Intermediate I Number Sense (3-4 new curriculum)

Question 1

By what multiple is 10 larger than 1?

Answer 1

10 times larger

Question 2

By what multiple is 100 larger than 10?

Answer 2

10 times larger

Question 3

For numbers in base 10, each place has how many times the value of the place to its right?

Answer 3

10

Question 4

the product of the digit and its place value

Answer 4

the value

for example in the number 23, 2 is the digit in the 10s place, so if you multiply 2 by 10, you get the resulting value of 20, which came from the product of the digit and the place value of that digit

Question 5

When can you round?

Answer 5

numbers can be rounded when an exact count is not needed

Question 6

Is 02 the same as 2?

Answer 6

yes, it does not change the value of the number

Question 7

Where do you place the dollar sign in comparison to the number?

Answer 7

before the number: $4 (in English, not in French)

Question 8

Where do you place the cent sign?

Answer 8

To the right of the number in both English and French

Question 9

Allows one to determine the value of a digit based on its place

Answer 9

place value

Question 10

By what multiple is 0.1 different than 1? If you had to say how much larger it is than 1, what could you say mathematically

Answer 10

1 is ten times larger than 0.1

mathematically you would need to say that 0.1 is one tenth of 1, or 1/10 to describe how much larger it is than 1.

Question 11

What do you look for to know that you have a decimal number?

Answer 11

. in English

, in French

You will see the word “and” used to describe the decimal point such as 2.3 is 2 and 3 tenths

Question 12

Is 0.2 the same as 0.20?

Answer 12

Yes

Question 13

What are the names of the places lower than the ones place?

What do we have here in terms of those names?
0.234

Answer 13

tenths, hundredths, thousandths

0.234

has 2 tenths, 3 hundredths, and 4 thousandths

Question 14

How can you represent a part of a whole?

Answer 14

fraction
decimal
percent

Question 15

Round this number to the nearest 10th:

0.12345

Answer 15

0.1

Question 16

Round this number to the nearest 10th:

0.56789

Answer 16

0.6

Question 17

How do you round?

Explain generally but also how to round this specific number to the tenths place:

0.36017

Answer 17

Look at the digit to the RIGHT of where you are told to round.

• If the digit is a 5, 6, 7, 8, or 9, then add one to the digit that you are supposed to round to, this will be the number to the left of the one you were just looking at
• If the digit is a 0, 1, 2, 3, or 4, then do not add one.

Then remove all numbers to the right of the place you are supposed to round to.

For example, to round to the nearest tenth, you will look at the hundredths place for if it is a 5 or greater. If it is a 6 for example, then you will add one to the tenths place.

0.36017 becomes 0.4 because the 6 is 5 or greater, so we add one to the 3 in the tenths place.

Question 18

What is in the hundredths place in this number?

0.4765

Answer 18

7

This represents seven hundredths or 7/100

Question 19

Describe how to add these two numbers using the standard addition algorithm:

67 + 89

Answer 19

Add the ones first:

7 + 9 = 16

Then write the ones down below, but carry the tens to the next addition. In this case you need to write a 6 down below in the ones place. Then you need to add 1 above the tens column because this symbolizes the 10 you had to make 16 from 10+6.

So now you have 6 + 8, but also +1 from the carry-over from adding the ones earlier on. Keep in mind that this is really 60 plus 80 plus 10 if you look at the place they are in, but make it easier on yourself and pretend they are ones for the moment so that you can use your memorized math facts.

6 + 8 + 1 = 15

So you will write the 5 in the tens column below (not the ones column below which is already filled anyway!) because this always represented 50 since we were adding tens.

You will write the 1 above the hundreds column if there are hundreds to add, but in this case, write it below since there are no more hundreds to add, so the number will not change and is your final answer.

In the end we will have 1 hundred, 5 tens, and 6 ones. This is 156 ones and your final answer. We just say it is one hundred fifty six.

Question 20

Describe how to subtract 57 from 63 by using the standard algorithm for subtraction.

Answer 20

63 - 57 is what the question is asking.

Start by subtracting the ones if possible. In this case we would get a negative number if we subtract 7 from 3, so instead, we do a trick where we borrow from the place to the left.

So we will borrow one ten from the tens place, so the 6 becomes a 5, and we now have 10 plus 3 in the ones place for the first number.

Take that 10 + 3, which you can see readily is 13, and subtract the 7 from it instead. This will give you a positive number and is what we want.

13 - 7 is easier to picture, since the numbers are smaller than the original question. You can do this subtraction because you already have memorized that 6 plus 7 is 13, or you know that you can count 6 between 7 and 13. Either way, 13 - 7 = 6, so we can write that.

Then we move to the tens column and see that 50 -50 can be pictured like a 5 - 5 which equals 0.

The final answer is just 6, since the leading zero does not influence the number to change.

Question 21

Use the addition algorithm to add the following three-digit numbers:

123 + 456

Answer 21

Question 22

Use the addition algorithm to add the following three-digit numbers:

987 + 654

Answer 22

Question 23

Use the subtraction algorithm to subtract:

567 - 389

Answer 23

Question 24

Use the subtraction algorithm to subtract:

4.05 - 3.97

Answer 24

Question 25

repeated addition

Answer 25

multiplication

Question 26

the inverse mathematical operation to multiplication

Answer 26

division

Question 27

multiplication

Answer 27

• equal groups
• array
• area
• repeated addition
• represented at this level by X
• follows commutative property just like addition, so the order they are multiplied does not affect the product

Question 28

division

Answer 28

• equal sharing
• equal grouping
• repeated subtraction
• represented at this level by ÷
• also the fraction bar can represent division! (so 16 / 4 means 16 divided by 4)
• the order in which two numbers are divided affects the quotient

Question 29

Name the numbers in a division problem

Answer 29

dividend / divisor = quotient

Question 30

Name the numbers in a multiplication problem

Answer 30

factor X factor = product

you can say a number is a “multiple” of any of its factors, so 6 is a multiple of 2, and 6 is also a multiple of 3

Question 31

prime number

Answer 31

has factors of only itself and one, but does not include 1

0 and 1 are neither prime, nor composite

Question 32

composite number

Answer 32

has factors other than one and itself

0 and 1 are neither prime, nor composite

Question 33

If a number is divided by one of its factors, what is the remainder?

Answer 33

0

Question 34

greatest common factor (also known as greatest common divisor)

provide an example with:

1. 7 and 14
2. 12 and 16

Answer 34

This is the largest number that can be divided into the numbers provided in the question without a remainder.

For example, the GCF of 14 and 7 is 7, since both 14 and 7 can be divided by 7 evenly and 7 is the largest number that can do this for this situation.

Another example:
The GCF of 12 and 16 is 4.

16 = 2x2x2x2
12 = 2x2x3

So what is common between both of them is 2x2

And 2x2 is really 4, so that is the GCF of 16 and 12.

Question 35

remainder

Answer 35

quantity left over after division

You may have 1 left over after dividing a number by 3, such as 13/3

12/3 = 4 but we are left with 1/3

The remainder here is 1.

The answer is 4 1/3 or four and one third.

So in the answer, you will use the remainder as the numerator of your fraction.

Question 36

fact families for multiplication and division

Provide your own example

Answer 36

3 x 4 = 12

4 x 3 = 12

12 / 3 = 4

12 / 4 = 3

Question 37

Explain how to multiply using the standard multiplication algorithm:

123 X 4

Answer 37

Multiply the one digit number by the ones in the three digit number. If the result is greater than 9 (so if it takes up two spots), then write the tens over the tens to be added to the product you will find next.

Then find the product of the one digit multiplied by the tens place. Just think of them as ones for the moment to make the multiplication easier. So in this case 4 multiplied by 2 is 8. The 9 below comes from adding the 8 to the 1 from before. This is really adding 80 to 10 since we are in the tens column. In this question, the result was not greater than 10, so we did not have to carry a number over to the hundreds place.

Finally multiply the one digit by the hundreds. Write the result in the hundreds place below.

If you were to multiply by two digits, this math would get more messy, so erase any numbers placed above the columns if you have another number to multiply later, because these are just temporary and should not be there for the next set. In this case, we are only multiplying by a one-digit number, so we are done!

Question 38

Explain how to divide using the standard division algorithm:

123 / 4

Answer 38

How many times does 4 go into 1? None, so ask a new question: how many times does 4 go into 12?

12 / 4 = 3 so we know that 4 goes into 12 3 times.

Write the 3 above.

Then multiply the 4 by the 3 and write the result under the 12. So in this case 12 is a nicely divisible number and we see that we have no remainder for this portion of the question (this is seen once we subtract 12 from 12, and get a result of 0).

Bring down the 3 so that you can see it better. Ask if 4 goes into 3? It doesn’t but you can say it goes into 3 zero times and write that number up top. Then multiply 4 by zero to get 0, then subtract 0 from 3 and you get a remainder of 3. There are no more numbers to work with, so this is our final remainder: 3

The answer to the division is 30 3/4

Question 39

What can you see in the name of a fraction? For example, what does 3/4 mean compared to a unit fraction with 4 parts designating the whole.

Answer 39

3/4 is 3 unit fractions of 1/4 added together. 1/4 + 1/4 + 1/4 = 3/4

Three quarters is the name of the fraction. There are 3 “quarter unit fractions”

Question 40

numerator

Answer 40

top part of a fraction; this is sometimes on the left if the fraction notation is typed

numerator / denominator

Question 41

denominator

Answer 41

bottom part of a fraction; this is sometimes on the right if the fraction notation is typed

numerator / denominator

Question 42

What is a fraction?

Answer 42

a/b where a is a number of equal parts, and b represents the total number of equal parts in the whole

Fractions represent part to whole relationships

Question 43

Which fraction is larger?

3/4 2/4

Answer 43

3/4

Question 44

Which fraction is larger?

6/7 6/8

Answer 44

6/7

Question 45

If the numerator is equal to the denominator, then what is the result?

Answer 45

one whole

Question 46

If the top number is the same as the bottom number in a fraction, then what is the result?

Answer 46

one whole

Question 47

What is anything divided by itself?

Answer 47

one whole

Question 48

Can fractions be plotted on the same number line as counting numbers?

Answer 48

Yes, fractions fit between the counting numbers

Question 49

If two differently written fractions are on the same point on the number line, what are they called in relation to each other?

Answer 49

equivalent fractions

Question 50

How can you model equivalent fractions?

Answer 50

partition each equal part of a fraction in the same way, what you have left is when you have made the smallest groups is the lowest terms fraction, or simplified fraction

Question 51

How do you simplify a fraction?

E.g. 12/16

Answer 51

This means to put the fraction in lowest terms, so divide the top and bottom of the fraction by a common divisor until you can no longer do so. The result is an equivalent fraction in lowest terms, or the simplified fraction.

If you divide by the greatest common factor, you will only need to divide once (on the top and bottom) to get the simplified fraction.

Divide by 4 and you get

3/4

But you may have divided by 2 twice to get there if you do not know the greatest common factor.

Question 52

Can fractions and decimals represent the same number?

Answer 52

Yes

Question 53

What are decimal numbers that terminate?

Answer 53

Numbers where a decimal is required to represent it rather than just being an integer and where the numbers after the decimal end rather than repeating forever.

Question 54

How do you convert a terminating decimal number into a fraction?

Show with this example:

0.34

Answer 54

There are 4 hundredths and 3 tenths in this number. To convert to a fraction, you need to look at how many hundredths there are if everything was represented as a hundredth and not use the word “tenths”.

So here we can say we have thirty four hundredths.

When you say it that way, it is much easier to picture the fraction that would represent this:

34/100

So decimals can be expressed as fractions with a denominator that is equivalent to the place value of the last non-zero digit of the decimal number. In this case the denominator is 100 because it is equivalent to the second decimal place’s place value: hundredths.

Make sure that you can convert back as well:

34/100 = 0.34

Question 55

How do you convert from a decimal to a percent? Show how with the following three decimals:

0.36

0.02

0.004

Answer 55

multiply by 100

36%

2%

0.4%

Question 56

How do you convert from a percent to a decimal? Show with the following three percents:

0.5%

6%

70%

Answer 56

divide by 100

0.005

0.06

0.7

Question 57

How do you convert from a percent to a simplified fraction? Show with the following three examples:

0.5%

6%

70%

Answer 57

0.005 = 5/1000 = 1/200

0.06 = 6/100 = 3/50

0.7 = 7/10

Question 58

Given a truthful equation: can you move the left side to the right side and the right side to the left side of an equation and still have the equation be a truthful statement?

Answer 58

Yes

2 + 5 = 7

and

7 = 2 + 5

Question 59

How do you represent an unknown value in an equation?

Answer 59

This is known as a variable and can be shown with
- a symbol such as a box to be filled in
- a letter (but some are not used since they are reserved for other purposes, so stick to w, x, y, z for variables for now and you will be fine)

So given 2 + x = 7, we could recognize that x = 5 at this moment. It was unknown so we used x to represent it. Then we could state that 2 + 5 = 7 and that we now know the value of x to be 5.

Note on letters to not use unless the situation permits it:
- a, b, and c represent constants that are unknown right now but they will have constants there and so they are reserved for math classes to show that a number that is known will go there when you start using that equation
- b represents the y-intercept and so it should be avoided
- d represents distance
- e should not be used because it represents a constant
- f can represent friction
- g can represent the gravitational constant
- h is reserved for height, also used in quadratic equations
- i denotes an imaginary number
- j might be okay to use but just isn’t used often so it might confuse the reader
- k reminds people of kilo as in the metric kilometer or kilogram so don’t use it when you are also using measurement units; also used in quadratic equations
- l looks like a one, or i so best to avoid it, also represents length
- m usually represents slope so that should be avoided
- I suppose you could use n, since it seems to not have another connection
- o usually refers to the center of a circle
- p and q are used in quadratic equations to show the vertex
- r is radius
- s represents seconds
- t represents time
- u could be used since it seems to not have another connection
- v represents speed

… just stick to w, x, y, z and you will do fine :)

Question 60

If you add a quantity to the left side of the equation, what must you do to the right side of the equation to keep the equality truthful?

(preservation of equality)

Answer 60

Add the same quantity to the right side of the equation so that the equation stays in balance.

Question 61

If you subtract a quantity from the right side of the equation, what must you do to the left side of the equation to keep the equality truthful?

(preservation of equality)

Answer 61

Subtract the same quantity from the left side of the equation so that the equation stays in balance.

Question 62

If you divide by a quantity on the left side of the equation, what must you do to the right side of the equation to keep the equality truthful?

(preservation of equality)

Answer 62

You must divide by that same quantity on the right side of the equation so that the equation stays in balance.

Question 63

If you multiply by a quantity on the right side of the equation, what must you do to the left side of the equation to keep the equality truthful?

(preservation of equality)

Answer 63

You must multiply by the same quantity on the left side of the equation so that the equation stays in balance.

Question 64

What does it mean to solve an equation?

Answer 64

Solving an equation means to find the value of the unknown variable(s), usually by adding, multiplying, subtracting, and dividing from both sides of the equation until you have the variable on one side, and what it is equal to on the other side.

Sometimes it will be equal to a constant, but other times this might still look like a regular equation and may equal a variable with some constants.

(preservation of equality)