

Mathematically yours, Students who count one

The authors of Think Mathl

at a time may need encour

agement and practice to

move to skip- -counting by

rows or by columns until

Student Handbook p. 17 they have mastered the

multiplication facts from

1 X 1 to 5 x 5

Dot Array Cards

Some students may count the dots Dot

one by one. Other students may Do

count the dots on the card one row

at a time (by 4s on the card shown Dot

above) or one column at a time Do

(by 3s on the card above). If no one

suggests a skip-counting method, try

to elicit this response

Activity Master 13

Activity Master 14

Activity Master 15

Activity Master 16

Activity Master 17

Use with Lesson Activity Book pp. 21-22 88 Chapter 2 Lesson 1

Demonstrate (or have a student demonstrate) how to count the rows and

columns in a rectangular array

2 3 4

000 2

3

3 rows 4 columns

(Activity Masters 13 through 17 will be used in the next activity and in future

lessons.)

Talk Math

What shape do all the dots on each card form? a rectangle

@ What are some ways you can find the number of dots on a Dot Array

Card? Possible answer: Count the dots one by one and skip- -count

Which way do you use? Explain why. Possible answer: Skip- counting row

by row or column by column, because it takes less time

89 Chapter 2 Lesson 1 Use with Lesson Activity Book pp. 21-22

Counting Hidden Dots whole class

20 0.0

MIN

Materials Purpose To determine the number of dots in a rectangular array when

For the teacher: some dots are hidden

AM13-AM17

blank sheet of paper Introduce Students practice finding the number of dots on the Dot Array

Cards when some of the dots are hidden NCTM Standards 1, 6, 8, 10

Task Hold up a Dot Array Card and ask, "How many rows?

How many columns? How many dots?

Show a few more cards, and have students continue finding the

number of rows, columns, and dots

Now cover part of an array with a blank sheet

of paper so students can only see a pair of

adjacent edges, as shown at right.

Ask the cass how many rows, columns, and

dots are on the card (lincluding the dots they

cannot see). Repeat this process for the rest

of the Dot Array Cards. Be sure to vary the

orientation of the card, the placement of the

blank paper, and the edges of the array that

you show

Differentiated Talk Math

Instruction How can you find the number of dots in the hidden columns or hidden

rows? Look at the column, or row, that is shown. The number of dots will Basic Level If students have

be the same in the hidden columns, or rows, because the dots form a difficulty finding the number

of dots when some dots rectangle

are hidden, demonstrate How did you figure out the number of dots on the card when some of the

skip-counting by rows or by dots were covered up? Possible answer: looked at the number of dots in

columns by gradually moving one row and skip- -counted by using the number of columns

the blank paper to reveal one

row or one column at a time

Use with Lesson Activity Book pp. 21-22 90 Chapter 2 Lesson 1

Recognizing Rectangular Arrays LAB PP. 21-22 individuals

20 MIN

Purpose To identify the number of dots in a rectangular aray NCTM Standards 1, 2, 6, 8, 10

Lesson Activity Book p. 22 Lesson Activity Book p. 21

grade classroom, there are 4 rows of Name Date/Time

chairs. Each row has 3 chairs. How many chairs are

in the classroom? Draw a picture to show how you Recognizing Rectangular Arrays solve the problem. Write a number sentence to

NCTM Standards 1, 2. 6. 8, 10

show your answer

How many dots? Drawings and number sentences 0 0 0 0 0 0000

may vary. Possible number 12

chairs in the classroom

sentence is given

4+4+ 4 = 12

0 0 0 0

00 How many dots?

0000

total D 0000

00 00000

total

000 0 0000

000 0000 0 000

total

in two problems Challenge above. List the pairs of problems that have the same

array and explain why the pairs match 00000 000000

2 and 5 3 and 9, 4 and 13, 7 and 10, 8 and 14, 12 and 15 00000 total They have the same number of dots, but are turned 000

0000 different ways

A xxI twenty-one 21 22 twenty-two XXI 2 x 11 3x7

ABOUT THE PAGE NUMBER 22 is the second two- ABOUT THE PAGE NUMBER 21 is a triangular number.

digit palindrome. It is a multiple of 11. What's the next How many weeks are 21 days? To multiply 21 by a

palindrome? Is it a multiple of 11, too? number, you can multiply 20 by that number and add the

number itself to the result. Why?

Teaching Notes for LAB page 22 Teaching Notes for LAB page 21

The problems on this page are arranged to mirror a standard Encourage students to draw a picture in the space to help

multiplication table. In Problem 10, students will find a 3-by-2 them solve the problem. In Problems 19- -21, students count the

number of dots of each color and find the total. The pictures are array. Encourage students to use skip- -counting or to find arrays

designed to help students begin to think about how two smaller that they know when working on this page. In Problem 17

arrays can be combined to make a larger array, or how to break students look for pairs of arrays that are the san ne, but are m

up large arrays into smaller, easier-to- -manage arrays viewed from two different orientations. For example, arrays

7 and 10 are the same Challenge Problem Students can find a square number if

they know the square number just before it. If they know that

5 x 5 = - 25, they can visualize adding one more row and one

more column of 5 dots to a 5- -by-5 array. By placing one more

dot to fill in the corner, students create a 6-by -6 array

Reflect and Summarize the Lessom

Write Use pictures, numbers, or words to describe some ways you can find the number of

Math dots in a rectangular array. Possible answer: I can count one by one, skip-count, and break the

array into smaller arrays and add to find the total

Chapter 2 Lesson 1 91 Use with Lesson Activity Book pp. 21-22

Differentiated Instruction

Leveled Problem Solving

Michael drew one row of 5 squares. Then he drew one row of 5 circles below the squares

Above Level Basic Level On Level

Michael draws a new row of How many shapes did Michael Michael makes each row longer

If his friend draws draw? Explain. 10 shapes 5 triangles by drawing the same number

3more shapes onto each of Possible answer: There are of each shape. There are

Michael 's three rows, how many 2 rows of 5 shapes 18 shapes in all. How many

shapes will be in the picture now? shapes did he add to each row?

Explain. 24 shapes; Possible 4 shapes

answer: Each row will have

8 shapes, and 3 rows of 8 is 24

Extension Imterventon Practce

Activity Combining Arrays Recognizing Rectangular Arrays Recognizing Rectangular Arrays O Fill in the total number of dots for each box or OIn each drde, write the total number of dots

group of boxes. in the two boxes touching the cirde.

Have pairs of students make an 8-by-2

m and an 8-by-1 array with counters 27

31 18

Ask students, "How 13 20 0g many groups of 8 17 16

40 14 36 are there in each 56

9 10 0

(19 56 and "How 19 14 12 56

array 16

many counters are there in each array? 21

O Fill in the total number of dots for each box and

the total.

Next, have students combine the Test Prep

O What is the missingnumber? Which does NOT equal 3217

arrays and write an addition sentence @2 hundreds,2 tens, 11 ones 23, 43, 53

B. 2 hundreds, 12 tens, 1 one

total 33

C 3 hundreds, 2 tens, 1 one

to find the number of counters in all D. 3 hundreds, I ten, 11 ones

ion E9 Practice

Extension Activity Spiral Review

Quick Counters

Provide students with at least Spiral Review Book page SR9 provides Measurement

Write the time shown on each clock. Use the

30 counters. Ask students to create review of the following previously learned words before and after to tell the time.

an array using the largest possible skills and concepts

number of counters. Challenge them telling time

39 minutes after 1 or before 4

21 minutes before 2 52 minutes after 3 43 minutes

to quickly find the number of counters finding sums

in their array. Have them explain to

You may wish to have students work with another student how they found the

a partner to complete the page 7 minutes after 23 minutes after 2 or 6 minutes after 5 or

number of counters 13 minutes before 7 37 minutes before 3 54 minutes before 6

Students who make an array Number and Operations

Find the sum.

containing only one row should be 04+ 11 - 09+8- 012+ 5 = 07+6-

15 17 13 17

encouraged to create an array with 015 + 6 = 08+ 10 = 07+ 12 =

21 20 18 19

more than one row and explain their

counting method Spiral Review SR9

Use with Lesson Activity Book pp. 21-22 92 Chapter 2 Lesson 1