TACHS: Matrices: track a turning figure
Learn to measure how far a figure turns across a row and down a column, then use both turns to build the missing cell, dot included.
How do you find the missing cell when an arrow and its dot keep turning across the matrix?
Short answer
Measure the turn across a row and the turn down a column in 45° notches, predict the missing arrow both ways, then turn the dot by the same amount as the arrow.
Know first: The eight compass directions (up, up-right, right and so on), Clockwise and counterclockwise
Step 1 of 5
Measure each turn in 45° notches
In a rotation matrix, every cell holds the same figure: an arrow with a small dot beside it. Only the direction changes. Moving one cell to the right turns the figure by one fixed amount, and moving one cell down turns it by another.
The whole figure turns as one piece. If the arrow makes a quarter turn, the dot makes the same quarter turn around the arrow's center.
| Notches | Turn | Example, starting from up |
|---|---|---|
| 1 | 45° | up to up-right (clockwise) or up to up-left (counterclockwise) |
| 2 | 90°, a quarter turn | up to right (clockwise) or up to left (counterclockwise) |
| 3 | 135° | up to down-right (clockwise) or up to down-left (counterclockwise) |
| 4 | 180°, a half turn | up to down, either way |
Clockwise order: up, up-right, right, down-right, down, down-left, left, up-left.
Step 2 of 5
Predict the arrow twice, then place the dot
| Column 1 | Column 2 | Column 3 | |
|---|---|---|---|
| Row 1 | Arrow up, dot lower left | Arrow left, dot lower right | Arrow down, dot upper right |
| Row 2 | Arrow up-right, dot left | Arrow up-left, dot below | Arrow down-left, dot right |
| Row 3 | Arrow right, dot upper left | Arrow up, dot lower left | ? |
Worked example: An arrow that turns two ways
Use the matrix above. Which figure belongs in the empty bottom-right cell?
- AArrow up-left, dot below
- BArrow left, dot upper right
- CArrow left, dot lower right
- DArrow right, dot upper left
- EArrow left, dot lower left
- 1
Measure the row turn
Top row: up, then left, then down. Up to left is 2 notches counterclockwise, and left to down is 2 more. The row turn is 90° counterclockwise.
- 2
Measure the column turn
First column: up, then up-right, then right. Each step is 1 notch clockwise. The column turn is 45° clockwise.
- 3
Predict from the bottom row
The bottom row shows right, then up. That is 90° counterclockwise, as expected. One more quarter turn counterclockwise from up gives left.
- 4
Check with the right column
The right column shows down, then down-left. One more notch clockwise from down-left gives left. Both routes agree.
- 5
Turn the dot
The cell to the left has the arrow up with the dot at its lower left. The missing cell is that figure turned a quarter turn counterclockwise. Picture a card turned a quarter turn to the left: its lower-left corner moves to the lower right. So the dot sits at the lower right.
- 6
See why the other choices are there
Arrow left, dot upper right is the mirror image of the answer. Arrow left, dot lower left turns the arrow but leaves the dot behind. Arrow up-left, dot below stops one notch short. Arrow right, dot upper left turns the bottom row clockwise, the wrong way.
Answer
Arrow left, dot lower right.
Find the wrong step
A student fills in this bottom row: arrow right, then arrow down, then the empty cell. The dots don't matter here.
- 1
Names the two arrows: right, then down.
- 2
Says right to down is a quarter turn counterclockwise.
What went wrong
Going clockwise, right is followed by down-right and then down. So right to down is a quarter turn clockwise.
- 3
Turns down a quarter turn counterclockwise and predicts an arrow pointing right.
The fix
Right to down is 90° clockwise. One more quarter turn clockwise from down gives left. Saying the clockwise order aloud (up, up-right, right, down-right, down) prevents the slip.
Step 3 of 5
Turn arrows and dots on your own
Check yourself · Question 1
In one row of a rotation matrix, the arrow points down-left in the first cell and left in the second. It turns the same way at each step. Which way does it point in the third cell?
Answer: B
Down-left to left is 1 notch clockwise, a 45° turn. One more notch clockwise from left is up-left.
| Column 1 | Column 2 | Column 3 | |
|---|---|---|---|
| Row 1 | Arrow down, dot right | Arrow up-right, dot upper left | Arrow left, dot below |
| Row 2 | Arrow left, dot below | Arrow down-right, dot upper right | Arrow up, dot left |
| Row 3 | Arrow up, dot left | Arrow down-left, dot lower right | ? |
Check yourself · Question 2
Each cell shows an arrow and a dot that turn together. Which figure belongs in the empty bottom-right cell?
Answer: C
Across each row the figure turns 135° counterclockwise (down to up-right is 3 notches counterclockwise). Down each column it turns 90° clockwise (down to left). Bottom row: down-left turned 3 notches counterclockwise is right. Right column: up turned a quarter turn clockwise is right. The neighbor on the left has its dot at the lower right. Turning that figure 3 notches counterclockwise carries the dot to the spot above the arrow.
Trap answer: Arrow right, dot below
Mirror image
- Why it looks right
- The arrow points the right way, and the dot is right next to it, so it looks finished.
- Why it's wrong
- A turn never flips a figure. Turning the bottom-middle figure carries its dot above the arrow. A dot below the arrow only appears if you flip the figure over.
- The right answer
- Arrow right, dot above, the figure turned 3 notches counterclockwise.
Common mistake
I got the arrow right, so I picked the first choice with an arrow pointing that way.
- Why it's tempting
- Several choices share the correct arrow, and the dot looks like a small detail.
- Do this instead
- Choices are built to share the arrow. Find the dot in a neighboring cell and turn it by the same notches as the arrow.
Step 4 of 5
What to remember
Remember
- Measure the row turn and the column turn in 45° notches, and name each as clockwise or counterclockwise.
- Predict the missing arrow from the bottom row and from the right column. The two predictions must agree.
- The dot turns with the arrow. A choice with the dot on the mirror side is a flip.
Step 5 of 5
Practice on a real question
Use what you just learned on this question, then check the explanation.
Try another one
In your own words
In the practice matrix, how many notches does the figure turn across a row and down a column, and how did you decide where the dot goes?