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Learning Math Home
Number and Operations Session 4: Solutions
 
Session 4 Part A Part B Part C Homework
 
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A B C 
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Solutions for Session 4, Part B

See solutions for Problems: B1 | B2 | B3 | B4 | B5 | B6


Problem B1

a. 

You would need 12 total pieces -- one flat, five longs, and six units -- arranged like this:

b. 

Notice that to do this we divided the 13 into (10 + 3) and the 12 into (10 + 2) when deciding what types of manipulatives to use. So this represents the model:

13 • 12 = (10 + 3) • (10 + 2) = (10 • 10) + (3 • 10) + (2 • 10) + (3 • 2)

This shows the division into one flat, five longs (split into 3 and 2), and six units (in a 3 • 2 arrangement).

<< back to Problem B1


 

Problem B2

The area model divides 24 into 2 tens and 4 ones, and 13 into 1 ten and 3 ones:

<< back to Problem B2


 

Problem B3

Following the example from Problem B2, we multiply step by step:

Note that the result of Problem B1 is a special case of the above expression, using the number x = 10.

<< back to Problem B3


 

Problem B4

a. 

To find the solution, you need to divide 187 into a number with 11 equal rows, or an 11-by-something rectangle. The solution will then be the quantity in each of the 11 rows, or the other dimension of the rectangle. The width of the rectangle is 17; therefore, 18711 = 17.

b. 

The problem we are solving is x • (10 + 1) = 100 + 80 + 7. Knowing that the result includes a 100, which is equal to 10 • 10, suggests that x is some number larger than 10 and can be written as (10 + y). So we then have the following:

(10 + y) • (10 + 1) = 100 + 80 + 7;
or,
(10 • 10) + (10 • 1) + (y • 10) + (y • 1) = 100 + 80 + 7.

It isn't too hard to justify that y should be 7 (especially with y • 1 being the only product in the above equation that can yield a single-digit number, in this case 7; thus, y • 1 = 7). So x is 17.

<< back to Problem B4


 

Problem B5

To make 182, you'll need one flat, eight longs, and two units. You will also need to break up one of the longs into 10 units. Then, form an area model of a rectangle with 13 rows, and count the number of columns. The result of 18213 is 14, since 14 is the number of columns when 13 equal rows are formed.

<< back to Problem B5


 

Problem B6

Yes, the algorithm matches the model. You first subtracted 10 thirteens and then subtracted 4 more thirteens, for a total of 14 thirteens. Thus, there are 14 thirteens in 182. The length of the rectangle is 14, and the answer to the problem 18213 is 14.

<< back to Problem B6


 

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