options: Operations & Algebra: students learn to solve problems using algebraic thinking and operations such as addition, subtraction, multiplication, and division. They may learn to identify and explain arithmetic patterns, evaluate or manipulate numerical expressions, and reason with equations and inequalities.

Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.

Modeling: students learn to choose and use appropriate mathematics and statistics to analyze real-world empirical situations, improve decisions, and report on their conclusions and the reasoning behind them.

Functions: students learn to define, use, and evaluate functions to model relationships between quantities.

Geometry: students learn to classify geometric figures by their properties, and understand and compare the relationships between them. Students develop and use formulas to compute lengths, areas, and volumes, and they use transformations to generate new shapes and compare existing ones. Geometry can be studied with and without coordinates.

Measurement & Data: students learn to work with data and measure attributes such as time, money, length, area, and volume. They may learn to compare measurements to operations and convert between different units of measure.

Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.

Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.

Statistics & Probability: students learn to analyze and produce data distributions, and build understanding of univariate and bivariate data. They learn to interpret data, make statistical inferences, justify conclusions, understand rules of probability, and use probability to make decisions.

Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.

Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.
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.
$$4,309\div7$$


correct_option_index: 7