Advanced I Number Sense (7-9 old curriculum)

Question 1

How do you determine if a number is divisible by a composite number?

Explain by using the example 210 and whether it is divisible by 30 or not.

Answer 1

Find the prime factors of the composite number, then make sure that the number is divisible by all the factors. Start by dividing by the factor that is easiest for you, continuing to divide by all the factors except the last factor. Then when you get to the last prime factor, you can just determine divisibility.

For example, to determine divisibility by 30:
30 can be broken down into its prime factors: 30=2x3x5

So given a number like 210, we can state if it is divisible by 30 by doing the following:

Divide 210 by 2:
210/2 = 105

Divide 105 by 3 (remember 1+5 = 6 so 105 is divisible by 3)
105/3 = 35

Check to see if 35 is divisible by 5:
35 ends in a 5 or 0 so it is divisible by 5

Therefore 210 is divisible by 30. You could perform the last division to determine the answer but it is not required to determine divisibility.

Question 2

What is a trick to determine divisibility by 4?

Answer 2

Ignore all the digits to the left of the tens column, and check that two digit number for divisibility by 4.

So 624 is divisible by 4 because 24 is divisible by 4.

Alternatively just divide by 2 and then check to see if the last digit ends in 0, 2, 4, 6, or 8.

Question 3

What are the prime factors of all composite numbers from 1-10?

Why is it useful to know these prime factors really well?

Answer 3

4 = 2x2
6 = 3x2
8 = 2x2x2
9 = 3x3
10 = 2x5

Knowing the prime factors of these numbers is usually useful when trying to determine divisibility by these numbers. For example, to determine if a number is divisible by 6, we would divide it by 2 and then check for divisibility by 3. (Alternatively divide it by 6 and then check divisibility by 2).

Question 4

Show four example problems to show four different operations on decimal numbers. Get your teacher to verify your answer, or practice with Smartermarks questions.

Answer 4

The four operations that you should know by now are addition, subtraction, multiplication, and division.

With addition and subtraction, line up your place values so that you are adding and subtracting within a specific place value and not across two different place values.

With multiplication it is easiest to just multiply your problem through by factors of ten so that there are no decimals while you do the multiplication. Then divide the answer by that same factor of ten.

In long division, the decimal point must be placed as soon as you need to start bringing down zeroes.

Make sure that you try your problems with a calculator to verify that your answer is correct if you do not have a teacher there or something to automatically correct your work.

Question 5

Terminating decimal

Answer 5

a decimal number that ends (so it does not repeat)

e.g.:
0.25
0.12434327

Divide fractions that you see commonly using long division or your calculator. You may be expected to have these memorized for your class so that you can recognize numbers such as 0.25 as their fraction (that is 1/4 ).

You can choose to memorize the unit fractions up to 1/10 and you will find that life is easier :)

1/1 = 1
1/2 = 0.5
1/3 = 0.33333333…. (this is not terminating but it is included here for you to start your memorization journey)
1/4 = 0.25
1/5 = 0.2
1/6 = 0.166666…. (not terminating, it is repeating the 6 forever)
1/7 = 0.142857142857142857142857…. (not terminating since the denominator is a prime number other than 2 or 5; the digits 1428571 get repeated infinitely)
1/8 = 0.125
1/9 = 0.1111…. (not terminating, it is repeating the 1 forever)
1/10 = 0.1

If you do not have a calculator, and you have not memorized the common fractions that you see most often and you are presented with a decimal such as 0.25, make sure that you can convert to a fraction using the longer methods.

For example 0.25 = 25/100 since there are 25 hundredths (look at the rightmost digit’s place value and that will tell you the denominator); but remember to reduce the fraction so 1/4 would be the equivalent of 0.25

I suggest you memorize 1/4 = 0.25 and then know how to skip count by 25s. This will allow you to understand that 2/4 = 0.50 (1/2 is also 0.5) and 3/4 = 0.75 and thus you do not have to memorize as much.

Question 6

Repeating decimal

Answer 6

a decimal number that does not end, also known as a non-terminating decimal

e.g.:
1/3 = 0.3333…
2/3 = 0.66666….

Usually you have non-terminating decimals when the denominator is a prime number, but not when it is a 2 or 5. So 3 is a prime number and therefore you can expect the numbers above to not terminate. Thus these are great examples of repeating decimals.

Notice how 3/5 = 0.6 and it terminates. That is because dividing by the prime number 5 is an exception to the prime number divisor rule.

Notice how 3/2 = 1.5 and it terminates. That is because dividing by the prime number 2 is an exception to the prime number divisor rule.

Question 7

What is a ratio?

Answer 7

A comparison of quantity by categories where the actual quantity is not shown, much like how a reduced fraction does not show the actual quantities. It is different from a fraction in the notation and in the fact that it is not required to have the total quantity of all objects in it.

a : b is the ratio of a to b

A ratio can compare more than two quantities.

a : b : c is the ratio of a to b to c

where a is one category, b is another category and c is another category for example a = quantity of apples, b = quantity of bananas and c = quantity of carrots

Notice that a : b is not the same as a/b since the bottom number of a fraction must always represent the total, but that b in this ratio is just another category within the set of objects.

The total is not usually expressed in a ratio unless it is a very specific ratio where the words say the ratio of a category to the total.

Ratio math can be similar to fractional math. Remember how you can multiply and divide the top and bottom of a fraction to get another fraction with the same spot on the number line? You can multiply and divide all quantities in a ratio by the same number in order to get another representation of the same proportions. For that reason, some textbooks will teach you to convert the ratio to look like a fraction in the hopes that it will teach you to manipulate the ratio as if it were a fraction.

Question 8

What is a rate?

Answer 8

A rate comes from dividing one number by another where the units are different from one number to the next.

A common example of a rate is speed, since speed comes from dividing distance by time. Distance could be measured in km and time could be measured in hours. So the rate of change in position is speed and it is expressed with the units of the top divided by the units of the bottom which is km / h

This same reasoning can be applied to other divisions, and you will see them quite a bit in science class.

When you divide one number by another you are left with one number, or a number over the number 1. This means that when you calculate a rate you are always calculating something per 1 something, as in 45 km / hour is 45 km over 1 hour.

Question 9

What is proportional reasoning?

Answer 9

This can be thought of as comparing two fractions that are meant to represent the same position on the number line. Proportional reasoning is what you do to solve for one unknown out of the four spots:

For example:

a/b = x/d

You could be solving for x for example in the above comparison of fractions. With numbers this may make more sense:

2/3 = x/9

Then use preservation of equality to isolate x. So you could multiply both sides by 9 to isolate x, because we know that 9/9 = 1 and 1 times x is just x. . Then compute the numbers on the left and you will have your answer.

There are many tricks that teachers teach here but it is far simpler to just label them all preservation of equality with the goal of isolating the variable and solving, rather than wasting time trying to figure out every last trick for each new problem you see.

In short, there is nothing new to learn here except to apply what you already know about preservation of equality, and to have a goal in mind about what you would like to see in the last step of your work (in this case x needs to be on one side of the equation with everything else on the other side).

Remember that you could have the x on the bottom of the fraction and that this should not change anything about your goal to preserve equality. Just think about the movement that is required by x to get it on top of the equation and it should follow that you recognize that you need to multiply both sides of the equation by x. It’s nothing new, just a goal to have x on the top of the fraction rather than below.

Question 10

How do you multiply two fractions together?

Answer 10

(a/b)(c/d) = (ac)/(bd)

The numerators are multiplied together.
The denominators are multiplied together.
Then reduce the resulting fraction by checking for divisibility and doing the division until it cannot reduce any further.

Since you can divide both the top and bottom of the fraction by the same number, you may find it keeps the numbers smaller if you notice a common factor on the top and bottom of your equation and then divide by that common factor both on the top and bottom. So don’t hurry to do a multiplication if you see this opportunity.

Question 11

How do you divide a fraction by a number?

Answer 11

(a/b) ÷ c = a/(bc) = (a/c) ÷ b

Thus if you have a division symbol, instead of dividing, multiply by something like 1/c in the above example instead. Then use your rules for multiplying two fractions together.

So a division sign followed by a number should make you think 1 divided by that number

When you divide a fraction by another fraction, it can look messy, so memorize this in order to be more efficient:

(a/b) ÷ (c/d) = (a/b) x (d/c)

and then use your multiplying by fractions rule to finish the question

In that example, we multiplied by the reciprocal instead of dividing

Question 12

reciprocal

Answer 12

the reciprocal to a/b is b/a

Question 13

What are the rules for multiplication and division of integers?

Answer 13

If the integers are both positive or both negative, then the result will be positive.

If the integers are one negative and one positive, the result will be negative.

The rules are the same for multiplication and division.

For a more conceptual understanding of this, try thinking of when the signs are different and think of it as you are in three times more debt than you had before, where your debt was $5 before:

(3)(-5) = -15

So now you are in debt $15.

Then remember that the negative sign means to do the opposite so if you are -15 from a positive and a negative, changing the positive to a negative will switch the direction and so you have the opposite 15.

It is not common in real life that we multiply negatives together so it is tougher to picture, but it does occur so we do need to know the rule.

Question 14

How do you evaluate (a^m)^n where a is a constant and m and n are exponents?

Answer 14

multiply the exponents together and then use that as the new exponent on the base a

a^(mn)

Notice how the question only showed “a” once? That is important to recognize since if it showed “a” more than once, as in (a^m)(a^n) then you would be adding the exponents! And if it was (a^m)/(a^n) you would be subtracting the exponents!

Question 15

How do you evaluate (ab)^m

Answer 15

(a^m)(b^m)

Apply the exponent to every number you see, but watch out for addition or subtraction, since this rule does not work with two terms or more.

Question 16

How do you evaluate (a/b)^m

b is not allowed to equal zero in this case since dividing by zero does not exist

Answer 16

(a^m)/(b^m)

Apply the exponent to every number you see, but watch out for addition or subtraction, since this rule does not work with two terms or more.

Question 17

What do exponents do?

Answer 17

They allow you to perform repeated multiplication.

E.g. 3^4 = 3 x 3 x 3 x 3 (notice that there are four threes)

Question 18

What is a^0?

Show how this is true by looking at the patterns of exponents that are decreasing.

What is a negative exponent?

Show this in the decreasing pattern as well.

Answer 18

1

Anything exponent zero is 1.

Use patterns to see that this is the case:

3^3 = 3 x 3 x 3
3^2 = 3 x 3
3^1 = 3
3^0 = ? = 3/3 since to get from one spot to the next we are dividing by 3 = 1
3^(-1) = 1/3
3^(-2) = 1/(3x3)
3^(-3) = 1/(3x3x3)

Question 19

What math rules interrupt us from computing from left to right?

Answer 19

multiplication and division must be solved before addition and subtraction, otherwise you would need to know other rules to bypass this.

brackets make you do the stuff inside them before anything else, otherwise you would need to know other rules to bypass this.

given an exponent on brackets where there are two or more terms in the brackets: exponents need to apply to all numbers in the term, and cannot be applied to two or more terms, so you will either need to find a way to combine the terms within the brackets first, or use other math rules to solve those complex situations such as FOIL to multiply two binomials together. Expanding bracketed stuff because of an exponent on the brackets is a large topic that is explored in depth in math 10C as well as sometimes introduced in math 9 in order to prepare you for math 10C.

You are only allowed to combine like terms, so do not add them together unless they have the exact same variable or are both constants. In this case you can just add like terms using the commutative property where things separated entirely by addition signs are allowed to move around the equation.

Multiplication is commutative, so if all you have is multiplication, you can perform this in any order you choose.

You can memorize the order of operations for now if you have a while before you will be learning math 10C since it will help you out for the moment:
1. Brackets (so deal with the stuff inside the brackets since you will only be given things that can actually combine if you are in a lower level math than grade 9; fraction bars that are really long work like brackets so watch out for those)
2. Exponents (apply the exponent to the one term that you see but if there are more terms you will need to learn how to multiply these in a different way)
3. Multiplication and Division from left to right (assuming that all brackets as well as fraction bars have been dealt with or will be dealt with, so watch out for fraction bars since these operate much like brackets; also assuming the exponents have also been dealt with)
4. Addition and subtraction from left to right assuming that the things are allowed to be added or subtracted (so combine or collect like terms)

Question 20

Combine like terms

Answer 20

Add terms that have the exact same variable together. You can add integers together, so this allows you to technically subtract like terms.

x^2 is not the same variable as x

They must be EXACTLY the same variable

The coefficients do not need to match

(coefficient is the number listed next to the variable, as in the one usually in front that is big, not the exponent)

This means that constants can be added together such as 2 + 7 can be combined making 9.

Question 21

distributive property

Answer 21

a(x+y) = ax + ay

Can use this property to prove that FOIL works later when you take math 10C.

FOIL stands for first, outside, inside, last and is like this:

(a + b) (x+y) = ax + ay + bx + by

So essentially, watch out for more than one term and if you see more than one term, ensure that you use distributive property, or just evaluate what is in the brackets first instead since that is already something you know how to do. In math 9 and math 10C you will be given problems where you cannot add what is in the brackets since they will not be like terms so then you will be forced to use distributive property.

Question 22

define polynomial

Answer 22

an expression of more than two algebraic terms where each term is a constant multiplied by one or more variables raised to a nonnegative integral power

e.g.:

x and x^2 might be present such as 3x + 4x^2

You can also have an exponent of 0 on x since 0 is a non negative integral exponent. This is how the constant term is made.

or x and y will be present such as 3xy + 4y

usually the sum of several terms that contain different powers of the same variable(s) as in the first example, but they do not have to be the same variables

you will never see the variables in the denominator of the fraction and still be called a polynomial since variables in the denominator have negative exponents when moved to the numerator.

Question 23

what is a square root?

Answer 23

Given an area, the square root of that area will return the side length of a square with that area.

Question 24

add rules for other divisibilities