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.

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 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.

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

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

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.

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

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: Activity
The purpose of this activity is for students to use rational approximations of irrational numbers and to match irrational numbers to equations they are solutions to. To do this, students sort cards into sets of three consisting of:
A square or cube root value
An equation of the form
\(x^2=p\)
or
\(x^3=p\)
that the value is a solution to
A number line showing the location of the value
Identify groups using clear explanations for how they chose to arrange their sets of 3, particularly when matching the cards with a value plotted on a number line.
Launch
Arrange students in groups of 2–4. Students should not use a calculator for this activity. Tell students that they will be given cards to sort into sets of three. Distribute 27 pre-cut slips from the blackline master to each group. Group work time followed by a whole-class discussion.
Representation: Internalize Comprehension.
Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity by beginning with fewer cards. For example, give students a subset of the cards (using multiples of 3) to start with and introduce the remaining cards once students have completed their initial set of matches.
Supports accessibility for: Conceptual processing; Organization
Student Facing
Your teacher will give your group a set of cards. For each card with a letter and value, find the two other cards that match. One shows the location on a number line where the value exists, and the other shows an equation that the value satisfies. Be prepared to explain your reasoning.
Student Response
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Anticipated Misconceptions
Some students may mix up cube and square roots. Encourage these students to pay close attention to the notation. They may want to fill in the “2” for the square roots similar to how the “3” is used for cube roots.
Activity Synthesis
Select 3–4 previously identified groups to share one of their sets of three cards.
Also discuss:
“What was a match that was hard to make?” (Two of the number lines have values between 4 and 5, and it took some extra reasoning to figure out that the one with the value closer to 4 was
\(\sqrt{18}\)
while the one with the point closer to 5 was
\(\sqrt[3]{100}\)
.
“If another class was going to sort these cards, what is something you would recommend they have or do that you found helpful?” (I would recommend they have a list of perfect squares and perfect cubes to help think about where the different roots are plotted on the number line.)
Speaking, Writing: MLR8 Discussion Supports.
Display sentence frames to support students as they justify their reasoning for each match. For example, “The solution to _________ (
\(x^3 = 25\)
) is _________ (
\(\sqrt[3]{25}\)
) and it is between ______ (2 and 3) on the number line, because . . .” The helps students place extra attention on language used to engage in mathematical communication, without reducing the cognitive demand of the task. It also emphasizes the importance of justification.
Design Principle: Maximize meta-awareness


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