standard_description: Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule “Add 3” and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.
problem_activity: Problem 1
a.   Shani calculated
$$81\div3$$
using an area model by finding the length of a rectangle whose area is 81 square units and width is 3 units.
[IMG]
The value being divided has 8 tens. The area model has only 6 tens in the leftmost region. Why?
Where did the 21 ones come from in the area model?
b.   Albie calculated
$$81\div3$$
using a method that was different from Shani’s.
[IMG]
Albie subtracted 60 from 81. What does the 60 represent?
Albie wrote 7 above the 20, and then subtracted 21 from 21. How is the 21 related to the 7?
What do the numbers 20 and 7 represent?
c.   Kimiko has a method of calculating quotients that is different from Shani’s method and Albie’s method. Here is how she found the quotient
$$81\div3$$
.
[IMG]
How is Kimiko’s method similar to and different from Shani’s method and Albie’s method?
Kimiko subtracted
$$3\times2$$
, then
$$3\times7$$
. Earlier, Albie subtracted
$$3\times20$$
, then
$$3\times7$$
. Why did they have the same quotient?
Why do you think Kimiko wrote 1 next to the remainder of 2 rather than adding 1 and 2 to get 3?
Problem 2
Estimate each quotient, then solve using any method. Then assess the reasonableness of your answer.
a.
$$97 \div2$$
b.
$$1,216\div4$$
c.
$$4,309\div7$$


aligns: no