options: 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.

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

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

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.

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.

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

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.

Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.
problem_activity: Optional activity
In this activity, students explore percentages that describe parts of a whole. They find both
\(B\)
and
\(C\)
, where
\(A\%\)
of
\(B\)
is
\(C\)
, in the context of available and consumed water on a hike.
Students who use a double number line may notice that the value of
\(B\)
is the same in both questions, so the same double number line can be used to solve both parts of the problem. To solve the second question, however, the diagram needs to be partitioned with more tick marks.
As in the previous task, students may solve using other strategies, including by simply multiplying or dividing, i.e.,
\((1.5) \boldcdot 2 = 3.0\)
and
\(\frac{80}{100} \boldcdot (3.0) = 2.4\)
. Encourage them to also explain their reasoning with a double number line, table, or tape diagram. Monitor for at least one student using each of these representations.
Launch
Give students quiet think time to complete the activity and then time to share their explanation with a partner. Follow with a whole-class discussion.
Student Facing
During the first part of a hike, Andre drank 1.5 liters of the water he brought.
If this is 50% of the water he brought, how much water did he bring?
If he drank 80% of his water on his entire hike, how much did he drink?
Student Response
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Student Facing
Are you ready for more?
Decide if each scenario is possible.
Andre plans to bring his dog on his next hike, along with 150% as much water as he brought on this hike.
Andre plans to drink 150% of the water he brought on his hike.
Student Response
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for free access to Extension Student Response.
Anticipated Misconceptions
After students create a double number line diagram with tick marks at 50% and 100%, some may struggle to know how to fit 80% in between. Encourage them to draw and label tick marks at 10% increments or work with a table instead. Some students may think that the second question is asking for the amount of water Andre drank on the second part of the hike. Clarify that it is asking for his total water consumption on the entire hike.
Activity Synthesis
Select 1–2 students who used a double number line, a table of equivalent ratios, and a tape diagram to share their strategies. As students explain, illustrate and display those representations for all to see.
Ask students how they knew what 100% means in the context. At this point it is not necessary for students to formally conceptualize the two ways percentages are used (to describe parts of whole, and to describe comparative relationships). Drawing their attention to concrete and contextualized examples of both, however, serves to build this understanding intuitively.
Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct.
Before students share their answers for the second question, present an incorrect answer and explanation. For example, “If Andre drank 80% of his water on his entire hike, then he drank 1.2 liters of water because 0.8 times 1.5 liters is 1.2 liters.” Ask students to identify the error, critique the reasoning, and write a correct explanation. As students discuss in partners, listen for students who identify and clarify the assumption in the statement. For example, the author assumed that Andre brought 1.5 liters for his entire hike; however, the problem states that Andre drank 1.5 liters of the water he brought. This will remind students to carefully identify the quantity they are finding a percentage of.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness


correct_option_index: 3