Question

build a model to show (\frac{4}{9} \times 2).

Explanation:

Step1: Understand the fraction multiplication

To model $\frac{4}{9} \times 2$, we can think of it as adding $\frac{4}{9}$ two times or representing 2 groups of $\frac{4}{9}$. First, let's recall that when multiplying a fraction by a whole number, we can use the formula $a\times\frac{b}{c}=\frac{a\times b}{c}$ (where $a = 2$, $b = 4$, $c=9$ in this case). But for the model, we can use a number line or an area model.

Step2: Using the number line model

The number line is from 0 to 2. We know that $\frac{4}{9}\times2=\frac{8}{9}$. Let's divide the interval from 0 to 2 into 9 equal parts? Wait, no. Wait, actually, when we multiply $\frac{4}{9}$ by 2, we can think of the unit as 1, but since we have a multiplier of 2, maybe it's better to consider the length. Alternatively, using an area model: draw a rectangle, divide it into 9 equal parts (since the denominator is 9), shade 4 parts to represent $\frac{4}{9}$, then do this twice (because we are multiplying by 2) and combine the shaded parts.

But let's calculate the value first to check. $\frac{4}{9}\times2=\frac{4\times2}{9}=\frac{8}{9}$. Now, on the number line from 0 to 2, we can mark the position of $\frac{8}{9}$. The number line has length 2, so each "1" unit can be divided into 9 parts. So from 0, moving $\frac{8}{9}$ units (since $\frac{4}{9}\times2=\frac{8}{9}$) will give the result.

Alternatively, using the area model:

• Draw a rectangle, represent 1 whole. Divide it into 9 equal columns (since denominator is 9). Shade 4 columns (to represent $\frac{4}{9}$).
• Now, since we are multiplying by 2, we can draw another identical rectangle, also divided into 9 columns and shade 4 columns.
• Now, combine the two shaded regions. The total number of shaded columns is $4 + 4=8$, and the total number of columns if we consider both rectangles as a single unit? Wait, no. Wait, when multiplying a fraction by a whole number, it's like repeated addition. So $\frac{4}{9}\times2=\frac{4}{9}+\frac{4}{9}=\frac{8}{9}$.

So the model (either number line or area) should show two times the fraction $\frac{4}{9}$, resulting in $\frac{8}{9}$.

Answer:

To model $\frac{4}{9} \times 2$:

• Area Model: Draw two identical rectangles, each divided into 9 equal parts (columns). Shade 4 parts in each rectangle. Combining the shaded parts, we have $4 + 4 = 8$ shaded parts out of 9 (per "whole" rectangle, but since we have 2 groups, the total shaded is $\frac{8}{9}$ of a single whole or 2 groups of $\frac{4}{9}$).
• Number Line Model: Divide the interval from 0 to 2 into 9 equal sub - intervals per unit (so 18 sub - intervals total, but focus on the first unit for the fraction). Mark $\frac{4}{9}$ and then $\frac{4}{9}+\frac{4}{9}=\frac{8}{9}$ on the number line (since $\frac{4}{9}\times2=\frac{8}{9}$).

The value of $\frac{4}{9} \times 2$ is $\boldsymbol{\frac{8}{9}}$.