Numbers & Operations
Tackling Math and Supermath Head Items
Math Head items are pretty simple, but it’s easy to fall into traps if you’re not careful when solving these items. The goal is to protect yourself against traps by following our four-step method. Supermath Head items are a bit more complicated, but you can use the same step method to solve these items. The only major difference between the two is that steps 2 and 3 are more intensive for Supermath Head items and may contain several different concepts and formulas.
Follow these four steps every time you approach a Math Head or Supermath Head item:
Step 1: Determine what the item is really asking you to solve.
Step 2: Determine which formulas and concepts to use.
Step 3: Do the math by writing out every step.
Step 4: Plug it in.
Math Head and Supermath Head Items in Slow Motion
Now let’s work out each step in slow motion. We’ll start out with a basic Math Head item:
3. If , what is the value of x?
(A)
(B)
(C)
(D)
(E)
Step 1: Determine what the item is really asking you to solve.
Carefully analyze the item. In this case, the ultimate goal is to find the value of x. Also notice that all the answers involve exponents.
Step 2: Determine which formulas and concepts to use.
To calculate the value of x, you first need to simplify the equation by getting rid of the parentheses and exponents so you can arrive at the answer. To simplify the equation, you need to follow PEMDAS on the left side of the equation.
Step 3: Do the math by writing out every step.
Make sure you take the time to write out your steps. If you try doing the math in your head, you run the risk of making a careless mistake. Furthermore, writing out your work reinforces the concepts you’ve already learned. The more explicit you make these concepts, the easier it will be to remember them.
First, solve the parentheses:
Now do the exponents:
Be very careful with signs changes depending on the odd or even exponent.
Then do the multiplication:
There is no division, so do the addition:
Now do a little algebra and divide both sides by 5:
Step 4: Plug it in.
Congratulations. You solved for x. Before giving your humble acceptance speech, look down at the answer choices. There’s no 35, so there’s one more step to go. Simplify each answer choice to see which one gives you 35.
Choice A:
Choice B:
Voilà! Answer B is correct.
Guided Practice
Try solving this Math Head item on your own:
6. If , what is the value of ?
(A)
(B)
(C)
(D)
(E)
Step 1: Determine what the item is really asking you to solve.
What is the stem asking you for? What concepts apply?
Step 2: Determine which formulas and concepts to use.
What steps do you have to follow? How can you simplify the equations?
Step 3: Do the math by writing out every step.
Be careful when you bring the numbers to the same base, and be attentive when you change the exponents.
Step 4: Plug it in.
Is the answer that you found in the form the item requires? If so, choose the correct answer. If not, what are you going to do about it?
Guided practice: Explanation
Step 1: Determine what the item is really asking you to solve.
The stem tells you that you have to figure out the value of . However, your real goal is to find the value of x. The item revolves around that unknown variable. Once you know x, you’ll be able to find .
Step 2: Determine which formulas and concepts to use.
Think back to what you learned about exponents and roots in the previous section. To solve for x in the first equation, you want to simplify the numbers so that you have the same bases on either side of the equation. Once the bases are the same on both sides, the exponents will be equal. After you find the value of x, you can plug it into . To make it easier to find the answer, you might have to change into exponential form.
Step 3: Do the math by writing out every step.
To solve , you need to bring both expressions to the same base. In this case, convert both bases to 5:
Now you can solve for x:
Once you find x, plug it into the second expression:
Step 4: Plug it in.
Now you have the answer, 4. But the stem doesn’t have a plain old 4 as an answer choice. Typical. Take a close look at the choices. Answer choice A works out to the value you need.
Independent Practice
Here’s a tougher Supermath Head item for you to solve on your own. Once you’ve completed your work, look at the following page to see how we solved it.
9. If , what is one possible value of a?
Independent Practice: Explanation
Step 1: Determine what the item is really asking you to solve.
There are no answer choices here, so this is a grid-in. The stem asks you to find out when will be greater than . You need to find a number whose value increases when the exponent decreases.
Step 2: Determine which formulas and concepts to use.
How do you solve such an item? Think back to what you’ve learned about fractions. When fractions are multiplied by themselves, their value decreases. So a fraction raised to a higher power is less than a fraction raised to a lower power. Of course, we are talking about only positive numbers here because you can’t grid in a negative number.
Step 3: Do the math by writing out every step.
Because the stem asks for one possible value of x, you know there’s going to be more than one correct answer. Plug in an easy fraction, such as , and check it:
That works.
Step 4: Plug it in.
There is more than one correct answer to this grid-in item, but works just fine. Make sure the answer is a decimal or a proper fraction.
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