Transcript Objective: Connect area models and the distributive property to

Objective: Connect area models and the
distributive property to partial products of
the standard algorithm with renaming.
Fluency
Draw an area model on your personal white board to solve
824 × 15
Answer
Model it for students on the whiteboard. 12,360
Application Problem
The length of a school bus is 12.6 meters. If 9 school buses
park end-to-end with 2 meters between each one, what’s
the total length from the front of the first bus to the end of
the last bus?
Note: This problem is designed to bridge to the current lesson with multi-digit multiplication while also
reaching back to decimal multiplication work from Module 1. Students should be encouraged to
estimate for a reasonable product prior to multiplying. Encourage students to use the most efficient
method to solve this problem.
Application Problem Solved
Concept Development Multiplying 2 digit by
2 digit
64 × 73
Let’s solve using an area model on our whiteboards with our math partners.
But first let’s represent units of 73. How many units of 73 do we have (or how
many 73s are we counting)?
64
Answer
Concept Development Multiplying 2 digit by
2 digit.
How can we decompose 64 to make our multiplication easier?
We can split it into 4 and 60.
Show this on your model.
Concept Development Multiplying 2 digit by
2 digit.
Now how could we decompose 73 to make finding these partial products easier
to solve?
Split it into 70 and 3.
4,672
Now add them up and what is the
answer?
Concept Development Multiplying 2 digit by
2 digit.
Now do the standard algorithm with your partner and we
will talk about it after.
Concept Development Multiplying 3 digits by
3 digits
524 × 136
How is this problem different than the last problem 64 x 73?
How will the area model for this problem be different than
previous models?
What would an area model look like if we multiplied 3 digits by 2
digits?
Concept Development Multiplying 3 digits by
3 digits
Remember we put our # of units on the side of the area model, and we put our
total units on top.
Let’s designate 524 as our unit and 136 as our # of units.
Partner A, draw an area model to find the product. Partner B, solve using the
standard algorithm.
Compare your solutions by matching your partial products and final product.
Then we will discuss together.
Concept Development Multiplying 3 digits by
3 digits
Concept Development Multiplying 4 digits by
3 digits
4,519 × 326
What is different about this problem than the last one 524 × 136?
Which factor will go on top of our area model? Is one more efficient to put on top
than the other? Turn and talk to your partner and or table.
Does the presence of the fourth digit change anything about how we multiply? Why
or why not?
We will have an extra column in the area model, but we just multiply the same way .
Concept Development Multiplying 4 digits by
3 digits
Before we solve this problem, let’s estimate our product. Round the factors and
make an estimate.
When we round the factors, what numbers do they round to?
5,000 × 300 = 1,500,000 So now we know what the actual answer should be
close to.
Concept Development Multiplying 4 digits by
3 digits
Now, solve this problem with your partner.
Partner B should do the area model this time, and Partner A should use the
algorithm.
As you work, explain to your partner how you organized your thoughts to make
this problem easier. (How did you decompose your factors?)
Concept Development Multiplying 4 digits by
3 digits
Pick a partnership to share 4,519 × 326 in an
area model and with the standard algorithm.
Concept Development Multiplying 4 digits by
3 digits
4,509 × 326.
First let’s quickly estimate the product.
What do we get when we estimate?
Concept Development Multiplying 4 digits by
3 digits
4,509 × 326.
5,000 x 300 = 15,000,000 This is the same
estimate as we got for our previous problem of
4,519 x 326
Concept Development Multiplying 4 digits by
3 digits
Compare 4,519 and 4,509. How are they
different?
In the second one, there’s a zero in the tens
place in 4,509.
Concept Development Multiplying 4 digits by
3 digits
What does 4,509 look like in expanded form?
4,000 + 500 + 9.
Remember expanded form is decomposing the number.
Concept Development Multiplying 4 digits by
3 digits
Can you imagine what the length of our rectangle will look like? How many
columns will we need to represent the total length?
We will need only three columns.
This is a four-digit number. Why only three columns?
Concept Development Multiplying 4 digits by
3 digits
The rectangle shows area. That’s why it is called an area model.
So, if we put a column in for the tens place, we would be drawing the rectangle
bigger than it really is.
We are chopping the length of the rectangle into three parts—4,000, 500, and
9. That is the total length already.
The width of the tens column would be zero, so it has no area.
Concept Development Multiplying 4 digits by
3 digits
Now with a partner, draw the area model out for 4,509 ×
326.
What does it look like? Show me on your whiteboards.
Now one partner erase the area model and both of you do
the standard algorithm. Then compare your area model on
the other whiteboard with the standard algorithm.
Concept Development Multiplying 4 digits by
3 digits
Does your example look like these?
Concept Development Multiplying 4 digits by
3 digits
Problem 4
4,509 × 306
If we estimate the product of this problem we will get 15,000,000 just like in the
last 2 problems.
But how is 4,509 x 306 different than 4,509 x 326?
306 is 20 units less of 4,506 than 326. And there is a zero in the 10s place in
both factors.
Concept Development Multiplying 4 digits by
3 digits
Thinking about the expanded forms of the factors, imagine the area
model for 5,409 x 306
How will the length and width be decomposed? How will it compare to the
last problem 4,509 × 326?
How many columns will we have in the area model for 5,404 x 306? How
many rows will we have?
Concept Development Multiplying 4 digits by
3 digits
The model doesn’t need three rows because there’s nothing in the tens place.
We only need to show rows for hundreds and ones.
Compare
the
difference
Concept Development Multiplying 4 digits by
3 digits
Teacher model it on the board to save time.
What two partial products do these two rows represent?
6 × 4,509 and 300 × 4,509
Concept Development Multiplying 4 digits by
3 digits
Let’s record what we just drew with the algorithm. We’ll begin with the first partial product 6 × 4,509.
Find that partial product. (circle it in the problem)
Now, let’s record 300 x 4,509. When we multiply a number by 100, what happens to the value and
position of each digit?
Each becomes 100 times as large and shifts two places to the left.
In the case of 4,509, when we multiply it by 300, what would need to be
recorded in the ones and tens place after the digits shift?
Zeros would go in those places.
Homework Problems
Homework
Do problems 1 (a, b, c), 2 (a, b), 3 (a, b)
(That is all of it because the teacher didn’t
print the last page of homework. So feel
lucky :) ).
Exit ticket
Do Exit ticket!
Debrief
Explain why a multiplication problem with a
three-digit multiplier will not always have three
partial products. Use Problems 1(a) and (b) as
examples.
How are the area models for Problems 2(a) and (b) alike, and how are they
different?
What pattern did you notice in Problem 3?