Intermediate II number sense (5-6 new curriculum)

Question 1

If you have a number expressed with more decimal places, is it more or less precise than a number with less decimal places?

Answer 1

It is more precise.

Question 2

Does a zero in the rightmost place of a decimal number change the value of the number?

Answer 2

No, it does not.

Question 3

How many decimal numbers can fit between any two decimal numbers?

Answer 3

infinitely many

Question 4

What is place value symmetry?

Answer 4

This states that the left and right of the ones place mirror each other. One over is tens and tenths, two over is hundreds and hundredths and so on infinitely.

Question 5

You need to be able to place any decimal number on a number line.

Place a decimal number on a number line of your choosing or do the question your teacher gives you.

Answer 5

Answers vary.

Concepts explored here are to be able to divide into tenths on the number line and follow the tenths place in the decimal to locate where to place the number. Of course, you will need to find the ones place first and then look at the tenths.

Question 6

You need to be able to round any decimal number to any place.

Round a sample number of your choosing or do the question your teacher gives you.

Answer 6

Answers vary.

Concepts explored here are that you must look one place over to the right of where you intend to round. If that number is 0, 1, 2, 3, or 4, then you remove that number and anything to the right. BUT if the number is 5, 6, 7, 8, or 9, then you must add one to the place to the left of that large number, and then remove the large number and anything to the right.

Question 7

Where are negative numbers placed on the number line?

What is the symbol to represent a negative number?

What are some real-life contexts of negative numbers?

Answer 7

To the left of zero if it is horizontal. Below zero if it is vertical.

negative sign -

temperature
debt
elevation

Question 8

Is zero positive?

Answer 8

no

Question 9

Is zero negative?

Answer 9

no

Question 10

What is magnitude?

Answer 10

The distance from zero.

Magnitude will not be negative because it is a distance. So the magnitude of -6 is just 6.

Every positive number has an opposite negative number with the same magnitude.

No direction is given if you have just magnitude. You cannot tell where to go, just the distance.

Question 11

What is an additive inverse?

Answer 11

A number and its opposite are called additive inverses.

For example 6 and -6 are additive inverses.

If you add them together they make 0.

They have the same magnitude.

Question 12

What signs are used to indicate direction?

Answer 12

positive and negative

+ -

Negative is the opposite direction as positive.

Question 13

Which number is larger?

7 or -8?

Answer 13

7

Question 14

Which number has a larger magnitude?

7 or -8?

Answer 14

-8 has a larger magnitude. Its magnitude is 8.

Question 15

What is an integer?

Answer 15

all natural numbers, their additive inverses, and zero

…-3, -2, -1, 0, 1, 2, 3, …

Question 16

Is the sum of two positive numbers positive or negative?

Answer 16

positive

Question 17

Is the sum of two negative numbers positive or negative?

Answer 17

negative

Question 18

What is the same as adding an additive inverse?

Answer 18

subtracting a number

for example adding -8 is the same as subtracting 8

Question 19

You need to be able to add any two integers. This explores if you are able to express a difference as a sum. Model this on a number line.

Make up a question or use the one your teacher provides.

Make sure to practice four types of questions: - two positives
- two negatives
- one positive that is greater in magnitude to one negative
- one positive that is smaller in magnitude to one negative.

Answer 19

Answers vary.

Concepts needed here are how to plot on an number line, as well as addition means to go to the right, and subtraction means to go to the left. If you add a negative number, you will go to the right a negative amount, which really just means to go in the opposite direction, so you will go to the left.

Question 20

What is an algorithm?

Answer 20

A process you can follow to more efficiently solve a problem. Someone else proved that it works, so you can use it even if you don’t quite understand how it works.

For example I use this algorithm to solve a certain step in the rubix cube:

R’DRD’

R = turn the right side clockwise
D = turn the down side clockwise
R’ = turn the right side counterclockwise

There are many algorithms that people use such as the addition, subtraction, multiplication, and division algorithms when working with two digit numbers.

Question 21

Add two numbers (mininum two-digit) together using the addition algorithm. Come up with a question or use one provided by your teacher.

Answer 21

Answers vary.

Make sure to line up the numbers by place value. When you get more than 9 as an answer, add one or more to the place to the left, depending on the number in the tens column in your answer.

Question 22

What is a divisibility test?

Answer 22

A method to determine the factors of a natural number.

Question 23

Is it possible to divide by 0?

Answer 23

no

Question 24

If a number is divisible by another number and you divide it by that number, what is the remainder?

Answer 24

0

Question 25

What is the divisibility test for 2?

Answer 25

If it ends in either 0, 2, 4, 6, or 8, then it is divisible by 2.

Question 26

What is the divisibility test for 3?

Answer 26

If you add up all the digits in the number and if the sum is divisible by 3, then the original number is also divisible by 3.

Keep in mind that with a number like this: 3966633933
It is possible to just make sure that every digit is divisible by 3 instead of adding them all together first. But, every single one of them must be divisible by 3!

Question 27

What is the divisibility test for 5?

Answer 27

If the number ends in 5 or 0 then it is divisible by 5.

Question 28

Be able to determine the factors of any natural number. Pick a natural number and determine all its factors, or have your teacher give you a number.

Answer 28

Answers vary.

The main concept here is that you should be able to split the number by two factors and then continue to make a prime factorization tree by further splitting each number into two factors.

For example:

18 = 9 x 2

But 9 = 3 x 3

So 18 = 3 x 3 x 2

This is usually shown more easily in a tree diagram.

Question 29

Associative Property for multiplication

Answer 29

The order in which three or more numbers are multiplied does not affect the product.

2 x 5 x 3 = 2 x 3 x 5

Question 30

What is a composite number?

Answer 30

A number that can be expressed as a product of smaller numbers other than itself and one.

Usually we talk about different natural numbers and we determine if they are prime or composite. 0 and 1 are neither prime nor composite.

Question 31

What is prime factorization?

Answer 31

A number represented as a product of prime numbers.

Question 32

How can you find a composite factor of a number?

Answer 32

If a number does have a composite factor, you can get this by mulitiplying the prime factors together of that number in various combinations.

Question 33

What is a power? (What is exponentiation)

Answer 33

The symbolic representation of repeated multiplication of identical factors.

Question 34

What is a base?

Answer 34

The bottom number of the power, that represents the repeated factor.

Question 35

What is an exponent?

Answer 35

This indicates the quantity of identical factors that are repeatedly multiplied.

Question 36

Is a power divisible by its base?

Answer 36

yes

Question 37

Complete the prime factorization of a composite number of your choice. Express your final answer in order of lowest to greatest prime and use exponents to represent your final answer.

Answer 37

Answers vary.

Try to learn to check for divisibility by 2, 3, 5, and 10 as this is the most efficient way. Then divide the number as you go to see what is left. Then check divisibility by other prime numbers such as 7, 11, 13, 17, 19 etc. You should have only prime numbers in your final answer.

Question 38

Use the standard algorithm for multiplication to multiply a three-digit number by a two digit number.

Answer 38

Answers vary.

Make sure that you remember to erase carried numbers between each of the bottom numbers so that you don’t double-count something from a previous process.

Your teacher will show you this on paper since it must be seen in real-time.

Question 39

Use the standard algorithm for division to divide a three digit number by a two digit number.

Answer 39

Answers vary.

Make sure that you are fluent in regular multiplication and division first, and then try this with the help of a teacher. It is important to know how to express the remainder if the number does not divide evenly.

Question 40

Can a fraction represent quantities greater than 1?

Answer 40

yes

Question 41

What is an improper fraction?

Answer 41

The numerator is greater than the denominator.

Question 42

How can you express a natural number as a fraction?

Answer 42

Use the natural number as the numerator, and 1 as the denominator.

Anything divided by 1 is just itself, so this means that this new representation is equal in value to the old representation.

Question 43

Can an improper fraction be represented in a different way? What way?

Answer 43

Yes, a mixed number in the form of A b/c where A is the number of wholes, and b/c is the fractional remainder or fractional part.

Question 44

Why do we need fractions?

Answer 44

They allow counting and measuring between whole quantities.

Question 45

Be able to compare fractions to benchmarks of 0, 1/2, and 1. Pick a fraction (including mixed, and improper) and plot it on a number line.

Answer 45

Answers vary. Make sure that you are thinking about what the fraction would be if it were over 10 or some other divisor that you can see on your number line. This will help you to more accurately plot your point.

Question 46

What is a fraction?

Answer 46

Numerator is the quantity to be shared.
Denominator is the number of shares or number of groups.

Numerator / Denominator

Question 47

Convert from a fraction to a decimal. Choose fractions that are mixed and improper as well.

Answer 47

Answers vary. Have your teacher help you to find a denominator of 10, 100, or 1000 to help convert the fractional part.

Question 48

Before adding or subtracting fractions, what must you do?

Answer 48

Find a common denominator by finding the least common multiple between the two denominators present. Alternatively you can just multiply the two denominators together and the product will be your new common denominator.

The denominators must be the same before you can add or subtract. If they are already the same, you do not need to find a LCM or change the denominators to match.

Question 49

What is a LCM?

Answer 49

Least Common Multiple

Given two or more numbers, you can determine the least common multiple by skip counting by the number itself, and recognizing when one of those results matches the other set.

For example: 6 and 8

6: 6, 12, 18, 24, 30, 36, 42, 48, ….

8: 8, 16, 24, 32, 40, 48, ….

You can see that 24 is the LCM here since it is the lowest of the common numbers between the two sets. 48 is just a common multiple and is higher than required.

When finding a LCM it is important to find the smallest solution so that the subsequent math steps are easier.

Question 50

Be able to add mixed number fractions together. Ask your teacher to give you a question, or make one up yourself.

Answer 50

Answers vary.

You can add the fractional parts first, and then remember to carry any wholes to your whole addition.

Question 51

Be able to reduce fractions and also be able to make equivalent fractions. Make up a question or ask your teacher for a question.

Answer 51

Answers vary. If you know your divisibility tests the process of reducing a fraction is much easier.

Question 52

What does “related denominator” mean?

Answer 52

If one denominator is a multiple of the other, then the two denominators are related.

It is useful to recognize when denominators are related since if they are related, you can just use the higher denominator as your common denominator. This is used when adding or subtracting fractions.

Question 53

Using 2, 3, and their additive inverses, make four addition problems and find their sums.

Answer 53

a) (+2) + (+3) = 5

b) (+2) + (-3) = -1

c) (-2) + (+3) = 1

d) (-2) + (-3) = -5

Question 54

Using 2, 3, and their additive inverses, make eight subtraction problems and find their differences.

Answer 54

a) (+2) - (+3) = -1

b) (+2) - (-3) = 5

c) (-2) - (+3) = -5

d) (-2) - (-3) = 1

e) (+3) - (+2) = 1

f) (+3) - (-2) = 5

g) (-3) - (+2) = -5

h) (-3) - (-2) = -1