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

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

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.

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

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.

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.

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.

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.

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.
problem_activity: Activity
The purpose of this activity is to understand that the division facts for rational numbers are simply a consequence of the multiplication done previously. Students work several numerical examples relating multiplication to division, and then articulate a rule for the sign of a quotient based on the signs of the dividend and divisor (MP8).
Monitor for students who identify and describe the rule clearly.
Launch
Remind students that we can rearrange division equations to be multiplication equations, and vice versa. It may be useful to demonstrate with positive numbers if students struggle to recall this. For example, ask how we could rewrite
\(10 \div 2 = 5\)
as a multiplication equation. Students can also reference the multiplication table from a previous lesson, if needed.
Arrange students in groups of 2. Give students 4 minute of quiet work time followed by 2 minutes of partner discussion, then follow with whole-class discussion.
Engagement: Develop Effort and Persistence.
Connect a new concept to one with which students have experienced success. For example, reference examples from the previous lesson on signed multiplication to provide an entry point into this activity.
Supports accessibility for: Social-emotional skills; Conceptual processing
Reading: MLR6 Three Reads.
Use this routine to support students’ reading comprehension of the final question. In the first read, students read the text with the goal of understanding what the situation is about (e.g., Han and Clare are walking toward each other). In the second read, ask students to name the important quantities (e.g., Han’s velocity, Clare’s velocity, each person's distance from 0 in feet, elapsed time in seconds). After the third read, ask students to brainstorm possible strategies to determine where each person will be 10 seconds before they meet up, and when each person will be
\(-10\)
feet from the meeting place. This will help students connect the language in the word problem with the reasoning needed to solve the problem.
Design Principle(s): Support sense-making
Student Facing
Find the missing values in the equations
\(\text-3 \boldcdot 4 = \text{?}\)
\(\text-3 \boldcdot \text{?} = 12\)
\(3 \boldcdot \text{?} = 12\)
\(\text{?} \boldcdot \text-4 = 12\)
\(\text{?} \boldcdot 4 = \text-12\)
Rewrite the unknown factor problems as division problems.
Complete the sentences. Be prepared to explain your reasoning.
The sign of a positive number divided by a positive number is always:
The sign of a positive number divided by a negative number is always:
The sign of a negative number divided by a positive number is always:
The sign of a negative number divided by a negative number is always:
Han and Clare walk towards each other at a constant rate, meet up, and then continue past each other in opposite directions. We will call the position where they meet up 0 feet and the time when they meet up 0 seconds.
Han's velocity is 4 feet per second.
Clare's velocity is -5 feet per second.
Where is each person 10 seconds before they meet up?
When is each person at the position -10 feet from the meeting place?
Student Response
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Student Facing
Are you ready for more?
It is possible to make a new number system using
only
the numbers 0, 1, 2, and 3. We will write the symbols for multiplying in this system like this:
\(1 \otimes 2 = 2\)
. The table shows some of the products.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "row"}, "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "43618"}, "_value": "\\(\\otimes\\)"}]}, {"_attributes": {"scope": "col"}, "_value": "0"}, {"_attributes": {"scope": "col"}, "_value": "1"}, {"_attributes": {"scope": "col"}, "_value": "2"}, {"_attributes": {"scope": "col"}, "_value": "3"}]}]}], "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "row"}, "_value": "0"}], "td": [{"_value": "0"}, {"_value": "0"}, {"_value": "0"}, {"_value": "0"}]}, {"th": [{"_attributes": {"scope": "row"}, "_value": "1"}], "td": [{}, {"_value": "1"}, {"_value": "2"}, {"_value": "3"}]}, {"th": [{"_attributes": {"scope": "row"}, "_value": "2"}], "td": [{}, {}, {"_value": "0"}, {"_value": "2"}]}, {"th": [{"_attributes": {"scope": "row"}, "_value": "3"}], "td": [{}, {}, {}, {}]}]}]}]}
In this system,
\(1 \otimes 3 = 3\)
and
\(2 \otimes 3 = 2\)
. How can you see that in the table?
What do you think
\(2 \otimes 1\)
is?
What about
\(3\otimes 3\)
?
What do you think the solution to
\(3\otimes n = 2\)
is?
What about
\(2\otimes n = 3\)
?
Student Response
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for free access to Extension Student Response.
Anticipated Misconceptions
Have students refer to their previous work with signed multiplication to help them with the division statements.
Activity Synthesis
Select students who can articulate and explain what they have learned about the sign of a quotient based on the signs of the dividend and divisor (note that students do not need to use these words, they can just give examples). Highlight that the division facts are simply a consequence of the multiplication done previously. Moving forward, we are going to take these facts to be true for all rational numbers.


correct_option_index: 0