TITLE:    Multiplying Round numbers (Adapted from Math Trailblazers: grade 4)

CONTENT  AREAS:    Math: Multiplying Round numbers

GRADE  LEVEL:    4^th Grade

MATERIALS   NEEDED:   

Tray of flowers (6 rows and 8 flowers in each row)

Handout One (either a transparency or written on the board)

Multiplying by Zeros (worksheet, copies for each student)

Calculators

KEY   CONCEPTS:   

1. Patterns of multiplying numbers that end in zero.

2. Estimating products

EALR'S (Make the connections clear and specific):   

1.1.3        Understand and apply concepts and procedures from number sense.

Estimation: use estimation to predict computation results and to determine the reasonableness of answers.

1.1.6   Apply procedures of multiplication and division on whole numbers with fluency

Write and solve problem situations with whole numbers using a combination of any two operations.
 

GOALS (Remember the difference between goals and objectives):   
 Practice estimation and introduce the concept of multiplying round numbers

OBJECTIVES:   
After watching and participating in a teacher demonstration of estimating, the students will create and solve problems for each other.

PROCEDURES:  (Label each step in the process:  Activating Prior Knowledge, Disequilibration, Elaboration, Crystallization)

Special Note: The term round is not a technical term in math. It is used here to mean the lesson is about multiplying numbers that produce easy straightforward answers. This lesson is designed to allow the student to grasp the basic concept of multiplication.

• Introduction/Pre-assessment

(Activating Prior Knowledge)

Today we are going to review estimating and learn about multiplying. Let us review the purpose of estimating. Why do we estimate? (Have students answer. Possible answers are: a reasonable guess at a possible answer/ use estimation as a reference point for comparison to the actual calculation). How do we choose the numbers for our estimation? (We want numbers that are close to the actual numbers in the problem).

(Disequilibration)

The Bark and Garden Center sells trays of flowers. A tray has 6 rows. There are 8 flowers in each row. Look at the example I brought in. We want to know how many flowers there are. First, we are going to estimate an answer. What numbers will we use? (6 is close to 5 and 10 is close to 8) 5 X 10 = 50. This is our estimation. Now let us calculate the answer using the true numbers 6 X 8 = 48. Let us check our answer with our estimation. It is very close, so I am confidant that 48 is right. If I really want to be sure, I can count the flowers here in the tray and yes, it is true.

Now in the Bark and Garden Center they have 28 trays per shelf. I wonder how many flowers are on each shelf. How would we estimate the total number of flowers on each shelf if each tray has 48 flowers and there are 28 trays per shelf?  ( Well 30 is close to 28 and 50 is close to 48) Wow, this is going to be a big number. One strategy we can use is to look for a pattern.

• Activity

(Elaboration)

In order to see patterns of multiplying go through these problem step by step on the board or overhead.

5 x 3 =                     15

5 x 30 =                   150

50 x 3 =                   150

50 x 30 =                 1500

50 x 300 =               15,000

Do you see a pattern? Can you predict the number of zeros in the product before doing the calculation?

What is the pattern? (For every zero in the initial problem numbers there are the same number of zeros in the answer). So back to the original question, what is our estimation of the number of flowers on one shelf in the Bark and Garden center? (about 1,500)

What are some other ways to estimate this answer? (skip counting by 50’s = 50, 100, 150, … 30 times)

(Crystallization)

I want you to pair up now and solve these problems in your head, then check with your partner. Handout One

I want to emphasis, that there are different methods to arrive at the solution. Be prepared to explain how you solved the problem. (Give students time to solve the problems. Have 3 or 4 pairs walk up front and put their answers on the board.)

* Pairs are a strategy to aid different ability. In addition, it can aid Second language learners as their partner can possible explain.

Do ya’ll see a pattern in these problems? (if you take the single digits and multiply them, then add the zeros, you have the answer ie: 80 x 2 : 8 x 2 = 16  then add one 0 = 160) What is the pattern with the zeros? (there are as many zeros in the answer as in the problem)

(Disequilibration)

However, there are cases where the pattern that you are seeing is not true. What happens when we multiply 40 x 500? We get 20,000. Now there is one more zero in the answer then in the problem. So our rule does not hold any more. Why is this tricky?  (The extra zero comes from the 4 x 5 = 20. Therefore, it is 4 x 5 = 20 and then add 3 zeros. This makes 4 zeros)

(Elaboration)

Class, I want look at a problem that will require combining these two strategies. Look at the problem 32 x 6.  What are some ways to solve it? (30 x 6 = and 2 x 6 = and then add them up 180 + 12 = 192) What about 4 x 67? (4 x 60 = 240 and 4 x 7 = 28 so 240 + 28 = 268). Notice that we are splitting up the numbers into more manageable equations.

You can write it like this:

(Crystallization)

Now try these with your partner: Be prepared to show your work and explain your answer. (write these on the board or overhead)

Class. Please get out a clean piece of paper. In your pairs, you are going to create 5 problems for another group to solve. What do you think are some guidelines for writing these problems for your classmates? (as students answer write them on the board. You as the teacher need to make sure that the following also are included)

1. Your names have to be on top so we can ask you questions if we do not understand.
2. We have to be able to read your writing. 
3. The problems have to be related to what we learned today. No trick questions.
4. You have to know the answer.

What are the concepts that you learned today? 

1. Estimation
2. Multiplying with zeros

When you are finished with your set of 5 problems and your answer key, place them up here on the desk until everyone is finished. When everyone is finished, we will pass them out and try them.

You are expected to work with your partner to solve the problems and when you are sure that you have the answers, go to the group that wrote them and have it checked.

Closure:

Before we wrap it all up let us review what we learned today. Why do we estimate? (reasonable guess at a possible answer/ use estimation as a reference point for comparison to the actual calculation) Can you predict the number of zeros in the product before doing the calculation? (yes) Is it always true? (no) What is an example of this? (40 x 500 The extra zero comes from the 4 x 5 = 20. Therefore, it is 4 x 5 = 20 and then add 3 zeros. This makes 4 zeros).

POST-ASSESSMENT   
    

Student groups will create a set of 5 problems, with an answer key, to be given to another student group to solve.

Handout One

Find the following answer in your head. You can check with your neighbor.

A. 80

 B.   80

  C.   20

    D.  20

  E.  50

 F.     50

   x 2

    x 20

       x 4

      x 40

      x 7

       x 70

 G.   90

H.   90

I.    70

J.    70

 K.   30

L.    30

      x 7

   x 70

      x 1

   x 10

      x 6

     x 60

M.  90 x 20 =

N.  30 x 50 =

O.  60 x 40 =