Exercises: Apply properties of operations to add, subtract, factor, and expand linear expressions with rational coefficients - Common Core Math - Grade 7

A

Recall / Warm-Up

1.

What is $\frac{3}{4} \times 8$ ?

A.

$6$

B.

$\frac{3}{32}$

C.

$\frac{11}{4}$

D.

$\frac{24}{4}$

2.

Which two terms are like terms?

A.

$5x$ and $5y$

B.

$3x$ and $7x$

C.

$4x$ and $4$

D.

$x^2$ and $x$

3.

Which expression is equivalent to $3(2x + 5)$ ?

A.

$6x + 5$

B.

$6x + 15$

C.

$5x + 8$

D.

$6x + 8$

B

Fluency Practice

C

Varied Practice

D

Word Problems

1.

A rectangular garden has a length of $(\frac{3}{4}x + 2)$ meters and a width of $(\frac{1}{4}x + 1)$ meters.

Write a simplified expression for the perimeter of the garden.

2.

A store is offering a 20% discount on all items. The original price of a jacket is $p$ dollars.

Write a simplified expression for the sale price of the jacket.

3.

Maya is planning a school event. She orders $n$ boxes of supplies. Each box costs $\frac{3}{2}$ dollars for materials and $\frac{1}{4}$ dollar for shipping. She also has a fixed setup fee of 5 dollars.

1.

Write an expression for the total cost for all $n$ boxes (materials plus shipping).

2.

Write a simplified expression for the total cost (boxes plus setup fee).

4.

A farmer notices that the total length of fencing needed for a rectangular pen can be written as $6x + 4$ .

Factor the expression $6x + 4$ . Then explain what your factored form tells you about the pen's dimensions.

E

Error Analysis

$Side-by-side comparison: Student A incorrectly writes (3/4)x + 2 = (11/4)x; Student B incorrectly writes (3/4)x + 2 = (3/8)x. Both marked with a red X.$

1.

Two students simplified the expression $\frac{3}{4}x + 2$ .

Student A wrote: $\frac{3}{4}x + 2 = \frac{11}{4}x$

Student B wrote: $\frac{3}{4}x + 2 = \frac{3}{8}x$

Which student made an error, and what mistake did they make?

A.

Student A only — combined unlike terms (added the constant 2 to the $x$ -coefficient)

B.

Student B only — multiplied the coefficients instead of adding

C.

Both students — Student A combined unlike terms; Student B multiplied instead of added

D.

Neither — both answers are correct equivalent forms

$Two-column comparison: Student Work shows incorrect sign (−4) in step 2, highlighted with red X; Correct Work shows the right sign (+4) circled in teal, leading to the correct answer 3x + 7.$

2.

A student simplified $(5x + 3) - (2x - 4)$ :

$= 5x + 3 - 2x - 4$
$= 3x - 1$

The student made an error in Step 1. Which of the following correctly identifies the mistake and gives the right answer?

A.

The student forgot to combine the $x$ -terms; the answer is $3x + 3$ .

B.

The student only distributed the minus sign to $2x$ but not to $-4$ . The correct simplification is $3x + 7$ .

C.

The student should have added the expressions, not subtracted. The answer is $7x - 1$ .

D.

The student subtracted $3$ from $4$ incorrectly; the correct answer is $3x + 1$ .

F

Challenge / Extension

1.

Factor the expression $\frac{2}{3}x - \frac{4}{9}$ completely by finding the GCF of the coefficients. Express the factored form as $\frac{a}{b}(cx + d)$ where $c$ and $d$ are integers.

2.

A school store has a profit formula written in two equivalent ways:

Form 1: $P = 1.05n - 0.6$

Form 2: $P = \frac{3}{20}(7n - 4)$