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

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.

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

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

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.

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.

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.

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

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

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

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.
problem_activity: Problem 1
a.   Here are two ways to find the area of a rectangle that is 23 units by 31 units.
[IMG]
What do the area models have in common? How are they alike?
How are they different?
If you were to find the area of a rectangle that is 37 units by 49 units, which of the two ways of decomposing the rectangle would you use? Why?
b.   You might be familiar with different ways to write multiplication calculations. Here are two ways to calculate
$$23\times31$$
.
[IMG]
In Calculation A, how are each of the partial products obtained? For instance, where does the 3 come from?
In Calculation B, how are the 23 and 690 obtained?
Look at the area models in Part (a). Which area model corresponds to Calculation A? Which one corresponds to Calculation B?
How are the partial products in Calculation A and the 23 and 690 in Calculation B related to the numbers in the area models?
Problem 2
a.   Label the area model to represent
$$37\times49$$
and to find that product.
[IMG]
Decompose each number into its base-ten units (tens, ones, etc.) and write them in the boxes on each side of the rectangle.
Label regions A, B, C, and D with their areas. Show your reasoning.
Find the product that the area model represents. Show your reasoning.
b.   Here are two ways to calculate 37 × 49. Each number with a box gives the area of one or more regions in the area model.
[IMG]
In the boxes next to each number, write the letter(s) of the corresponding region(s).
In Calculation B, which two numbers are being multiplied to obtain 333? Which two are being multiplied to obtain 1,480?
Problem 3
Estimate the following products. Then solve using the standard algorithm. If you get stuck, use an area model and/or the partial products algorithm to help.
a.
$$73\times84$$
b.
$$48\times67$$


correct_option_index: 8