Unit 1: Numbers, Variables, and Equations – Key Concepts

Question-and-answer flashcards covering digital roots, exponents, powers of 10, scientific notation, prime factorization, common factors/multiples, order of operations, solving equations, two-pan balances, and algebra tiles.

What is the digital root of a number?

The single-digit value obtained by repeatedly adding a number’s digits until only one digit remains.

What is the digital root of 168?

6 (1 + 6 + 8 = 15; 1 + 5 = 6).

What is the digital root of the product 8 × 4?

5 (8 × 4 = 32; 3 + 2 = 5).

In exponent notation, what does the exponent indicate?

How many times the base is multiplied by itself.

Write 5³ in expanded form.

5 × 5 × 5.

Write 81 as a power in two different ways.

9² and 3⁴.

Express one hundred thousand as a power of 10.

10⁵.

What is the standard form of 5 × 10³?

5 000.

State the standard order of operations.

Brackets first, then exponents, then multiplication and division left-to-right, then addition and subtraction left-to-right.

Name two strategies for solving simple equations without tiles.

Systematic trial and inspection.

Solve: 3x + 4 = 28.

x = 8.

Solve: 100 − 7x = 58.

x = 6.

Define a prime number.

A number with exactly two factors: 1 and itself.

Define a composite number.

A number with more than two factors.

Give the prime factorization of 36.

2² × 3².

Give the prime factorization of 144.

2⁴ × 3².

What are the common prime factors of 18 and 24?

2 and 3.

What is the greatest common factor (GCF) of 18 and 24?

6.

What is the lowest common multiple (LCM) of 18 and 24?

72.

Write 140 000 000 in scientific notation.

1.4 × 10⁸.

How is the exponent in scientific notation related to the number of digits in the standard form of a large number?

The exponent is one less than the total number of digits.

When is mental math an appropriate strategy?

When numbers are easy to handle and only a few simple or estimated steps are needed.

What is an inverse operation?

An operation that undoes another (e.g., subtraction undoes addition, division undoes multiplication).

In a two-pan balance model, what must you do to keep the scales balanced when you add or remove masses?

Perform the same operation on both pans (both sides of the equation).

Using inverse operations, solve x + 7 = 25.

x = 18.

Which algebra tile represents the value +1?

A small yellow unit tile.

What is a zero pair in algebra tiles?

A yellow +1 tile paired with a red −1 tile, representing 0.

Using algebra tiles, what is the solution to x − 3 = 10?

x = 13.

Using algebra tiles, solve 2x + 1 = 9.

x = 4.

Using algebra tiles, solve 2 = 3x − 4.

x = 2.

In the equation 2x + 1 = 9, which inverse operations isolate x algebraically?

Subtract 1 from both sides, then divide both sides by 2.

Why do we write very large numbers in scientific notation?

It shortens them and makes them easier to read, compare, and enter into calculators.