Describe the relationship between input and output for each flow diagram and calculate the missing values.

Here are the solutions for Activity 1 and Activity 2.

Activity 1:

a) The relationship in words is: The output value is obtained by multiplying the input value by two, and then subtracting one.

b) To complete the flow diagram, we apply the rule $2x + (-1)$ to each input value:

• For input 10: $2(10) + (-1) = 20 - 1 = 19$
• For input 5: $2(5) + (-1) = 10 - 1 = 9$
• For input $-3$: $2(-3) + (-1) = -6 - 1 = -7$
• For input $-2$: $2(-2) + (-1) = -4 - 1 = -5$
• For input $-1$: $2(-1) + (-1) = -2 - 1 = -3$
• For input 0: $2(0) + (-1) = 0 - 1 = -1$

The completed output values for the flow diagram are:
Input value $\rightarrow$ Output value
$10 \rightarrow \boxed{19}$
$5 \rightarrow \boxed{9}$
$-3 \rightarrow \boxed*{-7*}$
$-2 \rightarrow \boxed*{-5*}$
$-1 \rightarrow \boxed*{-3*}$
$0 \rightarrow \boxed*{-1*}$

Activity 2:

a) Using number sentences to find all the output values:

• For input 3: $3 + (-2) = \boxed{1}$
• For input $-2$: $-2 + (-2) = \boxed*{-4*}$
• For input $-1$: $-1 + (-2) = \boxed*{-3*}$
• For input 0: $0 + (-2) = \boxed*{-2*}$
• For input 1: $1 + (-2) = \boxed*{-1*}$

b) The completed output values for the flow diagram are:
Input (m) $\rightarrow$ Output (n)
$3 \rightarrow \boxed{1}$
$-2 \rightarrow \boxed*{-4*}$
$-1 \rightarrow \boxed*{-3*}$
$0 \rightarrow \boxed*{-2*}$
$1 \rightarrow \boxed*{-1*}$

c) The input is represented by $m$ and the output by $n$.

d) The formula of the relationship between $m$ and $n$ is:

$$n = m + (-2)$$

or

$$\boxed{n = m - 2}$$