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INVESTIGATION
CHANGING THE ORDER OF OPERATIONS
This investigation is about the results of two calculations when you change the order of operations.
Each calculation uses three integers, $x$, $y$, and $z$.
Example
$x = 3$, $y = 5$, $z = 7$.
In this question $x = 2$, $y = 4$, $z = 6$.
Complete calculation 1 and calculation 2.
Calculation 1
$y + z = 4 + 6 = ............$
$ ............ \times x = ............ \times 2 = ............$
Result = ............
Calculation 2
$y \times z = 4 \times 6 = ............$
$ ............ + x = ............ + 2 = ............$
Result = ............
542
570
549
(a) Work out the results of calculation 1 and calculation 2 when $x = 3$, $y = 4$, $z = 9$.
Result of calculation 1: .......................................................
Result of calculation 2: ....................................................... [2]
(b) What do you notice about your results in part (a)?
................................................................................................................. [1]
553
541
519
In this question $x$ and $z$ do not change in each part.
(a) In this part $x = 2$ and $z = 4$.
Complete the table.
| | | | Calculation 1 | | Calculation 2 | |
|-----|--------|--------|--------------|--------------|--------------|--------------|
| $x$ | $y$ | $z$ | $y + z$ | $\times x$ = result | $y \times z$ | $+ x$ = result |
| 2 | 1 | 4 | 5 | | 4 | |
| 2 | 2 | 4 | | 12 | | 10 |
| 2 | 3 | 4 | | | | |
[3]
(b) In this part $x = 3$ and $z = 6$.
$y$ increases by 1 each time.
Continue the table until the two results are the same.
You may not need all the rows.
| | | | Calculation 1 | | Calculation 2 | |
|-----|--------|--------|--------------|--------------|--------------|--------------|
| $x$ | $y$ | $z$ | $y + z$ | $\times x$ = result | $y \times z$ | $+ x$ = result |
| 3 | 1 | 6 | 7 | 21 | 6 | 9 |
| 3 | 2 | 6 | 8 | 24 | 12 | 15 |
| 3 | 3 | 6 | | | | |
| | | | | | | |
| | | | | | | |
[3]
(c) In this part $x = 4$ and $z = 8$.
Find the value of $y$ that makes the results of calculation 1 and calculation 2 the same.
Use this table to help you.
| | | | Calculation 1 | | Calculation 2 | |
|-----|--------|--------|--------------|--------------|--------------|--------------|
| $x$ | $y$ | $z$ | $y + z$ | $\times x$ = result | $y \times z$ | $+ x$ = result |
| | | | | | | |
| | | | | | | |
| | | | | | | |
| | | | | | | |
| | | | | | | |
| | | | | | | |
$y = \text{...............................................................}$ [3]
552
501
522
(a) Complete the table.
Use your answers to Question 3 and any patterns you notice to help you.
[Table_1]
\begin{tabular}{|c|c|c|}
\hline
x & y & z \\
\hline
2 & & 4 \\
3 & & 6 \\
4 & & 8 \\
5 & & \\
6 & & \\
\hline
& 17 & \\
\hline
\end{tabular}
[3]
(b) Find expressions in terms of $x$ for $y$ and for $z$.
$y = \text{..............................................}$
$z = \text{..............................................}$ [2]
533
589
509
In this question $x = 1$ and $y = 1$.
(a) Write down expressions for calculation 1 and for calculation 2 in terms of $z$. Show that these results are the same. [2]
(b) What do your results in part (a) tell you about the value of $z$? [1]
501
502
575
In this question $x$, $y$ and $z$ are consecutive. For example 15, 16 and 17.
(a) Complete the table.
| x | y | z |
|---|---|---|
| 15 | 16 | 17 |
| 12 | ||
| 20 |
(b) Write $y$ and $z$ in terms of $x$.
$$y = ext{..................................................}$$
$$z = ext{..................................................}$$ [1]
(c) Show that the result of calculation 1 is $2x^2 + 3x$. [2]
(d) Find the result of calculation 2 in terms of $x$. Give your answer in its simplest form.
.................................................. [3]
(e) The results of calculation 1 and calculation 2 are the same. Show that $x^2 = x + 2$. [1]
(f) The results of calculation 1 and calculation 2 are the same.
$x$, $y$ and $z$ are consecutive.
$x$, $y$ and $z$ are all between $-5$ and 5.
Find two sets of values for $x$, $y$ and $z$.
.................................................. [4]
500
522
541
