ES Math Resources

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Accuracy

Accuracy: The student uses time effectively, checks work for errors, and ensures solutions make sense.

Accuracy and Precision

Using Time Effectively

As we strive to create mathematicians who can solve problems correctly and consistently, we have noticed that those who rush and those who are off-task will often make errors. These errors can be as simple as incorrect calculations, but can also be a false conceptual understanding of the problem, perhaps due to not reading the problem wording carefully. While it is important for students to be able to recall basic facts quickly, students are encouraged to take as much time as needed (but no more!) to solve problems. Completing problems quickly does not mean one is smarter than another. On the other hand, if students are distracted or off-task, this typically means they are not focused and will generate "careless" errors or take a lot more time to solve a problem than is necessary. Thus, it is important for mathematicians to focus and take an appropriate amount of time in order to solve problems accurately, precisely, and efficiently.

Note: Some students may have learning differences that make certain aspects of math difficult. Our teachers use accommodations to help these students learn and work as effectively as possible.

Checking Work

Students can better ensure correct solutions by checking their work and checking their solutions make sense.

During the problem-solving phase

Write down all steps needed to solve the problem or show the work using manipulatives. Mental calculations may not need to be fully worked out on paper, but should be documented. For instance, a student may be able to calculate 14 × 8 in her head without using a written algorithm, but the student should still write down 14 × 8 = 112 on her paper to show this step. This provides evidence (also known as proof) that a solution is correct or incorrect.

Examples of showing work can be found in the bar model example slides on this site's Communication and Visualization pages. The examples on the Visualization page are typed, but the general structure of the work is what our students should aim for.

Note: In Kindergarten and the beginning of Grade 1, many students may rely on verbal explanations of work if they are not yet able to write out their work.

After the problem has been solved

For one-step problems, we can use the inverse (opposite) operation to prove our calculation is correct. Addition and subtraction are inverses; multiplication and division are inverses.

An example with subtraction and addition:

An example with multiplication and division:

For multi-step problems, go back through each step of the problem, making sure each calculation or step is correct. If you notice an error, fix it, and then make sure to fix any numbers or steps after the error.

For example:

There are errors in the second calculation, as 27 + 15 = 42, not 33. If the student only corrects the second calculation (which often happens!), his final answer will still be incorrect:

The student should carry the corrected calculation forward and redo the rest of the problem:

Note: Students can always check their calculations using the inverse operation, as needed.

Ensuring Solutions Make Sense

Estimation plays a key role in ensuring solutions make sense. We can round numbers in the problem so they are easier to work with, carry out any operations, then check if the actual answer is reasonably close to our estimated answer. If so, our answer makes sense. If not, we have a clue that there may be something wrong in our solution.

In the bracelet problem above, the student can use estimation to check if his answer is reasonable in the following way:

Quick Notes on Rounding

We introduce rounding using number lines in order to visualize which benchmarks a given number is closest to.

As the curriculum progresses, students are expected to use an algorithm for rounding, exemplified below:

Accuracy Tips

Fact fluency plays an important role in accuracy. See our Fact Fluency page to learn more.
If a student is rushing, have the student read the problem aloud at a normal pace, then read it to themselves again. This process puts emphasis on reading carefully.
Encourage showing calculations and steps for every problem, otherwise there won't be any way to check where in the process the error occurred. This has the added benefit of slowing the student down to help their thinking catch up with their problem solving process.
After each problem is solved, students should check their work for errors and to see if their answer is reasonable. It is better to take a little more time and get the correct answer than to rush and leave errors behind.
Give students problems that contain errors (or not!) and have them hunt for all the mistakes. Some teachers have had success setting up a system of earning bonus points on assessments if students catch the teacher making an error. Sometimes these occur naturally, other times the teacher may intentionally write a wrong answer to keep students actively looking for errors.
While we want students to be accurate and precise, it is also important to remember that mistakes are a normal part of the learning process; doing everything perfectly is a sign that we are no longer learning!

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