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.

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.

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

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.

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

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

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.

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.

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.
problem_activity: Activity
In general,
\(P\%\)
of something is
\(\frac{P}{100}\)
times that thing. The purpose of this activity is to make this explicit. In this activity, students are asked to find 80% of several different values, generalize their process, and express their generalization in different ways. As they make repeated calculations, students look for and express regularity in their work (MP8) and see more explicitly that
\(P\%\)
of a number is
\(\frac{P}{100}\)
times that number. Once they arrive at one or more generalizations, students practice articulating why each generalization always works (MP3).
Launch
Give students quiet think time to complete the activity and then time to share their explanation with a partner.
Representation: Internalize Comprehension.
Activate or supply background knowledge about equivalent fractions. Incorporate explicit opportunities for review and practice if necessary. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing
Writing, Speaking, Listening: MLR1 Stronger and Clearer Each Time.
Use this with successive pair shares to give students a structured opportunity to revise and refine their own strategies to determine 80% of any value to find a sale price. Make sure students include a table as well as an expression in their explanations. Ask each student to meet with 2–3 other partners in a row for feedback. Provide students with prompts for feedback that will help them strengthen their ideas and clarify their language (e.g., “Can you explain how…”, “Will this expression always work? How do you know?”, etc.). Students can borrow ideas and language from each partner to refine and clarify their original explanation. This will help students refine their own strategies and learn about other ways to find the sale price of an item.
Design Principles(s): Optimize output (for explanation); Maximize meta-awareness
Student Facing
During a sale, every item in a store is 80% of its regular price.
If the regular price of a T-shirt is
$
10, what is its sale price?
The regular prices of five items are shown here. Find the sale price of each item.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}}, {"_attributes": {"scope": "col"}, "_value": "item 1"}, {"_attributes": {"scope": "col"}, "_value": "item 2"}, {"_attributes": {"scope": "col"}, "_value": "item 3"}, {"_attributes": {"scope": "col"}, "_value": "item 4"}, {"_attributes": {"scope": "col"}, "_value": "item 5"}]}]}], "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "regular price"}], "td": [{"span": [{"_value": "$"}], "_value": "1"}, {"span": [{"_value": "$"}], "_value": "4"}, {"span": [{"_value": "$"}], "_value": "10"}, {"span": [{"_value": "$"}], "_value": "55"}, {"span": [{"_value": "$"}], "_value": "120"}]}, {"th": [{"_attributes": {"scope": "col"}, "_value": "sale price"}], "td": [{}, {}, {}, {}, {}]}]}]}]}
You found 80% of many values. Was there a process you repeated over and over to find the sale prices? If so, describe it.
[IMG]
Select
all
of the expressions that could be used to find 80% of
\(x\)
. Be prepared to explain your reasoning.
\(\frac{8}{100} \boldcdot x\)
\(\frac{80}{100} \boldcdot x\)
\(\frac{8}{10} \boldcdot x\)
\(\frac{4}{10} \boldcdot x\)
\(\frac85 \boldcdot x\)
\(\frac45 \boldcdot x\)
\(80 \boldcdot x\)
\(8 \boldcdot x\)
\((0.8) \boldcdot x\)
\((0.08) \boldcdot x\)
Student Response
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Activity Synthesis
Make sure students have the correct values in the table for the second question. Then, discuss the expressions, starting with
\(\displaystyle \frac{80}{100} \boldcdot x.\)
Guide students to see that if we know the value of 100%, dividing that value by 100 (or equivalently, multiplying it by
\(\frac{1}{100}\)
) tells us the corresponding value for 1%. We can then multiply that value by the desired percentage. Consider using a table to organize this argument, as shown below:
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "percentage"}, {"_attributes": {"scope": "col"}, "_value": "regular price"}]}]}], "tbody": [{"tr": [{"td": [{"_value": "100"}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "50"}, "_value": "\\(x\\)"}]}]}, {"td": [{"_value": "1"}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "3264"}, "_value": "\\(\\frac{1}{100} \\boldcdot x\\)"}]}]}, {"td": [{"_value": "80"}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "3266"}, "_value": "\\(\\frac{80}{100} \\boldcdot x\\)"}]}]}]}]}]}
After everyone understands where this expression comes from, ask students to discuss the remaining expressions with a partner. This is a good opportunity for just-in-time review on equivalent fractions, if needed.


correct_option_index: 7