Question

use the explore tool to model the multiplication expression 3(3x + 1).
step 1: drag and drop three (+x)-tiles and one (+1)-tile into the horizontal section at the top. this represents the expression 3x + 1.
step 2: next drag and drop three (+1)-tiles into the vertical section at the left. this represents multiplying the expression 3x + 1 by 3 or 3(3x + 1).
step 3: press the multiply button. the result is shown in the model.
there are 9 (+x)-tiles and square (+1)-tiles.
write the product as an expression.
square x + square
use the explore tool to model the multiplication expression 3(3x − 1).
to do this, build the expressions in the same way as the previous multiplication expression, 3(3x + 1), but use one (−1)-tile in the horizontal section. then press the multiply button. the result is shown in the model.
there are square (+x)-tiles and square (−1)-tiles.

Explanation:

Step1: Analyze the multiplication

We are multiplying 3 by the expression \(3x + 1\). Using the distributive property of multiplication over addition, \(a(b + c)=ab+ac\), here \(a = 3\), \(b=3x\) and \(c = 1\).

Step2: Apply the distributive property

First, multiply 3 with \(3x\): \(3\times3x=9x\). Then multiply 3 with 1: \(3\times1 = 3\).

Step3: Combine the terms

The product is \(9x+3\).

For the second part, when we have \(3(3x - 1)\), using the distributive property \(a(b - c)=ab - ac\), where \(a = 3\), \(b = 3x\) and \(c=1\). Multiply 3 with \(3x\): \(3\times3x = 9x\), and multiply 3 with 1: \(3\times1=3\), so the product is \(9x-3\). The number of \((+x)\)-tiles is 9 and the number of \((- 1)\)-tiles is 3 (for \(3(3x - 1)\)).

Answer:

For \(3(3x + 1)\), the product is \(9x + 3\). For \(3(3x-1)\), there are 9 \((+x)\)-tiles and 3 \((-1)\)-tiles, and the product is \(9x - 3\).