Real Number Vault

D: \(2.75\)

E: \(\sqrt{3} \approx 1.73\)

A (-3.5)

C (-0.5)

E (√3≈1.7)

D (2.75)

B (4)

-4

-3

-2

-1

0

1

2

3

4

Real Number Explorer • Answer Key Page 1 of 2

P2

Mathematical Reasoning & Applications Key

Section Score: 14 Points

4 Part 4: True or False & Justify Solutions

4 Points (1 pt each)

1. All whole numbers are integers. TRUE

Key Explanation: The set of integers includes all whole numbers \(\{0, 1, 2, ...\}\) plus their negative opposites \(\{-1, -2, ...\}\).

2. Every rational number is an integer. FALSE

Counterexample: \(\frac{1}{2}\) or \(0.75\) are rational numbers, but neither is an integer because they represent non-whole parts.

3. The number \(0\) is a rational number. TRUE

Key Explanation: \(0\) can be expressed as a fraction of two integers with a non-zero denominator: \(\frac{0}{1}\) or \(\frac{0}{5}\).

4. Square roots are always irrational. FALSE

Counterexample: The square root of any perfect square is rational: \(\sqrt{25} = 5\) or \(\sqrt{0.36} = 0.6\).

5 Part 5: Detective Error Analysis Solutions

4 Points (2 pts per prompt)

1. The Student's Misconception:

The student judges numbers strictly by their outward notation format rather than their evaluated quantitative value. They assume any fraction can't be an integer and any radical symbol must be irrational.

2. Fully Corrected Proof:

• \(-\frac{30}{6} = -5\), which is both an Integer and a Rational number.
• \(\sqrt{64} = 8\), which is a Natural, Whole, Integer, and Rational number (not irrational!).

6 Part 6: Real-World Context Solutions

3 Points (1 pt each)

Winter Chill

Value: \(-8\) or \(-8^\circ\)F

Most Specific: Integer (\(\mathbb{Z}\))

Below zero indicates negative whole units.

Baking Slices

Value: \(2\frac{3}{4}\) or \(2.75\) or \(\frac{11}{4}\)

Most Specific: Rational (\(\mathbb{Q}\))

Contains a non-integer fractional part.

Class Roll Call

Value: \(28\)

Most Specific: Natural (\(\mathbb{N}\))

A positive counting number of people.

Vault Master Challenge Solution (3 Points):

The unique number is 0 (Zero). Zero is in Whole (\(\mathbb{W}\)), Integer (\(\mathbb{Z}\)), and Rational (\(\mathbb{Q}\)), but was excluded when natural numbers were defined as counting objects starting at 1.

0

Real Number Explorer • Answer Key Page 2 of 2

Distribute the Real Number Explorer Worksheet. Model Page 1 Part 1 using think-aloud strategies. Contrast rational numbers (\(\frac{a}{b}\) format, terminating or repeating) with irrational numbers (\(\pi\), \(\sqrt{2}\) that never settle into a repeating pattern).

20 MIN

Collaborative Investigation & Check Table:

Students work in pairs on Part 2 (Checklist) and Part 3 (Number Line). Circulate to check for the trap in \(-\frac{24}{6}\). Transition to Page 2 for Part 4 (True/False justifications) and Part 5 (Error Analysis).

15 MIN

Synthesis, Vault Riddle & Wrap-Up:

Review the Error Analysis case study as a whole group. Have students attempt the Vault Master Challenge riddle independently before collecting papers as a formative check.

Real Number Vault • Instructional Guide Page 1 of 2

P2

Pedagogical Frameworks & Differentiation

Real Number System • Grades 6–7

Diagnostic Misconceptions & Teacher Countermeasures

Purposeful Questioning Stems

To Prompt Simplification:

• "Is that number wearing a disguise? How can we evaluate it first?"
• "What integer or whole number does that expression simplify to?"

To Check Nested Logic:

• "If a number is in the Integer box, what other outer boxes MUST hold it?"
• "Can you provide a counterexample to prove that statement is false?"

Tiered Differentiation Strategies

Support & ELL

• Provide a colored physical set of nesting cups labeled with \(\mathbb{N}, \mathbb{W}, \mathbb{Z}, \mathbb{Q}\).
• Use the mnemonic: Never Walk Into Quicksand.
• Give students a pre-printed simplified scratch column beside each problem.

On-Grade Level

• Emphasize complete written justifications for the True/False items on Page 2.
• Require plotting decimal and fraction equivalents on the number line simultaneously.
• Encourage peer partner dialogue before finalizing the Error Analysis defense.

Extension & Challenge

• Challenge: Convert repeating decimals algebraically to fractions (e.g. \(x = 0.\overline{7} \rightarrow 10x - x = 7 \rightarrow x = \frac{7}{9}\)).
• Investigate why \(\sqrt{2}\) is irrational and explore decimal approximations.
• Have students create their own two-truth-and-a-lie number classification riddles.

Recommended 3-Minute Exit Ticket Prompt:

"Classify \(-\frac{18}{3}\) and \(\sqrt{20}\). Explain why \(-\frac{18}{3}\) belongs in the integer subset even though it is written as a fraction."

Check For: Simplification to -6

Real Number Vault • Instructional Guide Page 2 of 2