Tables, coordinates and straight-line patterns

A table row records input x = 2 and output y = 7. What does this row tell you?

Use the same rule for every input

A table of values records inputs and the outputs produced by a rule

The input variable is often called x and the output y. The output depends on the input through the stated rule

For each row, replace x with that row's input and calculate y

Do not use the previous output as the next input unless the question specifically defines that process

Build a table by separate substitutions

Use the rule

Calculate the missing output for each listed input

Substitute every required input independently into the same rule

For \(x=0\)

Keep this input paired with its own output

For \(x=1\)

Keep this input paired with its own output

For \(x=2\)

Keep this input paired with its own output

For \(x=3\)

Keep this input paired with its own output

Include negative and zero inputs

Preserve the rule's operation order

For a rule such as y = 2x + 1, multiply the input by two before adding one

Adding one to the input before multiplying would use a different rule

A negative output is possible even when the input is nonnegative, if the rule subtracts a larger fixed amount

Multiply each input before adding

Use the rule

Calculate the missing output for each listed input

Substitute every required input independently into the same rule

For \(x=0\)

Keep this input paired with its own output

For \(x=1\)

Keep this input paired with its own output

For \(x=2\)

Keep this input paired with its own output

For \(x=3\)

Keep this input paired with its own output

Keep the subtracted constant in every row

Read the actual input gap

Rows do not have to use consecutive inputs. In y = 2x + 1, compare inputs zero, two and four

Here the input increases by two and the output increases by four

Adding the coefficient two to the previous output would be wrong for this input gap

Independent substitution works whether the inputs are consecutive or spaced farther apart

Substitute the listed inputs, not their row numbers

A rule can produce smaller outputs as inputs increase

For y = 6 − x, subtract each input from six

The output decreases by one when the input increases by one

Keep the subtraction in the stated order. Six minus x is not x minus six

Generate a decreasing set of outputs

Extend a table using the same rule and input step

Use the rule

Calculate the missing output for each listed input

Also calculate the output for the next input \(x=6\)

Substitute every required input independently into the same rule

For \(x=0\)

Keep this input paired with its own output

For \(x=2\)

Keep this input paired with its own output

For \(x=4\)

Keep this input paired with its own output

For \(x=6\)

Keep this input paired with its own output

Use the next input even when the table starts below zero

Separate the growing part from the fixed part

A tile pattern has two blue tiles for each stage number and three red tiles that stay fixed

At stage n, the blue count is two times n and the red count is three

The stage number is the input and the total tile count is the output

This rule comes from the stated construction, not just a guess based on a few totals

The blue group has twice the stage number of tiles. The red group always has three tiles. Change n from one to four and compare the growing and fixed parts

At stage three, how does the construction give the total tile count?

Record the visual pattern as a table

Predict the next stage from the stated construction

Write each row as an ordered pair

An ordered pair lists the horizontal input coordinate first and the vertical output coordinate second

A row with input two and output seven becomes

The pair with the numbers reversed usually represents a different point

The comma separates the two coordinates. The pair is not a fraction or a multiplication

A table row gives x = 3 and y = 8. Which ordered pair represents that row?

Use two perpendicular number lines

The horizontal axis is the x-axis and the vertical axis is the y-axis. They meet at the origin

Positive x is to the right and negative x to the left. Positive y is upward and negative y downward

To plot a point, start at the origin, move to its x-coordinate horizontally, then move to its y-coordinate vertically

For a point on an axis, one coordinate is zero. The origin has both coordinates zero

Read the numbered scale on each axis. Do not assume every drawn grid interval always represents one

The grid has unit spacing. Which ordered pair gives the position of P?

The grid has unit spacing. Which ordered pair gives the position of R?

Which point lies on the vertical y-axis, two units below the origin?

Construct a relationship from its table

Write each table row as an ordered pair and plot the pairs on the grid

When the question states that the relationship is a straight line, two distinct points determine that line

Use a third supplied point to check the construction. If it is not on your line, recheck the pair order, signs and scales

For the plotting tasks, drag the two marked points or enter their coordinates. They must be different points on the required line

The constructed line extends in both directions across the grid. These tasks describe an abstract straight-line relationship, not a count restricted to separate stages

Plot a relationship whose outputs increase

The table gives three pairs from a straight-line relationship

These pairs belong to one straight-line relationship. Plot two distinct table points and extend the line through them in both directions. Check that it also passes through the third point

Read each row as an ordered pair, keeping its input and output together

Plot the horizontal x-coordinate first, then the vertical y-coordinate

Draw a straight line through two distinct points and check that the remaining point lies on it

Other pairs of distinct integer-coordinate points on this same line are also valid

Place two distinct points from the table

Plot a relationship whose outputs decrease

The table gives three pairs from a straight-line relationship

These pairs belong to one straight-line relationship. Plot two distinct table points and extend the line through them in both directions. Check that it also passes through the third point

Read each row as an ordered pair, keeping its input and output together

Plot the horizontal x-coordinate first, then the vertical y-coordinate

Draw a straight line through two distinct points and check that the remaining point lies on it

Other pairs of distinct integer-coordinate points on this same line are also valid

Plot negative outputs in their correct positions

Keep a constant output at the same height

The table gives three pairs from a straight-line relationship

These pairs belong to one straight-line relationship. Plot two distinct table points and extend the line through them in both directions. Check that it also passes through the third point

Read each row as an ordered pair, keeping its input and output together

Plot the horizontal x-coordinate first, then the vertical y-coordinate

Draw a straight line through two distinct points and check that the remaining point lies on it

Other pairs of distinct integer-coordinate points on this same line are also valid

Draw a horizontal relationship below the x-axis

A straight-line relationship contains the pairs (0, 3), (1, 2) and (2, 1). Which description matches it?

Do not invent intermediate stages in a counting pattern

The tile construction has whole-number stage numbers. Its first four stage-and-count pairs are

These points follow a straight-line pattern, but the stated construction does not define a stage one and a half

Keep separate points for the whole-number stages rather than claiming every point between them is an allowed tile stage

Whether points should be joined depends on the stated relationship, not merely on whether they appear lined up

For the tile construction, n is a whole-number stage. Why are its recorded stage points kept separate?

Connect the rule, table and graph

Use the same rule for every listed input and calculate each output independently

Explain a visual pattern by separating its growing and fixed parts, then record stage and count together

Write each table row as an ordered pair with x first and y second

Plot horizontal position before vertical position, keeping signs and axis scales

For a stated straight-line relationship, construct the line through distinct points and check another supplied pair

Describe whether outputs increase, decrease or stay constant as inputs increase. Keep separate stages separate when the context defines only whole-number inputs