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.

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

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.

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.

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.

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

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.

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

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.
problem_activity: Narrative
This activity develops students’ understanding of the vertical method of recording partial quotients and their ability to use it to perform division. One of the quotients here involves a remainder, prompting students to interpret it. Students are reminded that the term “remainder” is used to describe “leftovers” when dividing.
MLR2 Collect and Display.
Synthesis: Direct attention to words collected and displayed from the previous lessons. Invite students to borrow language from the display as needed, and update it throughout the lesson.
Advances: Conversing, Reading
Launch
Groups of 2
Activity
3 minutes: independent work time on the first 2 problems.
Pause after problem 2 to discuss students’ responses.
Display the different ways that students decompose 389 to divide it by 7.
“Most other calculations we’ve seen so far end with a 0, but this one ends with a 4. What does the 4 tell us?” (We cannot make a group of 7 with 4 leftover. 389 is not a multiple of 7, and there are leftovers.)
“When we divide and end up with leftovers we call them remainders, because they represent what is remaining after we divide into equal groups.”
Display:
\(389 = 7 \times 55 + 4\)
“How does this equation show that
\(389 \div 7\)
has a remainder?” (It shows that 389 is not a multiple of 7. It also shows that 7 and 55 make a factor pair for 385, and 389 is 4 more than that.)
3 minutes: independent work time on the last 2 problems.
As students work on the last two problems monitor for students who:
start with the largest multiple of 3 and 10 within 702 that they can think of to decompose the dividend (690, 600).
use the fewest steps to find the quotient.
Student Facing
Jada used partial quotients to find out how many groups of 7 are in 389.
Analyze Jada’s steps in the algorithm.
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Look at the three numbers above 389. What do they represent?
Look at the three subtractions below 389. What do they represent?
What is another way you can decompose 389 to divide by 7?
Is 389 a multiple of 7? Explain your reasoning.
Use an algorithm that uses partial quotients to find out how many groups of 3 are in 702.
Is 702 a multiple of 3? Explain your reasoning.
Student Response
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Activity Synthesis
Display responses that show variation in how students decomposed 702 in problem 3.
Highlight that there are countless ways of using partial quotients to divide a number, but it may be more efficient to divide larger groups than smaller groups (for example, removing 350 once, as opposed to removing 70 five times. Removing smaller groups is just as valid, however.)
To help students see the relationship between corresponding partial products and partial quotients, consider illustrating them visually. Some examples:
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