Minds On

Working out math steps

Consider the following two math problems with every step worked out:

Version 1	Version 2
2 + 3 × 4 = 5 × 4 = 20	2 + 3 × 4 = 2 + 12 = 14

They both started with the same problem, but which one is right?

If possible, discuss with a partner or record your thoughts in a notebook or using another method of choice.

Action

BEDMAS

Let's think about the two math problems from the Minds On again.

Version 1	Version 2
2 + 3 × 4 = 5 × 4 = 20	2 + 3 × 4 = 2 + 12 = 14

It can be tempting to think that the first option was correct, since it was calculated in order from left to right, which is how we read text in English (but not every language!) However, in mathematics we can use a convention called BEDMAS. This convention is an agreed upon method of reading mathematics. Like languages have certain conventions that allow people to read and understand the same thing, so does math.

Version 2 and Version 3 in the following example result in the same answer even when we have reversed the order of the numbers. Therefore, multiplication needs to come before addition.

Version 1	Version 2	Version 3
2 + 3 × 4 = 5 × 4 = 20	2 + 3 × 4 = 2 + 12 = 14	3 × 4 + 2 = 12 + 2 = 14

In general, we follow a specific order with the acronym BEDMAS:

Brackets – do any operations in brackets first

Exponents – (eg. 4³)

Division

Multiplication

Addition

Subtraction

Why do we need this?

Why BEDMAS? Why not ASMEDB? Or MDASBE? Or any other combination? Consider the following example:

1 = 2 × 1 − 1 = 2 × 0 here we did 1 − 1 first instead of the multiplication first = 0 So, 1 = 0 without a standard rule system!!

If you would like, you can complete this activity using TVO Mathify. You can also use your notebook or the fillable document.

Construct your own way to “break” math by breaking the rules of BEDMAS. You can break it in all sorts of ways, like:

make one number equal to another (1 = 0, 4 = 3, etc.)
make rules that we know about operations break
show how not following BEDMAS breaks other parts of mathematics (eg. geometry: calculating perimeter, area, etc.)

Press the ‘TVO Mathify' button to access this interactive whiteboard and the ‘Activity’ button for your note-taking document. You will need a TVO Mathify login to access this resource.

Activity

Spot the mistake

If you would like, you can complete this activity using TVO Mathify. You can also use your notebook or the fillable document.

For each example given, a full solution has been provided. Your task is to find where BEDMAS was not followed.

Circle or indicate what BEDMAS step was not followed correctly.
Hint: every solution goes one step at a time, so you should try to find what step was done and determine if that was an allowed step, according to BEDMAS.
Write a corrected version from the step that went wrong, showing every step.

$14 + (25 - 9 \times 2) + 7$ $= 14 + (16 \times 2) + 7$ $= 14 + 32 + 7$ $= 53$	$(64 \div 8 \times 3) \div 6 + 6$ $= (8 \times 3) \div 6 + 6$ $= 24 \div 12 = 2$
$= \frac{1}{2} \times \frac{5}{3} + \frac{4}{3}$ $= \frac{1}{2} \times \frac{9}{3}$ $= \frac{9}{6}$ $= \frac{3}{2}$	$= (\frac{8}{12} + \frac{3}{12}) + 2 \times 5 - 10$ $= \frac{11}{12} + 2 \times 5 - 10$ $= \frac{11}{12} + \frac{24}{12} \times 5 - 10$ $= \frac{35}{12} \times \frac{5}{1} - 10$ $= \frac{175}{12} - 10$ $= \frac{175}{12} - \frac{120}{12}$ $= \frac{55}{12}$
$0 \times \frac{1}{2} + \frac{1}{8} - 4 \times 0$ $= 0 \times \frac{4}{8} + \frac{1}{8} - 4 \times 0$ $= 0 \times \frac{5}{8} - 4 \times 0$ $= 0 - 4 \times 0 = 0 - 0$ $= 0$	$(\frac{1}{2} + (4 + 3) \times \frac{3}{2}) + 5$ $= (\frac{1}{2} + 4 + \frac{9}{2}) + 5$ $= (\frac{1}{2} + \frac{8}{2} + \frac{9}{2}) + 5$ $= (\frac{9}{2} + \frac{9}{2}) + 5$ $= \frac{18}{2} + 5$ $= \frac{18}{2} + \frac{10}{2}$ $= \frac{28}{2}$ $= 14$

Press the ‘TVO Mathify' button to access this interactive whiteboard and the ‘Activity’ button for your note-taking document. You will need a TVO Mathify login to access this resource.

Activity

Consolidation

Constructing your own examples

Construct two examples of Spot the Mistake. Things to keep in mind:

try to build each example so that there are at least 5 unique steps
after the “mistake” has been done, continue to solve the problem as if you were still following BEDMAS
you can choose to make it as challenging as you would like

When you have your completed examples, if possible exchange them with a partner and try to spot their mistakes or record the steps to solving your questions in a notebook or recording.

Reflection

As you read the following descriptions, select the one that best describes your current understanding of the learning in this activity. Press the corresponding button once you have made your choice.

I feel...

I have an excellent understanding of the concepts I have a very good understanding of the concepts I have some understanding but need to review some more my understanding of the concepts needs a lot more work

Now, expand on your ideas by recording your thoughts using a voice recorder, speech-to-text, or writing tool.

When you review your notes on this learning activity later, reflect on whether you would select a different description based on your further review of the material in this learning activity.

Press ‘Discover More’ to extend your skills.

Discover More

Exponents and square roots

Exponents and square roots are also part of BEDMAS and must be done in a certain order. You can treat a square root like a “bracket,” where everything underneath the square root needs to be calculated first. For example:

\sqrt{18 + 7} = \sqrt{25} = 5

Exponents should be treated the same way: complete any operations inside the exponent (noted with a bracket) first, if possible. For example:

How to deal with exponents:	How NOT to deal with exponents: This is one of the most commonly made mistakes in mathematics! It even has a name: the freshman’s dream
(3 + 2)³ = 5³ = 125	(3 + 2)³ = 3³ + 2³ = 27 + 8 = 35

Practice

If you would like, you can solve the next series of questions using TVO Mathify. You can also use your notebook or the following fillable and printable document.

$\sqrt{10 + 3 \times 5} \times 2$	$5 - \frac{1}{5} \times {(1 + \frac{3}{2} + \frac{4}{8})}^{4}$
${(\sqrt{7 \times 8 + 4 \times 2})}^{2} - 44$	$\frac{1}{3} + \sqrt{\frac{17}{3} + \frac{62}{6}} - \frac{1}{9} \times 1^{45}$

Press the ‘TVO Mathify' button to access this interactive whiteboard and the ‘Activity’ button for your note-taking document. You will need a TVO Mathify login to access this resource.

Activity

Connect with a TVO Mathify tutor

Think of TVO Mathify as your own personalized math coach, here to support your learning at home. Press ‘TVO Mathify’ to connect with an Ontario Certified Teacher math tutor of your choice. You will need a TVO Mathify login to access this resource.

Learning goals

Success criteria

Minds On

Working out math steps

Action

BEDMAS

Why do we need this?

Spot the mistake

Consolidation

Constructing your own examples

Reflection

I feel...

Exponents and square roots

Practice

Connect with a TVO Mathify tutor