Sensemaking is an important step in problem solving and critical thinking. Physicists use it to check their work while they analyze systems. The basical premise is to evaluate whether procedures or results pass a number of sensemaking tests. Very closely related to these tests are the different multiple representations that are used to analyze problems. Below is an overview of the various representations and sensemaking techniques used to help improve your problem solving skills.

Sensemaking | Problem Solving (2 min)

Sensemaking | Problem Solving (2 min)

## Multiple Representations

Sensemaking | Multiple Representations (2 min)

Sensemaking | Multiple Representations (2 min)

Multiple Representations is the concept that a physical phenomena can be expressed in different ways.

## Sense-making Techniques

Problem Evaluation Tests | Overview of Sense-making Techniques (5 min)

Problem Evaluation Tests (5 min)

### Sign

** Sign:** Check that the sign of the quantity (e.g. if you are asked for a mass, your answer should be a positive number since there is no such thing as an object with negative mass.)

Sense-making | Check the Sign (2 min)

Sense-making Check the Sign (2 min)

**What to do: **

In solution evaluation sensemaking we should...

- State what we expect our answer to be
*and explain why using a technique* - Compare our answer to our expectation

For sign sensemaking, this means we should state why we expect our answer to be positive or negative given information in the problem. Finally, we need to compare our expectation to our result. Does our final answer have the sign we expected? Yes, or no.

### Dimensionality

** Dimensionality:** Check the dimensionality and units of the quantity make sense (e.g. a length should be in m, not m

^{2})

Sense-making | Units and Dimensions (2 min)

Sense-making | Units and Dimensions (2 min)

**What to do: **

In solution evaluation sensemaking we should...

- State what we expect our answer to be
*and explain why using a technique* - Compare our answer to our expectation

For dimensionality sensemaking, this means we should state what we expect the units of our answer to be given information in the problem (e.g. does the problem ask for a velocity? If so, our answer should have units of m/s). We need to show, by keeping track of the units throughout our solution, what the units of our final answer actually are. Finally, we need to compare our expectation to our result. Does our final answer have the units we expected? Yes, or no.

### Order of Magnitude

* Order of Magnitude:* Make a very rough estimate (is it 1, 10, 1000?) of what you expect your answer to be from the given information in the question. After you have calculated a more precise answer, check if this more precise number is within a factor of ten of your estimate. Each factor of ten is called an "order of magnitude". 4500 is three orders of magnitude bigger than 5.

Order of Magnitude (3min)

**What to do: **

In solution evaluation sensemaking we should...

- State what we expect our answer to be
*and explain why using a technique* - Compare our answer to our expectation

For order of magnitude sensemaking, this means we should state what we expect the order of magnitude of our answer to be, given information in the problem (e.g. do we expect an answer of 1 m/s? 10 m/s? 100 m/s?). The order of magnitude essentially means rounding a number to the nearest power of 10 (e.g. 1/100th, 1/10th, 1, 10, 100, 1000, etc). You should explain why you expect your final answer to be of a certain order of magnitude. Finally, we need to compare our expectation to our result. Does our final answer have the order of magnitude we expected? Yes, or no.

### Graphical Analysis

* Graphical Analysis:* Check that the graph of the function behaves as expected (e.g. the speed increases as time increases).

**What to do: **

In solution evaluation sensemaking we should...

- State what we expect our answer to be
*and explain why using a technique* - Compare our answer to our expectation

Graphical analysis sensemaking has several options as a solution evaluation sensemaking technique. We could state what we expect the shape of a graph to look like and then explain why we expect that shape. Alternatively, we could explain why the graph given in the problem statement leads us to expect our answer to have a certain property, like being positive or of a certain order of magnitude (another sensemaking technique may be used in conjunction here). Once you have made a prediction based on graphical analysis, we need to compare our prediction to our result. Do they match? Yes, or no.

### Proportionality

* Proportionality:* Using a symbolic solution, check the behavior of the answer when you change a given quantity on which it is dependent. Does the answer vary proportionally to what you expect?

Sense-making | Proportionality (2 min)

Sense-making | Proportionality (2 min)

**What to do: **

In solution evaluation sensemaking we should...

- State what we expect our answer to be
*and explain why using a technique* - Compare our answer to our expectation

In order to use proportionality sensemaking, it helps to have a symbolic solution for your problem. This very powerful sensemaking technique is one of the reasons physicists like to solve problems using symbols and then plug in numbers at the very last step (this also helps with dimensionality sensemaking!). For this technique, we should make a prediction for how we expect our answer to change if we were to adjust one of the given quantities in the problem statement (e.g. how do you expect the final velocity to change if we increase the initial velocity?). We should also explain why we expect this type of relationship between the quantities (without using our symbolic solution yet). Finally, we need to compare our predicted relationship with the relationship we observe in our symbolic solution to the problem.

### Special Cases

* Special Cases:* Check the behavior of a derived equation in limiting (special) cases makes sense, e.g. as x goes to 90 degrees in sin(x).

Sense-making | Special Cases (4 min)

Sense-making Special Cases (4 min)

**What to do: **

In solution evaluation sensemaking we should...

- State what we expect our answer to be
*and explain why using a technique* - Compare our answer to our expectation

For special cases sensemaking we should pick out a particular value for a given quantity in our problem's scenario. We then should explain conceptually what we expect to happen to the given situation if the given quantity has the particular value and why. For example, if you a ball is on a ramp, what would you expect the motion of the ball to be if the ramp were held horizontal (an angle of theta = 0 degrees with the ground)? What about if the ramp were held vertically (an angle of theta = 90 degrees)? Finally, we should compare our conceptually predicted behavior to what happens to our solution if the same quantity is made to be the chosen value. Common values to pick are limiting values such as 0 degrees or 90 degrees for an angle.

### Self-consistency

* Self-consistency:* Check derived equations, functions, or values, are self-consistent, e.g. check that the slope of a derived position plot matches the values of the given velocity plot.

**What to do: **

In solution evaluation sensemaking we should...

- State what we expect our answer to be
*and explain why using a technique* - Compare our answer to our expectation

For self-consistency sensemaking, we should take the final solution we found and plug it back into a previous step or parallel step in the solution process. We can then work out the algebra to make sure that we arrive at an expected result. Commonly this will result in an equation that shows 0 = 0, or that a different quantity equals itself. Make sure to explain what you are doing with a brief explanation.

### Known Values

* Known Values:* Compare given or derived quantities with known values (e.g. solving for the speed of a car gives a value between 0-100 mph).

**What to do: **

In solution evaluation sensemaking we should...

- State what we expect our answer to be
*and explain why using a technique* - Compare our answer to our expectation

For this sensemaking technique, we should compare our answer with a known value and explain why we expect our answer to be similar or different from this known value. You should also show that your answer is reasonably close or different from this value. Also, don't forget to cite your sources for information that is not common knowledge!

### Related Quantities

* Related Quantities:* Compare relative value of two related quantities (e.g. a vector at 70 degrees above the x-axis should have a larger y-component than x-component).

**What to do: **

In solution evaluation sensemaking we should...

- State what we expect our answer to be
*and explain why using a technique* - Compare our answer to our expectation

For related quantities sensemaking, we should compare two quantities in our solution path. We need to explain what relationship we conceptually expect there to be between the two quantities. Finally, we need to show that the quantities do have the expected relationship. For example, you would expect that the other two sides of a right triangle should be shorter than the hypotenuse.